Boolean Valued Analysis: Selected Topics

Semen Kutateladze

RUSSIAN ACADEMY OF SCIENCES VLADIKAVKAZ SCIENTIFIC CENTER MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION NORTH OSSETIAN STATE UNIVERSITY SOUTHERN MATHEMATICAL INSTITUTE TRENDS IN SCIENCE • THE SOUTH OF RUSSIA A MATHEMATICAL MONOGRAPH Issue 6 BOOLEAN VALUED ANALYSIS: SELECTED TOPICS by A. G. Kusraev and S. S. Kutateladze Vladikavkaz 2014 ББК 22.16 УДК 517.98 K 94 Editor A. E. Gutman Reviewers: S. A. Malyugin and E. Yu. Emelyanov Editors of the series: Yu. F. Korobeinik and A. G. Kusraev Kusraev A. G. and Kutateladze S. S. Boolean Valued Analysis: Selected Topics / Ed. A. E. Gutman.—Vladikavkaz: SMI VSC RAS, 2014.—iv+400 p.—(Trends in Science: The South of Russia. A Mathematical Monograph. Issue 6). The book treats Boolean valued analysis. This term signiﬁes the technique of studying properties of an arbitrary mathematical object by means of comparison between its representations in two diﬀerent set-theoretic models whose construction utilizes principally distinct Boolean algebras. As these models, we usually take the classical Cantorian paradise in the shape of the von Neumann universe and a speciallytrimmed Boolean valued universe in which the conventional set-theoretic concepts and propositions acquire bizarre interpretations. Exposition focuses on the fundamental properties of order bounded operators in vector lattices. This volume is intended for the classical analyst seeking new powerful tools and for the model theorist in search of challenging applications of nonstandard models of set theory. Кусраев А. Г., Кутателадзе С. С. Булевозначный анализ: Избранные темы / отв. ред. А. Е. Гутман.— Владикавказ: ЮМИ ВНЦ РАН и РСО-А, 2014.—iv+400 с.—(Итоги науки. Юг России. Математическая монография. Вып. 6). Монография посвящена булевозначному анализу. Так называют аппарат исследования произвольных математических объектов, основанный на сравнительном изучении их вида в двух моделях теории множеств, конструкции которых основаны на принципиально различных булевых алгебрах. В качестве этих моделей фигурируют классический канторов рай в форме универсума фон Неймана и специально построенный булевозначный универсум, в котором теоретикомножественные понятия и утверждения получают весьма нетрадиционные толкования. Основное внимание уделено фундаментальным свойствам порядково ограниченных операторов в векторных решетках. Книга ориентирована на широкий круг читателей, интересующихся современными теоретико-модельными методами в их приложении к функциональному анализу. ISBN 978-5-904695-24-8 c Southern Mathematical Institute VSC RAS & RNO-A, 2014 c A. G. Kusraev, 2014 c S. S. Kutateladze, 2014 PREFACE Humans deﬁnitely feel truth but cannot deﬁne truth properly. That is what Alfred Tarski explained to us in the 1930s. Mathematics pursues truth by way of proof, as wittily phrased by Saunders Mac Lane. Boolean valued analysis is one of the vehicles of the pursuit, resulting from the fusion of analysis and model theory. Analysis is the technique of diﬀerentiation and integration. Differentiation discovers trends, and integration forecasts the future from trends. Analysis opens ways to understanding of the universe. Model theory evaluates and counts truth and proof. The chase of truth not only leads us close to the truth we pursue but also enables us to nearly catch up with many other instances of truth which we were not aware nor even foresaw at the start of the rally pursuit. That is what we have learned from Boolean valued models of set theory. These models stem from the famous works by Paul Cohen on the continuum hypothesis. They belong to logic and yield a profusion of the surprising and unforeseen visualizations of the ingredients of mathematics. Many promising opportunities are open to modeling the powerful habits of reasoning and veriﬁcation. Logic organizes and orders our ways of thinking, manumitting us from conservatism in choosing the objects and methods of research. Logic of today is a ﬁne instrument of pursuing truth and an indispensable institution of mathematical freedom. Logic liberates mathematics, providing nonstandard ways of reasoning. Some model of set theory is nonstandard if the membership between the objects of the model diﬀers from that of the originals. In fact, the nonstandard tools of today use a couple of set-theoretic models simultaneously. Boolean valued models reside within the most popular logical tools. Boolean valued analysis is a blending of analysis and Boolean valued models which originated and distinguishes itself by ascending and descending, mixing, cycling hulls, etc. iv Preface In this book we show how Boolean valued analysis transforms the theory of operators in vector lattices. We focus on the recent results that were not reﬂected in the monographic literature yet. In Chapter 1 we collect the Boolean valued prerequisites of the further analysis. Chapter 2 provides the presentation of the reals and complexes within Boolean valued models. In Chapter 3 we give the Boolean valued interpretations of order bounded operators with the emphasis on lattice homomorphisms and disjointness preserving operators. Chapter 4 contains the solution of the Wickstead problem as well as other new results on band preserving operators. Chapter 5 deals with various applications of order continuous operators to injective Banach lattices, Maharam operators, and related topics. Adaptation of the ideas of Boolean valued models to functional analysis projects among the most important directions of developing the synthetic methods of mathematics. This approach yields the new models of numbers, spaces, and types of equations. The content expands of all available theorems and algorithms. The whole methodology of mathematical research is enriched and renewed, opening up absolutely fantastic opportunities. We can now transform matrices into numbers, embed function spaces into a straight line, yet having still uncharted vast territories of new knowledge. The chase of truth not only leads us close to the truth we pursue but also enables us to nearly catch up with many other instances of truth which we were unaware nor even foresaw at the start of the rally pursuit. That is what we have learned from Boolean valued analysis. Quite a long time had passed until the classical functional analysis occupied its present position of the language of continuous mathematics. Now the time has come of the new powerful technologies of model theory in mathematical analysis. Not all theoretical and applied mathematicians have already gained the importance of modern tools and learned how to use them. However, there is no backward traﬃc in science, and the new methods are doomed to reside in the realm of mathematics for ever and they will shortly become as elementary and omnipresent in analysis as Banach spaces and linear operators. A. Kusraev S. Kutateladze CHAPTER 1 BOOLEAN VALUED REQUISITES In this chapter we brieﬂy present some prerequisites of the theory of Boolean valued models. All missing details may be found in Bell [43], Jech [184], Kusraev and Kutateladze [248, 249], Takeuti and Zaring [388]. We mainly keep the notation of [248] and [249]. The most important feature of Boolean valued analysis consists in comparative analysis of the standard and nonstandard (Boolean valued) models under consideration which uses the special technique of ascending and descending. Moreover, it is often necessary to carry out some syntactic comparison of formal texts. Therefore, before we launch into the ascending and descending machinery, we have to grasp a clearer idea of the status of mathematical objects in the framework of a formal set theory, the construction of a Boolean valued universe, and the way of assigning the Boolean truth value to each sentence of the language of set theory. We use several notations for implication: ⇒ presents the Boolean operation, =⇒ stands for the logical connective, but often in a set theoretic formula we use → instead of =⇒ indicating that this formula will be interpreted in some Boolean valued model. We also use ↔, ⇔, and ⇐⇒ with the similar meaning. The proof-theoretic consequence relation ⊢ is applied alongside with the semantic (or model-theoretic) consequence relation |=. Observe that, speaking of a formal set theory, we will freely (because this is in fact unavoidable) adhere to the level of rigor which is current in the mainstream of mathematics and introduce abbreviations by means of the definor, i.e. the assignment operator, := without specifying any subtleties. 2 Chapter 1. Boolean Valued Requisites 1.1. Zermelo–Fraenkel Set Theory At present, the most widespread axiomatic foundation for mathematics is Zermelo–Fraenkel set theory. We will brieﬂy recall some of its concepts, outlining the details we need in the sequel. 1.1.1. The alphabet of Zermelo–Fraenkel theory ZF or ZFC, if the presence of choice AC is stressed, comprises the symbols of variables; the parentheses ( and ); the propositional connectives (i.e., the signs of propositional calculus) ∨, ∧, →, ↔, and ¬; the quantiﬁers ∀ and ∃; the equality sign =; and the symbol of the special binary predicate of containment or membership ∈. In general, the domain of the variables of ZF is thought as the world or universe of sets. In other words, the universe of ZF contains nothing but sets. We write x ∈ y rather than ∈ (x, y) and say that x is an element of y. 1.1.2. The formulas of ZF are deﬁned by the routine procedure. In other words, the formulas of ZF are ﬁnite texts resulting from the atomic formulas x = y and x ∈ y, where x and y are variables of ZF, by reasonably placing parentheses, propositional connectives, and quantiﬁers ϕ ∨ ψ, ϕ ∧ ψ, ¬ ϕ, ϕ → ψ, ϕ ↔ ψ, (∀ x) ϕ, (∃ x) ϕ. So, if ϕ1 and ϕ2 are formulas of ZF and x is a variable then the texts ϕ1 → ϕ2 and (∃ x) (ϕ1 → (∀ y) ϕ2 ) ∨ ϕ1 are formulas of ZF, whereas ϕ1 ∃ x and ∀ (x∃ ϕ1 ¬ ϕ2 are not. We attach the natural meaning to the terms free and bound variables and the term domain of a quantifier. For instance, in the formula (∀ x) (x ∈ y) the variable x is bound and the variable y is free, whereas in the formula (∃ y) (x = y) the variable x is free and y is bound (for it is bounded by a quantiﬁer). Henceforth, in order to emphasize that the only free variables in a formula ϕ are the variables x1 , . . . , xn , we write ϕ(x1 , . . . , xn ). Sometimes such a formula is considered as a “function”; in this event, it is convenient to write ϕ(·, . . . , ·) or ϕ = ϕ(x1 , . . . , xn ), implying that ϕ(y1 , . . . , yn ) is a formula of ZF obtained by replacing each free occurrence of xk by yk for k := 1, . . . , n. 1.1.3. Studying ZF, it is convenient to use some expressive tools absent in the formal language. In particular, in the sequel it is worthwhile employing the concepts of class and definable class and also the corresponding symbols of classifiers like Aϕ := Aϕ(·) := {x : ϕ(x)} and Aψ := Aψ(·,y) := {x : ψ(x, y)}, where ϕ and ψ are formulas of ZF and y is 1.1. Zermelo–Fraenkel Set Theory 3 a distinguished collection of variables. If it is desirable to clarify or eliminate the appearing records then we can assume that the use of classes and classiﬁers is connected only with the conventional agreement on introducing abbreviations. This agreement, sometimes called the Church schema, reads: z ∈ {x : ϕ(x)} ↔ ϕ(z), z ∈ {x : ψ(x, y)} ↔ ψ(z, y). 1.1.4. Working within ZF, we will use some notations that are widely spread in mathematics. We start with the most frequent abbreviations: x = y := ¬ x = y, x∈ / y := ¬ x ∈ y; (∀ x ∈ y) ϕ(x) := (∀ x) (x ∈ y → ϕ(x)); (∃ !z) ϕ(z) := (∃ z) ϕ(z) ∧ ((∀ x) (∀ y) (ϕ(x) ∧ ϕ(y) → x = y)); (∃ x ∈ y) ϕ(x) := (∃ x) (x ∈ y ∧ ϕ(x)). The empty set ∅, the pair {x, y}, the singleton {x}, the ordered pair (x, y), and the ordered n-tuple (x1 , . . . , xn ) are deﬁned as ∅ := {x : x = x}; {x, y} := {z : z = x ∨ z = y}, {x} := {x, x}, (x, y) := {x, {x, y}}; (x1 , . . . , xn ) := ((x1 , . . . , xn−1 ), xn ). The inclusion ⊂, the union , the intersection , the powerset P(·), and the universe of sets V are introduced as follows: x ⊂ y := (∀ z) (z ∈ x → z ∈ y); x := {z : (∃ y ∈ x) z ∈ y}; x := {z : (∀ y ∈ x) z ∈ y}; P(x) := “the class of all subsets of x” := {z : z ⊂ x}; V := “the class of all sets” := {x : x = x}. 4 Chapter 1. Boolean Valued Requisites Note also that in the sequel we accept more complicated descriptions in which much is presumed: Fnc(f ) := “f is a function”; dom(f ) := “the domain of f ”; im(f ) := “the range of f ”; ϕ ⊢ ψ := ϕ → ψ := “ψ is derivable from ϕ”; “a class A is a set” := A ∈ V := (∃ x) (∀ y) (y ∈ A ↔ y ∈ x). Similar simpliﬁcations will be used in rendering more complicated formulas without further stipulation. For instance, instead of some rather involved formulas of ZF we simply write f : x → y ≡ “f is a function from x to y”; “X is a vector lattice”; U ∈ L∼ (X, Y ) ≡ “U is an order bounded linear operator from X to Y .” 1.1.5. In ZFC, we accept the usual axioms and rules of a ﬁrst-order theory with equality which ﬁx the standard means of classical reasoning. Recall the equality axioms: (1) (∀ x) x = x (reflexivity); (2) (∀ x) (∀ x) x = y → y = x (symmetry); (3) (∀ x) (∀ y) (∀ y) x = y ∧ y = x → x = z (transitivity); (4) (∀ x) (∀ y) (∀ u) (∀ v) (x = y ∧ u = v) → (x ∈ u → y ∈ v) (substitution). 1.1.6. The classical ﬁrst-order logic CL has the following axiom schemas (ϕ, ψ, and ω are arbitrary formulas of CL): (1) ϕ → (ϕ ∧ ϕ); (2) (ϕ ∧ ψ) → (ψ ∧ ϕ); (3) (ϕ → ψ) → ((ϕ ∧ ω) → (ψ ∧ ω)); (4) ((ϕ → ψ) ∧ (ψ → ω)) → (ϕ → ω); (5) ψ → (ϕ → ψ); (6) (ϕ ∧ (ϕ → ψ)) → ψ; (7) ϕ → (ϕ ∨ ψ); (8) (ϕ ∨ ψ) → (ψ ∨ ϕ); 1.1. Zermelo–Fraenkel Set Theory 5 (9) ((ϕ → ω) ∧ (ψ → ω)) → ((ϕ ∧ ψ) → ω); (10) ¬ ϕ → (ϕ → ψ); (11) ((ϕ → ψ) ∧ (ϕ → ¬ ψ)) → ¬ ϕ; (12) ϕ ∨ ¬ ϕ. The only rule of inference in CL is modus ponens: (MP) If ϕ and ϕ → ψ are provable in CL then so is ψ. 1.1.7. Moreover, the following special or proper axioms are accepted in ZFC as a correct formalization of the principles of most mathematicians working with sets: (1) Axiom of Extensionality. If x and y have the same elements then x = y: (∀ x) (∀ y) (x ⊂ y ∧ y ⊂ x → x = y). (2) Axiom of Union. To each x there exists a set y = x: (∀ x) (∃ y) y = x . (3) Axiom of Powerset. To each x there exists a set y = P(x): (∀ x) (∃ y) (y = P(x)). (4) Axiom Schema of Replacement. If a class Aϕ is a function then to each x there exists a set v = Aϕ (x) = {Aϕ (z) : z ∈ x}: (∀ x) ((∀ y) (∀ z) (∀ u) ϕ(y, z) ∧ ϕ(y, u) → z = u) → (∃ v) (v = {z : (∃ y ∈ x) ϕ(y, z)}). (5) Axiom of Foundation. Each nonempty set has an ∈-minimal element: (∀ x) (x = ∅ → (∃ y ∈ x) (y ∩ x = ∅)). (6) Axiom of Inﬁnity. There exists an inductive set: (∃ ω) (∅ ∈ ω) ∧ (∀ x ∈ ω) (x ∪ {x} ∈ ω). (7) Axiom of Choice. Each family of nonempty sets has a choice function: (∀ F ) (∀ x) (∀ y) ((x = ∅ ∧ F : x → P(y)) → ((∃ f ) f : x → y ∧ (∀ z ∈ x) f (z) ∈ F (z)). 6 Chapter 1. Boolean Valued Requisites 1.1.8. Grounding on the above axiomatics, we acquire a clear idea of the class of all sets, the von Neumann universe V. As the initial object of all constructions we take the empty set. The elementary step of introducing new sets consists in taking the union of the powersets of the sets already available. Transﬁnitely repeating thesesteps, we exhaust the class of all sets. More precisely, we assign V := α∈On Vα , where On is the class of all ordinals and Vβ := V0 := ∅, Vα+1 := P(Vα ), Vα (β is a limit ordinal). α<β 1.1.9. The pair (V, ∈) is a standard model of ZFC. 1.2. Boolean Valued Universes Everywhere below B is a complete Boolean algebra with supremum (join) ∨, meet (inﬁmum) ∧, complement (·)∗ , unit (top) 1, and zero (bottom) 0. The necessary information on Boolean algebras can be found in Givant and Halmos [130], Sikorski [365], and Vladimirov [399]. 1.2.1. Let B be a complete Boolean algebra. Given an ordinal α, put (B) Vα := x : Funct(x) ∧ (∃ β) (β < α ∧ dom(x) (B) ⊂ Vβ ∧ im(x) ⊂ B) . Thus, in more detail we have (B) V0 := ∅, (B) (B) and range in B}; Vα+1 := {x : x is a function with domain in Vα (B) (B) := (β is a limit ordinal). Vβ Vα β<α The class V(B) := α∈On (B) Vα 7 1.2. Boolean Valued Universe is a Boolean valued universe. An element of V(B) is a B-valued set. Observe that V(B) consists only of functions. In particular, ∅ is the function with domain ∅ and range ∅. Hence, the “lower” levels of V(B) are organized as follows: (B) (B) (B) V0 = ∅, V1 = {∅}, V2 = ∅, {∅}, b : b ∈ B . (B) (B) 1.2.2. It is worth stressing that α β =⇒ Vα ⊂ Vβ is valid for all ordinals α and β. Moreover, we have the induction principle for V(B) : ∀ x ∈ V(B) (∀ y ∈ dom(x)) ϕ(y) =⇒ ϕ(x) =⇒ ∀ x ∈ V(B) ϕ(x) , where ϕ is a formula of ZFC. 1.2.3. Take an arbitrary formula ϕ = ϕ(u1 , . . . , un ) of ZFC. If we replace u1 , . . . , un by x1 , . . . , xn ∈ V(B) then we obtain some statement about the objects x1 , . . . , xn . It is to this statement that we intend to assign some Boolean truth value. Such a truth value [[ψ]] must be an element of B. Moreover, we desire naturally that the theorems of ZFC be true; i.e., they attain the greatest truth value 1 ∈ B, the unity of B. We must obviously deﬁne truth values by double induction, taking into consideration the way in which formulas are built up from atomic formulas and assigning truth values to the atomic formulas x ∈ y and x = y, where x, y ∈ V(B) in accord with the way in which V(B) is constructed. It is clear that if ϕ and ψ are evaluated formulas of ZFC and [[ϕ]] ∈ B and [[ψ]] ∈ B are their truth values then we should put (1) [[ϕ ∧ ψ]] := [[ϕ]] ∧ [[ψ]]; (2) [[ϕ ∨ ψ]] := [[ϕ]] ∨ [[ψ]]; (3) [[ϕ → ψ]] := [[ϕ]] ⇒ [[ψ]]; (4) [[¬ ϕ]] := [[ϕ]]∗ ; (5) [[(∀ x) ϕ(x)]] := x∈V(B) [[ϕ(x)]]; (6) [[(∃ x) ϕ(x)]] := x∈V(B) [[ϕ(x)]]; where the right-hand sides involve the Boolean operations that correspond to the logical connectives and quantiﬁers on the left-hand sides: ∧ is the meet of two elements, ∨ is the join of two elements, ∗ is the 8 Chapter 1. Boolean Valued Requisites taking of the complement of an element, and the operation ⇒ is intro duced as follows: a ⇒ b := a∗ ∨ b (a, b ∈ B). Moreover, E and E stand for the supremum and inﬁmum of a subset E ⊂ B. Only these deﬁnitions provide the value “unit” for the classical tautologies. The elements x1 ∨ · · · ∨ xn and x1 ∧ · · · ∧ xn may alternatively be denoted by n n x and k=1 k k=1 xk . 1.2.4. We turn to evaluating the atomic formulas x ∈ y and x = y for x, y ∈ V(B) . The intuitive idea consists in the fact that a B-valued set y is a “(lattice) fuzzy set,” i.e., a “set that contains an element z in dom(y) with probability y(z).” With this in mind and intending to preserve the logical tautology of x ∈ y ↔ (∃ z ∈ y) (x = z) as well as the axiom of extensionality, we arrive at the deﬁnition by recursion: [[x ∈ y]] := [[x = y]] := y(z) ∧ [[z = x]], z∈dom(y) x(z) ⇒ [[z ∈ y]] ∧ z∈dom(x) y(z) ⇒ [[z ∈ x]]. z∈dom(y) 1.2.5. We are able now to attach some meaning to the formal expressions of the form ϕ(x1 , . . . , xn ), where x1 , . . . , xn ∈ V(B) and ϕ is a formula of ZFC; i.e., we can deﬁne exactly in which sense the set-theoretic proposition ϕ(u1 , . . . , un ) is valid for x1 , . . . , xn ∈ V(B) . Namely, we say that the formula ϕ(x1 , . . . , xn ) is valid within V(B) or the elements x1 , . . . , xn possess the property ϕ if [[ϕ(x1 , . . . , xn )]] = 1. In this event we write V(B) ϕ(x1 , . . . , xn ). It is easy to check that the axioms and theorems of the ﬁrst-order predicate calculus are valid in V(B) . In particular (cp. 1.1.5), (1) [[x = x]] = 1, (2) [[x = y]] = [[y = x]], (3) [[x = y]] ∧ [[y = z]] [[x = z]], (4) [[x = y]] ∧ [[z ∈ x]] [[z ∈ y]], (5) [[x = y]] ∧ [[x ∈ z]] [[y ∈ z]]. 1.2.6. It is worth observing that for each formula ϕ we have V(B) |= x = y ∧ ϕ(x) → ϕ(y), 1.3. Transformations of the Boolean Valued Universe 9 i.e., in terms of Boolean truth values, [[x = y]] ∧ [[ϕ(x)]] [[ϕ(y)]]. 1.2.7. In a Boolean valued universe V(B) , the relation [[x = y]] = 1 in no way implies that the functions x and y (considered as elements of V) (B) coincide. For example, the function equal to zero on each layer Vα , where α 1, plays the role of the empty set in V(B) . This circumstance may complicate some constructions in the sequel. In this connection, (B) often we pass from V(B) to the separated Boolean valued universe V (B) (B) (B) preserving for the latter the same symbol V ; i.e., we put V := V . (B) Moreover, to deﬁne V , we consider the relation {(x, y) : [[x = y]] = 1} on the class V(B) which is obviously an equivalence. Choosing an element (a representative of least rank) in each class of equivalent functions, we arrive at the separated universe V (B) . Note that [[x = y]] = 1 =⇒ [[ϕ(x)]] = [[ϕ(y)]] is valid for an arbitrary formula ϕ of ZF and elements x and y in V(B) . Therefore, in the separated universe we can calculate the truth values of formulas paying no attention to the way of choosing representatives. Furthermore, working with the separated universe, for the sake of convenience we often consider (exercising due caution) a concrete representative rather than a class of equivalence as it is customary, for example, while dealing with function spaces. Concluding the section we state a very useful exhaustion principle for Boolean algebras. A subset of a Boolean algebra is said to be disjoint or antichain if the meet of its every two elements is 0. 1.2.8. Exhaustion Principle. Let B be a Boolean algebra. To each nonempty set B ⊂ B having the least upper bound, there is an antichain A ⊂ B such that A = B and, given x ∈ A, we may ﬁnd y in B with x y. 1.3. Transformations of the Boolean Valued Universe Each homomorphism of a Boolean algebra B induces a transformation of the Boolean valued universe V(B) . The topic to be discussed in 10 Chapter 1. Boolean Valued Requisites this section is the behavior of these transformations and, in particular, the manner in which they change the Boolean truth values of formulas. 1.3.1. Assume that π is a homomorphism of B in a complete Boolean algebra D. By recursion on a well-founded relation y ∈ dom(x), we deﬁne the mapping π ∗ : V(B) → V(D) using the formulas dom(π ∗ x) := {π ∗ y : y ∈ dom(x)} and π ∗ x : v → π(x(z)) : z ∈ dom(x), π ∗ z = v . 1.3.2. If σ is a complete homomorphism of a complete Boolean algebra A to B then (π ◦ σ)∗ = π ∗ ◦ σ ∗ . Moreover, IB∗ is the identity mapping on V(B) . If π is injective then π ∗ is also injective. Moreover, π ∗ x : π ∗ y → π(x(y)) (y ∈ dom(x)). A formula is called bounded or restricted if each bound variable in it is restricted by a bounded quantiﬁer; i.e., each of its quantiﬁers occures in the form (∀ x ∈ y) or (∃ x ∈ y) for some y (cp. 1.1.2 and 1.1.4), or it is equivalent in ZFC to a formula of this kind. A formula is of class Σ1 (or Σ1 -formula) if it is built up from atomic formulas and their negations using only the logical operations ∧, ∨, ∀ x ∈ y, ∃ x, or if it is equivalent in ZFC to such a formula. 1.3.3. Let π be a complete homomorphism from B to D, let ϕ(x1 , . . . , xn ) be a formula of ZFC, and let u1 , . . . , un ∈ V(B) . Then (1) if ϕ is a formula of class Σ1 and π is arbitrary then π([[ϕ(u1 , . . . , un )]]B ) [[ϕ(π ∗ u1 , . . . , π∗ un ]]D ; (2) if ϕ is a restricted formula and π is arbitrary, or π is an epimorphism and ϕ is arbitrary; then π([[ϕ(u1 , . . . , un )]]B ) = [[ϕ(π ∗ u1 , . . . , π∗ un )]]D . 1.3.4. Assume that π, ϕ, and u1 , . . . , un are the same as in 1.3.3 assume further that one of the following is fulfilled: (1) ϕ(x1 , . . . , xn ) is a formula of class Σ1 , and π is arbitrary; (2) π is an epimorphism and ϕ(x1 , . . . , xn ) is arbitrary. Then V(B) |= ϕ(u1 , . . . , un ) =⇒ V(D) |= ϕ(π ∗ u1 , . . . , π∗ un ). 1.3. Transformations of the Boolean Valued Universe 11 1.3.5. Assume that π, ϕ, and u1 , . . . , un are the same as in 1.3.3. Assume further that one of the following is fulfilled: (1) ϕ is restricted and π is a monomorphism; (2) π is an isomorphism and ϕ is arbitrary. Then V(B) |= ϕ(u1 , . . . , un ) ⇐⇒ V(D) |= ϕ(π ∗ u1 , . . . , π∗ un ). We now consider the two important particular cases: 1.3.6. Let B0 be an order closed subalgebra of a complete Boolean algebra B. Then B0 is itself a complete Boolean algebra and the least upper bound and the greatest lower bound of every subset of B0 are the same in B0 and in B. In these circumstances V(B0 ) ⊂ V(B) . Moreover, denoting by ı the identical embedding of B0 into B, we then see that ı is a complete monomorphism and ı∗ is an embedding of V(B0 ) into V(B) . Thus, the following is immediate from 1.3.5 (1). If ϕ(x1 , . . . , xn ) is a restricted formula and u1 , . . . , un ∈ V(B0 ) then V(B0 ) |= ϕ(u1 , . . . , un ) ⇐⇒ V(B) |= ϕ(u1 , . . . , un ). Since the two-valued algebra 2 := {0, 1} may be viewed as a complete subalgebra of the Boolean algebra B, the above is also valid for the universe V(2) . As can easily be seen from 1.4.5 (2,3) below, V(2) is naturally isomorphic to the von Neumann universe V. 1.3.7. Fix a nonzero b ∈ B and consider the relative subalgebra B̄ := [0, b] ⊂ B with unit 1̄ := b. The mapping πb : x → b ∧ x (x ∈ B) is a complete Boolean epimorphism from B onto B̄. Given u ∈ V(B) , the element b ∧ u := πb∗ (u) ∈ V(B̄) is deﬁned by recursion according to 1.3.1: dom(b ∧ u) := {b ∧ v : v ∈ dom(u)}; (b ∧ u)(v) = {π(u(z)) : z ∈ dom(u), b ∧ z = v}. Applying 1.3.3 (2) we get b ∧ [[ϕ(u1 , . . . , un )]]B = [[ϕ(b ∧ u1 , . . . , b ∧ un )]]B̄ . In particular, if ϕ(x1 , . . . , xn ) is a formula of ZFC and u1 , . . . , un ∈ V(B) then V(B) |= ϕ(u1 , . . . , un ) =⇒ V(B̄) |= ϕ(b ∧ u1 , . . . , b ∧ un ). 12 Chapter 1. Boolean Valued Requisites 1.4. Principles of Boolean Valued Set Theory The most important properties of a Boolean valued universe V(B) are stated in the three principles: 1.4.1. Transfer principle. If ϕ(x1 , . . . , xn ) is a theorem of ZFC then (∀x1 , . . . , xn ∈ V(B) ) V(B) |= ϕ(x1 , . . . , xn ) is also a theorem of ZFC. The transfer principle is established by rather laboriously checking that all axioms of ZFC have truth value 1 and all applications of the rule of inference increase the truth value of each formula. Sometimes, the transfer principle is worded as follows: “V(B) is the Boolean valued model of ZFC,” or “all theorems of ZFC are true in V(B) ,” or another simile. Using the transfer principle, we will often simplify the reference and say “by transfer.” 1.4.2. Maximum Principle. For each set-theoretic formula ϕ(u, x1 , . . . , xn ) the following is provable in ZFC: for every collection x1 , . . . , xn ∈ V(B) there exists x0 ∈ V(B) such that [[(∃ x) ϕ(x)]] = [[ϕ(x0 )]]. In particular, if it is true in V(B) that there is x for which ϕ(x) then there is an element x0 in V(B) (in the sense of V) for which [[ϕ(x0 )]] = 1. In symbols, the following is provable in ZFC: V(B) |= (∃ x) ϕ(x) =⇒ (∃ x0 ) V(B) ϕ(x0 ) . In other words, the maximum principle (∃ x0 ∈ V(B) ) [[ϕ(x0 )]] = [[ϕ(x)]] x∈V(B) is valid for every formula ϕ of ZFC. The last equality accounts for the origin of the term maximum principle. The proof of the principle is a simple consequence of mixing. A partition of unity in a Boolean algebra B is a family (bξ )ξ∈Ξ of elements of B such that (∀ ξ, η ∈ Ξ) (ξ = η =⇒ bξ ∧ bη = 0) and {bξ : ξ ∈ Ξ} = 1. 1.4. Principles of Boolean Valued Set Theory 13 1.4.3. Mixing Principle. Given a family (xξ )ξ∈Ξ in V(B) and a partition of unity (bξ )ξ∈Ξ in B, there exists a (unique) mixture of (xξ ) by (bξ ); i.e., the unique x ∈ V(B) such that bξ [[x = xξ ]] for all ξ ∈ Ξ. The mixture x of a family (xξ ) by (bξ ) is denoted as follows: x = mixξ∈Ξ (bξ xξ ) = mix{bξ xξ : ξ ∈ Ξ}. A set A of elements of V(B) is called cyclic if the family of elements of A is closed under mixing. The least cyclic set that includes A is the cyclic hull of A, and we denote it by cyc(A). 1.4.4. The comparative analysis, mentioned above, presumes that there is some close interconnection between the universes V and V(B) . In other words, we need a rigorous mathematical technique that would allow us to reveal the interplay between the interpretations of one and the same fact in the two universes V and V(B) . The base for the technique is constituted by the operations of canonical embedding, descent, and ascent. We start with the canonical embedding of the von Neumann universe, while the operations will be presented below. Given x ∈ V, denote by x∧ the standard name of x in V(B) , i.e., the element deﬁned by the recursion schema: ∅∧ := ∅, dom(x∧ ) := {y ∧ : y ∈ x}, im(x∧ ) := {1}. 1.4.5. Observe some simple properties of standard names we need in the sequel. Slightly abusing the language, we will call the passage from a set to its standard name canonical embedding. (1) Given x ∈ V and a formula ϕ of ZF, we have [[(∃ y ∈ x∧ ) ϕ(y)]] = {[[ϕ(z ∧ )]] : z ∈ x}, [[(∀ y ∈ x∧ ) ϕ(y)]] = {[[ϕ(z ∧ )]] : z ∈ x}. (2) The canonical embedding is injective. Moreover, for all x, y ∈ V we have x ∈ y ⇐⇒ V(B) |= x∧ ∈ y ∧ , x = y ⇐⇒ V(B) |= x∧ = y ∧ . (3) The canonical embedding sends V onto V(2) : (∀ u ∈ V(2) ) (∃ !x ∈ V) V(B) |= u = x∧ . 14 Chapter 1. Boolean Valued Requisites 1.4.6. If π is a complete homomorphism from B to a Boolean al∧ ∧ gebra D then π ∗ x∧ = x∧ for all x ∈ V, where ( · )∧ is the canonical (D) embedding of V to V . 1.4.7. Restricted Transfer Principle. For each restricted settheoretic formula ϕ the following is provable in ZFC: ϕ(x1 , . . . , xn ) ⇐⇒ V(B) |= ϕ(x∧1 , . . . , x∧n ) for all collection x1 , . . . , xn ∈ V. (B) Henceforth, working in the separated universe V , we agree to pre∧ serve the symbol x for the distinguished element of the class corresponding to x. 1.4.8. A correspondence from X to Y is a triple (X, Y, F ) with F ⊂ X × Y . The domain dom(Φ) and the image im(Φ) of Φ are introduced by dom(Φ) := {x ∈ X : (∃ y ∈ Y ) (x, y) ∈ F }; im(Φ) := {y ∈ Y : (∃ x ∈ X) (x, y) ∈ F }. The correspondence Φ is often identiﬁed with the point-to-set mapping x → Φ(x) := F (x) := {y ∈ Y : (x, y) ∈ F }. Consider another set Z and a correspondence Ψ := (Y, Z, G) from Y to Z. Put F −1 := {(y, x) ∈ Y × X : (x, y) ∈ F }; G ◦ F := {(x, z) ∈ X × Z : (∃ y ∈ Y )(x, y) ∈ F ∧ (y, z) ∈ G}. The correspondences Φ−1 := (Y, X, F −1 ) from Y to X and Ψ ◦ Φ := (X, Y, G ◦ F ) from X to Z are called the inverse Φ and the composite of Φ and Ψ. If A ⊂ X then Φ(A) := x∈A Φ(x) is the image of A under Φ. In particular, dom(Φ) = Φ−1 (Y ) and im(Φ) = Φ(X). Observe by way of example that the restricted transfer principle yields “Φ is a correspondence from X to Y ” ⇐⇒ V (B) |= “Φ∧ is a correspondence from X ∧ to Y ∧ ”; V(B) |= (Ψ ◦ Φ)∧ = Ψ∧ ◦ Φ∧ ∧ (Φ−1 )∧ = (Φ∧ )−1 ∧ Φ(A)∧ = Φ∧ (A∧ ); V(B) |= dom(Φ)∧ = dom(Φ∧ ) ∧ im(Φ)∧ = im(Φ∧ ); “f is a function from X to Y ” ⇐⇒ V(B) |= “f ∧ is a function from X ∧ to Y ∧ ” (moreover, f (x)∧ = f ∧ (x∧ ) within V(B) for every x ∈ X). 1.5. Descents 15 Thus, the standard name can be considered as a covariant functor of the category of sets (or correspondences) in V to the appropriate subcategory of V(2) in the separated universe V(B) . 1.4.9. A set X is finite if X coincides with the image of a function on a ﬁnite ordinal. In symbols, this is expressed as ﬁn(X); hence, ﬁn(X):= (∃ n)(∃ f )(n ∈ ω ∧ Fnc (f ) ∧ dom(f ) = n ∧ im(f ) = X). Obviously, the above formula is not bounded. Nevertheless there is a simple transformation rule for the class of ﬁnite sets under the canonical embedding. Denote by Pfin (X) the class of all ﬁnite subsets of X: Pfin (X):= {Y ∈ P(X) : ﬁn(Y )}. 1.4.10. For an arbitrary set X ∈ V we have V(B) |= Pfin (X ∧ ) = Pfin (X)∧ . 1.5. Descents In this section we deﬁne the mapping that assigns to each element x ∈ V(B) some subclass of V(B) which is a set in the sense of V. 1.5.1. Given an arbitrary element x of the (separated) Boolean valued universe V(B) , we deﬁne the descent x↓ of x as x↓ := y ∈ V(B) : [[y ∈ x]] = 1 . The class x↓ is a set; i.e., x↓ ∈ V for each x ∈ V(B) . If [[x = ∅]] = 1 then x↓ is a nonempty set by the maximum principle. If [[a ⊂ x ∧ b ⊂ x]] = 1 then (a ∩ b)↓ = a↓ ∩ b↓. We list the simplest properties of descending: 1.5.2. Let z ∈ V(B) and [[z = ∅]] = 1. Then for every formula ϕ of ZFC we have [[(∀ x ∈ z) ϕ(x)]] = {[[ϕ(x)]] : x ∈ z↓}, [[(∃ x ∈ z) ϕ(x)]] = {[[ϕ(x)]] : x ∈ z↓}. Moreover, there exists x0 ∈ z↓ such that [[ϕ(x0 )]] = [[(∃ x ∈ z) ϕ(x)]]. 16 Chapter 1. Boolean Valued Requisites 1.5.3. Let Φ be a correspondence from X to Y within V(B) . Thus, Φ, X, and Y are elements of V(B) and, moreover, [[Φ ⊂ X × Y ]] = 1. There is a unique correspondence Φ↓ from X↓ to Y ↓ such that Φ↓(A↓) = Φ(A)↓ for every nonempty subset A of X within V(B) . The correspondence Φ↓ from X↓ to Y ↓ involved in the above proposition is called the descent of the correspondence Φ from X to Y in V(B) . 1.5.4. The correspondence Φ↓ is extensional; i.e., it satisﬁes the condition y1 ∈ Φ↓(x1 ) =⇒ [[x1 = x2 ]] [[y1 = y2 ]] y2 ∈Φ(x2 ) for all x1 , x2 ∈ dom(Φ↓) = dom(Φ)↓. 1.5.5. (1) The descent of the composite of correspondences within (B) is the composite of their descents: V (Ψ ◦ Φ)↓ = Ψ↓ ◦ Φ↓. (2) The descent of the inverse correspondence within V(B) is the inverse of its descent: (Φ−1 )↓ = (Φ↓)−1 . (3) If IX ∈ V(B) is the identity mapping on X within V(B) then (IX )↓ = IX↓ . 1.5.6. Suppose that X, Y, f ∈ V(B) are such that [[X = ∅]] = 1, [[Y = ∅]] = 1, and [[f : X → Y ]] = 1; i.e., f is a mapping from X to Y within V(B) . Then there is a unique mapping f ↓ from X↓ to Y ↓ for which [[f ↓(x) = f (x)]] = 1 (x ∈ X↓). The descent of a function is extensional in the sense that (cp. 1.5.4) [[x1 = x2 ]] [[f ↓(x1 ) = f ↓(x2 )]] (x1 , x2 ∈ X↓). By 1.5.5 we can consider the descent as a functor from the category of B-valued sets and mappings (correspondences) to the category of the usual (i.e., in the sense of V) sets and mappings (correspondences). 17 1.6. Ascents 1.5.7. Given x1 , . . . , xn ∈ V(B) , denote by (x1 , . . . , xn )B the corresponding ordered n-tuple within V(B) . Assume that P is an n-ary rela∧ tion on X within V(B) ; i.e., X, P ∈ V(B) and [[P ⊂ X n ]] = 1 (n ∈ ω). ′ Then there exists an n-ary relation P on X↓ such that (x1 , . . . , xn ) ∈ P ′ ⇐⇒ [[(x1 , . . . , xn )B ∈ P ]] = 1. Slightly abusing notation, we denote the relation P ′ by the same symbol P ↓ and call it the descent of P . 1.5.8. Suppose that X ∈ V, X = ∅; i.e., X is a nonempty set. Let ι denote the canonical embedding x → x∧ (x ∈ X). Then ι(X)↑ = X ∧ and X = ι−1 (X ∧ ↓). Using the above relations, we can extend the descent operation to the case in which [[Ψ is a correspondence from X ∧ to Y ]] = 1, where Y ∈ V(B) and [[Y = ∅]] = 1. Namely, we put Ψ↓ := Ψ↓ ◦ ι. In this case, Ψ↓ is called the modified descent of the correspondence Ψ. (If the context excludes ambiguity then we simply speak of descents using simple arrow.) It is easy to see that Ψ↓ is the unique correspondence from X to Y ↓ satisfying the equality Ψ↓(x) = Ψ(x∧ )↓ (x ∈ X). If Ψ := g is a function then g↓ is a function from X to Y ↓ uniquely determined by [[g↓(x) = g(x∧ )]] = 1 (x ∈ X). 1.5.9. Let [[X ∧ → Y ]] stand for the set of all members g ∈ V(B) with [[g : X ∧ → Y ]] = 1, and [X → Y ↓] denote the set of all functions f : X → Y ↓. The mapping g → g↓ is a bijection between [[X ∧ → Y ]] and [X → Y ↓]. The converse mapping f → f ↑ is deﬁned in the next section (see 1.6.8). 1.6. Ascents We now consider some transformation acting in the reverse direction, i.e. sending each subset x ⊂ V(B) into an element of V(B) . 1.6.1. Let x ∈ V and x ⊂ V(B) ; i.e., let x be some set composed of B-valued sets or, in other words, x ∈ P(V(B) ). Put ∅↑ := ∅ and dom(x↑) = x, im(x↑) = {1} 18 Chapter 1. Boolean Valued Requisites if x = ∅. The element x↑ (of the separated universe V(B) , i.e. the distinguished representative of the class {y ∈ V(B) : [[y = x↑]] = 1}) is called the ascent of x. 1.6.2. The following hold for every x ∈ P(V(B) ) and every formula ϕ: [[(∀ z ∈ x↑) ϕ(z)]] = [[ϕ(y)]], y∈x [[(∃ z ∈ x↑) ϕ(z)]] = [[ϕ(y)]]. y∈x Introducing the ascent of a correspondence Φ ⊂ X × Y , we have to bear in mind a possible diﬀerence between the domain of departure X and the domain dom(Φ) := {x ∈ X : Φ(x) = ∅}. This diﬀerence is inessential for our further goals; therefore, we assume that, speaking of ascents, we always consider the correspondences Φ that are deﬁned everywhere; i.e., dom(Φ) = X. 1.6.3. Let X, Y, Φ ∈ V(B) , and let Φ be a correspondence from X to Y . There exists a unique correspondence Φ↑ from X↑ to Y ↑ within V(B) such that Φ↑(A↑) = Φ(A)↑ is valid for every subset A of dom(Φ) if and only if Φ is extensional; i.e., Φ satisﬁes the condition y1 ∈ Φ(x1 ) → [[x1 = x2 ]] [[y1 = y2 ]] y2 ∈Φ(x2 ) for all x1 , x2 ∈ dom(Φ). In this event, Φ↑ = Φ′ ↑, where Φ′ := {(x, y)B : (x, y) ∈ Φ}. The element Φ↑ is called the ascent of Φ. 19 1.6. Ascents 1.6.4. The composite of extensional correspondences is extensional. In addition, the ascent of a composite is equal to the composite of the ascents (within V(B) ): Assuming that dom(Ψ) ⊃ im(Φ) we have V(B) (Ψ ◦ Φ)↑ = Ψ↑ ◦ Φ↑. Note that if Φ and Φ−1 are extensional then (Φ↑)−1 = (Φ−1 )↑. But in general the extensionality of Φ in no way guarantees the extensionality of Φ−1 . 1.6.5. It is worth mentioning that if an extensional correspondence f is a function from X to Y then the ascent f ↑ is a function from X↑ to Y ↑. Moreover, the extensionality property can be stated as follows: [[x1 = x2 ]] [[f (x1 ) = f (x2 )]] (x1 , x2 ∈ X). It is immediate from the last property that for an extensional function f : X → Y , a family (xξ )ξ∈Ξ in X, and a partition of unity (bξ )ξ∈Ξ in B we have f mix bξ xξ = mix bξ f (xξ ). ξ∈Ξ ξ∈Ξ 1.6.6. Given X ⊂ V(B) , we denote by mix(X) the set of all mixtures of the form mix(bξ xξ ), where (xξ ) ⊂ X and (bξ ) is an arbitrary partition of unity. The following are referred to as the rules for canceling arrows or the Escher rules. Let X and X ′ be subsets of V(B) and let f : X → X ′ be an extensional mapping. Suppose that Y, Y ′ , g ∈ V(B) are such that [[ Y, Y ′ = ∅]] = [[ g : Y → Y ′ ]] = 1. Then X↑↓ = mix(X), Y ↓↑ = Y ; f = (f ↑↓)|X , g = g↓↑. Observe that mix(X) = cyc(X) (cp. 1.4.3). 1.6.7. Moreover, the mapping f → f ↑ is a one-to-one embedding of Ext(X, Y ) into Y X ↓, where Ext(X, Y ) is the set of all extensional mappings from X to Y and Y X is the set of all mappings from X to Y within V(B) ; i.e., Y X is a member of V(B) deﬁned as φ∈YX ↔φ:X →Y. This embedding is a bijection whenever X = mix(X) and Y = mix(Y ). 20 Chapter 1. Boolean Valued Requisites 1.6.8. Let X ∈ V, Y ∈ V(B) , and let ι be as in 1.5.8. By analogy with 1.5.8 we can extend the ascent operations to the case that Φ is a correspondence from X to Y ↓. We need only to put Φ↑ := (Φ ◦ ι−1 )↑. In this case, Φ↑ is called the modified ascent of Φ. (Again, when there is no ambiguity, we simply speak of ascents and use simple arrows.) Clearly, Φ↑ is the unique correspondence from X ∧ to Y within V(B) satisfying [[Φ↑(x∧ ) = Φ(x)↑]] = 1 (x ∈ X). Moreover, the correspondence Φ ◦ ι−1 is extensional, and consequently we have [[Φ↑(A∧ ) = Φ(A)↑]] = 1 for every nonempty A ⊂ X. If Φ := f is a function then f ↑ is a function from X ∧ to Y within V(B) uniquely determined by [[f ↑(x∧ ) = f (x)]] = 1 (x ∈ X). 1.6.9. The following useful fact is immediate from 1.4.10: Pfin (X ↑) = {θ ↑ : θ ∈ Pfin (X)} ↑ . 1.7. Algebraic B-systems In this section we describe a category of algebraic systems comprised of descents of Boolean algebraic systems. 1.7.1. A Boolean set or a B-set is a pair (X, d), where X ∈ V, X = ∅, and d is a B-metric on X; i.e., d is a mapping from X ×X to the Boolean algebra B which satisﬁes the following conditions for all x, y, z ∈ X: (a) d(x, y) = 0 ⇐⇒ x = y; (b) d(x, y) = d(y, x); (c) d(x, y) d(x, z) ∨ d(z, y). Each ∅ = X ⊂ V(B) gives an example of a B-set if we put d(x, y) := [[x = y]] = [[x = y]]∗ (x, y ∈ X). Another example is a nonempty X ∈ V with the “discrete B-metric” d; i.e., d(x, y) = 1 if x = y and d(x, y) = 0 if x = y. Given x ∈ X, a family (xξ ) in X, and a partition of unity (bξ ) in B, we write x = mix(bξ xξ ) provided that bξ ∧ d(x, xξ ) = 0 for all ξ. As 21 1.7. Algebraic B-systems in 1.4.3, x is called the mixture of (xξ ) by (bξ ). The mixture, if existent, is unique. A B-set X is called mix-complete if mix(bξ xξ ) exists in X for all families (xξ ) in X and partitions of unity in B. 1.7.2. Let (X, d) be some B-set. There exist an element X ∈ V(B) and an injection ι : X → X ′ := X ↓ such that d(x, y) = [[ιx = ιy]] (x, y ∈ X) and X ′ = mix(ιX). Thus, every x′ ∈ X ′ admits the representation x′ = mixξ∈Ξ (bξ ιxξ ), where (xξ )ξ∈Ξ ⊂ X and (bξ )ξ∈Ξ is a partition of unity in B. The element X ∈ V(B) is referred to as the Boolean valued representation of the B-set X. If X is a discrete B-set then X = X ∧ and ιx = x∧ (x ∈ X). If X ⊂ V(B) then ι↑ is an injection from X↑ to X (within V(B) ). A mapping f from a B-set (X, d) to a B-set (X ′ , d′ ) is said to be nonexpanding or contracting if d(x, y) d′ (f (x), f (y)) for all x, y ∈ X. 1.7.3. We exhibit some example of a B-set that is important for the sequel. Let X be a vector lattice and B := P(X). Put d(x, y) := {|x − y|}⊥⊥ (x, y ∈ X). Clearly, d satisﬁes 1.7.1 (b, c). At the same time, 1.7.1 (a) is valid only provided that X is Archimedean (cp. 2.1.3). Thus, (X, d) is a B-set if and only if the vector lattice X is Archimedean. 1.7.4. Recall that a signature is a 3-tuple σ := (F, P, a), where F and P are some (possibly, empty) sets and a is a mapping from F ∪ P to ω. If the sets F and P are ﬁnite then σ is a finite signature. In applications we usually deal with algebraic systems of ﬁnite signature. An n-ary operation and an n-ary predicate on a B-set A are contractive mappings f : An → A and p : An → B, respectively. By deﬁnition, f and p are contractive mappings provided that d(f (a0 , . . . , an−1 ), f (a′0 , . . . , a′n−1 )) n−1 k=0 n−1 ds p(a0 , . . . , an−1 ), p(a′0 , . . . , a′n−1 ) d(ak , a′k ), d(ak , a′k ) k=0 for all a0 , a′0 , . . . , an−1 , a′n−1 ∈ A, where d is the B-metric of A, and ds is the symmetric diﬀerence on B; i.e., ds (b1 , b2 ):= b1 △ b2 := (b1 ∧ b∗2 ) ∨ (b∗1 ∧ b2 ). 22 Chapter 1. Boolean Valued Requisites Clearly, the above deﬁnitions depend on B and it would be cleaner to speak of B-operations, B-predicates, etc. We adhere to a simpler practice whenever this entails no confusion. 1.7.5. An algebraic B-system A of signature σ is a pair (A, ν), where A is a nonempty B-set, the underlying set or carrier or universe of A, and ν is a mapping such that (a) dom(ν) = F ∪ P ; (b) ν(f ) is an a(f )-ary operation on A for all f ∈ F ; and (c) ν(p) is an a(p)-ary predicate on A for every p ∈ P . It is in common parlance to call ν the interpretation of A, in which case the notations f ν and pν are substitutes for ν(f ) and ν(p). The signature of an algebraic B-system A := (A, ν) is often denoted by σ(A); while the universe A of A, by |A|. Since A0 = {∅}, the nullary operations and predicates on A are mappings from {∅} to the set A and to the algebra B respectively. We agree to identify a mapping g : {∅} → A ∪ B with the element g(∅). Each nullary operation on A thus transforms into the unique member of A. Analogously, the set of all nullary predicates on A turns into the Boolean algebra B. If F := {f1 , . . . , fn } and P := {p1 , . . . , pm } then an algebraic B-system of signature σ is often written down as (A, ν(f1 ), . . . , ν(fn ), ν(p1 ), . . . , ν(pm )) or even (A, f1 , . . . , fn , p1 , . . . , pm ). In this event, the expression σ = (f1 , . . . , fn , p1 , . . . , pm ) is substituted for σ = (F, P, a). 1.7.6. We now address the B-valued interpretation of a ﬁrst-order language. Consider an algebraic B-system A:= (A, ν) of signature σ:= σ(A):= (F, P, a). Let ϕ(x0 , . . . , xn−1 ) be a formula of signature σ with n free variables. Assume given a0 , . . . , an−1 ∈ A. We may readily deﬁne the truth value |ϕ|A (a0 , . . . , an−1 ) ∈ B of a formula ϕ in the system A for the given values a0 , . . . , an−1 of the variables x0 , . . . , xn−1 . The deﬁnition proceeds as usual by induction on the complexity of ϕ: Considering propositional connectives and quantiﬁers, we put |ϕ ∧ ψ|A (a0 , . . . , an−1 ) := |ϕ|A (a0 , . . . , an−1 ) ∧ |ψ|A (a0 , . . . , an−1 ); |ϕ ∨ ψ|A (a0 , . . . , an−1 ) := |ϕ|A (a0 , . . . , an−1 ) ∨ |ψ|A (a0 , . . . , an−1 ); |¬ ϕ|A (a0 , . . . , an−1 ) := |ϕ|A (a0 , . . . , an−1 )∗ ; |ϕ|A (a0 , a1 , . . . , an−1 ); |(∀ x0 ) ϕ|A (a1 , . . . , an−1 ) := a0 ∈A 1.7. Algebraic B-systems |(∃ x0 ) ϕ|A (a1 , . . . , an−1 ) := 23 |ϕ|A (a0 , a1 , . . . , an−1 ). a0 ∈A 1.7.7. Now, the case of atomic formulas is in order. Suppose that p ∈ P symbolizes an m-ary predicate, q ∈ P is a nullary predicate, and t0 , . . . , tm−1 are terms of signature σ assuming values b0 , . . . , bm−1 at the given values a0 , . . . , an−1 of the variables x0 , . . . , xn−1 . By deﬁnition, we let |ϕ|A (a0 , . . . , an−1 ) := ν(q), |ϕ|A (a0 , . . . , an−1 ) := d(b0 , b1 )∗ , ν A |ϕ| (a0 , . . . , an−1 ) := p (b0 , . . . , bm−1 ), if ϕ := q ν ; if ϕ := (t0 = t1 ); if ϕ := pν (t0 , . . . , tm−1 ), where d is a B-metric on A. 1.7.8. Say that ϕ(x0 , . . . , xn−1 ) is valid in A at the given values a0 , . . . , an−1 ∈ A of x0 , . . . , xn−1 and write A |= ϕ(a0 , . . . , an−1 ) provided that |ϕ|A (a0 , . . . , an−1 ) = 1B . The alternative expressions are as follows: a0 , . . . , an−1 ∈ A satisfies ϕ(x0 , . . . , xn−1 ), or ϕ(a0 , . . . , an−1 ) holds true in A. In case B := {0, 1}, we arrive at the conventional deﬁnition of the validity of a formula in an algebraic system. Recall that a closed formula ϕ of signature σ is a tautology if ϕ is valid on every algebraic 2-system of signature σ. 1.7.9. Consider algebraic B-systems A := (A, ν) and C := (C, μ) of the same signature σ. The mapping h : A → C is a homomorphism of A to C provided that, for all a0 , . . . , an−1 ∈ A, the following are valid: (1) dB (h(a1 ), h(a2 )) dA (a1 , a2 ); (2) h(f ν ) = f ν , a(f ) = 0; (3) h(f ν (a0 , . . . , an−1 )) = f ν (h(a0 ), . . . , h(an−1 )), 0 = n := a(f ); (4) pν (a0 , . . . , an−1 ) pμ (h(a0 ), . . . , h(an−1 )), n := a(p). A homomorphism h is called strong if (5) a (p) := n = 0 for all p ∈ P , and for all c0 , . . . , cn−1 ∈ C we have pμ (c0 , . . . , cn−1 ) pν (a0 , . . . , an−1 ) ∧ dC (c0 , h(a0 )) a0 ,...,an−1 ∈A ∧ · · · ∧ dC (cn−1 , h(an−1 )) . 24 Chapter 1. Boolean Valued Requisites 1.7.10. If a homomorphism h is injective and 1.7.9 (1, 4) are fulﬁlled with equality holding, then h is said to be an isomorphism from A to C. Undoubtedly, all surjective isomorphisms h and, in particular, the identity mapping IA : A → A are strong homomorphisms. The composite of (strong) homomorphisms is a (strong) homomorphism. Clearly, if h is a homomorphism and h−1 is a homomorphism too, then h is an isomorphism. Note again that in the case of the two element Boolean algebra 2 := {0, 1} we come to the conventional notions of homomorphism, strong homomorphism, and isomorphism. 1.8. Boolean Valued Algebraic Systems Before giving the general deﬁnition of descent of an algebraic system, consider the descent of a very simple but important algebraic system, the two element Boolean algebra. Choose two arbitrary elements, say 0, 1 ∈ V(B) , satisfying [[0 = 1]] = 1B . We may for instance assume that 0 := 0∧B and 1 := 1∧B . 1.8.1. The descent C of the two-element Boolean algebra {0, 1}B ∈ (B) V is a complete Boolean algebra isomorphic to B. The formulas [[χ(b) = 1]] = b, [[χ(b) = 0]] = b∗ (b ∈ B) yield the isomorphism χ : B → C. 1.8.2. Let X and Y be some B-sets, let X and Y be their Boolean valued representations, and let ι and κ be the corresponding embeddings X → X ↓ and Y → Y ↓. If f : X → Y is a contracting mapping then there is a unique element g ∈ V(B) such that [[g : X → Y ]] = 1 and f = κ −1 ◦ g↓ ◦ ι. We also accept the denotations X := F ∼ (X) := X ∼ and g := F ∼ (f ) := f ∼ . 1.8.3. The following are valid: (1) V(B) |= f (A)∼ = f ∼ (A∼ ) for A ⊂ X. (2) If g : Y → Z is a contraction then g ◦ f is a contraction and V(B) |= (g ◦ f )∼ = g ∼ ◦ f ∼ . (3) V(B) |= “f ∼ is injective” if and only if f is a B-isometry. (B) ∼ (4) V |= “f is surjective” if and only if for every y ∈ Y we have {d(f (x), y) : x ∈ X} = 1. 1.8. Boolean Valued Algebraic Systems 25 1.8.4. Consider an algebraic system A of signature σ ∧ within V(B) , and let [[A = (A, ν)B ]] = 1 for some A, ν ∈ V(B) . The descent of A is the pair A↓ := (A ↓, μ), where μ is the function determined from the formulas: μ : f → (ν↓(f ))↓ (f ∈ F ), μ : p → χ−1 ◦ (ν↓(p))↓ (p ∈ P ). Here χ is the above isomorphism of the Boolean algebras B and {0, 1}B↓. In more detail, the modiﬁed descent ν↓ is the mapping with domain dom(ν↓) = F ∪ P . Given p ∈ P , observe [[a (p)∧ = a∧ (p∧ )]] = 1, [[ν↓(p) = ν(p∧ )]] = 1 and so V(B) |= ν↓(p) : Aa (f )∧ → {0, 1}B. It is now obvious that (ν↓(p))↓ : (A↓)a (f ) → C := {0, 1}B↓ and we can put μ(p) := χ−1 ◦ (ν↓(p))↓. 1.8.5. Let ϕ(x0 , . . . , xn−1 ) be a ﬁxed formula of signature σ in n free variables. Write down the formula Φ(x0 , . . . , xn−1 ,A) in the language of set theory which formalizes the proposition A |= ϕ(x0 , . . . , xn−1 ). Recall that the formula A |= ϕ(x0 , . . . , xn−1 ) determines an n-ary predicate on A or, which is the same, a mapping from An to {0, 1}. By the maximum and transfer principles, there is a unique element |ϕ|A ∈ V(B) such that ∧ [[|ϕ|A : An → {0, 1}B]] = 1, [[|ϕ|A (a↑) = 1]] = [[Φ(a(0), . . . , a(n − 1), A)]] = 1 for every function a : n → A↓. Instead of |ϕ|A (a↑) we will write |ϕ|A (a0 , . . . , an−1 ), where al := a(l). Therefore, the formula V(B) |= “ϕ(a0 , . . . , an−1 ) is valid in A” holds true if and only if [[Φ(a0 , . . . , an−1 , A)]] = 1. 1.8.6. Let A be an algebraic system of signature σ∧ within V(B) . Then A↓ is a laterally complete algebraic B-system of signature σ. In this event χ ◦ |ϕ|A↓ = |ϕ|A ↓ for each formula ϕ of signature σ. An algebraic system is laterally complete whenever its universe is mix-complete. 1.8.7. Let A and B be algebraic systems of the same signature σ ∧ within V(B) . Put A′ := A↓ and B′ := B↓. Then, if h is a homomorphism (strong homomorphism) within V(B) from A to B then h′ := h↓ is a homomorphism (strong homomorphism) of the B-systems A′ and B′ . 26 Chapter 1. Boolean Valued Requisites Conversely, if h′ : A′ → B′ is a homomorphism (strong homomorphism) of algebraic B-systems then h := h′↑ is a homomorphism (strong homomorphism) from A to B within V(B) . 1.8.8. Let A := (A, ν) be an algebraic B-system of signature σ. Then there are A and μ ∈ V(B) such that the following are fulfilled: (1) V(B) |= “(A , μ) is an algebraic system of signature σ ∧ ”. (2) If A′ := (A′ , ν ′ ) is the descent of (A , μ) then A′ is a laterally complete algebraic B-system of signature σ. (3) There is an isomorphism ı from A to A′ such that A′ = mix(ı(A)). (4) For every formula ϕ of signature σ in n free variables, we have ′ |ϕ|A (a0 , . . . , an−1 ) = |ϕ|A (ı(a0 ), . . . , ı(an−1 )) ∼ = χ−1 ◦ (|ϕ|A )↓(ı(a0 ), . . . , ı(an−1 )) for all a0 , . . . , an−1 ∈ A and χ the same as in 1.8.1. 1.9. Boolean Valued Ordinals and Cardinals Now we dwell for a while on the properties of ordinals and cardinals within the Boolean valued universe. 1.9.1. A set x is transitive (not to be confused with a transitive relation) if each member of x is also a subset of x: Tr(x) := (∀ y) (y ∈ x → y ⊂ x). An ordinal is a transitive set well-ordered by membership. The record Ord(x) means that x is ordinal. The terms ordinal number or transfinite number are also in common parlance. Denote by On the class of all ordinals. We often let lowercase Greek letters stand for ordinals. Moreover, we use the abbreviations: α < β := α ∈ β, α β := (α ∈ β) ∨ (α = β), α + 1 := α ∪ {α}. If α < β then we say that α precedes β and β succeeds α. 1.9.2. If x ⊂ On is a set then x is the least upper bound of x in the class On ordered by the membership relation ∈. The least upper 1.9. Boolean Valued Ordinals and Cardinals 27 bound of a set of ordinals x is usually denoted by lim(x). An ordinal α is a limit ordinal if α = ∅ and lim(α) = α. In other words, α is a limit ordinal provided that α cannot be written down as α = β + 1 with β ∈ On. The least limit ordinal whose existence is ensured by the axiom of inﬁnity is denoted by ω (or ω0 ; see 1.9.4 (2)). The least ordinal, the zero set 0 := ∅, belongs to ω. The successor 1 := 0 + 1 = 0 ∪{0} = {∅} contains the only element 0. Furthermore, 2 := 1 ∪ {1} = {0} ∪ {1} = {0, 1} = {0, {0}}, 3 := 2 ∪ {2} = {0, {0}, {{0, {0}}}, etc. Thus, ω := {0, {0}, {0, {0}}, . . .} = {0, 1, 2, . . .}. The following notation is also used: N := ω \ {0} = {1, 2, . . .}. The members of ω are finite ordinals or positive integers. The elements of N are called natural numbers or simply naturals by historical reasons. But the whole of ω is called the naturals rather often too (since 0 seems very common today). 1.9.3. Two sets are equipollent, or equipotent, or of the same cardinality if there is a bijection of one of them onto the other. An ordinal that is equipotent to no preceding ordinal is a cardinal. Each natural is a cardinal. A cardinal not in ω is an infinite cardinal. Therefore, ω is the least inﬁnite cardinal. Given an ordinal α, we denote by ωα an inﬁnite cardinal such that the ordered set of all inﬁnite cardinals less than ωα is similar to α. If such a cardinal exists then it is unique. 1.9.4. Cardinal Comparability Principle. The following are valid: (1) Inﬁnite cardinals form a well-ordered proper class. (2) To each ordinal α there is a cardinal ωα so that the mapping α → ωα is a similarity between the class of ordinals and the class of inﬁnite cardinals. (3) There is a mapping | · | from the universal class U onto the class of all cardinals such that the sets x and |x| are equipollent for all x ∈ U. 1.9.5. Clearly, Ord(x) is a bounded formula. Since lim(α) α for every ordinal α, the formula Ord(x) ∧ x = lim(x) may be rewritten as 28 Chapter 1. Boolean Valued Requisites Ord(x) ∧ (∀ t ∈ x)(∃ s ∈ x)(t ∈ s). Hence, Ord(x) ∧ x = lim(x) is a bounded formula as well. Finally, the record Ord(x) ∧ x = lim(x) ∧ (∀ t ∈ x)(t = lim(t) → t = 0) convinces us that the “least limit ordinal” is a bounded formula too. Hence α is the least limit ordinal if and only if V(B) |= “α∧ is the least limit ordinal” by the restricted transfer principle. Since ω is the least limit ordinal in V, we have V(B) |= “ω ∧ is the least limit ordinal.” 1.9.6. It can be demonstrated that V(B) |= “On∧ is the unique ordinal class that is not an ordinal” (with On∧ deﬁned in an appropriate way). Given x ∈ V(B) , we thus have [[Ord(x)]] = [[x = α∧ ]]. α∈On This yields the convenient formulas for quantiﬁcation over ordinals: [[(∀ x) Ord(x) → ψ(x) ]] = [[ψ(α∧ )]], α∈On [[(∃ x) Ord(x) ∧ ψ(x) ]] = [[ψ(α∧ )]]. α∈On 1.9.7. Each ordinal within V(B) is a mixture of some set of standard ordinals. In other words, given x ∈ V(B) , we have V(B) |= Ord(x) if and only if there are an ordinal β ∈ On and a partition of unity (bα )α∈β ⊂ B such that x = mixα∈β bα α∧ . 1.9.8. By transfer every Boolean valued model enjoys the classical cardinal comparability principle. In other words, there is a V(B) -class Cn whose elements are only cardinals. Let Card(α) denote the formula that declares α a cardinal. Within V(B) we then see that α ∈ Cn ↔ Card(α). Clearly, the class of ordinals On∧ is similar to the class of inﬁnite cardinals, and we denote the similarity from On∧ into Cn by α → ℵα . In particular, to each standard ordinal α ∈ On there is a unique inﬁnite cardinal ℵα∧ within V(B) since [[Ord(α∧ )]] = 1. 1.9.9. Recall that it is customary to refer to the standard names of ordinals and cardinals as standard ordinals and standard cardinals within V(B) . 1.9. Boolean Valued Ordinals and Cardinals 29 (1) The standard name of the least inﬁnite cardinal is the least inﬁnite cardinal: V(B) |= (ω0 )∧ = ℵ0 . (2) Within V(B) there is a mapping |·| from the universal class UB into the class Cn such that x and |x| are equipollent for all x. The standard names of equipollent sets are of the same cardinality: (∀ x ∈ V) (∀ y ∈ V) |x| = |y| =⇒ [[|x∧ | = |y ∧ |]] = 1 . 1.9.10. (1) If the standard name of an ordinal α is a cardinal then α is a cardinal too: (∀ α ∈ On) V(B) |= Card(α∧ ) =⇒ Card(α). (2) The standard name of a ﬁnite cardinal is a ﬁnite cardinal too: (∀ α ∈ On) α < ω =⇒ V(B) |= Card(α∧ ) ∧ α∧ ∈ ℵ0 . 1.9.11. Given x ∈ V(B) , we have V(B) |= Card(x) if and only if there are nonempty set of cardinals Γ and a partition of unity (bγ )γ∈Γ ⊂ B such that x = mixγ∈Γ bγ γ ∧ and V(Bγ ) |= Card(γ ∧ ) with Bγ := [0, bγ ] for all γ ∈ Γ. In other words, each Boolean valued cardinal is a mixture of some set of relatively standard cardinals. 1.9.12. A σ-complete Boolean algebra B is said to be σ-distributive if B satisﬁes one of the following equivalent conditions (cp. [365, 19.1]): (1) n∈N m∈N bnm = m∈NN n∈N bnm(n) for all (bnm )n,m∈N in B; (2) n∈N m∈N bnm = m∈NN n∈N bnm(n) for all (bnm )n,m∈N in B; (3) ε∈{1,−1}N n∈N ε(n)bn = 1 for all (bn )n∈N in B. Here 1bn := bn and (−1)bn is the complement of bn . It is worth noting that σ-distributive Boolean algebras are often referred to as (ω, ω)-distributive Boolean algebras. This term is related to a more general notion, (α, β)-distributivity, where α and β are arbitrary cardinals. 1.9.13. If B is a complete Boolean algebras then the following are equivalent: (1) B is σ-distributive. (2) V(B) |= (ℵ0 )ℵ0 = (ω ω )∧ . (3) V(B) |= P(ℵ0 ) = P(ω)∧ . 30 Chapter 1. Boolean Valued Requisites The latter is the result by Scott on (α, β)-distributive Boolean algebras which was formulated in the case α = β = ω (cp. [43, 2.14]). More details and references are collected in [249]. 1.10. Boolean Algebras In this section we specify the general results of 1.8.6–1.8.8 on algebraic B-systems for Boolean algebras. 1.10.1. Let B be a complete Boolean algebra and let ı be a Boolean homomorphism from B to a Boolean algebra D. Deﬁne the mapping d : D × D → B by putting d(x, y) := {b ∈ B : ı(b∗ ) ∧ x = ı(b∗ ) ∧ y} (x, y ∈ D). It can easily be seen that d is a Boolean (or B-valued ) semimetric; i.e., d satisﬁes 1.7.1 (b, c) and d(x, x) = 0 for all x ∈ D. Moreover, d is a B-metric whenever ı is a complete homomorphism. The results of Section 1.8 are applicable to D: If ı : B → D is a complete Boolean homomorphism then D is an algebraic B-system of signature (∨, ∧, ∗, 0, 1). This B-system is laterally complete whenever D is complete. 1.10.2. Let D be a Boolean algebra within V(B) and D := D↓. Then D is a Boolean algebra and there exists a complete monomorphism ı : B → D such that for all x, y ∈ D and b ∈ B we have b [[x y]] ⇐⇒ ı(b) ∧ x ı(b) ∧ y. Moreover, D is order complete if and only if so is D within V(B) . ⊳ In view of 1.8.6 D is a laterally complete algebraic B-system of signature (∨, ∧, ∗, 0, 1). The fact that D is a Boolean algebra follows also from 1.8.6. ⊲ 1.10.3. Let D1 and D2 be complete Boolean algebras in V(B) . Put Dk := Dk ↓ and denote by ık : B → Dk (k := 1, 2) the monomorphism from 1.10.2. If h ∈ V(B) is an internal isomorphism from D1 to D2 , then H := h↓ is a Boolean isomorphism from D1 to D2 such that the diagram 1.11. Applications to Boolean Homomorphisms 31 commutes: B❄ ⑧⑧ ❄❄❄ ⑧ ❄❄ı2 ı1 ⑧⑧ ❄❄ ⑧ ⑧ ❄❄ ⑧⑧ ❄ ⑧ ⑧ / D2 D1 H Conversely, if H : D1 → D2 is an isomorphism of Boolean algebras and the above diagram commutes, then h := H↑ is a Boolean isomorphism from D1 to D2 within V(B) . ⊳ All can be deduced from 1.8.7 and 1.10.2. ⊲ 1.10.4. Assume that D is a complete Boolean algebra and j : B → D is a complete monomorphism. Then there are a complete Boolean algebra D within V(B) and an isomorphism H from D onto D′ := D↓ such that the diagram commutes: B ⑧ ❄❄❄ ⑧ ❄❄ ′ ⑧ j ⑧⑧ ❄❄ı ⑧ ❄❄ ⑧ ⑧ ❄❄ ⑧ ⑧ ⑧ / D′ D H where ı′ is the monomorphism from B to D′ defined as in 1.10.2. ⊳ According to 1.10.1 D is a laterally complete algebraic B-system of signature σ := {∨, ∧, ∗, 0, 1}. By 1.8.8 we can assume without loss of generality that D coincides with D↓ and j = ı for some algebraic system D within V(B) of signature σ ∧ . If a formula ϕ formalizes the axioms of a complete Boolean algebra, then we can check by direct calculation of Boolean truth values that |ϕ|D = 1. From 1.8.8 we deduce [[|ϕ|D = 1]] = 1. Hence, D is a complete Boolean algebra within V(B) . ⊲ 1.11. Applications to Boolean Homomorphisms In this section we demonstrate that some Hahn–Banach type extension results for Boolean homomorphisms can be deduced by Boolean valued interpretation of the properties of ﬁlters and ultraﬁlters. 32 Chapter 1. Boolean Valued Requisites 1.11.1. Let X be a set, and let B be a complete Boolean algebra. Given σ ∈ V(B) with [[σ ⊂ X ∧ ]] = 1, deﬁne hσ : X → B as hσ (x) := [[x∧ ∈ σ]] (x ∈ X). The mapping σ → hσ is a bijection between P(X ∧ )↓ and BX . ⊳ This mapping is clearly injective. Take h : X → B. Let η stand for the modiﬁed ascent of χ ◦ h : X → {0, 1}B↓, with χ the same as in 1.8.1. By the maximum principle, we can deﬁne σ ∈ P(X ∧ ) as σ := {x ∈ X ∧ : η(x) = 1}. Then we derive from 1.8.1 that h(x) = [[χ(h(x) = 1]] = [[η(x∧ ) = 1]] = [[x∧ ∈ σ]]. So, h = hσ . ⊲ 1.11.2. Take another Boolean algebra A. A mapping p : A → B is called a submorphism (supermorphism), provided that p(1A ) = 1 and p(x ∨ y) = p(x) ∨ p(y) ( p(0A ) = 0 and p(x ∧ y) = p(x) ∧ p(y) respectively) for all x, y ∈ A. If h∗ : x → h(x)∗ (x ∈ A) is a Boolean homomorphism then we call h : A → B a Boolean antimorphism. The fact that A is a Boolean algebra can be expressed by a restricted formula. Consequently, V(B) |= “A∧ is a Boolean algebra.” 1.11.3. Assume that σ ∈ P(A∧ )↓. Then the following hold: (1) V(B) |= “σ is an ideal” ⇐⇒ h∗σ is a submorphism. (2) V(B) |= “σ is a filter” ⇐⇒ hσ is a supermorphism. (3) V(B) |= “σ is an ultrafilter” ⇐⇒ hσ is a Boolean homomorphism. (4) V(B) |= “σ is a maximal ideal” ⇐⇒ hσ is a Boolean antimorphism. ⊳ A subset A of a Boolean algebra is a ﬁlter (an ideal) if and only if A does not contain 0 (1), and the meet (join) of two elements of B belongs to A if and only if each of the two elements belongs to A. The same fact holds for the Boolean valued universe by transfer. Therefore, the formulas V(B) |= “σ is an ideal” and V(B) |= “ρ is a ﬁlter” amount to the two groups of equalities: [[1∧A ∈ σ]] = 0, [[x∧ ∨ y ∧ ∈ σ]] = [[x∧ ∈ σ]] ∧ [[y ∧ ∈ σ]]; [[0∧A ∈ ρ]] = 0, [[x∧ ∧ y ∧ ∈ ρ]] = [[x∧ ∈ ρ]] ∧ [[y ∧ ∈ ρ]]. This yields (1) and (2) by 1.11.1. Furthermore, a ﬁlter in a Boolean algebra is an ultraﬁlter if and only if each element or its Boolean complement 1.11. Applications to Boolean Homomorphisms 33 belongs to the ﬁlter. Interpreting this criterion in the Boolean valued universe we see that V(B) |= “σ is an ultraﬁlter” if and only if hσ is a supermorphism and [[x∗ ∈ σ]] ∨ [[x ∈ σ]] = 1 (x ∈ A∧ ), or, equivalently, hσ (x∗ ) ∨ hσ (x) = 1 (x ∈ A). Observe that hσ (x∗ ) ∧ hσ (x) = [[(x∗ )∧ ∈ σ]] ∧ [[x∧ ∈ σ]] = [[0 ∈ σ]] = 0 amounts to the identity hσ (x∗ ) = hσ (x)∗ . These arguments prove (3), while (4) is easy from (3). ⊲ 1.11.4. Let Hom(A, B) be the set of all Boolean homomorphisms from A to B. By U(A∧ ) we denote the element of V(B) such that [[U(A∧ ) is the set of all ultraﬁlters in the Boolean algebra A∧ ]] = 1. The mapping ψ → hψ is a bijection between U(A∧ )↓ and Hom(A, B). ⊳ The claim follows from 1.11.1 and 1.11.3(3). ⊲ 1.11.5. Sandwich Theorem. Let p, q : A → B be such that p is a submorphism and q is a supermorphism. Assume that q(x) p(x) for all x ∈ A. Then there is h ∈ Hom(A, B) satisfying q(x) h(x) p(x) (x ∈ A). ⊳ By 1.11.1 there are ρ, σ ∈ P(A∧ )↓ such that q = hσ and p∗ = hρ . By 1.11.3 V(B) |= “σ is a ﬁlter” and V(B) |= “ρ is an ideal.” Moreover, [[x∧ ∈ σ]] = q(x) p(x) = [[x∧ ∈ / ρ]] and so V(B) |= “σ ∩ ρ is empty.” By the transfer and maximum principles we see that the ﬁlters σ and ρ∗ := {x∗ : x ∈ ρ} lie in some common ﬁlter within V(B) . Otherwise, there would exist x ∈ σ and y ∈ ρ such that x ∧ y ∗ = 0 or, equivalently, x y. But this would imply that x ∈ ρ, contradicting the condition σ ∩ ρ = ∅. We now choose some ultraﬁlter ψ ⊂ A∧ within V(B) that includes both σ and ρ∗ . Put h := hψ and note that h is a Boolean homomorphism by 1.11.3 (3). Clearly, σ ⊂ ψ and ψ ∩ ρ = ∅. Thus x∈σ→x∈ψ→x∈ / ρ for all x ∈ A∧ . Calculating the Boolean truth value of the latter formula yields q(x) h(x) p(x). ⊲ Deriving corollaries to the Sandwich Theorem, we mention two facts about extension of Boolean homomorphisms. The ﬁrst is analogous to the Hahn–Banach Extension Theorem for linear functionals. 1.11.6. Hahn–Banach Theorem for Boolean Homomorphisms. Let A0 be a subalgebra of a Boolean algebra A and let p : A → B be a submorphism. Assume that a Boolean homomorphism h0 : A0 → B satisfies the inequality h0 (x0 ) p(x0 ) for all x0 ∈ A0 . Then there exists a Boolean homomorphism h : A → B such that h(x) p(x) (x ∈ A) and h(x0 ) = h0 (x0 ) (x0 ∈ A0 ). 34 Chapter 1. Boolean Valued Requisites ⊳ Introduce the mapping q : A → B by letting q(x) := {h0 (a) : a ∈ A0 , a x} (x ∈ A). Clearly, q is a supermorphism, q p, and q|A0 = h0 . By 1.11.5 there is h ∈ Hom(A, B) satisfying q h p. In particular, h0 |A0 h. Given x ∈ A0 , we hence see that h(x) = h(x∗ )∗ h0 (x∗ )∗ = h0 (x). Therefore, h|A0 = h and h is a desired homomorphism. ⊲ 1.11.7. Sikorski Extension Theorem. Each Boolean homomorphism h0 from a subalgebra A0 of an arbitrary Boolean algebra A to a complete Boolean algebra B admits an extension to a Boolean homomorphism h defined on the whole of A. ⊳ Let p(0A ) = 0 and p(x) = 1 for 0A = x ∈ A. Then p is a submorphism and h0 p|A0 . So, the claim follows from 1.11.6. We may proceed otherwise not appealing to 1.11.6, but recalling 1.11.1 and 1.11.3. Indeed, [[A∧0 is a subalgebra of the algebra A∧ ]] = 1, and by 1.11.1 h0 = hσ for some σ ∈ P(A∧0 )↓. By 1.11.3 (3) [[σ is an ultraﬁlter in A∧0 ]] = 1. The claim follows now from the fact that σ, presenting (within V(B) ) a ﬁlterbase in A∧ , admits extension to some ultraﬁlter ψ ⊂ A∧ , so that h = hψ is a sought homomorphism. ⊲ 1.12. Variations on the Theme The purpose of this section is to present brieﬂy intuitionistic set theory and quantum set theory as counterparts of Boolean valued set theory. This is done by constructing universes based respectively on a complete Heyting algebra and a complete orthomodular lattice, which are reasonable models of set theory. Intuitionistic propositional calculus is based on Heyting algebras and quantum propositional calculus is based on orthomodular lattices, just as classical propositional calculus is based on Boolean algebras. 1.12.A. Heyting Algebras and Orthomodular Lattices In this section we give a brief overview of the elementary properties of Heyting algebras and quantum logics. 1.12.A.1. Consider some lattice L. The relative pseudocomplement of x ∈ L with respect to y ∈ L is the top of the set {z ∈ L : x ∧ 1.12. Variations on the Theme 35 z y}. The pseudocomplement of x with respect to y, if existent, is denoted by x ⇒ y. The following easy property can be viewed as another deﬁnition of relative pseudocomplement: z x ⇒ y ⇐⇒ x ∧ z y. 1.12.A.2. A lattice Ω with zero 0 and unity 1 is called a Heyting algebra provided that the relative pseudocomplement x ⇒ y exists for every two elements x, y ∈ Ω. A Heyting algebra is also referred to as a pseudo-Boolean algebra or Brouwer lattice. Each distributive Heyting algebra is a distributive lattice. The lattice O(X) of all open subsets of a topological space X ordered by inclusion is a complete Heyting algebra. If A, B, Bξ ∈ O(X) then B = ξ∈Ξ ξ ξ∈Ξ Bξ and A ⇒ B coincides with the interior of (X\A)∪B. 1.12.A.3. Given elements x, y, and z of a Heyting algebra, we have (1) x ⇒ y = 1 ⇐⇒ x y; x ⇒ 1 = 1; 1 ⇒ y = y. (2) (x =⇒ y) ∧ y = y; x ∧ (x ⇒ y) = x ∧ y. (3) x1 x2 =⇒ x2 ⇒ y x1 ⇒ y. (4) y1 y2 =⇒ x ⇒ y1 x ⇒ y2 . (5) (x ⇒ y) ∧ (x ⇒ z) = x ⇒ (y ∧ z). (6) (x ⇒ z) ∧ (y ⇒ z) = (x ∨ y) ⇒ z. (7) (x ⇒ y) ∧ (y ⇒ z) (x ⇒ z). (8) (x ⇒ y) ((x ∧ z) ⇒ (y ∧ z)). (9) x ⇒ (y ⇒ z) = (x ∧ y) ⇒ z = y ⇒ (x ⇒ z). (10) x ⇒ (y ⇒ z) (x ⇒ y) ⇒ (x ⇒ z). ⊳ See [344, Theorem I.12.2]. ⊲ 1.12.A.4. The pseudocomplement of x in a lattice L with zero is the top of the set {y ∈ L : x ∧ y = 0}. Clearly, if L is a Heyting algebra then each x ∈ L has the pseudocomplement x∗ := x ⇒ 0. Therefore, the properties of pseudocomplements follow from the corresponding properties of relative pseudocomplements. 1.12.A.5. Given elements x, y, and z of a Heyting algebra, we have (1) x y =⇒ y ∗ x∗ ; x ∧ x∗ = 0. (2) x∗ = 1 ⇐⇒ x = 0; x∗ = 0 ↔ x = 1. 36 Chapter 1. Boolean Valued Requisites (3) x x∗∗ ; x∗ = x∗∗∗ ; (x ∨ x∗ )∗∗ = 1. (4) (x ∨ y)∗ = x∗ ∧ y ∗ ; (x ∧ y)∗ x∗ ∨ y ∗ . (5) x ⇒ y ∗ = y ⇒ x∗ = (x ∧ y)∗ . (6) x ⇒ y y ∗ ⇒ x∗ ; (x ⇒ y) ∧ (x ⇒ y ∗ ) = x∗ . ⊳ See [344, Theorem I.12.3]. ⊲ 1.12.A.6. An element x of a Heyting algebra Ω is regular provided that x∗∗ = x. The set of all regular elements of a Heyting algebra Ω with the order induced from Ω will be denoted by R(Ω). Note that x ∈ Ω is regular if and only of x = y ∗ for some y ∈ Ω. The following holds (cp. [344, Theorem IV.6.5]): The ordered set R(Ω) is a Boolean algebra for each Heyting algebra Ω. 1.12.A.7. An ortholattice or orthocomplemented lattice is a lattice L with some bottom 0 and top 1, together with the unary operation ( · )⊥ : L → L, called orthocomplementation, such that for all x, y ∈ L we have x ∧ x⊥ = 0, x ∨ x⊥ = 1; x⊥⊥ := (x⊥ )⊥ = x; (x ∨ y)⊥ = x⊥ ∧ y ⊥ , (x ∧ y)⊥ = x⊥ ∨ y ⊥ . An ortholattice L is a Boolean algebra if and only if L satisﬁes the distributive law (x, y, z ∈ L): x ∧ (y ∨ z) = (x ∨ y) ∧ (x ∨ z). We say that some elements x and y of an ortholattice are orthogonal and write x ⊥ y whenever x y ⊥ or, equivalently, y x⊥ . 1.12.A.8. If L is an ortholattice then for all x, y, z ∈ L the following are equivalent: (1) If x y then there exists u ∈ L with x ⊥ u and y = x ∨ u. (2) x y implies y = x ∨ (y ∧ x⊥ ). (3) (x ∧ y) ∨ (x⊥ ∧ y ⊥ ) = 1 implies x = y. (4) (x ∨ (x⊥ ∧ (x ∨ y)) = x ∨ y. (5) If x = (x ∧ y) ∨ (x ∧ y ⊥ ) and x = (x ∧ z) ∨ (x ∧ z ⊥ ), then x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). ⊳ See [117, 187, 332]. ⊲ 1.12. Variations on the Theme 37 1.12.A.9. An ortholattice L is said to be an orthomodular lattice or quantum logic if one of (and hence all) the conditions 1.12.A.8 (1–5) is satisﬁed. An ortholattice is orthomodular if and only if it does not include a subalgebra isomorphic to hexagon 06, which is deﬁned as the set 06 := {a, b, a⊥ , b⊥ , 0, 1} with the order 0 < a < b < 1, 0 < a⊥ < b⊥ < 1 (cp. [188, p. 22]). 1.12.A.10. A quantum logic will be denoted by Q. Let H = (H, ·, ·) be a complex Hilbert space and let M be a von Neumann algebra on H. Denote by P(M ) the set of all orthogonal projections in M with the induced order: P Q ⇐⇒ (∀ x ∈ H)(x, P x x, Qx) (P, Q ∈ P(M )). Then Q := P(M ) is a quantum logic with P ⊥ = IH − P and P ∧ Q = limn→∞ (P ◦ Q)n . 1.12.A.11. For all x and y of a complete orthomodular lattice, the following are equivalent: (1) The sublattice generated by {x, x⊥ , y, y ⊥ } is distributive. (2) (x ∧ y) ∨ (x⊥ ∧ y) ∨ (x ∧ y ⊥ ) ∨ (x⊥ ∧ y ⊥ ) = 1. (3) (x ∧ y) ∨ (x⊥ ∧ y) = y. (4) (x ∨ y ⊥ ) ∧ y = x ∧ y. ⊳ See [336, Theorems 2.15, 2.17, and 2.19]. ⊲ 1.12.A.12. Elements x and y of a complete orthomodular lattice Q are said to be compatible, in symbols x ◦| y, if one of (and hence all) the equivalent assertions 1.12.A.11 (1–4) is fulﬁlled. For a subset C of Q and x ∈ Q we put x ◦| C, whenever x ◦| y for all y ∈ C. The set of elements compatible with all other elements, called the center, of Q is a complete Boolean algebra. 1.12.A.13. Let Q be a complete orthomodular lattice. Assume that x ∈ Q, (xξ )ξ∈Ξ is a family in Q, and x ◦| xξ for all ξ ∈ Ξ. Then the following hold: (1) x ∧ ξ∈Ξ xξ = ξ∈Ξ x ∧ xξ . (2) x ∨ ξ∈Ξ xξ = ξ∈Ξ x ∨ xξ . (3) x ◦| ξ∈Ξ xξ . (4) x ◦| ξ∈Ξ xξ . ⊳ See [336, Theorems 2.21 and 2.24]. ⊲ 38 Chapter 1. Boolean Valued Requisites 1.12.A.14. Given a nonempty subset of a complete orthomodular lattice Q, put S(A) = x ∈ Q : x ◦| A (∀ p, q ∈ A) (p ∧ x ◦| q ∧ x) . The Boolean domain ⊥ ⊥(A) of A ⊂ Q is deﬁned as ⊥ ⊥(A) = S(A). Also, put ⊥ ⊥(x1 , . . . , xn ) :=⊥ ⊥(A) whenever A = {x1 , . . . , xn }. It is easily seen from 1.12.A.13 that ⊥ ⊥(A) ◦| A and p ∧ ⊥ ⊥(A) ◦| q ∧ ⊥ ⊥(A) for all p, q ∈ A (cp. [326, § 2] and [382, Proposition 4 and Corollary 1]). 1.12.B. Intuitionistic Set Theory In this section we present an intuitionistic set theory ZFI based on intuitionistic logic IL. 1.12.B.1. Intuitionistic predicate calculus IL is a formal deductive system with a set of logical axioms and a set of rules of deduction. The logical axioms are the same as the classical excluding the axiom scheme 1.1.6 (12). Thus, the logical axioms of IL comprise the axioms schemes 1.1.6 (1–11) and the following axiom schemes: if ϕ(x) is a formula and t is a term then we have (∀ x) ϕ → ϕ(t) and ϕ(t) → (∃ x) ϕ. We only have the three rules of the predicate calculus: modus ponens and the two quantification laws: (MP) If ϕ and ϕ → ψ are theorems of CL then ψ is a theorem of CL too. (∀) If x is not free in ϕ then ϕ → ψ implies that ϕ → (∀ x) ψ. (∃) If x is not free in ψ then ϕ → ψ implies that (∃ x) ϕ → ψ. By deﬁnition, all theorems of IL are theorems of CL. The converse is obviously false: the CL-theorems ¬(¬ ϕ) → ϕ and (¬ ϕ) ∨ ¬(¬ ϕ) are not theorems of IL. But ϕ → ¬ (¬ ϕ) and ¬¬(ϕ ∨ ¬ ϕ) are IL-theorems. Note that neither of the logical connectives ∨, ∧, and → can be expressed through the others in IL. 1.12.B.2. The system ZFI of intuitionistic set theory is the ﬁrst order theory with the nonlogical symbols ∈, =, E, where E is a predicate symbol with one argument place and Ex := x ∈ E is interpreted as “x exists.” 1.12. Variations on the Theme 39 There are two groups of nonlogical axioms: the equality axioms and the ZF type axioms. First, we present the four equality axioms: u = u, u = v → v = u, u = v ∧ ϕ(u) → ϕ(v), (Eu ∨ Ev → u = v) → u = v. 1.12.B.3. To formulate the ZF type nonlogical axioms, we use the notation ∀˙ x . . . and ∃˙ x . . . to abbreviate ∀ x (Ex → . . .) and ∃ x (Ex∧. . .), respectively. Extension: ∀˙ z (z ∈ u ↔ z ∈ v) ∧ (Eu ↔ Ev) → u = v. Pair: ∃˙ z ∀˙ x (x ∈ z ↔ x = u ∨ x = v). Union: ∃˙ v ∀˙ x (x ∈ v ↔ ∃˙ y ∈ u (x ∈ y)). Power: ∃˙ v ∀˙ x (x ∈ v ↔ x ⊆ u). Infinity: ∃˙ v (∃˙ x ∈ v ∧ ∀˙ x ∈ v ∃˙ y ∈ v (x ∈ y)). Separation: ∃˙ v ∀˙ x (x ∈ v ↔ x ∈ u ∧ ϕ(x)). Foundation: ∀˙ x (∀˙ y ∈ xϕ(y) → ϕ(x)) → ∀˙ x ϕ(x). Replacement: ∃˙ v (∀˙ x ∈ u ∃˙ y ϕ(x, y) → ∀˙ x ∈ u ∃˙ y ∈ v ϕ(x, y)). To the end of this section, we write ∀ x and ∃ x instead of ∀˙ x and ∃˙ x, since ∀ x and ∃ x always appear in the form ∀˙ x and ∃˙ x. 1.12.B.4. A model of a theory consists of a universe M , a set Ω of truth values, a function E : M → Ω, and a function [[ · ]] that assigns some truth value [[ϕ(x1 , . . . , xn )]] ∈ Ω to each sentence ϕ(u1 , . . . , un ) and all x1 , . . . , x ∈ M . We say that (Ω, M, E , [[ · ]]) is a model of ZFI , if the operations ∧, ∨, , , ⇒, and (·)∗ , corresponding to the logical operations ∧, ∨, ∀, ∃, →, and ¬, are deﬁned on Ω and satisfy the following conditions for all sentences ϕ, ψ and an arbitrary formula ϕ(u) with one variable: (1) {[[ϕ]] : ϕ is a sentence} = Ω; (2) [[ϕ ∧ ψ]] = [[ϕ]] ∧ [[ψ]]; (3) [[ϕ ∨ ψ]] = [[ϕ]] ∨ [[ψ]]; (4) [[¬ ϕ]] = [[ψ]]∗ ; (5) [[∀ x ϕ(x)]] = x∈M (E x ⇒ [[ϕ(x)]]); (6) [[∃ x ϕ(x)]] = x∈M (E x ∧ [[ϕ(x)]]); 40 Chapter 1. Boolean Valued Requisites (7) [[ϕ → ψ]] = [[ϕ]] ⇒ [[ψ]]; (8) if ⊢ ϕ → ψ then [[ϕ]] = [[ψ]]∗ . It is easy to see that if (Ω, M, E , [[ · ]]) is a model of ZFI then Ω is a Heyting algebra. Conversely, if Ω is a complete Heyting algebra then we can deﬁne a universe M and function [[ · ]] such that (Ω, M, [[·]]) is a model of ZFI , as follows. (Ω) 1.12.B.5. Let V be a standard universe of ZFC. Deﬁne Vα ⊆ V for (Ω) all α ∈ Ord by transﬁnite induction. Assume that Vβ is deﬁned already (Ω) for β < α and each element u of Vβ is of the form (D(u), ⌊u⌋, E u), (Ω) Vγ for some γ < β, ⌊u⌋ is a function of D(u) into Ω where D(u) ⊂ and E u ∈ Ω. For convenience we write u(x) instead of ⌊u⌋(x). Now we (Ω) deﬁne Vα by (Ω) V(Ω) α = u = (D(u), ⌊u⌋, E u) : (∃ β < α) D(u) ⊂ Vβ ⌊u⌋ : D(u) → Ω ∧ E u ∈ Ω ∧ ∀ x ∈ D(u) (u(x) E u ∧ E x) . Finally, we deﬁne the Heyting valued universe as V(Ω) V(Ω) = α . α∈On 1.12.B.6. The Heyting truth value [[ϕ]] is deﬁned by induction on the number of logical symbols in ϕ. An atomic sentence over V(Ω) is of the from u = v, u ∈ v or Eu, where u, v ∈ V(Ω) . Now, [[u = v]] and [[u ∈ v]] are deﬁned by recursion as follows (cp. 1.1.8 and 1.2.4): = x∈D(u) [[u = v]] (u(x) ⇒ [[x ∈ v]]) ∧ (v(y) ⇒ [[y ∈ u]]) ∧ (E u ⇔ E v), y∈D(v) [[u ∈ v]] = (v(y) ∧ [[u = y]]), y∈D(v) [[Eu]] = E u. Note that for all x ∈ D(u) and y ∈ D(v) we have max(rank(x), rank(y)) < max(rank(u), rank(v)). 1.12. Variations on the Theme Hence, [[x ∈ v]] = [[u = y]] = x∈D(v) y∈D(v) 41 v(y) ∧ [[x = v]] , y(t) ⇒ [[t ∈ u]] ∧ E u ⇔ E y u(x) ⇒ [[x ∈ y]] ∧ t∈D(v) are deﬁned at an earlier stage. For a sentence with logical symbols [[ · ]] is deﬁned as in 1.2.3: [[ϕ ∧ ψ]] = [[ϕ]] ∧ [[ψ]], [[ϕ ∨ ψ]] = [[ϕ]] ∨ [[ψ]], [[ϕ → ψ]] = [[ϕ]] ⇒ [[ψ]], [[¬ϕ]] = [[ϕ]] ⇒ 0 = [[ϕ]]∗ , E x ∧ [[ϕ(x)]], [[∃ xϕ(x)]] = x∈V(Ω) [[∀ xϕ(x)]] = (E x ⇒ [[ϕ(x)]]). x∈V(Ω) 1.12.B.7. If Ω is a complete Heyting algebra then (Ω, V(Ω) , E , [[ · ]]), defined above, is a model of ZFI . ⊳ See [149, 150, 386]. ⊲ 1.12.C. Quantum Set Theory 1.12.C.1. Quantum predicate calculus QL is a formal deductive system, and so it deﬁned as a language consisting of propositions and connectives and the axioms and a rule of inference. Just as classical propositional calculus bases on Boolean algebras, quantum propositional calculus bases on orthomodular lattices. We will avoid going into the details of the quantum propositional calculus. The interested reader is referred to [188, 336]. 1.12.C.2. The system ZFQ of quantum set theory is the ﬁrst order theory with the nonlogical symbols ∈, =, ∨ , where ∨ := ∨ (x0 , . . . , xn ) is an n-ary predicate symbol for all n = 2, 3, . . . interpreted as “x0 , . . . , xn are compatible.” The implication can be deﬁned as the Sasaki hook: ϕ → ψ := ¬ ϕ ∨ (ϕ ∧ ψ). Consider the equality axioms: (1) u = u. (2) u = v → v = u. 42 Chapter 1. Boolean Valued Requisites (3) ∨ (u, v, u′ ) ∧ u = u′ ∧ u ∈ v → u′ ∈ v. (4) ∨ (u, v, u′ ) ∧ u ∈ v ∧ v = v ′ → u ∈ v ′ . (5) ∨ (u, v, w) ∧ u = v ∧ v = w → u = w. 1.12.C.3. Consider the special axioms of quantum set theory. (1) Axiom of Pair: (∀ u, v) ∨ (u, v) → (∃ x) ∨ (u, v, x) ∧ ∀ y (y ∈ x ↔ y = u ∨ y = v) . (2) Axiom of Union: (∀ u) ∨ (u) → (∃ v) ∨ (u, v) ∧ (∀ x) ∨ (x, u) → (x ∈ v ↔ (∃ y ∈ u) (x ∈ y)) . (3) Axiom of Powerset: (∀ u) ∨ (u) → (∃ v) ∨ (u, v) ∧ (∀ t) ∨ (u, v, t) → (t ∈ v ↔ (∀ x ∈ t) (x ∈ u)) . (4) Axiom of Replacement: (∀ u) (∀ x ∈ u) (∃ y) ϕ(x, y) → (∃ v) (∀ x ∈ u) (∃ y ∈ v) ϕ(x, y) . (5) Axiom of Foundation: (∀ u) (∨ (u) ∧ (∃ x ∈ u)(x ∈ u)) → ((∃ x ∈ u) (∀ y ∈ x)¬(y ∈ u)) . (6) Axiom of Infinity: (∅ ∈ ω ∧ ) ∧ (∀ x ∈ ω ∧ ) (x ∪ {x} ∈ ω ∧ ). (7) Axiom of Choice: (∀ u) ∨ (u) → (∃ v) ∨ (u, v) ∧ (∀ x ∈ u) (∃ y ∈ x) (∃!z ∈ u) (y ∈ z) → (∃!y ∈ x) (y ∈ v) . 43 1.12. Variations on the Theme 1.12.C.4. Let Q be a quantum logic. Given an ordinal α, put (Q) (Q) . = u : D(u) → Q and D(u) ⊆ Vβ Vα β<α The Q-valued universe V(Q) is deﬁned as V(Q) V(Q) = α . α∈On For every u ∈ V(Q) , the rank of u, denoted by rank(u), is the least α (Q) such that u ∈ Vα . Clearly, if u ∈ D(v) then rank(u) < rank(v). 1.12.C.5. Given u ∈ V(Q) , deﬁne the support of u, denoted by L(u), by transﬁnite recursion on the rank of u: L(u) := L(x) ∪ u(x) : x ∈ D(u) . x∈D(u) For A ⊂ V(Q) we write L(A) := u∈A L(u) and deﬁne the Boolean domain ∨(A) of A by the formula ∨(A) :=⊥ ⊥ L(A). We also put L(u1 , . . . , un ) := L({u1 , . . . , un }), ∨(u1 , . . . , un ) := ∨({u1 , . . . , un }) for all u1 , . . . , un ∈ V(Q) . Put x ⇒ y := x⊥ ∨ (x ∧ y) for all x, y ∈ Q. Deﬁne the Q-valued truth values for the atomic formulas [[u = v]] ∈ Q and [[u ∈ v]] ∈ Q with u, v ∈ V(Q) as follows (cp. 1.2.4): (v(y) ⇒ [[y ∈ u]]), (u(x) ⇒ [[x ∈ v]]) ∧ [[u = v]] := y∈D(v) x∈D(u) [[u ∈ v]] := (v(y) ∧ [[u = y]]). y∈D(v) 1.12.C.7. To each statement ϕ of ZFC we assign the Q-valued truth value [[ϕ]] just as in 1.2.3 with the only diﬀerence that [[¬ φ]] = [[φ]]⊥ is taken instead of [[¬ φ]] = [[φ]]∗ and the following additional rule is included [[∨ (x0 , . . . , xn )]] = ∨ (u0 , . . . , un ) (u0 , . . . , un ∈ V(Q) ). 44 Chapter 1. Boolean Valued Requisites 1.12.C.8. We say that ϕ holds within V(Q) and write V(Q) |= ϕ, whenever [[ϕ]] = 1. The axioms of equality 1.12.C.2 hold within V(Q) (cp. [382, Theorem 1] and [326, Theorem 3.3]). At the same time the classical axioms of transitivity 1.1.4 (3) and substitution 1.1.4 (4) fail within V(Q) (cp. [382, pp. 313, 314]). 1.12.C.9. Given v ∈ V, deﬁne v ∧ ∈ V(2) ⊂ V(Q) by putting D(v) := ∧ {x : x ∈ v} and v ∧ (x∧ ) = 1 (x ∈ v). Then, for every bounded formula ϕ(x1 , . . . , xn ) of ZFC and all u1 , . . . , un ∈ V, we have V |= ϕ(u1 , . . . , un ) ⇐⇒ V(Q) |= ϕ(u∧1 , . . . , u∧n ). The following two results were obtained under the assumption that Q = P(M ), where M is a von Neumann algebra (cp. 1.12.A.10). 1.12.C.10. All axioms 1.12.C.3 (1–7) are true within the universe V(Q) ; i.e., V(Q) |= ZFQ . ⊳ See [382, pp. 315–321]. ⊲ 1.12.C.11. Transfer Principle. Given a bounded formula ϕ(x1 , . . . , xn ) of ZFC and u1 , . . . , un ∈ V(Q) , the implication holds ZFC ⊢ ϕ(x1 , . . . , xn ) =⇒ V(Q) |= ∨(u1 , . . . , un ) → ϕ(u1 , . . . , un ). ⊳ See [326, Theorem 4.6]. ⊲ 1.13. Comments 1.13.1. (1) The ﬁrst system of axioms for set theory, existing alongside the Russell type theory and suggested by Zermelo in 1908, coincides essentially with the collection of 1.1.7 (1–4, 6) in which the axiom schema of replacement is replaced by the two of its consequences: the axiom of separation—(∀ x)(∃ y)(∀ z)(z ∈ y ↔ (z ∈ x) ∧ ϕ(z)) where ϕ is a formula of ZF and the axiom of pairing—(∀ x)(∀ y)(∃ z)(∀ u)(u ∈ z ↔ (u = x ∨ u = y)). The axioms of extensionality 1.1.7 (1) and union 1.1.7 (2) had been previously proposed by Frege in 1883 and Cantor in 1899. The idea of the axiom of inﬁnity 1.1.7 (6) stems from Dedekind. (2) Zermelo set theory appeared in the beginning of the 1920s. It terminated the important stage of formalizing the language of set theory which eliminated the ambiguous descriptions of the tricks for distinguishing sets. But Zermelo’s axioms did not make it possible to allow 1.13. Comments 45 for the heuristic view of Cantor which asserts that the one-to-one image of a set is a set too. This shortcoming was eliminated by Fraenkel in 1922 and Skolem in 1923 who suggested versions of the axiom schema of replacement 1.1.7 (4). These achievements may be considered as the birth of ZFC. (3) The axiom of foundation 1.1.7 (5) was propounded by Gödel and Bernays in 1941. It replaces the axiom of regularity which was proposed by von Neumann in 1925. The axiom of foundation is independent of the rest of the axioms of ZFC. (4) The axiom of choice AC 1.1.7 (7) seems to have been used implicitly since long ago (for instance, Cantor used it in 1887 while proving that each inﬁnite set includes a countable subset), whereas it was distinguished by Peano in 1890 and by B. Levi in 1902. The axiom of choice had been propounded by Zermelo in 1904 and remained most disputable and topical for quite a few decades. But the progress of “concrete” mathematics has showed that the possibility of virtual choice is perceived as an obvious and indispensable part of many valuable fragments of modern mathematics. There is no wonder that the axiom of choice is accepted by most of scientists. The discussion of the place and role of the axiom of choice in various areas of mathematics can be found in Gödel [145], Jech [184], Cohen [92], Lévy [279], and Fraenkel, Bar-Hillel, and Lévy [120]. (5) The concept of continuum belongs to the most important general tools of science. The mathematical views of the continuum relate to the understanding of a straight line in geometry and time and timedependent processes in physics. The set-theoretic stance revealed a new enigma of the continuum. Cantor demonstrated that the set of the naturals is not equipollent with the simplest mathematical continuum, the real axis. This gave an immediate rise to the problem of the continuum which consists in determining the cardinalities of the intermediate sets between the naturals and the reals. The continuum hypothesis reads that the intermediate subsets yield no new cardinalities. 1.13.2 (1) Boolean valued models were invented for research into the foundations of mathematics. Many delicate properties of the objects of V(B) depend essentially on the structure of the initial Boolean algebra B. The diversity of opportunities together with a great stock of information on particular Boolean algebras ranks Boolean valued models among the most powerful tools of foundational studies; see Bell [43], Jech [184], and Takeuti and Zaring [388]. 46 Chapter 1. Boolean Valued Requisites (2) Boolean valued analysis stems from the brilliant results of Gödel and Cohen who demonstrated the independence of the continuum hypothesis from the axioms of ZFC. Gödel proved the consistency of the continuum hypothesis with the axioms of ZFC by inventing the universe of constructible sets [145]. Cohen [92] demonstrated the consistency of the negation of the continuum hypothesis with the axioms of ZFC by forcing, the new method he invented for changing the properties of available or hypothetical models of set theory. Boolean valued models made Cohen’s diﬃcult result simple demonstrating to the working mathematician the independence of the continuum hypothesis with the same visuality as the Poincaré model for non-Euclidean geometry. Those who get acquaintance with this technique are inclined to follow Cohen [92] and view the continuum hypothesis as “obviously false.” (3) The book [344] by Rasiowa and Sikorski is devoted to the basics of Boolean valued models for the predicate calculus. The ideas of using Boolean valued models for simplifying the method of forcing by Cohen had independently been suggested by Solovay [369] and Vopěnka [400, 401] in 1965. Somewhat later Scott and Solovay, as well as Vopěnka in the research of his own, draw the conclusion that the topics of forcing should be addressed within the objects of a Boolean valued universe from the very beginning. The Boolean valued models whose construction was not considered as adverse by the majority of the “traditional” mathematicians have gained much popularity after it was revealed that they allow for deriving the same results as the method of forcing. (4) The Boolean valued universe V(B) is used for proving relative consistency of some set theoretic propositions by the following scheme: Assume that the theories T and T ′ are some enrichments of ZF such that the consistency of ZF entails the consistency of T ′ . Assume further that we can deﬁne B so that T ′ |= “B is a complete Boolean algebra” and T ′ |= [[ϕ]]B = 1 for every axiom ϕ of T . Then the consistency of ZF will imply the consistency T (cp. Bell [43]). 1.13.3. (1) We now exhibit another interesting Boolean valued model of set theory. Let G be a subgroup of the automorphism group of a complete Boolean algebra B, and let Γ be a ﬁlter of subgroups of G; i.e., Γ is a nonempty set of subgroups of G such that H, K ∈ Γ implies H ∩K ∈ Γ while H ∈ Γ and H ⊂ K imply K ∈ Γ for all subgroups H and K of G. Say that Γ is a normal filter if g ∈ G and H ∈ Γ imply gHg −1 ∈ Γ. Each g ∈ G induces the automorphism g ∗ of V(B) which is in 1.3.1. The sta- 47 1.13. Comments bilizer stab(x) of x ∈ V(B) is deﬁned as stab(x) := {g ∈ G : g ∗ (x) = x}. It is easy from 1.3.2 that stab(x) is a subgroup of G. We deﬁne the sets V(Γ) recursively as follows: (B) V(Γ) α := {x : Funct(x) ∧ (∃ β) (β < α ∧ dom(x) ⊂ Vβ ∧ im(x) ⊂ B) ∧ stab(x) ∈ Γ}. (Γ) Put V(Γ) := {x : (∃ α ∈ On)x ∈ Vα } and deﬁne Boolean truth values by [[x ∈ y]](Γ) = [[x ∈ y]](B) , and [[x = y]](Γ) = [[x = y]](B) for x, y ∈ V(Γ) . Deﬁne [[ϕ]] ∈ B as in 1.2.3 and 1.2.4. (2) Scott established that all axioms and so all theorems of ZF are true in V(Γ) ; see Bell [43, Theorem 3.19]. Scott succeeded in choosing B, G, and Γ in such a way that V(Γ) |= ¬AC. Then it follows that the consistency of ZF implies the consistency of ZF +¬AC. So the model V(Γ) is eﬀective in proving consistency. It seems reasonable to suppose that these models will be useful in Boolean valued analysis, but we are unaware of any applications of the sort yet. 1.13.4. (1) Scott established the maximum and transfer principles of Section 1.4 together with many other properties of Boolean valued models. He also gave the schematic exposition of the models. But the manuscript of 1967 remained unpublished although it was rather widely used by specialists. The literature on Boolean valued models has references to the nonexistent paper by Scott and Solovay which was intended to be an extension of the Scott manuscript. These and other details of the creation and development of the theory of Boolean valued models are disclosed in Scott’s introduction to Bell’s book [43] (2) The restricted transfer principle is often referred to as as the Boolean valued version of absoluteness of bounded formulas; see [184, Lemma 14.21] 1.13.5, 1.13.6. (1) Various versions of the tricks of Sections 1.5 and 1.6 are common for studying Boolean valued models. In Kusraev [218, 222] and Kutateladze [267, 268] they appeared as the technique of descents and ascents which proved to be convenient in applications to analysis (cp. Kusraev and Kutateladze [244, 245, 246, 248]). The terminology of “descents and ascents” was suggested by Kutateladze [267, 268] in memory of Escher (whose life and achievements are reﬂected, for instance, by Locher [285] and Hofstedter [172]). The arrow cancellation rules in 1.6.6 are also called Escher rules. 48 Chapter 1. Boolean Valued Requisites (2) The same symbols ↓ and ↑ are used for various operators having the same nature. Therefore, the records like X ↓ and X↑ can be properly understood only with extra information about the object that is ascended to or descended from a Boolean valued universe. The situation here is pretty similar to that with using the plus sign for recording completely diﬀerent group operations: addition of numbers, vectors, linear operators, etc. The precise meaning is always reconstructible from the context. We use the symbols ↑ and ↓ by analogy. 1.13.7. Boolean valued interpretations have a long history. It seems that the ﬁrst Boolean valued model (for the theory of types) was suggested by Church in 1951. Since then many authors have considered Boolean valued models for ﬁrst order propositions and theories; for instance, Halmos, Mostowski, and Tarski. But it was Rasiowa and Sikorski who advanced the technique substantially; cp. [344]. In regard to the theory of algebraic systems look at the deﬁnitive monograph by Maltsev [304]. 1.13.8. The descent and embedding of an algebraic B-system to a Boolean valued model were accomplished in the articles by Kusraev [218] and Kutateladze [267] on using the method of Solovay and Tennenbaum [370] which they applied to proving Theorem 1.10.4. The descents of various particular algebraic systems were performed by many authors. Part of these results is collected in the books [248, 249] by Kusraev and Kutateladze. 1.13.9. (1) A Boolean algebra is said to satisfy the countable chain condition if its every disjoint family of nonzero elements is at most countable. If the complete Boolean algebra B satisﬁes the countable chain condition, then cardinals in V retain their true size in V(B) ; i.e., α ∈ V is a cardinal if and only if α∧ is a cardinal within V(B) and, consequently α∧ = |α∧ |; see Bell [43, Theorem 1.51]. Clearly, if B does not satisfy this condition, then it becomes possible for two inﬁnite cardinals κ < λ have the coinciding standard names κ ∧ and λ∧ . More precisely, for inﬁnite cardinals κ < λ there exists a complete Boolean algebra B such that V(B) |= |κ ∧ | = |λ∧ |. In this event we say that λ has been collapsed to κ in V(B) ; see Bell [43, Theorem 5.1, Corollary 5.2, and 5.4]. (2) A Boolean algebra B is said to be (κ, λ)-distributive if for every family (bα,β )α<κ,β<λ in B we have bα,β = bα,φ(α) . α<κ β<λ φ∈λκ α<κ 1.13. Comments 49 It can be shown that (κ, λ)-distributivity of B is equivalent to the relation ∧ V(B) |= (λκ )∧ = (λ∧ )κ ; see Bell [43, 2.14]. The monographs [43] by Bell and [184] by Jech are excellent sources of the facts concerning Boolean valued cardinals. 1.13.10. (1) Let B and D be Boolean algebras and let B ⊗ D be their free product. That is, B ⊗ D is isomorphic to the Boolean algebra of clopen sets of the Cartesian product of the Stone spaces of B and D; the Dedekind see Koppelberg [204, Subsection 11.1]. Denote by B⊗D completion of B ⊗ D; see Koppelberg [204, Section 4.3]. Given a Boolean algebra B and an element D ∈ V(B) satisfying V(B) |= “D is the Dedekind completion of the Boolean algebra D∧ ,” the Boolean algebras D↓ and are isomorphic (see Solovay and Tennenbaum [370]). B⊗D (2) The results by Solovay and Tennenbaum [370] (Theorems 1.10.2– 1.10.4) can serve as a basis for iterating the construction of a Boolean valued model. Assume that D ∈ V(B) and V(B) |= “D is a complete Boolean algebra.” Using the scheme of Section 1.3 we can construct within V(B) a few V(B) -classes: the Boolean valued universe (V(B) )(D) , the corresponding Boolean truth values [[ · = · ]]D and [[ · ∈ · ]]D together with the canonical embedding ( · )∧ of the universal class UB to (V(B) )D . Put D := D↓, W(D) := (V(B) )(D) ↓, [] · = · []D := ([[ · = · ]]D )↓, [] · ∈ · []D := ([[ · ∈ · ]]D )↓, and j := ( · )∧ ↓. Assume that ı : B → D is the canonical isomorphism, and ı∗ : V(B) → V(D) is the corresponding injection (see Section 1.3). Then there is a unique bijection h : V(D) → W(D) such that [[x = y]]D = []h(x) = h(y)[]D and [[x ∈ y]]D = []h(x) ∈ h(y)[]D for all x and y ∈ V(B) . In this event the diagram commutes: V(B)❄ ⑧ ❄❄ ⑧⑧ ❄❄ j ı∗ ⑧⑧ ❄❄ ⑧ ⑧ ❄❄ ⑧ ❄ ⑧⑧ (D) / W(D) V h See details in Solovay and Tennenbaum [370]. (3) Further iterations of the above construction lead to a transﬁnite collection of Boolean valued enrichments. This approach leads to the iterated forcing which was used for instance in establishing the relative consistency of the Suslin hypothesis and ZFC which was done in Solovay and Tennenbaum [370]. 50 Chapter 1. Boolean Valued Requisites 1.13.11. (1) The Sandwich and Hahn–Banach Theorems for Boolean homomorphisms (i.e. Theorems 1.11.5 and 1.11.6) were obtained by Monteiro [313] who used another method. Some analogous results for distributive lattices were demonstrated by Cignoli [91]. The proofs in Section 1.11 show that the results about extension of Boolean homomorphisms are simply the existence theorem of a nontrivial ultraﬁlter modulo translation into a Boolean valued model. For instance, Theorem 1.11.4 is a Boolean valued interpretation of the Stone Theorem: If an ideal I and a filter F are disjoint in a Boolean algebra, then there is a maximal ideal I including I and disjoint from F as well as there is an ultrafilter F including F and disjoint from I. 1.13.12. (1) The system ZFI of intuitionistic set theory and construction of a Heyting valued model V(Ω) within the theory, as presented in 1.12.C, are due to Grayson [149, 150]. Takeuti and Titani in [386], using Grayson’s ZFI , extended the Solovay and Tennenbaum’s results on iterated Cohen extensions in [370] to Heyting valued universes. More precisely, a complete Heyting algebra Ω and the corresponding Heyting ′ valued universe V(Ω) are considered in a universe V(Ω ) with Ω′ another complete Heyting algebra. For more detail on complete Heyting algebras, refer to Fourman and Scott [119] in which some related subjects are also discussed. (2) Let 2 = {0, 1} be the complete least subalgebra of a complete Heyting algebra Ω. Just as in 1.4.4 and 1.4.5 the universe V is equivalent to V(2) ⊂ V(Ω) , so that V is embedded in V(Ω) as a submodel. But the copy V̂ := {x∧ : x ∈ V} of V is not expressible in the language of ZFI on V(Ω) , since the concepts expressible in the language of ZFI on V(Ω) are local, whereas V̂ is a global concept. In Takeuti and Titani [387] a modiﬁcation of ZFI , the global intuitionistic set theory GIZF in which the global concepts are expressible, are presented. (3) The idea of quantum logic stems from von Neumann’s 1932 book on the mathematical foundations of quantum mechanics. In [394, p. 253] he wrote: “As can be seen, the relation between the properties of a physical system on the one hand, and the projections on the other, makes possible a sort of logical calculus with these.” A systematic attempt to propose a “propositional calculus” for quantum logic was made in the seminal joint paper [59] by Birkhoﬀ and von Neumann which marked the birth of quantum logic. As regads the history and the main ideas of quantum logic, see Dalla Chiara, Giuntini, and Rédei [102] as well as Foulis, Greechie, Dalla Chiara, and Giuntini [118]. The mathematical 1.13. Comments 51 and logical investigation of various aspects of quantum mechanics is the topic of the Handbook of Quantum Logic and Quantum Structures edited by Engesser, Gabbay, and Lehmann [115]; see also Piron [336] and Ptak and Pulmannova [339]. (4) Quantum set theory was introduced by Takeuti in [382] as the quantum counterpart of Boolean valued set theory. In [382, p. 303] he wrote: “Since quantum logic is an intrinsic logic, i.e. the logic of the quantum world (cp. Birkhoﬀ and von Neumann [59]), it is an important problem to develop mathematics based on quantum logic, more speciﬁcally set theory based on quantum logic. It is also a challenging problem for logicians since quantum logic is drastically diﬀerent from the classical logic or the intuitionistic logic and consequently mathematics based on quantum logic is extremely diﬃcult. On the other hand, mathematics based on quantum logic has a very rich mathematical content.” (5) In [390] Titani presented the lattice valued logic and lattice valued set theory by introducing the basic implication. The completeness of the lattice valued logic was proved in Takano [378]. For an arbitrary complete lattice L, the L-valued universe V(L) is a model of lattice valued set theory based on the lattice valued logic. (6) The possibilities are open for deﬁning implication in the quantum logics that satisfy the order known as the Birkhoﬀ–von Neumann requirement (cp. Pavičić and Megill [334]): x ⇒ y := x⊥ (x ∧ y) (Sasaki); x ⇒ y := y ∨ (x⊥ ∧ y ⊥ ) (Dishkant); x ⇒ y := (((x⊥ ∧ y) ∨ (x⊥ ∧ y ⊥ )) ∨ (x ∧ (x⊥ ∨ y))) (Kalmbach); x ⇒ y := (((x ∧ y) ∨ (xp erp ∧ y)) ∨ ((x⊥ (x ∨ y) ∧ y ⊥ ))) (nontollens); x ⇒ y := (((x ∧ y) ∨ (x⊥ ∧ y)) ∨ (x⊥ ∧ y)) (relevance). In a Boolean algebra, all reduce to the classical implication x ⇒ y := (x⊥ ∨ y). (7) An ortholattice L is called weakly orthomodular provided that x ≡ y = 1 =⇒ x = y and L is called a weakly distributive ortholattice whenever (x ≡ y) ∨ (x ≡ y ⊥ ) = (x ∧ y ⊥ ) ∨ (x⊥ ∧ y) ∨ (x⊥ ∧ y ⊥ ) = 1 for all x, y ∈ Ω, where x ≡ y := (x ∧ y) ∨ (x⊥ ∧ y ⊥ ). There exist weakly distributive ortholattices that are not orthomodular and therefore not distributive, weakly orthomodular ortholattices that are not orthomodular, ortholattices that are neither weakly orthomodular nor weakly distributive, and weakly orthomodular ortholattices that are not weakly distributive (cp. Pavičić and Megill [332, 333]). 52 Chapter 1. Boolean Valued Requisites (8) Surprisingly, the quantum propositional calculuses and the classical propositional calculuses are noncategorical. Recall that a formal system is called categorical and if all its models are isomorphic with one another. More precisely, quantum logic can be modeled by an orthomodular lattice as well as a weakly orthomodular lattice, and the classical logic can be modeled by a Boolean algebra as well as a weakly distributive lattice (cp. Pavičić and Megill [333]). CHAPTER 2 BOOLEAN VALUED NUMBERS Boolean valued analysis stems from the fact that the image of the reals in each Boolean valued model presents a universally complete vector lattice. Therefore, the theorems about real numbers can be “externalized” by transfer so as to yield results about universally complete vector lattices. Depending on which Boolean algebra B (the algebra of measurable sets, regular open sets, or projections in a Hilbert space, etc.) forms the base for constructing the Boolean valued model V(B) , we obtain various vector lattices (the spaces of measurable functions, continuous functions, selfadjoint operators, etc.). Thereby the remarkable opportunity opens up to expand the treasure-trove of knowledge about the reals to a profusion of classical objects of analysis. In this chapter we show that the most important structure properties of Dedekind complete vector lattices such as representation as function spaces, the Freudenthal Spectral Theorem, functional calculus, etc. are some translations of the properties of the reals in an appropriate Boolean valued model. As in Chapter 1, to simplify the simultaneous work with two universes, we agree to some extra pedantry in notation. Denoting implication and equivalence in the sequel, we will use =⇒ and ⇐⇒ outside V(B) and → and ↔ inside V(B) , while ⇒ and ⇔ we reserve for the Boolean operations: x ⇒ y := x∗ ∨ y and x ⇔ y := (x ⇒ y) ∧ (y ⇒ x). Throughout the sequel N, Z, Q, R, and C symbolize the naturals, the integers, the rationals, the reals, and the complexes. 2.1. Vector Lattices In this section we give some preliminaries to the theory of vector lattices; a more explicit exposition can be found elsewhere (cp. Akilov and Kutateladze [22], Kusraev [228], Luxemburg and Zaanen [297], 54 Chapter 2. Boolean Valued Numbers Meyer-Nieberg [311], Schaefer [356], Schwarz [361], Vulikh [403], and Zaanen [427]). 2.1.1. Let F be a linearly ordered ﬁeld. An ordered vector space over F is a pair (X, ), where X is a vector space over F and is an order on X satisfying the conditions: (1) if x y and u v then x + u y + v for all x, y, u, v ∈ X; (2) if x y then λx λy for all x, y ∈ X and 0 λ ∈ F. Informally speaking, we can “sum inequalities in X and multiply them by positive members of F.” This circumstance is worded as follows: is an order compatible with the vector space structure or, brieﬂy, is a vector order. 2.1.2. The subset X+ := {x ∈ X : x 0} of an ordered vector space X is called the positive cone of X. The elements of X+ are called positive. The positive cone X+ of an ordered vector space X has the properties: X+ + X+ ⊂ X+ , λX+ ⊂ X+ (0 λ ∈ F), X+ ∩ −X+ = {0}. Moreover, if X+ is a subset of a vector space X over F satisfying the above properties, then X transforms into an ordered vector space over F by letting x y ⇐⇒ y − x ∈ X+ (x, y ∈ X). 2.1.3. A vector lattice is an ordered vector space that is also a lattice. Thereby each ﬁnite set {x1 , . . . , xn } ⊂ X of a vector lattice has the join or the least upper bound sup{x1 , . . . , xn } := x1 ∨ · · · ∨ xn as well as the meet or the greatest lower bound inf{x1 , . . . , xn } := x1 ∧ · · · ∧ xn . In particular, each element x of a vector lattice has the positive part x+ := x ∨ 0, the negative part x− := (−x)+ := −x ∧ 0, and the modulus |x| := x ∨ (−x). A vector lattice X is called Archimedean if for every pair of elements x, y ∈ X from (∀ n ∈ N) nx y it follows that x 0. In the sequel, all ordered vector spaces are assumed to be Archimedean. 2.1.4. Let X be a vector lattice. If x, y, z ∈ X then the following hold: (1) x = x+ − x− , |x| = x+ + x− = x+ ∨ x− . (2) x y ⇐⇒ (x+ y + and y − x− ). 2.1. Vector Lattices 55 (3) x ∨ y = 12 (x + y + |x − y|), x ∧ y = 21 (x + y − |x − y|). (4) |x| ∨ |y| = 21 (|x + y| + |x − y|), |x| ∧ |y| = 12 (|x + y| − |x − y|). (5) x + y = x ∨ y + x ∧ y, |x − y| = x ∨ y − x ∧ y. (6) x + y ∨ z = (x + y) ∨ (x + z), x + y ∧ z = (x + y) ∧ (x + z). (7) x, y, z ∈ X+ =⇒ (x + y) ∧ z (x ∧ z) + (y ∧ z). (8) |x − y| = |x ∨ z − y ∨ z| + |x ∧ z − x ∧ z|. 2.1.5. Let (xα ) and (yα ) be families in X for which sup(xα ) and inf(yα ) exist. Then the infinite distributive laws are valid (z ∈ X): (1) z ∧ supα (xα ) = supα (z ∧ xα ), z ∨ inf α (yα ) = inf α (z ∨ yα ). Moreover, for every z ∈ X we have the following: (2) z + supα (xα ) = supα (z + xα ); (3) z + inf α (yα ) = inf α (z + yα ); (4) supα (xα ) = − inf α (−xα ). 2.1.6. An order interval in X is a set of the form [a, b] := {x ∈ X : a x b}, where a, b ∈ X. In a vector lattice we have the very useful Riesz decomposition property: (1) [0, x + y] = [0, x] + [0, y] (x, y ∈ X+ ). Note the two corollaries of (1): (2) (x1 + · · · + xn ) ∧ y x1 ∧ y + · · · + xn ∧ y (xk , y ∈ X+ ); (3) If xk,l ∈ X+ (k n, l m) and J is the set of all functions from {1, . . . , n} to {1, . . . , m}, then n m k=1 l=1 xk,l x1,j(1) ∧ · · · ∧ xn,j(n) . j∈J 2.1.7. Two elements x, y ∈ X are called disjoint if |x| ∧ |y| = 0. The disjointness of x and y is denoted by x ⊥ y. Say that two subsets M and N of X are disjoint and write M ⊥ N if x ⊥ y for all x ∈ M and y ∈ N . The properties of disjointness are easy from 2.1.4: x ⊥ y ⇐⇒ |x + y| = |x − y| ⇐⇒ |x| ∨ |y| = |x| + |y|; x+ ⊥ x− ; (x − x ∧ y) ⊥ (y − x ∧ y); x ⊥ y =⇒ |x + y| = |x| + |y|, (x + y)+ = x+ + y + , (x + y)− = x− + y − . 56 Chapter 2. Boolean Valued Numbers The disjoint complement M ⊥ of M ⊂ X, M = ∅, is deﬁned as M ⊥ := {x ∈ X : (∀ y ∈ M ) x ⊥ y}. Put M ⊥⊥ := (M ⊥ )⊥ . The disjoint complement has the properties: M ⊂ M ⊥⊥ , M ⊥ = M ⊥⊥⊥ , M ⊥ ∩ M ⊥⊥ = {0}. 2.1.8. A nonempty set K in X meeting the identity K = K ⊥⊥ is called a band (a component in the Russian literature) of X. Every band of the form {x}⊥⊥ with x ∈ X is called principal. The inclusion-ordered set of all bands of X is denoted by B(X) and presents a complete Boolean algebra. The Boolean operations on B(X) take the shape: L ∧ K = L ∩ K, L ∨ K = (L ∪ K)⊥⊥ , L∗ = L⊥ (L, K ∈ B(X)). Let u ∈ X+ and e ∧ (u − e) = 0 for some 0 e ∈ X. Then e is said to be a component or a fragment of u. The set C(u) of all components of u with the order induced by X is a Boolean algebra. The lattice operations in C(u) are taken from X, while the Boolean complement has the form e∗ := u − e (e ∈ C(u)). If an order unit 1 is ﬁxed in X then the notation C(X) := C(X, 1) := C(1) is also in use. 2.1.9. A band B in a vector lattice X is said to be a projection band if X = B ⊕ B ⊥ . It can easily be seen that B is a projection band if and only if B is an order ideal and for every x ∈ X+ the supremum of B+ ∩ [0, x] exists in X and belongs to B. The projection from X onto B along B ⊥ is called a band projection or an order projection and denoted by [B] or PB . A linear operator P : X → X is a band projection if and only if P 2 = P and 0 P x x for all x ∈ X+ . Moreover, [B]x := sup(B+ ∩ [0, x]) + − [B]x := [B]x − [B]x (x ∈ X+ ), (x ∈ X). (1) The set P(X) of all band projections ordered by π ρ ⇐⇒ π ◦ ρ = π is a Boolean algebra. The Boolean operations on P(X) take the shape π ∧ ρ = π ◦ ρ, π ∨ ρ = π + ρ − π ◦ ρ, π ∗ = IX − π (π, ρ ∈ P(X)). 2.1. Vector Lattices 57 The band projection onto a principal band is called principal. (2) The principal projection πu := [u] := [u⊥⊥ ], where 0 u ∈ X, can be calculated by the rule simpler than that above: πu x = sup{x ∧ (nu) : n ∈ N}. A vector lattice X is said to have the projection property (principal projection property) if each band (each principal band) in B(X) is a projection band. 2.1.10. (1) A linear subspace J of a vector lattice X is called an order ideal or o-ideal (or, ﬁnally, just an ideal, when it is clear from the context what is meant) if the inequality |x| |y| implies x ∈ J for all x ∈ X and y ∈ J. Each order ideal of a vector lattice is a vector lattice. If an ideal J possesses the additional property J ⊥⊥ = X (or, which is the same, J ⊥ = {0}) then J is referred to as an order dense ideal of X (the term “foundation” is also used in the Russian literature). (2) A vector sublattice is a vector subspace X0 ⊂ X such that x ∧ y, x ∨ y ∈ X0 for all x, y ∈ X0 . We say that a sublattice X0 is order dense or minorizing if, for every 0 = x ∈ X+ , there exists x0 ∈ X0 satisfying 0 < x0 x. We say that X0 is a majorizing or massive sublattice if, for every x ∈ X, there exists x0 ∈ X0 such that x x0 . Thus, X0 is a minorizing or a majorizing sublattice if and only if X+ \ {0} = X+ + X0 + \ {0} or X = X+ + X0 , respectively. (3) A set in X is called (order ) bounded (or o-bounded ) if it is included in some order interval. The o-ideal generated by the element ∞ 0 u ∈ X is the set X(u) := n=1 [−nu, nu]; clearly, X(u) is the least o-ideal in X containing u. If X(u) = X then we say that u is a strong unit or strong order unit . If X(u)⊥⊥ = X then we say that u is an order unit or weak order unit. It is evident that an element u ∈ X+ is an order unit if and only if {u}⊥⊥ = X; i.e., if there is X no nonzero element in X disjoint from u. Recall that the term unit is often replaced with unity by tradition. This leads to no confusion. (4) An element x 0 of a vector lattice is called discrete if [0, x] = [0, 1]x; i.e., if 0 y x implies y = λx for some 0 λ 1. A vector lattice X is called discrete or atomic if, for every 0 = y ∈ X+ , there exists a discrete element x ∈ X such that 0 < x y. If X lacks nonzero discrete elements then X is said to be continuous or diﬀuse. 58 Chapter 2. Boolean Valued Numbers 2.1.11. A vector lattice X is said to be Dedekind complete or order complete whenever each nonempty upper bounded subset of X has the least upper bound (or, equivalently, whenever each nonempty lower bounded subset of X has the greatest lower bound). If, in a vector lattice X, the least upper bounds (or equivalently the greatest lower bounds) exist only for countable bounded sets, then X is called Dedekind σ-complete or order σ-complete. A Dedekind complete vector lattice and a Dedekind σ-complete vector lattice are frequently referred to in the Russian literature as K-space (= Kantorovich space) and a Kσ -space, respectively. 2.1.12. Let X be a Dedekind complete vector lattice. Then X has the projection property and the mapping B → [B] is an isomorphism of the Boolean algebras B(X) and P(X). If there is an order unit 1 in X then the mappings P → P (1) from P(X) in C(X) and e → {e}⊥⊥ from C(X) in B(X) are isomorphisms of Boolean algebras, too. 2.1.13. Recall that a vector lattice is called laterally complete whenever each disjoint set positive vectors in it has a supremum. A vector lattice that is at the same time laterally complete and Dedekind complete is referred to as a universally complete vector lattice. A linear operator T : X → Y is a lattice homomorphism provided that T preserves the joins and meets of nonempty ﬁnite sets. If X is an Archimedean vector lattice then there exists a unique (up to lattice isomorphism) universally complete vector lattice X u (called the universal completion of X) such that X is lattice isomorphic to an order dense sublattice of X u . Identifying X with its copy in X u , we have the vector sublattice inclusion X ⊂ X u with X order dense in X u . In particular, the Dedekind completion X δ of X can be identiﬁed with the ideal generated by X in X u , and so we have the lattice isomorphisms δ X ⊂ X ⊂ X u with X δ order dense in X u . 2.2. Gordon’s Theorem In this section we will demonstrate that the externalization of the Boolean valued reals is a universally complete vector lattice. 2.2.1. By a field of reals we mean an algebraic system that satisﬁes the axioms of an Archimedean ordered ﬁeld (with distinct zero and unity) and the axiom of completeness. Recall the two well-known assertions: 2.2. Gordon’s Theorem 59 (1) There exists a field of reals R that is unique up to isomorphism. (2) If P is an Archimedean ordered field then there is an isomorphic embedding h of the field P into R such that the image h(P) is a subfield of R containing a subfield of rationals. In particular, h(P) is dense in R. 2.2.2. Successively applying the transfer and maximum principles of Boolean valued analysis to 2.2.1(1), we ﬁnd an element R ∈ V(B) for which [[ R is a ﬁeld of reals ]] = 1. Moreover, if an arbitrary R ′ ∈ V(B) satisﬁes the condition [[ R ′ is a ﬁeld of reals ]] = 1 then it also satisﬁes [[ the ordered ﬁelds R and R ′ are isomorphic]] = 1. In other words, there exists a ﬁeld of reals R in V(B) and such a ﬁeld is unique up to isomorphism. We call R the reals in V(B) . Note also that ϕ(x), formally presenting the expressions of the axioms of an Archimedean ordered ﬁeld x, is restricted; therefore, [[ ϕ(R∧ ) ]] = 1; i.e., [[ R∧ is an Archimedean ordered ﬁeld ]] = 1. “Pulling” 2.2.1(2) by transfer we conclude that [[ R∧ is isomorphic to a dense subﬁeld of the ﬁeld R ]] = 1. In this regard, we further assume that R is the reals in V(B) and R∧ is a dense subﬁeld of R. It is easy to note that the elements 0 := 0∧ and 1 := 1∧ are the zero and unit of R. 2.2.3. Let us consider the descent R↓ of the algebraic system R := (|R|, ⊕, ⊙, , 0, 1) (see 1.5.6, 1.5.7, and 1.8.4). In other words, we consider the descent of the universe |R| of the system R together with the descended operations + := ⊕↓, · := ⊙↓ and order := ↓. In more detail, we introduce addition, multiplication, and order on R↓ by the formulas z = x + y ⇐⇒ [[ z = x ⊕ y ]] = 1, z = x · y ⇐⇒ [[ z = x ⊙ y ]] = 1, x y ⇐⇒ [[ x y ]] = 1 (x, y, z ∈ |R|↓). Also, we can introduce multiplication by the standard reals in R↓ by y = λx ⇐⇒ [[ y = λ∧ ⊙ x ]] = 1 (λ ∈ R, x, y ∈ |R|↓). For simplicity, in the sequel we identify R and |R| and denote the operations and order on R and R↓ by the same symbols +, · , and . 2.2.4. Gordon Theorem. Let R be the reals in V(B) . Then R↓ (with the descended operations and order) is a universally complete vector lattice with a weak order unit 1 := 1∧ . Moreover, there exists a Boolean 60 Chapter 2. Boolean Valued Numbers isomorphism χ of B onto the Boolean algebra of band projections P(R↓) (or components of the unit C(1)) such that χ(b)x = χ(b)y ⇐⇒ b [[ x = y ]], χ(b)x χ(b)y ⇐⇒ b [[ x y ]] (G) for all x, y ∈ R↓ and b ∈ B. ⊳ We omit an elementary veriﬁcation of the fact that R↓ is a vector space over R and an ordered set. The remaining part is given in 2.2.5 and 2.2.6 below. ⊲ 2.2.5. The algebraic system R↓ is a universally complete vector lattice with weak order unit 1 := 1∧ . ⊳ Show that the operations and order agree on R↓. Take elements x, y ∈ R↓ such that x y. This means that V(B) |= “x and y are reals and x y.” Let u := x+z, v := y+z, x′ := λx, and y ′ := λy, where z ∈ R↓ and λ ∈ R, λ 0. By the deﬁnition of operations and order on R↓, we have V(B) |= “x′ , y ′ , u, and v are reals; moreover, u = x + z, v = y + z, x′ = λ∧ x, and y ′ = λ∧ y.” The inequality λ 0 implies V(B) |= λ∧ 0∧ = 0. Using the requested properties of real numbers within V(B) , we obtain V(B) |= “u v and x′ y ′ .” Thereby u v and x′ y ′ by 2.2.3. Let us show that the supremum of a nonempty bounded set A exists. Suppose that A ⊂ R↓ is bounded above by y ∈ R↓. By deﬁnition, [[ x y ]] = 1 for every x ∈ A. Then V(B) |= “A↑ is a set of reals bounded above by y” or, in view of 1.6.2, [[ (∀ x ∈ A↑) (x y) ]] = [[ x y ]] = 1. x∈A The completeness of R yields [[ (∃a ∈ R)(a = sup(A↑)) ]] = 1. By the maximum principle we ﬁnd a ∈ V(B) such that [[ a ∈ R ]] = [[ a = sup(A↑) ]] = 1. Thereby a ∈ R↓ and if z ∈ R↓ is an upper bound of A then, as was already shown, [[ z is an upper bound of A↑ ]] = 1; therefore, [[ a z ]] = 1 or a z. Consequently, a is the supremum of A in R↓. Incidentally, we have established that a = sup(A) if and only if 2.2. Gordon’s Theorem 61 [[ a = sup(A↑) ]] = 1. In particular, given arbitrary x, x1 , x2 ∈ R↓, we have x = x1 ∨ x2 if and only if [[ x = sup{x1 , x2 } = x1 ∨ x2 ]] = 1, since [[ {x1 , x2 }↑ = {x1 , x2 } ]] = 1. Of course, an analogous assertion is valid for meets. Finally, take an arbitrary disjoint set A ⊂ R↓ of positive elements. We may see from the above remarks and 1.5.2 that [[ (∀ x1 ∈ A↑)(∀ x2 ∈ A↑)x1 ∧ x2 = 0 ]] = [[ x1 ∧ x2 = 0 ]] = 1. x1 ,x2 ∈A Hence, the numerical set A↑ (within V(B) ) consists of pairwise disjoint positive elements. For such a set we have only the two possibilities open: either [[ A↑ = {0} ]] = 1 and then A ⊂ A↑↓ = {0}, or [[ A↑ = {0, a} ]] = 1 for some 0 < a ∈ R↓ (by the maximum principle) and then [[ sup(A↑) = a ]] = 1. As was mentioned above, the latter is equivalent to the equality a = sup A. We may conclude now that R↓ is a universally complete vector lattice. Recalling that 1 := 1∧ is the unity of the ﬁeld R within V(B) and using the formulas of 1.5.2, we ﬁnd [[ x ∧ 1 = 0 ]] ⇒ [[ x = 0 ]]. 1 = [[ (∀ x ∈ R)(x ∧ 1 = 0 → x = 0) ]] = x∈R↓ Hence, we see that [[ x ∧ 1 = 0 ]] [[ x = 0 ]] for each x ∈ R↓. If x ∧ 1 = 0 then [[ x ∧ 1 = 0 ]] = 1 and so [[ x = 0 ]] = 1; i.e., x = 0. Thereby 1 is a weak order unit of the vector lattice R↓. ⊲ 2.2.6. There exists an isomorphism χ of the Boolean algebra B onto P(R↓) such that for all x, y ∈ R↓ and b ∈ B the equivalences of 2.2.4(G) hold. ⊳ Let us introduce some mapping χ : B → P(R↓). Take an arbitrary element b ∈ B and put χ(b)x := mix{bx, b∗ 0} for x ∈ R↓. In other words, the element χ(b)x ∈ R↓ is uniquely determined by the relations (cp. 1.4.3): b [[ χ(b)x = x ]], b∗ [[ χ(b)x = 0 ]]. This implies that π := χ(b) : R↓ → R↓ is an extensional mapping. Indeed, given x, y ∈ R↓ we have (cp. 1.2.5 (3)): [[ x = y ]] ∧ b [[ x = y ]] ∧ [[ x = πx ]] ∧ [[ y = πy ]] [[ πx = πy ]], [[ x = y ]] ∧ b∗ [[ x = y ]] ∧ [[ πx = 0 ]] ∧ [[ πy = 0 ]] [[ πx = πy ]]. 62 Chapter 2. Boolean Valued Numbers If ρ := π↑ then [[ ρ : R → R ]] = 1 by 1.6.5 and ρ = mix{bIR , b∗ 0} by deﬁnition. Since 0 and IR are idempotent positive linear mappings from R to R, so is π. Moreover, [[ (∀ x ∈ R+ )ρx x ]] = 1; therefore, πx x for all x ∈ R↓+ . Thus, π = χ(b) is a band projection. Since ρ is positive, we have [[ x y → ρx ρy ]] = 1 for x, y ∈ R↓ and so [[ x y ]] [[ ρx ρy ]] = [[ πx πy ]]. Assume that πx πy. Then [[ πx = πy ]] = 1 and by 1.2.6 b [[ πx πy ]] ∧ [[ πx = x ]] ∧ [[ πy = y ]] [[ x y ]]. Conversely, if we assume that b [[ x y ]] then b [[ πx πy ]] by the above observation. Moreover, b∗ [[ πx = 0 ]] ∧ [[ πy = 0 ]] ∧ [[ 0 0 ]] [[ πx πy ]]; consequently, [[ πx πy ]] = 1 or πx πy. Thereby we have established the second of the required equivalences 2.2.4 (G). The ﬁrst ensues from that by the formula u = v ⇐⇒ u v ∧ v = u. It remains to demonstrate that the mapping χ is an isomorphism between the Boolean algebras B and P(R↓). Take an arbitrary band projection π ∈ P(R↓) and put b := [[ π↑ = IR ]]. The fact that a band projection is extensional (and so the ascent π↑ of π is well-deﬁned) follows from 2.2.4 (G), because c = [[ x = y ]] =⇒ χ(c)x = χ(c)y =⇒ πχ(c)x = πχ(c)y =⇒ χ(c)πx = χ(c)πy =⇒ c [[ πx = πy ]]. Since π is idempotent, π↑ as well is an idempotent mapping in R; i.e., either π↑ = IR or π↑ = 0. Hence, we derive b∗ = [[ π↑ = IR ]] = [[ π↑ = 0 ]] and thereby π↑ = mix{bIR , b∗ (0)}. The mixture is unique; therefore, π↑ = χ(b)↑; i.e., π = χ(b). Thus, χ is a bijection between B and P(R↓). Take arbitrary b1 , b2 ∈ B and put ρk := χ(bk )↑ (k := 1, 2). Recalling that ρk = mix{bk IR , b∗k 0}, we derive [[ χ(b1 ∧ b2 )↑ = IR ]] = b1 ∧ b2 = [[ ρ1 = IR ∧ ρ2 = IR ]] [[ ρ1 ◦ ρ2 = IR ]], [[ χ(b1 ∧ b2 )↑ = 0 ]] = (b1 ∧ b2 )∗ = [[ ρ1 = 0 ∨ ρ2 = 0 ]] [[ ρ1 ◦ ρ2 = 0 ]]. 2.3. Gordon’s Theorem Revisited 63 From this, using 1.2.5 (3), we obtain b1 ∧ b2 [[ χ(b1 ∧ b2 )↑ = IR ]] ∧ [[ ρ1 ◦ ρ2 = IR ]] [[ χ(b1 ∧ b2 )↑ = ρ1 ◦ ρ2 ]], (b1 ∧ b2 )∗ [[ χ(b1 ∧ b2 )↑ = 0 ]] ∧ [[ ρ1 ◦ ρ2 = 0 ]] [[ χ(b1 ∧ b2 )↑ = ρ1 ◦ ρ2 ]]. Thus, [[ χ(b1 ∧ b2 )↑ = ρ1 ◦ ρ2 ]] = 1 and taking into account the identity [[ ρ1 ◦ ρ2 = χ(b1 )↑ ◦ χ(b2 )↑ = (χ(b1 ) ∧ χ(b2 ))↑ ]] = 1 (see 1.6.4) we arrive at the desired property [[ χ(b1 ∧ b2 )↑ = ρ1 ◦ ρ2 = (χ(b1 ) ∧ χ(b2 ))↑ ]] = 1 or, equivalently, χ(b1 ∧ b2 ) = χ(b1 ) ∧ χ(b2 ). In particular, 0 = χ(b)∧χ(b∗ ) for χ(0) = 0. Given elements ρ := χ(b)↑ and ρ′ := χ(b∗ ), we have [[ ρ, ρ′ ∈ {0, IR }; ρ = 0 or ρ′ = 0; and ρ and ρ′ do not vanish simultaneously ]] = 1. Hence, we see that [[ ρ + ρ′ = IR ]] = 1 and thereby χ(b) + χ(b∗ ) = IR↓ . Summarizing, we conclude that χ preserves meets and complements; i.e., χ is an isomorphism. ⊲ 2.3. Gordon’s Theorem Revisited In this section we examine some additional properties of Boolean valued reals: multiplicative structure, complexiﬁcation, and some absoluteness. 2.3.1. An ordered algebra over an ordered ﬁeld F is an ordered vector space X over F which is simultaneously an algebra over the same ﬁeld and satisﬁes the condition: if x 0 and y 0 then xy 0 for all x, y ∈ X. To characterize the positive cone X+ of an ordered algebra X, we must add to what was said in 2.1.2 the property X+ · X+ ⊂ X+ . We say that X is a lattice ordered algebra if X is a vector lattice and an ordered algebra simultaneously. A lattice-ordered algebra is an f -algebra if, for all a, x, y ∈ X+ , the condition x ⊥ y = 0 implies that (ax) ⊥ y and (xa) ⊥ y. The multiplication on every (Archimedean) f -algebra is commutative and associative. An f -algebra is called semiprime if xy = 0 implies x ⊥ y for all x and y. Clearly, an f -algebra is semiprime if and only if it do not contain nonzero nilpotent elements. The semiprimness of an f -algebra is equivalent to saying that there is no strictly positive element with zero square in it. A multiplicative unit vector of an 64 Chapter 2. Boolean Valued Numbers Archimedean f -algebra is a weak order unit. Moreover, an f -algebra with unit is semiprime. 2.3.2. Theorem. The universally complete vector lattice R↓ with the descended multiplications is a semiprime f -algebra with ring unit 1 := 1∧ . Moreover, for every b ∈ B the band projection χ(b) acts as multiplication by χ(b)1. ⊳ The multiplicative structure on R↓ was deﬁned in 2.2.3. As in the Gordon Theorem, we establish that R↓ is a semiprime f -algebra. Take x ∈ R↓ and b ∈ B. By the deﬁnition of χ(b), we have b [[χ(b)x = x]] and b∗ [[χ(b∗ )x = 0]]. Applying these to x := 1∧ and appealing to the deﬁnition of multiplication on R↓, we obtain b [[x = x·1∧ = x·χ(b)1∧ ]] and b∗ [[ 0 = x · 0 = x · χ(b)1∧ ]]. Thereby [[χ(b)x = x · χ(b)1∧ ]] [[χ(b)x = x]] ∧ [[x = x · χ(b)1∧ ]] b. In a similar way, b∗ [[χ(b)x = χ(b)1∧ · x]]. Hence, [[χ(b)x = x · χ(b)1∧ ]] = 1. ⊲ We see from the above that the mapping b → χ(b)1∧ (b ∈ B) is a Boolean isomorphism between B and the algebra C(R↓) := C(1∧ ) of the components of the weak order unit 1∧ . This isomorphism is denoted by the same letter χ. Thus, depending on the context, x → χ(b)x is either the appropriate band projection or the operator of multiplication by χ(b) 1. 2.3.3. A complex vector lattice is deﬁned to be the complexiﬁcation XC := X ⊕ iX (with i standing for the imaginary unit) of a real vector lattice X; i.e., the additive group of X × X is endowed additionally with the scalar multiplication (α + iβ)(x, y) = (αx − βy, αy + βx) for all α, β ∈ R and x, y ∈ X. Identifying x ∈ X with (x, 0) ∈ X and iy with (0, y), we will write x + iy instead of (x, y). Often it is additionally required that the modulus |z| := sup{(cos θ) x + (sin θ) y : 0 θ < 2π} exists for every element z := x + iy ∈ X ⊕ iX. This requirement is automatically satisﬁed in a uniformly complete vector lattice, so that a Dedekind complete complex vector lattice is the complexiﬁcation of a Dedekind complete real vector lattice. The Riesz decomposition property remains valid in every complex vector lattice: For all z, z1 , z2 ∈ XC with |z| |z1 | + |z2 | there exist v1 , v2 ∈ XC satisfying z = v1 + v2 and |vk | |zk | (k = 1, 2). 2.3. Gordon’s Theorem Revisited 65 Speaking about the order properties of the complex vector lattice XC , we mean its real part X. The concepts of sublattice, ideal, band, projection, etc. are naturally translated to the case of a complex vector lattice by appropriate complexiﬁcation. For example, a subset A from XC is said to be order bounded if the set {|z| : z ∈ A} is order bounded in X. The the disjointness relation ⊥ in XC is deﬁned as z1 ⊥ z2 ⇐⇒ |z1 |∧|z2 | = 0, etc. A complex f -algebra is deﬁned as the complexiﬁcation A⊕iA of a real f -algebra A and is denoted by AC . The multiplication on AC is given by (x + iy)(u + iv) = (xu − yv) + i(xv + yu) for all x, y, u, v ∈ A. The modulus |z| of z = x + iy is introduced by the above formula. Then |z1 z2 | = |z1 ||z2 | and z1 ⊥ z2 implies wz1 ⊥ z2 for all z1 , z2 , w ∈ AC . Evidently, AC has the same identity element as A. 2.3.4. Theorem. Let C be the complexes in V(B) . Then the algebraic system C ↓ is a universally complete complex f -algebra. Moreover, C ↓ the complexification of the universally complete real f -algebra R↓; i.e., C ↓ = R↓ ⊕ iR↓. ⊳ Since C = R ⊕ iR symbolizes a bounded formula, we have [[ C∧ = R ⊕R∧ ]] = 1 (cp. 1.4.7), where i is the imaginary unit and the element i∧ is denoted by the same letter i. From 2.2.2 we see that [[ C∧ is a dense subﬁeld of the ﬁeld C ]] = 1 and, in particular, [[ i is the imaginary unit of the ﬁeld C ]] = 1. If z ∈ C ↓ then z is a complex number within V(B) ; therefore, [[ (∃ !x ∈ R) (∃ !y ∈ R) z = x + iy ]] = 1. ∧ The maximum principle implies that we have the unique pair of elements x, y ∈ V(B) such that [[ x, y ∈ R ]] = [[ z = x + iy ]] = 1. Hence, we obtain x, y ∈ R↓ such that z = x+iy, and so C ↓ = R↓⊕iR↓. Appealing to the Gordon Theorem and 2.3.2 completes the proof. ⊲ Consider complete Boolean algebras B and D and a complete epimorphism h : B → D. Denote by h∗ : V(B) → V(D) the corresponding epimorphism of Boolean valued universes (cp. Section 1.3). 2.3.5. Let R be the reals within V(B) . Then h∗ (R) is the reals within V and the mapping h∗ : x → h∗ (x) is an order continuous lattice homomorphism from R↓ onto h∗ (R)↓. Moreover, if χ : B → P(R↓) (D) 66 Chapter 2. Boolean Valued Numbers and χ̄ : D → P(h∗ (R)↓) are Boolean isomorphisms from the Gordon Theorem then the diagram χ B −−−−→ P(R↓) ⏐ ⏐ ⏐ ⏐ ∗ h h χ̄ D −−−−→ P(h∗ (R)↓) commutes. In particular, if ker(h) = [0, b] for some b ∈ B and πb := χ(b) then ker(h∗ ) = ker(πb ), so that h∗ (R)↓ can naturally be identified with the band πb (R↓) in the universally complete vector lattice R↓. ⊳ It follows from 1.3.3 (2) that h∗ (R) is the reals within VD and h∗ is a lattice homomorphism from R↓ onto h∗ (R)↓. If x ∈ R↓, 1̄ := 1D is the unit of D and πb := χ(b) then by 1.3.3 (2) and 1.4.6 we have ∧ h∗ (x) = 0 ⇐⇒ [[h∗ (x) = 0∧ ]] = 1̄ ⇐⇒ h([[x = 0∧ ]]) = h(b∗ ) ⇐⇒ b [[x = 0∧ ]] ⇐⇒ πb x = 0 and so ker(h∗ ) = ker(χ(b)). To prove the commutativity of the above diagram it is enough to estimate Boolean truth values for all a ∈ B: ∧ [[χ̄(h(a)) = h∗ (χ(a))]]D [[χ̄(h(a)) = 1∧ ]]D ∧ [[h∗ (χ(a)) = h∗ (1∧ )]]D = b ∧ a ∧ b ∧ [[χ(a) = 1∧ ]]B = b ∧ a, h(a) ∧ h([[χ(a) = 1∧ ]]B ) = h(a), ∧ [[χ̄(h(a)) = h∗ (χ(a))]]D [[χ̄(h(a)) = 0∧ ]]D ∧ [[h∗ (χ(a)) = h∗ (0∧ )]]D = b∧a∗ ∧b∧[[χ(a) = 0∧ ]]B = b∧a∗ , h(a∗ )∧h([[χ(a) = 0∧ ]]B ) = h(a∗ ), whence [[χ̄(h(a)) = h∗ (χ(a))]]D h(a) ∨ h(a∗ ) = 1̄. It follows that χ̄(h(a)) = h∗ (χ(a)) for all a ∈ D and the proof is complete. ⊲ 2.3.6. Consider the relative Boolean algebra B̄ := [0, b] ⊂ B with 0 = b ∈ B. Then b ∧ R is the reals within VB̄ and the mapping hb : x → b ∧ x is an order continuous lattice homomorphism from R↓ onto (b ∧ R)↓. Moreover, if χ : B → P(R↓) and χ̄ : B̄ → P((b ∧ R)↓) are Boolean isomorphisms from the Gordon Theorem then χ̄(b ∧ a) = b ∧ χ(a) for all a ∈ B. In particular, ker(hb ) = ker(πb ), πb := χ(b), so that (b ∧ R)↓ can naturally be identiﬁed with the band πb (R↓). 2.3.7. Let h : B → D be an isomorphism of Boolean algebras. Then h∗ (R) is the reals within V(D) and the mapping h∗ : x → h∗ (x) is a lattice 2.4. Boolean Valued Reals Translated 67 isomorphism of R↓ onto h∗ (R)↓. Moreover, if χ and χ̄ are the same as in 2.3.5 then h∗ ◦ χ = χ̄ ◦ h. 2.4. Boolean Valued Reals Translated Henceforth, R denotes the reals within V(B) . We will clarify the meaning of the least upper and greatest lower bounds, order limits, carriers, and spectral systems in the vector lattice R↓. 2.4.1. First, we will introduce a few deﬁnitions we need. The order on a vector lattice generates various types of convergence. Let (A, ) be an upward directed set. A net (xα ) := (xα )α∈A in X is called increasing (decreasing) if xα xβ (xβ xα ) for α β (α, β ∈ A). We say that a net (xα ) in a vector lattice X o-converges to x ∈ X if there exists a decreasing net (eβ )β∈B in X such that inf{eβ : β ∈ B} = 0 and for each β ∈ B there is α(β) ∈ A with |xα − x| eβ for all α α(β). In this event, we call x the o-limit of the net (xα ) and write x = o-lim xα (o) or xα −→ x. If a net (eβ ) in this deﬁnition is replaced by a sequence (λn e)n∈N , where 0 v ∈ X+ and (λn )n∈N is a numerical sequence with limn→∞ λn = 0, then we say that a net (xα )α∈A converges relatively uniformly or more precisely e-uniformly to x ∈ X. The elements e and x are called the regulator of convergence and the r-limit of (xα ), respec(r) tively. The notations x = r-limα∈A xα and xα −→ x are also frequent. A net (xα )α∈A is called o-fundamental (r-fundamental with regulator e) provided that the net (xα − xβ )(α,β)∈A×A o-converges (respectively, r-converges with regulator e) to zero. A vector lattice is said to be (relatively) uniformly complete if every r-fundamental sequence in it is r-convergent. 2.4.2. Deﬁne the sum of an inﬁnite family (xξ )ξ∈Ξ in a vector lattice X. Given θ := {ξ1 , . . . , ξn } ∈ Pfin (Ξ), put yθ := xξ1 + · · · + xξn . So, we arrive at the net (yθ )θ∈Θ , where Θ := Pfin (Ξ) is ordered by inclusion. Assuming that there is some x satisfying x = o-limθ∈Θ yθ , we call the family (xξ ) summable in order or order summable or o-summable. The element x is the o-sum of (xξ ) and we write x = o- ξ∈Ξ xξ . Obviously, if xξ 0 (ξ ∈ Ξ) then for the o-sum of the family (xξ ) to exist it is necessary and suﬃcient that the net (yθ )θ∈Θ has join, in which case 68 o- Chapter 2. Boolean Valued Numbers ξ∈Ξ xξ = supθ∈Θ yθ . If (xξ ) is a disjoint family then o- − xξ = sup x+ ξ − sup xξ . ξ∈Ξ ξ∈Ξ ξ∈Ξ 2.4.3. Let (bξ )ξ∈Ξ be a partition of unity in B and let (xξ )ξ∈Ξ be a family in R↓. Then mix(bξ xξ ) = oχ(bξ )xξ . ξ∈Ξ ξ∈Ξ ⊳ If x := mixξ∈Ξ (bξ xξ ) then bξ [[ x = xξ ]] (ξ ∈ Ξ) (cp. 1.4.3). According to 2.2.4 (G), χ(bξ )xξ = χ(bξ )x for all ξ ∈ Ξ. Summing the last equalities over ξ, we arrive at what was required. ⊲ 2.4.4. For a nonempty set A ⊂ R↓ and all a ∈ R and b ∈ B the equivalences hold: b [[ a = sup(A↑) ]] ⇐⇒ χ(b)a = sup χ(b)(A); b [[ a = inf(A↑) ]] ⇐⇒ χ(b)a = inf χ(b)(A). ⊳ We will prove only the ﬁrst equivalence. In view of 2.2.4 (G), the equality χ(b) a = sup{χ(b) x : x ∈ A} holds if and only if b [[ x a ]] for all x ∈ A and the formula (∀ x ∈ A)(b [[ x y ]]) implies b [[ a y ]] for each y ∈ R↓. Using the rules for calculating the truth values for quantiﬁers (cp. 1.2.3), we can represent the conditions under consideration in equivalent form: b [[ (∀ x ∈ A↑) x a ]], b [[ (∀ y ∈ R) (A↑ y → a y) ]]. This system of inequalities is equivalent to b [[ a = sup(A↑) ]]. ⊲ 2.4.5. Let A be an upward directed set and let s : A → R↓ be a net in R↓. Then A∧ is directed upward and σ := s↑ : A∧ → R is a net in R (within V(B) ). Moreover, b [[ x = lim σ ]] ⇐⇒ χ(b)x = o-lim χ(b) ◦ s for arbitrary x ∈ R↓ and b ∈ B. 2.4. Boolean Valued Reals Translated 69 ⊳ The assertion “A is an upward directed set” is a bounded formula. By 1.4.7, we have V(B) |= “A∧ is an upward directed set.” The equality χ(b)x = o-lim χ(b) ◦ s means that there exists a net d : A → R↓ for which the following system of conditions is compatible: α β → d(α) d(β) (α, β ∈ A), |χ(b)x − χ(b)s(α)| d(α) inf d(α) = 0, α∈A (α ∈ A). Taking account of the easy formula [[ s(A)↑ = σ(A∧ ) ]] = 1 (see 1.6.9) and putting δ := d ↑, we see that the system of conditions is equivalent to the simultaneous inequalities: b [[ inf σ(A∧ ) = 0 ]], b [[ (∀ α, β ∈ A∧ ) (α β → σ(α) σ(β)) ]], b [[ (∀ α ∈ A∧ ) (|x − σ(α)| < δ(α)) ]], whose short form is just as follows: b [[ x = lim σ ]]. ⊲ 2.4.6. Suppose that A and σ ∈ V(B) are such that [[ A is directed upward and σ : A → R ]] = 1. Then A↓ is an upward directed set and so the mapping s := σ↓ : A↓ → R↓ is a net in R↓. Moreover, b [[ x = lim σ ]] ⇐⇒ χ(b)x = o-lim χ(b) ◦ s for all x ∈ R↓ and b ∈ B. ⊳ The proof is similar to that of 2.4.5. ⊲ 2.4.7. Let f be a mapping from a nonempty set Ξ to R↓ and g := f ↑. Then b x= g(ξ) ⇐⇒ χ(b)x = oχ(b)f (ξ) ξ∈Ξ∧ ξ∈Ξ for all x ∈ R↓ and b ∈ B. ⊳ First of all observe that the required equivalence holds for a ﬁnite set Ξ0 ⊂ Ξ. Afterwards, apply 2.4.5 to the net s : Pfin (Ξ) → R↓, where Pfin (Ξ) is the set of ﬁnite subsets of Ξ and s(θ) := ξ∈θ f (ξ), and use the formula [[ Pfin (Ξ)∧ = Pfin (Ξ∧ ) ]] = 1 (cp. 1.4.10). ⊲ 2.4.8. Let X be a vector lattice with the principal projection property and a weak order unit 1. We call the projection of 1 to the band {x}⊥⊥ the trace of x and denoted it by ex . Thus, ex := sup{1 ∧ (n|x|) : n ∈ N}. 70 Chapter 2. Boolean Valued Numbers Clearly, the trace ex serves both as a weak order unit of {x}⊥⊥ and a component of 1. Given a real λ, denote the trace of the positive part of λ1 − x by exλ : exλ := e(λ1−x)+ = sup{1 ∧ (n(λ1 − x)+ ) : n ∈ N}. The function λ → exλ (λ ∈ R) arising in this case is called the spectral system or characteristic of x. 2.4.9. The following hold for every x ∈ R↓: ex := χ([[ x = 0 ]]), exλ = χ([[ x < λ∧ ]]) (λ ∈ R). ⊳ A real t is distinct from zero if and only if the join of the set {1 ∧ (n|t|) : n ∈ ω} is equal to 1. Consequently, for x ∈ R↓ the transfer principle yields b := [[ x = 0 ]] = [[ 1∧ = sup A ]], where A ∈ V(B) is determined by the formula A := {1∧ ∧ (n|x|) : n ∈ ω ∧ }. If C := {1∧ ∧ (n|x|) : n ∈ ω} then we prove that [[ C↑ = A ]] = 1 using the second formula of 1.6.2 and the representation ω ∧ = (ιω)↑ of 1.5.8. Hence, [[ sup(A) = sup(C↑) ]] = 1. Using 2.4.4, we derive b = [[ sup(C↑) = 1∧ ]] = [[ sup(C) = 1∧ ]] = [[ ex = 1∧ ]]. On the other hand, [[ ex = 0 ]] = [[ ex = 1∧ ]]∗ = b∗ . By 2.2.4 (G), we can write down χ(b)ex = χ(b)1∧ = χ(b), χ(b∗ )ex = 0 =⇒ χ(b)ex = ex . Finally, χ(b) = ex . Take λ ∈ R and put y := (λ1 − x)+ . Since [[ λ∧ = λ1 ]] = 1, we have [[ y = (λ∧ − x)+ ]] = 1. Consequently, exλ = ey = χ([[ y = 0 ]]). It remains to observe that within V(B) the number y = (λ∧ − x) ∨ 0 is distinct from zero if and only if λ∧ − x > 0; i.e., [[ y = 0 ]] = [[ x < λ∧ ]]. ⊲ 2.5. Vector Lattices Within Boolean Valued Reals The aim of this section is to demonstrate that an Archimedean vector lattice is represented as a vector sublattice of the internal reals R in an appropriate Boolean valued universe considered as a vector lattice over the ﬁeld of standard reals. 2.5. Vector Lattices Within Boolean Valued Reals 71 2.5.1. Representation Theorem. Let X be an Archimedean vector lattice, let R be the reals within V(B) , and let j be an isomorphism of B onto the Boolean algebra B(X). Then there exists an element X ∈ V(B) satisfying the conditions: (1) V(B) |= “X is a vector sublattice of the field R considered as a vector lattice over the subfielf R∧ ”. (2) X ′ := X ↓ is a laterally complete vector sublattice of R↓ which is majorizing and invariant under each band projection χ(b) (b ∈ B). (3) ι(X) is an order dense sublattice in R↓ for some o-continuous lattice isomorphism ι : X → X ′ . (4) For every b ∈ B the band projection in R↓ onto {ι(j(b))}⊥⊥ coincides with χ(b). ⊳ Put d(x, y) := j−1 ({|x − y|}⊥⊥ ). Let X be the Boolean valued representation of the B-set (X, d) and X ′ := X ↓ (cp. 1.7.1 and 1.7.2). By 1.7.2, without loss of generality we can assume that X ⊂ X ′ , d(x, y) = [[ x = y ]] (x, y ∈ X), and X ′ = mix(X). Further, furnish X ′ with a vector lattice structure. To this end, take λ ∈ R and x, y ∈ X ′ of the form x := mix(bξ xξ ) and y := mix(bξ yξ ), where (xξ ) ⊂ X, (yξ ) ⊂ X, and (bξ ) is a partition of unity in B. Put x + y := mix(bξ (xξ + yξ )), λx := mix(bξ (λxξ )), x y ⇐⇒ x = mix(bξ (xξ ∧ yξ )). (B) Within V , we deﬁne the addition ⊕, multiplication ⊙, and order on X as the ascents of the corresponding objects on X ′ . More precisely, the operations ⊕ : X × X → X and ⊙ : R∧ × X → X and the predicate ⊂ X × X are determined from the formulas: [[ x ⊕ y = x + y ]] = 1, [[ λ ⊙ x = λx ]] = 1 (x, y ∈ X ′ , λ ∈ R), [[ x y ]] = [[ x = x′ ]] ∧ [[ y = y ′ ]] : x′ , y ′ ∈ X ′ , x′ y ′ . ∧ Thus, we can claim that X is a vector lattice over the ﬁeld R∧ and, in particular, a lattice ordered group within V(B) . Also, it is clear that the Archimedean axiom is valid on X , since X ′ is an Archimedean lattice. Note that if x ∈ X+ then {x}⊥⊥ = d(x, 0) = [[ x = 0 ]]; i.e., {x}⊥ = [[ x = 0 ]]. Consequently, we have [[ x = 0 ]] ∨ [[ y = 0 ]] = {x}⊥ ∨ {y}⊥ = 1B 72 Chapter 2. Boolean Valued Numbers for every pair of disjoint x, y ∈ X. Hence, we easily derive that [[ X is linearly ordered ]] = 1, for [[ (∀ x ∈ X )(∀ y ∈ X ) (|x| ∧ |y| = 0 → x = 0 ∨ y = 0) ]] = 1. It is well known that an Archimedean linearly ordered group is isomorphic to an additive subgroup of the reals. Applying this assertion to X within V(B) , without loss of generality we can assume that X is an additive subgroup of R. Furthermore, we presume that 1∧ ∈ X , since otherwise X could be replaced by the isomorphic group e−1 X with 0 < e ∈ X . The multiplication ⊙ represents a continuous R∧ -bilinear mapping from R∧ × X to X . Let β : R × R → R be its extension by continuity. Then β is R-bilinear and β(1∧ , 1∧ ) = 1∧ ⊙ 1∧ = 1∧ . Consequently, β coincides with the usual multiplication on R; i.e., X is a vector sublattice of the ﬁeld R considered as a vector lattice over R∧ . Thereby X ′ ⊂ R↓. The fact that X ′ is majorizing in R↓ ensues obviously, since [[ X is dense in R ]] = 1 and by the maximum principle for each y ∈ R↓ there exists x ∈ X ↓ with [[y x]] = 1, whence y x. Prove that X is minorizing in X ′ . It follows from the properties of the isomorphism χ (cp. 2.2.6) that χ(b)ιx = 0 ⇐⇒ j(b) {x}⊥ ⇐⇒ x ∈ j(b⊥ ), whatever b ∈ B and x ∈ X+ might be. Hence, χ(b) is the band projection onto the band in R↓ generated by ι(j(b)). Moreover, if χ(b)x = 0 for all x ∈ X+ then b = {0}. Thus, for every b ∈ B we can ﬁnd a positive element y ∈X for which y = χ(b)y. Take 0 < z ∈ X ′ . The representation z = o- ξ∈Ξ χ(bξ )xξ is valid, where (bξ ) is a partition of unity in B and (xξ ) ⊂ X+ . We see that χ(bξ )xξ = 0 at least for one index ξ. Let π := χ(bξ )◦χ([[ xξ = 0 ]]) and let y be a strictly positive element in X such that y = πy. Then for x0 := y ∧ xξ we have 0 < x0 πxξ χ(bξ )xξ z and x0 ∈ X. Thereby X is minorizing in X ′ . ⊲ Observe some corollaries to Theorems 2.2.4 and 2.5.1 with the same denotations B, X, X ′ , X , ι, and R. 2.5.2. A few additional remarks are in order. (1) For every x′ ∈ X ′ there exist a family (xξ ) ⊂ X and a partition of unity (πξ ) in P(R↓) such that πξ ιxξ . x′ = oξ∈Ξ 2.5. Vector Lattices Within Boolean Valued Reals 73 ⊳ By 1.7.2 X ′ = mix(ı(X)) and 2.4.3 yields the result. ⊲ (2) For every x ∈ R↓ and ε > 0 there is xε ∈ X ′ such that |x − xε | ε1. ⊳ This is a consequence of the fact that [[ X is dense in R ]] = 1. ⊲ (3) X is laterally complete if and only if X = X ′ . ⊳ The suﬃciency is obvious. If X is laterally complete then X has the projection property (see Veksler and Geyler [398]) and the claim follows from (1). ⊲ 2.5.3. The element X ∈ V(B) arising in 2.5.1 is called the Boolean valued representation of X. Thus, the Boolean valued representations of Archimedean vector lattices are vector sublattices of the reals R considered as a vector lattice over the ﬁeld R∧ . The vector lattice X ′ = X ↓ is called the lateral completion of X and denoted by X λ . We identify X and its lattice isomorphic image ι(X) in X λ , so that we consider X as a sublattice of its lateral completion X λ . Given a vector sublattice L of a laterally complete vector lattice X ′ , denote by λ(L) the laterally complete sublattice in X ′ generated by L, i.e. the smallest laterally complete sublattice in X ′ including L. It is easy to check that λ(L) = L↑↓ and hence, by 1.6.6 and 2.4.3, λ(L) comprises all x ∈ X of the form x = o- πξ yξ with an arbitrary family (xξ ) in L and partition of unity (πξ ) in P(X ′ ). The lateral completion of a vector lattice is essentially unique: If Z is a laterally complete vector lattice and X is lattice isomorphic to an order dense sublattice Y of Z, then X λ is lattice isomorphic to the sublattice Y ′ of Z consisting of all y ∈ Z representable as y = πξ yξ with an arbitrary family (yξ ) in Y and a partition of unity in P(Y ). 2.5.4. If h : X → R↓ is a lattice isomorphism and for every b ∈ B the band projection onto the band in R↓ generated by h(j(b)) coincides with χ(b) then there exists a ∈ R↓ such that hx = a · ι(x) (x ∈ X). If there exists a weak order unit 1 in X then the isomorphism ι is uniquely determined by the requirement ι1 = 1. ⊳ Indeed, if X0 := im ι and h0 := h ◦ ι−1 then the isomorphism h0 : X0 → R↓ is extensional; therefore, for τ := h0 ↑ we have [[ the mapping τ : X → R is isotonic, injective, and additive ]] = 1. Consequently, h0 is continuous and has the form τ (α) = a · α (α ∈ R), where a is a ﬁxed element in R↓. Hence, we derive that h0 (y) = a · y (y ∈ X0 ) or h(x) = a · ι(x) (x ∈ X). ⊲ 74 Chapter 2. Boolean Valued Numbers 2.5.5. If X is a Dedekind complete vector lattice then X = R, X ′ = R↓, and ι(X) is an order dense ideal in R↓. Moreover, ι−1 ◦ χ(b)◦ ι is the band projection onto j(b) for every b ∈ B. ⊳ If X is order complete then so is X ′ . From 2.4.4 we see that the order completeness of X ′ is equivalent to the axiom of existence of suprema (inﬁma) for bounded sets in X . By 2.2.1 (1), X = R and X ′ = R↓. Let e ∈ X+ , y ∈ R↓, and |y| ιe. Since ι(X) is an order dense sublattice in R↓, we have y + = sup ι(A), where A := {x ∈ X+ : ιx y + }. But the set A is bounded in X by e; therefore, sup A ∈ X and y + = ι(sup A) ∈ ιX. Similarly, y − ∈ ι(X) and, ﬁnally, y ∈ ι(X). ⊲ 2.5.6. The image ι(X) coincides with the whole R↓ if and only if X is a universally complete vector lattice. ⊳ If X is a Dedekind complete vector lattice then X = R by 2.5.5 and, hence, R↓ = X ↓ = mix ι(X). But for the universally complete vector lattice X we have mix ι(X) = ι(X). The converse is obvious. ⊲ 2.5.7. Universally complete vector latices are isomorphic if and only if so are their bases. ⊳ If X and Y are universally complete vector lattices and the Boolean algebras B(X) and B(Y ) are isomorphic then by 2.5.6 X and Y are isomorphic to the same vector lattice R↓ with R ∈ V(B) and B ≃ B(X) ≃ B(Y ). On the other hand, if h is an isomorphism from X onto Y then the mapping K → h(K) (K ∈ B(X)) is an isomorphism of the Boolean algebras B(X) and B(Y ). ⊲ 2.5.8. Let X be a universally complete vector lattice with a weak order unit 1. Then we can uniquely define the multiplication on X so as to make X into a semiprime f -algebra and 1, into a ring unit. ⊳ By 2.5.4 and 2.5.6, we can assume that X = R↓ and 1 = 1∧ . The existence of the required multiplication on X follows from 2.3.2. Assume that there is another multiplication ⊙ : X × X → X on X and (X, +, ⊙, ) is a semiprime f -algebra with unity 1. The semiprimeness of the f -algebra implies that ⊙ is an extensional mapping. But then the ascent × := ⊙↑ is a multiplication on R. By uniqueness of the multiplicative structure on R, we conclude that × = · . Hence, we derive that ⊙ coincides with the original multiplication on X (cp. 2.3.2). ⊲ 2.6. Order Convergence 75 2.6. Order Convergence Interpreting the concept of convergent numerical net within V(B) and using 2.4.5 and 2.5.5, we obtain some useful tests for o-convergence in a Dedekind complete vector lattice. Recall that [x] stands for the band projection onto the band generated by x. 2.6.1. Let (xα )α∈A be an order bounded net in a Dedekind complete vector lattice X and x ∈ X. The following are equivalent: (1) (xα ) o-converges to x. (2) o-limα∈A [e] (|xα − x| − e)+ = 0 in P(X) for all positive e ∈ X. (3) For every e ∈ X+ there exists a partition (πα )α∈A of [e] in the Boolean algebra P(X) such that πα |x − xβ | e (α, β ∈ A, β α). (4) For every e ∈ X+ there exists an increasing net (ρα )α∈A in the Boolean algebra P(X) such that ρα |x − xβ | e (α, β ∈ A, β α). ⊳ Without loss of generality we can assume that X is an order dense ideal of the universally complete vector lattice R↓ (cp. 2.5.5). (1) ⇐⇒ (2): Let σ be the modiﬁed ascent of the mapping s : α → xα . Then [[σ is a net in R]] = 1. By 2.4.5, o-lim s = x if and only the identity if [[ lim σ = x]] = 1 holds. We may rewrite this identity as 1 = [[(∀ ε ∈ R+ )(ε > 0 → (∃ α ∈ A∧ )(∀ β ∈ A∧ ) (β α → |x − xβ | ε))]]. (∗) Calculating the Boolean truth values for the quantiﬁers, we ﬁnd the following equivalent form (∀ ε ∈ X+ ) [[ε = 0]] [[|x − xβ | − ε 0]] α∈A β∈A βα which in turn amounts to the formula [[ε = 0]] ∧ [[(|x − xβ | − ε)+ = 0]] . (∀ ε ∈ X+ ) [[ε = 0]] = α∈A β∈A βα 76 Chapter 2. Boolean Valued Numbers Since χ([[(|x − xβ | − ε)+ ]]) = (|xα − x| − ε)+ and χ([[ε = 0]]) = [ε] (o) (cp. 2.4.9), we see from the above that xα → x if and only if 0 =[[ε = 0]] ∧ [[ε = 0]] ∗ + [[ε = 0]] ∧ [[(|x − xβ | − ε) = 0]] =[[ε = 0]] ∧ α∈A β∈A βα =[[ε = 0]] ∧ [[ε = 0]] ∨ [[(|x − xβ | − ε)+ = 0]] α∈A β∈A βα [[ε = 0]] ∧ [[(|x − xβ | − ε)+ = 0]] α∈A β∈A βα = o-lim[ε] (|xα − x| − ε)+ . α∈A (1) ⇐⇒ (3): Arguing as in (1) ⇐⇒ (2) and putting b := χ−1 ([ε]) and cα := [[|x − xβ | ε]] : β ∈ A, β α , we ﬁnd that the equality o-lim xα = x is equivalent to the formula (∀ ε ∈ X+ ) ∃ (cα )α∈A ⊂ B cα = b α∈A ∧ (∀ β ∈ A) β α ⇒ cα [[|xα − x| ε]] . By the exhaustion principle for Boolean algebras, there exist a partition of unity (dξ )ξ∈Ξ in B and a mapping δ : Ξ → A such that dξ cδ(ξ) (ξ ∈ Ξ). Put bα := b ∧ {dξ : α = δ(ξ)} if α ∈ δ(Ξ) and bα = 0 if α∈ / δ(Ξ). We see that (bα )α∈A is a partition of b and bα cα (α ∈ A). Thus, if xα → x then for every ε ∈ X+ there is a partition of unity (bα ) such that bα [[|x − xβ | ε]] (α, β ∈ A, β α). As follows from 2.2.4 (G), the latter means that πα |x − xβ | πα ε ε (α, β ∈ A, β α), where πα := χ(bα ). Since (πα ) is a partition of [ε] in P(X), the necessity is proven. 77 2.6. Order Convergence To prove the suﬃciency, observe that if the conditions are satisﬁed and a := lim sup |xα − x| then |xβ − x| πα ε πα a βα for all α ∈ A. Consequently, 0a= πα a πα ε = ε. Since ε ∈ X+ is arbitrary, we have a = 0 and o-lim α xα = x. (3) ⇐⇒ (4): We only have to put ρα := {πβ : β ∈ A, α β} in (3). ⊲ 2.6.2. Corollary. Assume that X has a weak order unit 1, (xα )α∈A is an order bounded net in X, and x ∈ X. The following are equivalent: (1) The net (xα ) o-converges to x. y(α) (2) For every 0 < ε ∈ R the equality o-limα∈A eε = 1 with y(α) := |x − xα | holds in P(X). (3) For every n ∈ N the equality o-limα∈A (|x − xα | − 1/n)+ = 0 holds in P(X). (4) For every 0 < ε ∈ R there exists a partition of unity (πα )α∈A in P(X) such that πα |x − xβ | ε1 for all α, β ∈ A, β α. (5) For every 0 < ε ∈ R there exists an increasing net (ρα )α∈A in P(X) such that ρα |x − xβ | ε1 for all α, β ∈ A, β α. ⊳ The proof proceeds along the same lines as before. Since [[R∧ is dense in R]] = 1, we can rewrite (∗) in equivalent form: 1 = [[(∀ ε ∈ R∧ )(ε > 0 → (∃ α ∈ A∧ )(∀ β ∈ A∧ ) (β α → xβ < ε))]]. In further arguments we should replace [[|xα − x| ε]] by [[|xα − x| ε∧ ]] ∧ and take it into account that [[(ε1) = ε∧ 1∧ = ε∧ 1]] = 1 and χ [[xβ < xβ ∧ −1 + (|xα − x| − ε) (cp. 2.4.9). ⊲ ε ]] = eε = χ 2.6.3. Corollary. Suppose that A is an order bounded set in a Dedekind complete vector lattice X. Then the following hold: (1) x = inf(A) if and only if for every ε ∈ X+ there exists a partition (πa )a∈A of the band projection [ε] in P(X) such that πa (a − x) ε (a ∈ A). 78 Chapter 2. Boolean Valued Numbers (2) x = sup(A) if and only if for every ε ∈ X+ there exists a partition (πa )a∈A of the band projection [ε] in P(X) such that πa (x − a) ε (a ∈ A). ⊳ Suﬃce it to interpret the deﬁnitions of the least upper bound and the greatest lower bound of a bounded set of reals within V(B) with B := P(X). ⊲ 2.7. Freudenthal Spectral Theorem In the present section we will show that the properties of a spectral system can be deduced from the properties of reals. We start with several useful remarks to be applied below without further speciﬁcation. 2.7.1. Take a Dedekind σ-complete vector lattice X. By 2.5.1, we can assume that X is a vector sublattice of the universally complete vector lattice R↓, where, as usual, R is the reals within V(B) and := I(X) generated by X in R↓ is B := B(X). Moreover, the ideal X an order dense ideal of R↓ and an o-completion of X. Each weak order unit in X is also a weak order unit in R↓. The countable joins and meets in X are inherited from R↓. In more detail, if the least upper (greatest lower) bound x of a sequence (xn ) ⊂ X exists in R↓ then x is also the least upper (greatest lower) bound in X, provided that x ∈ X. So, it does not matter whether the o-limit (o-sum) of a sequence in X is calculated in X or R↓, provided the result belongs to X. The same is true for the r-limit and r-sums. In particular, C(X) is a σ-subalgebra of a complete Boolean algebra C(R↓), while the trace ex and the spectral system R ∋ λ → exλ of an element x ∈ X calculated in R↓ are an element of C(X) and a mapping from R to C(X) respectively. By an easy application of the Boolean valued approach we prove the properties of a spectral system. According to the above remarks, we lose no generality in assuming that the Dedekind σ-complete vector lattice under consideration coincides with R↓. But then the claims can easily be derived from the elementary properties of reals with the help of 2.4.9. In 2.7.2–2.7.5, X is an arbitrary Dedekind σ-complete vector lattice with a weak order unit 1 and P is a dense subﬁeld of R. 2.7. Freudenthal Spectral Theorem 79 2.7.2. The spectral system λ → exλ (λ ∈ R) of x ∈ X has the properties: (1) (∀ λ, μ ∈ R) (λ μ =⇒ exλ exμ ). (2) ex+∞ := μ∈P exμ = 1 and ex−∞ := μ∈P exμ = 0. (3) exλ = {exμ : μ ∈ P, μ < λ} (λ ∈ R). ⊳ Observe ﬁrst that P∧ is a dense subﬁeld of R within V(B) . (1): If λ, μ ∈ P, λ μ, and x ∈ R; then, obviously, trivially x < λ → x < μ. By transfer [[x < λ∧ ]] ⇒ [[x < μ∧ ]] = 1 or equivalently [[x < λ∧ ]] [[x < μ∧ ]], and the result follows from 2.4.9. (2): Take x ∈ R↓ and consider the two formulas ϕ(x, P) := (∃ t ∈ P) (x < t) and ψ(x, P) := (∀ t ∈ P) (x < t). For a real x the formula ϕ(x, P) is true and ψ(x, P) is false. Consequently, by transfer [[ϕ(x, P∧ )]] = 1 and [[ψ(x, P∧ )]] = 0. Calculating the Boolean truth values for the quantiﬁers by the rules of 1.4.5 (1) yields [[x < t∧ ]] = 1, t∈P [[x < t∧ ]] = 0, t∈P which is equivalent to (2) by 2.4.9. (3): Applying the transfer principle to x < λ ↔ (∃ μ ∈ P) x < μ < λ and taking it into account that by 1.4.7 [[μ∧ < λ∧ ]] = 1 whenever μ < λ and [[μ∧ < λ∧ ]] = 0 otherwise, we deduce [[x < λ∧ ]] = [[x < μ∧ ]] ∧ [[μ∧ < λ∧ ]] μ∈P = {[[x < μ∧ ]] : μ ∈ P, μ < λ}. It remains to appeal to 2.4.9. ⊲ 2.7.3. Given x, y ∈ X, we have (1) eλx+y = {exμ ∧ eyν : μ, ν ∈ P, μ + ν = λ}. x {eμ ∧ eyν : 0 μ, ν ∈ P, μν = λ} (x 0, y 0). (2) ex·y λ = ⊳ We conﬁne demonstration to (2). Take positive elements x, y ∈ R↓ and 0 < t ∈ P. Then x, y, and t∧ are reals in V(B) . For reals we have x 0 ∧ y 0 → (xy < t∧ ↔ (∃ r, s ∈ P∧+ )(x < r ∧ y < s ∧ rs = t)). 80 Chapter 2. Boolean Valued Numbers By transfer and the rules of 1.4.5 (1) for Boolean truth values, we obtain [[x < r∧ ]] ∧ [[y < s∧ ]]. [[xy < t∧ ]] = 0r,s∈P rs=t Hence, (2) ensues if we apply χ to both sides of the preceding equality (cp. 2.4.9). ⊲ 2.7.4. If x, y ∈ X and ∅ = A ⊂ X then the assertions hold: (1) x y ⇐⇒ (∀ λ ∈ P) (eyλ exλ ). = exλ ∧ eyλ for all λ ∈ R. (2) ex∨y λ = exλ ∨ eyλ for all λ ∈ R. (3) ex∧y λ (4) x = inf(A) ⇐⇒ (∀ λ ∈ P) (exλ = {eaλ : a ∈ A}). ⊳ Clearly, (2) is immediate from the equivalence (∀x, y, λ ∈ R) x ∨ y < λ ↔ (x < λ) ∧ (y < λ) and (3) is a particular case of (4). Prove (1) and (4). (1): Observe ﬁrst that x y ↔ (∀ t ∈ P)(y < t → x < t). By transfer and the properties of Boolean truth values [[x y]] = [[y < λ∧ ]] ⇒ [[x < λ∧ ]] λ∈P for all x, y ∈ X. Since the formulas x y and [[x y]] = 1 are equivalent, x y is fulﬁlled if and only if [[y < λ∧ ]] [[x < λ∧ ]] or equivalently eyλ exλ for all λ ∈ P. (4): If A is a nonempty subset of X then A↑ is a set of reals within V(B) and inf(A) < t ↔ (∃ a ∈ A↑)(a < t). By 1.6.2 and 2.4.4, we obtain the chain of equivalent formulas: x = inf(A) ⇐⇒ [[x = inf(A↑)]] = 1 ⇐⇒ [[(∀ t ∈ P∧ )(x < t ↔ inf(A↑) < t)]] = 1 ⇐⇒ (∀ t ∈ P)[[x < t∧ ]] = [[(∃ a ∈ A↑)(a < t∧ )]] [[a < t∧ ]]. ⇐⇒ (∀ t ∈ P)[[x < t∧ ]] = a∈A Appealing to 2.4.9 completes the proof of (4). ⊲ 2.7. Freudenthal Spectral Theorem 81 2.7.5. Given x, y ∈ X, α ∈ R, and c ∈ C(1), for all λ ∈ R the following are valid: −x x αx (1) eαx λ = eλ/α (α > 0), eλ = e−λ/α (α < 0). {1 − ex−μ : μ ∈ P, μ < λ} = (1 − ex−λ ) · e(x+λ1) . (2) e−x λ = |x| |x| (3) eλ = exλ ∧ (1 − ex−λ ) ∧ ex+λ1 (λ 0), eλ = 0 (λ < 0). x ∗ cx x (4) ecx λ = c ∧ eλ + c (λ > 0), eλ = c ∧ eλ (λ 0). ⊳ Note that (1) is easily seen from (λ/α)∧ = λ∧ /α∧ and (3) is immediate from (2) and 2.7.4 (2). Turn to proving (2) and (4). (2): The inequality −x < λ can be written in the two equivalent forms: −x < λ ⇐⇒ (¬(x < −λ)) ∧ (x + λ = 0), −x < λ ⇐⇒ (∃ μ ∈ P)(¬(x < −μ)) ∧ (μ < λ). Applying transfer and using the equivalence [[λ∧ < μ∧ ]] = 1 ⇐⇒ λ < μ (cp. 1.4.7), we get [[−x < λ∧ ]] = [[x < −λ∧ ]]∗ ∧ [[x + λ∧ 1 = 0]]; [[−x < λ∧ ]] = {[[x < −μ∧ ]]∗ : μ ∈ P, μ < λ}. The desired result follows from 2.4.9. (4): Take c ∈ C(X) and choose b ∈ B with c = χ(b). If a ∈ {0, 1} then for all x ∈ R and λ ∈ P we evidently have ax < λ ↔ (a = 1 ∧ x < λ) ∨ (a = 0 ∧ 0 < λ). Since [[c ∈ {0∧ , 1∧ }B ]] = 1, the transfer principle together with 1.2.3 (1, 2, 4) and 2.2.6 yields [[cx < λ∧ ]] = ([[c = 1∧ ]] ∧ [[x < λ∧ ]]) ∨ ([[c = 0∧ ]] ∧ [[0∧ < λ∧ ]]) = (b ∧ [[x < λ∧ ]]) ∨ (b∗ ∧ [[0∧ < λ∧ ]]. ∧ x ∗ If λ > 0 then [[0∧ < λ∧ ]] = 1 and ecx λ = χ([[cx < λ ]]) = c ∧ eλ + c ; if ∧ ∧ cx ∧ x λ < 0 then [[0 < λ ]] = 0 and eλ = χ([[cx < λ ]]) = c ∧ eλ . ⊲ 2.7.6. Sometimes it is important to have an estimate rather than knowing the exact values of the spectral system. For example, if 82 Chapter 2. Boolean Valued Numbers x = sup(A) then exλ is calculated by a more complicated formula than 2.7.4 (4): x = sup(A) ⇐⇒ (∀ λ ∈ P) exλ = {eaν : a ∈ A} . ν<λ At the same time for every 0 < ε ∈ R the following hold: sup(A) a∈A eaλ . (1) a∈A eaλ−ε eλ x (2) 1 − exε−λ e−x λ 1 − e−λ . |x| (3) exλ ∧ (1 − exε−λ ) eλ exλ ∧ (1 − ex−λ ) (λ ∈ R+ ). −1 (4) 1 − exε+1/λ exλ 1 − ex1/λ (x ∈ X+ , 0 < λ ∈ R). Applying 2.7.3, 2.7.4 (1–3), and 2.7.5 (1) to the inequalities 2(x∧y) x + y 2(x ∨ y) and (x ∧ y)2 xy (x ∨ y)2 yields the estimates: (5) exλ/2 ∧ eyλ/2 ex+y exλ/2 ∨ eyλ/2 (x, y ∈ X; λ ∈ R). λ y x √ √ (6) ex√λ ∧ ey√λ exy λ e λ ∨ e λ (x, y ∈ X+ ; λ ∈ R+ ). 2.7.7. Freudenthal Spectral Theorem. Let X be an arbitrary Dedekind σ-complete vector lattice with order unit 1. Every element x ∈ X admits the representation x= !∞ λ dexλ , −∞ where the integral is understood to be the 1-uniform limit of the integral sums x(β) := τn (extn+1 − extn ), tn τn tn+1 , n∈Z as δ(β) := supn∈Z (tn+1 − tn ) → 0, with β := (tn )n∈Z being a partition of the real line. ⊳ We may assume that R↓ is a universal completion of X and X ⊂ R↓. Let x ∈ X, β := (tn )n∈Z be a partition of R, and tn < τn < tn+1 (n ∈ Z). Put bn := extn+1 − extn . Then bn = [[t∧n x < t∧n+1 ]] ∧ [[t∧n τn∧ < t∧n+1 ]] ∧ [[t∧n+1 − t∧n δ(β)∧ ]] [[|x − τn∧ | δ(β)∧ ]]. 2.8. Representation of Vector Lattices 83 Since x(β) := mixn∈Z (bn τn∧ ), we derive [[|x − x(β)| δ(β)∧ ]] = 1 or |x − x(β)| δ(β)1. It remains to recall the remarks of 2.7.1. ⊲ 2.7.8. In particular, the Freudenthal Spectral Theorem states that if X is a Dedekind σ-complete vector lattice and e ∈ X+ then every x ∈ X(e) can be e-uniformly approximated by the linear n combinations of components of e; i.e., by the elements of the form k=1 λk ek , where λ1 , . . . , λn ∈ R and e1 , . . . , en ∈ C(e). In the case when the latter holds in a vector lattice X we say that X possesses the weak Freudenthal property. It may happen that every n x ∈ X(e) can be e-uniformly approximated by linear combinations k=1 λk πk e, where λ1 , . . . , λn ∈ R and π1 , . . . , πn ∈ P(X). Then a vector lattice X is said to possesses the strong Freudenthal property. Clearly, a vector lattice with the principal projection property possesses the strong Freudenthal property. The converse is false. 2.8. Representation of Vector Lattices By the Freudenthal Spectral Theorem, the mapping that assigns to each element of a Dedekind σ-complete vector lattice with weak order unit its spectral system is one-to-one and transforms the vector lattice structure in a deﬁnite way. This circumstance suggests that an arbitrary Dedekind σ-complete vector lattice with weak order unit can be represented as some space of “abstract spectral system.” We will expatiate upon this. 2.8.1. A spectral system or resolution of the identity in a Boolean algebra B is deﬁned as a mapping e : R → B satisfying the conditions (1) s t → e(s) e(t) (s, t ∈ R); (2) t∈R e(t) = 1, t∈R e(t) = 0; (3) s∈R,s<t e(s) = e(t) (t ∈ R). Let S(B) be the set of all spectral systems in B. Introduce some order by the formula e′ e′′ ⇐⇒ (∀ t ∈ R)(e′′ (t) e′ (t)) (e′ , e′′ ∈ S(B)). 2.8.2. Further, suppose that B is a σ-algebra and choose some countable dense subﬁeld P of R. By 2.8.1 (3), every spectral function is uniquely determined by its values on P. 84 Chapter 2. Boolean Valued Numbers Given e′ , e′′ ∈ S(B), we can deﬁne the mapping e : t → {e′ (r) ∧ e′′ (s) : r, s ∈ P, r + s = t} e : t → {e(s) : s ∈ P, s < t} (t ∈ R) (t ∈ P), which is obviously a spectral function in B. Putting e′ + e′′ := e, we obtain the structure of a commutative group in S(B). In particular, the zero element 0̄ is deﬁned as 0̄(t) := 1 if t > 0 and 0̄(t) := 0 if t 0, while −e(t) = {1 − e(−s) : s ∈ P, s < t}. Set 1̄(t) := 1 if t > 1 and 1̄(t) := 0 if t 1. Finally, the scalar multiplication (α, e) → αe (α ∈ R, e ∈ S(B)) is deﬁned as (αe)(t) := e(t/α) (α > 0, t ∈ R), (αe)(t) := (−e)(−t/α) (α < 0, t ∈ R). To each element b ∈ B we assign the spectral system b̄ that is deﬁned as ⎧ ⎪ for t > 1, ⎨1, ∗ b̄(t) := b := 1 − b, for 0 < t 1, ⎪ ⎩ 0, for t 0. 2.8.3. Theorem. Let B be a complete Boolean algebra. The set S(B) with the above operations and order is a universally complete vector lattice with a weak order unit 1̄. The mapping h assigning to each x ∈ R↓ the spectral system t → [[x < t∧ ]] (t ∈ R) is a lattice isomorphism from R↓ onto S(B). The mapping b → b̄ is a Boolean isomorphism of B onto C(1̄). ⊳ By 2.4.9 and 2.7.2 h(x) ∈ S(B). According to 2.7.3, and 2.7.4 h preserves addition, multiplication, and lattice operations. Moreover, h is one-to-one, since the equality h(x) = h(y) means [[x < t∧ ]] = [[y < t∧ ]] (t ∈ R) or equivalently (cp. 1.4.5 (1)) [[(∀ t ∈ R∧ ) (x < t ↔ y < t)]] = 1 and the latter amounts to the coincidence of x and y within V(B) . By Gordon’s Theorem, it remains to establish that h is surjective. Take an arbitrary spectral system e : R → B. Let β := (tn )n∈Z be a partition of 85 2.8. Representation of Vector Lattices the real axis; i.e., tn < tn+1 (n ∈ Z), limn→∞ tn = ∞, and limn→−∞ tn = −∞. The disjoint sum x̄(β) := tn+1 (χ(e(tn+1 )) − χ(e(tn ))) n∈Z exists in the universally complete vector lattice R↓; here χ is the isomorphism of B onto C(R↓) (cp. 2.2.4 and 2.3.2). Denote by A the set of all elements x̄(β). Each element of the form x(β) := tn (χ(e(tn+1 )) − χ(e(tn ))) n∈Z is a lower bound of A. Therefore, there exists x := inf(A) := inf{x̄(β)}. It is easy to observe that x̄(β) eλ = {χ(e(tn )) : tn < λ}. Hence, from 2.7.4 (4) we infer eaλ = exλ = a∈A χ(e(t)) = χ(e(λ)) (λ ∈ R). t∈R,t<λ Using 2.4.9, we conclude that h(x)(t) = χ−1 ([[x < t∧ ]]) = e(t). ⊲ Let us derive several important corollaries to 2.8.3. 2.8.4. Corollary. A universally complete vector lattice X with unit 1 is isomorphic to the Dedekind complete vector lattice S(B), where B := C(1). The isomorphism is established by the mapping that assigns to each x ∈ X the spectral system λ → exλ (λ ∈ R). ⊳ It suﬃces to compare 2.5.6 and 2.8.3. ⊲ 2.8.5. Corollary. For an arbitrary σ-algebra B, the set S(B) (with the structure defined as in 2.8.2) is a universally σ-complete vector lattice with order unit. Conversely, every universally σ-complete vector lattice X with order unit is isomorphic to S(B), where B := C(X). be an o-completion of the σ-algebra B. According to 2.8.3, ⊳ Let B is a universally complete vector lattice. The set S(B) lies in S(B). S(B) Moreover, it is easily seen from 2.7.3–2.7.5 and 2.8.4 that S(B) is a vector and the countable suprema and inﬁma in S(B) are subspace of S(B) inherited from S(B). Consequently, S(B) is a Dedekind σ-complete 86 Chapter 2. Boolean Valued Numbers vector lattice with order unit. The same arguments imply that every countable disjoint set of elements in S(B) is bounded. Take an arbitrary Dedekind σ-complete vector lattice X with order its universal completion. If B = C(X) and B := unit and denote by X C(X) then B is an o-completion of B. By 2.8.4, the spaces X and S(B) are isomorphic; moreover, S(B) is the image of X by 2.7.6. ⊲ 2.8.6. We proceed now to the functional representation of vector lattices. Some additional deﬁnitions and facts are needed for this purpose. Let Q be a topological space. Recall that a topological space Q is called extremally (quasiextremally) disconnected or simply extremal (quasiextremal ) if the closure of an arbitrary open set (open Fσ -set) in Q is open or, which is equivalent, the interior of an arbitrary closed set (closed Gδ -set) is closed. Clearly, an extremal (quasiextremal) space is totally disconnected. If Q is Hausdorﬀ compact then the respective terms Stonean and quasi-Stonean are in common parlance as well. Let Λ ⊂ R := R ∪ {±}. Given a function f : Q → R and λ ∈ R, put {f < λ} := {q ∈ Q : f (q) < λ}, {f λ} := {q ∈ Q : f (q) λ}. Consider a mapping λ → Uλ ⊂ Q that is assumed to be increasing: λ μ implies Uλ ⊂ Uμ . This mapping is said to be strictly increasing if and only if cl(Uλ ) ⊂ int(Uμ ) for all λ, μ ∈ Λ with λ < μ. Say that a function f : Q → R calibrates the mapping λ → Uλ whenever {f < λ} ⊂ Uλ ⊂ {f λ} for all λ ∈ Λ. 2.8.7. Assume that Q is a topological space and Λ is a dense subset of R. A mapping U : λ → Uλ from Λ to P(Q) is strictly increasing if and only if there is a unique continuous function f : Q → R that calibrates U . ⊳ See [228, 1.4.1 (1)]. ⊲ 2.8.8. Let Q be a quasiextremal compact space. Assume that Q0 is an open dense Fσ -set in Q and f : Q0 → R is a continuous function. Then there is a unique continuous function f¯ : Q → R such that f (t) = f¯(t) (t ∈ Q0 ). ⊳ Indeed, if Uμ := cl({f < μ}) then the mapping μ → Uμ (μ ∈ R) is strictly increasing. Therefore, by 2.8.7, there is a unique function f¯ : Q → R satisfying {f¯ < μ} ⊂ Uμ ⊂ {f¯ μ} (μ ∈ R). 2.8. Representation of Vector Lattices 87 Obviously, the restriction of f¯ to Q0 coincides with f . ⊲ 2.8.9. Let Q be a quasiextremal compact space. Denote by C∞ (Q) the set of all continuous functions x : Q → R assuming the values ±∞ possibly on a rare set. The order on C∞ (Q) is deﬁned by putting x y whenever x(t) y(t) for all t ∈ Q. Take x, y ∈ C∞ (Q) and put Q0 := {|x| < +∞} ∩ {|y| < +∞}. In this case Q0 is open and dense in Q. According to 2.8.8, we have the unique continuous functions u, v : Q → R such that u(t) = x(t) + y(t) and v(t) = x(t) · y(t) for t ∈ Q0 . So we can deﬁne addition and multiplication on C∞ (Q) by putting x + y := u and xy := v. The identically one function 1 is an order and ring unit in C∞ (Q). The scalar multiplication on C∞ (Q) is deﬁned as λx := (λ1)x. The space C∞ (Q) with the above algebraic operations and order is vector lattice and a semiprime f -algebra. The following result tells us that C∞ (Q) is universally σ-complete. 2.8.10. Let Q be the Stone space of a σ-algebra B. The vector lattices C∞ (Q) and S(B) are lattice isomorphic. In particular, C∞ (Q) is a universally σ-complete vector lattice with unit for every quasiextremal compact space Q. ⊳ Take e ∈ S(B). Let Gt be a clopen set in Q corresponding to the element e(t) ∈ B. The mapping t → Gt (t ∈ R) is strictly increasing, so that by 2.8.6, there exists a unique continuous function ê : Q → R such that {ê < t} ⊂ Gt ⊂ {ê t} (t ∈ R). It follows from 2.7.2 (2, 3)that the closed set {Gt : t ∈ R} has empty interior and the open set {Gt : t ∈ R} is dense in Q. Hence, the function ê is ﬁnite everywhere, except possibly the points of a nowhere dense set; therefore, ê ∈ C∞ (Q). It is easy to check that the mapping e → ê is the sought lattice isomorphism. ⊲ 2.8.11. Theorem. Let Q be the Stone space of a complete Boolean algebra B, and let R be the reals within V(B) . The vector lattice C∞ (Q) is isomorphic to the universally complete vector lattice R↓. The isomorphism is established by assigning to an element x ∈ R↓ the function x̂ : Q → R by the formula x̂(q) := inf{t ∈ R : [[x < t∧ ]] ∈ q} (q ∈ Q). ⊳ The proof is immediate from 2.8.10 and 2.8.3. ⊲ 88 Chapter 2. Boolean Valued Numbers 2.8.12. Let X be an Archimedean vector lattice and let Q be the Stone space of the Boolean algebra B(X). Then X is isomorphic to a mi- 2.9. Spectral Measure and Integral 89 norizing sublattice X0 ⊂ C∞ (Q). Moreover, X is an order dense ideal of C∞ (Q) (coincides with C∞ (Q)) if and only if X is a Dedekind complete vector lattice (a universally complete vector lattice). ⊳ See 2.8.10, 2.5.1, 2.5.5, and 2.5.6. ⊲ 2.8.13. Theorem. Let X be a universally σ-complete vector lattice with an order unit 1 and let Q be the Stone space of the Boolean algebra C(X, 1). Then X is lattice isomorphic to C∞ (Q). Moreover, X can uniquely be equipped with an f -algebra multiplication with 1 as ring unit; in this event X and C∞ (Q) are f -algebra isomorphic. ⊳ Immediate from Corollary 2.8.5 and 2.8.10. ⊲ 2.9. Spectral Measure and Integral In the sequel, we need the concept of integral with respect to a spectral measure. 2.9.1. Suppose that (Ω, Σ) is a measure space; i.e., Ω is a nonempty set and Σ is a ﬁxed σ-algebra of subsets of Ω. A spectral measure is deﬁned to be a σ-continuous Boolean homomorphism μ from Σ into the Boolean σ-algebra B. More precisely, a mapping μ : Σ → B is a spectral measure if μ(Ω \ A) = 1 − μ(A) (A ∈ Σ) and μ ∞ An n=1 = ∞ μ(An ) n=1 for each sequence (An ) of elements of Σ. Let B := C(X, 1) be the Boolean algebra of components of 1 in a Dedekind σ-complete vector lattice X with a ﬁxed unit 1. Take a measurable function f : Ω → R. Given an arbitrary partition of the real axis β := (λk )k∈Z , λk < λk+1 (k ∈ Z), lim λn = ±∞, n→±∞ put Ak := f −1 ([λk , λk+1 )) and compose the integral sums σ(f, β) := ∞ −∞ λk μ(Ak ), σ(f, β) := ∞ −∞ λk+1 μ(Ak ), 90 Chapter 2. Boolean Valued Numbers where the sums are calculated in X. It is clear that σ(f, β) ∞ f (tk )μ(Ak ) σ(f, β) −∞ for every choice of tk ∈ Ak (k ∈ Z). Also, it is evident that σ(f, β) increases and σ(f, β) decreases as we reﬁne the partition β. If there exists an element x ∈ X such that sup{σ(f, β)} = x = inf{σ(f, β)}, where the suprema and inﬁma are calculated over all partitions β := (λk )k∈Z of the real axis, then we say that f is integrable with respect to μ or the spectral integral Iμ (f ) exists; in this event we write ! ! Iμ (f ) := f dμ := f (t) dμ(t) := x. T T 2.9.2. The spectral integral Iμ (f ) exists for every bounded measurable function f . If X is a universally σ-complete vector lattice then every almost everywhere finite measurable function is integrable with respect to each spectral measure. ⊳ Note that Ak ∩ Al = ∅ (k = l) and k∈Z Ak = Ω; therefore, (μ(Ak ))k∈Z is a partition of unity in the Boolean algebra B. Putting δ := supk∈Z {λk+1 − λk }, we can write down δμ(Ak ) = δ1. 0 σ(f, β) − σ(f, β) k∈Z Consequently, a measurable function f is integrable with respect to μ if and only if σ(f, β) and σ(f, β) exist at least for one partition β. If f is bounded then the sums σ(f, β) and σ(f, β) contain at most ﬁnitely many nonzero summands. If X is a universally σ-complete vector lattice and a measurable function f is arbitrary then the sums also make sense, since in this case they involve at most countably many pairwise disjoint elements. ⊲ 2.9.3. Theorem. Let X := R↓ and let μ be a spectral measure with values in B := C(X) := C(1∧ ). Then for each measurable function f the integral Iμ (f ) is the unique element of the Dedekind complete vector lattice X satisfying the condition [[Iμ (f ) < λ∧ ]] = μ({f < λ}) (λ ∈ R). 2.9. Spectral Measure and Integral 91 ⊳ Take an arbitrary real λ ∈ R and a partition of the real axis β := (λk )k∈Z such that λ0 = λ. If b := [[Iμ (f ) < λ∧ ]] then b = [[(∃ t ∈ R∧ ) (Iμ (f ) < t ∧ t < λ∧ )]]. By the exhaustion principle, there exist a partition (bξ )ξ∈Ξ of b and a family (tξ )ξ∈Ξ ⊂ R such that tξ < λ and bξ [[Iμ (f ) t∧ξ ]] for all ξ. Hence, applying 2.2.4 (G), we derive bξ σ(f, β) tξ bξ < λbξ (ξ ∈ Ξ) and further λk bξ μ(Ak ) tξ bξ μ(Ak ) < λbξ μ(Ak ) (ξ ∈ Ξ, k ∈ Z). For k 1 we have λk > λ; therefore, bξ μ(Ak ) = 0. So, ∞ ∞ ∗ b= μ(Ak ) = μ Ω − Ak = μ({f < λ}). bξ k=1 ξ∈Ξ k=1 On the other hand, b∗ = [[Iμ (f ) λ∧ ]] and, by 2.2.4 (G), we again infer that λb∗ b∗ Iμ (f ) b∗ σ(f, β) or λb∗ μ(Ak ) b∗ λk μ(Ak ) (k ∈ Z). For k < 0 we have λk < λ; therefore, b∗ μ(Ak ) = 0. Consequently, −∞ −∞ ∗ ∗ b μ(Ak ) = μ Ω − Ak = μ({f λ}). k=−1 k=−1 This implies b μ({f < λ}) and we ﬁnally obtain b = μ({f < λ}). Assume that [[x < λ∧ ]] = μ({f < λ}) (λ ∈ R) for some x ∈ R↓. Then by what was established above we have [[x < λ∧ ]] = [[Iμ (f ) < λ∧ ]] for all λ ∈ R. This is equivalent to [[(∀ λ ∈ R∧ ) (x < λ ↔ Iμ (f ) < λ)]] = 1. Hence, recalling that R∧ is dense in R, we get the equality [[x = Iμ (f )]] = 1 or x = Iμ (f ). ⊲ 92 Chapter 2. Boolean Valued Numbers 2.9.4. Take a measurable function f : Ω → R and a spectral measure μ : Σ → B := C(X), where X is a Dedekind complete vector lattice. If the integral Iμ (f ) ∈ X exists then λ → μ({f < λ}) (λ ∈ R) coincides with the spectral system of Iμ (f ). ⊳ Suﬃce it to compare 2.4.9 with 2.9.3. ⊲ 2.9.5. Theorem. Let X be a universally σ-complete vector lattice with an order unit 1 and let M (Ω, Σ) stand for the unital f -algebra of measurable real functions on Ω. Given a spectral measure μ : Σ → B0 , B0 := C(X, 1), the spectral integral Iμ (·) is a sequentially o-continuous f -algebra homomorphism from M (Ω, Σ) to X. ⊳ Without loss of generality we can assume that X ⊂ R↓ and R↓ is a universal completion of X (cp. 2.5.1 (3)). Here R is the ﬁeld of the reals in V(B) , where B is a completion of the algebra B0 . It is obvious that the operator Iμ is linear and positive. Prove its sequential o-continuity. Take a decreasing sequence (fn )n∈N of measurable functions such that limn→∞ fn (t) = 0 for all t ∈ Ω, and let xn := Iμ (fn ) (n ∈ N) and ∞ 0 < ε ∈ R. If we assign An := {t ∈ Ω : fn (t) < ε} then Ω = n=1 An . By 2.9.4, we can write down o-lim exε n = o-lim μ(An ) = n→∞ n→∞ ∞ μ(An ) = 1. n=1 Appealing to the test for o-convergence 2.6.2 (2), we obtain o-limn→∞ xn = 0. Further, given arbitrary measurable functions f, g : Ω → R, we derive from 2.7.4 (2) and 2.9.4 that I(f ∨g) eλ = μ({f ∨ g < λ}) I(f ) = μ({f < λ}) ∧ μ({g < λ}) = eλ I(g) ∧ eλ I(f )∨I(g) = eλ (with I := Iμ ); consequently, I(f ∨ g) = I(f ) ∨ I(g). It means that Iμ is a lattice homomorphism. In a similar way, for f 0 and g 0 it follows from 2.7.3 (2) and 2.9.4 that ⎛ ⎞ ⎜ ⎟ I(f ·g) eλ = μ({f · g < λ}) = μ ⎝ {f < r} ∩ {g < s}⎠ = r,s∈Q+ rs=λ r,s∈Q+ rs=λ μ({f < r}) ∧ μ({g < s}) = r,s∈Q+ rs=λ I(f )·I(g) erI(f ) ∧ esI(g) = eλ 93 2.10. Functional Calculus for 0 < λ ∈ Q, with Q the rationals. Thus, I(f · g) = I(f ) · I(g). The validity of the latter equality for arbitrary functions f and g ensues from the above-established properties of the spectral integral: I(f · g) = I(f + g + ) + I(f − g − ) − I(f + g − ) − I(f − g + ) = I(f )+ I(g)+ + I(f )− I(g)− − I(f )+ I(g)− − I(f )− I(g)+ = I(f ) · I(g). ✄ 2.10. Functional Calculus In a universally σ-complete vector lattice X with an order unit we can deﬁne φ(x1 , . . . , xn ) ∈ X, given a ﬁnite collection x1 , . . . , xN ∈ X and a Borel measurable function φ : RN → R. To this end, we need the auxiliary result: 2.10.1. Loomis–Sikorski Theorem. Let Q be the Stone space of a Boolean σ-algebra B. Denote by Clopσ (Q) the σ-algebra of subsets of Q generated by the collection Clop(Q) of all clopen subsets of Q. Let ∆ stand for the σ-ideal of Clopσ (Q) comprising all meager sets. If ı is an isomorphism of B onto Clop(Q) and ϕ is the quotient mapping of Clopσ (Q) onto the quotient algebra Clopσ (Q)/∆ then the mapping h := ϕ ◦ ı is an isomorphism of B onto Clopσ (Q)/∆. ⊳ Observe that h is a homomorphism as the composite of two homomorphisms. If h(b) = 0 then ı(b) ∈ ∆ and ı(b) = ∅, since no nonempty clopen set is meager. Thus h is injective. To prove that h is surjective put F := {A ∈ Clopσ (Q) : (∃ b ∈ B) ϕ(A) = h(b)}. Since Clop(Q) ⊂ F ⊂ Clopσ (Q), it suﬃces to observe that F is a σalgebra. If A ∈ F with ϕ(A) = h(b) then ϕ(Q \ A) = h(b∗ ), so that Q \ A ∈ F . Consider a sequence (An ) ofF and choose a sequence (bn ) ∞ of B such that ϕ(An ) = h(bn ). Put A := n=1 ı(bn ) and A0 := cl(A) \ A. Since Q is quasiextremal, ∞ cl(A) is clopen and A0 is rare. Thus, we have the representation ı ( n=1 bn ) = A0 ∪ A from which we easily deduce ∞ ∞ ∞ ı(bn ) ϕ An = ϕ A0 ∪ An = ϕ A0 ∪ n=1 =ϕ ı n=1 ∞ n=1 and the result follows. ⊲ bn =h n=1 ∞ n=1 bn 94 Chapter 2. Boolean Valued Numbers 2.10.2. Let e1 , . . . , eN : R → B be a finite collection of spectral systems with values in a σ-algebra B. Then there exists a unique Bvalued spectral measure μ defined on the Borel σ-algebra Bor(RN ) of the space RN such that N N , (−∞, λk ) = ek (λk ) μ k=1 k=1 for all λ1 , . . . , λN ∈ R. ⊳ Without loss of generality we can assume that B = Clop(Q), where Q is the Stone space of B. According to 2.8.7, there are continuous functions xk : Q → R such that ek (λ) = cl{xk < λ} for all λ ∈ R and k := 1, . . . , N . Put f (t) := (x1 (t), . . . , xN (t)) if all xk (t) are ﬁnite and f (t) := ∞ if xk (t) = ±∞ at least for one index k. Thereby we have deﬁned some continuous mapping f : Q → RN ∪ {∞} (the neighborhood ﬁlterbase of the point ∞ is composed of the complements to various balls with center the origin). It is clear that f is measurable with respect to the Borel algebras Bor(Q) and Bor(RN ). Let Clopσ (Q) and ϕ be the same as in 2.10.1. Deﬁne the mapping μ : Bor(RN ) → B by the formula μ(A) := ϕ f −1 (A) (A ∈ Bor(RN )). It is obvious that μ is a spectral measure. If A := f −1 (A) = N -N k=1 (−∞, λk ) then {xk < λk }, k=1 and so μ(A) = e1 (λ1 ) ∧ · · · ∧ eN (λN ). If ν is another spectral measure with the same properties as μ then the set B := {A ∈ Bor(RN ) : ν(A) = μ(A)} is a σ-algebra containing all sets of the form N , (−∞, λk ) (λ1 , . . . , λN ∈ R). k=1 Hence, B = Bor(RN ). ⊲ 2.10.3. Let us take an ordered collection of elements x1 , . . . , xN in a Dedekind σ-complete vector lattice X with unity 1. Let exk : R → 2.10. Functional Calculus 95 B := C(1) denote the spectral system of the element xk . By 2.10.2, there exists a spectral measure μ : Bor(RN ) → B such that N N , (−∞, λk ) = exk (λk ). μ k=1 k=1 We may see that the measure μ is uniquely determined by the ordered collection x := (x1 , . . . , xN ) ∈ X N . For this reason, we write μx := μ and say that μx is the spectral measure of the collection x. The following denotations are accepted for the integral of a measurable function f : RN → R with respect to the spectral measure μx : x̂(f ) := f (x) := f (x1 , . . . , xN ) := Iμ (f ). If x = (x) then we also write x̂(f ) := f (x) := Iμ (f ) and call μx := μ the spectral measure of an element of x. Recall that the space B(RN , R) of all Borel functions in RN is a universally complete Dedekind σ-complete vector lattice and a semiprime f -algebra. 2.10.4. Theorem. The spectral measures of the element f (x) and a collection x := (x1 , . . . , xN ) satisfy the equality μf (x) = μx ◦ f , where f : Bor(R) → Bor(RN ) is the homomorphism acting by the rule A → f −1 (A). In particular, (f ◦ g)(x) = g(f (x)) for f ∈ B(RN , R) and g ∈ B(R, R) whenever f (x) and g(f (x)) exist. ⊳ By 2.9.4, we have f (x) μf (x) (−∞, t) = et = [[f (x) < t]] = μx ◦ f −1 (−∞, t) for every t ∈ R. Hence, the spectral measures μf (x) and μx ◦f on Bor(R) coincide on the intervals of the form (−∞, t). Reasoning in a standard manner, we then conclude that the measures coincide everywhere. To complete the proof, it suﬃces to observe that (g ◦ f ) = f ◦ g and apply what was established above twice. ⊲ 2.10.5. Theorem. For every ordered collection x := (x1 , . . . , xN ) of elements of a universally σ-complete vector lattice Xwith unit 1, the 96 Chapter 2. Boolean Valued Numbers mapping x̂ : f → x̂(f ) is the unique sequentially o-continuous f -algebra homomorphism from B(RN , R) to X satisfying the conditions x̂(1) = 1, x̂(dtk ) = xk (k := 1, . . . , N ), where 1 is the identically one function on RN and dtk : (t1 , . . . , tN ) → tk stands for the kth coordinate function on RN . ⊳ As was established in 2.9.5, the mapping f → x̂(f ) is a sequentially o-continuous homomorphism of f -algebras. From 2.10.4 we have μdtk (x) = μx ◦ (dtk )← = μxk . Consequently, the elements x̂(dtk ) = dtk (x) and xk coincide, for they have the same spectral function. If h : B(RN , R) → X is another homomorphism of f -algebras with the same properties as x̂(·), then h and x̂(·) coincide on all polynomials. Afterwards, we infer that h and x̂(·) coincide on the whole B(RN , R) due to o-continuity. ⊲ 2.11. Boolean Valued Vector Lattices In this section we will show that a vector lattice arises as the Boolean valued interpretation of a vector lattice if and only if the latter admits the structure of an f -module. 2.11.1. Let A be an f -algebra. Recall that every f -algebra is commutative. A vector lattice X is said to be an f -module over A if the following are satisﬁed: (1) X is a module over A (with respect to the multiplication A×X ∋ (a, x) → ax ∈ X). (2) ax 0 for all a ∈ A+ and x ∈ X+ . (3) x ⊥ y implies ax ⊥ y for all a ∈ A+ and x, y ∈ X. A vector lattice X has the natural f -module structure over Orth(X): πx := π(x) (x ∈ X, π ∈ Orth(X)). Clearly, X is an f -module over an arbitrary f -submodule A ⊂ Orth(X) and, in particular, over Z (X). If a Dedekind complete vector lattice Y 2.11. Boolean Valued Vector Lattices 97 is an f -module over an f -algebra A then the space L∼ (X, Y ) of regular operators from X to Y also has the natural f -module structure: (aT ) : x → a(T x) (x ∈ X) (a ∈ A, T ∈ L∼ (X, Y )). 2.11.2. Let X be an f -module over an f -algebra A. Then the following hold: (1) (a ∨ b)x = (ax) ∨ (vx) and (a ∧ b)x = (ax) ∧ (vx) for all a, b ∈ A and x ∈ X+ . (2) |ax| = |a||x| for all a ∈ A and x ∈ X. (3) a ⊥ b implies ax ⊥ by for all a, b ∈ A and x, y ∈ X. It is clear that if a ∈ A and the operator πa in X is deﬁned as πa x := ax then πa ∈ Orth(X). Moreover, the mapping h : a → πa is a positive algebra homomorphism from A to Orth(X) and so h is a lattice homomorphism, since Orth(X) is a semiprime f -algebra. Conversely, if X is a vector lattice and h is an f -algebra homomorphism from an f algebra A to Orth(X), then h induces an f -algebra structure over A on X by putting ax := h(a)x (x ∈ X). If the f -algebra A has a unit element e ∈ A then πe is a band projection in X. An f -module X is called unital if πe = IX . 2.11.3. Let X be a vector lattice, B a complete Boolean algebra and j a complete homomorphism from B into B(X). Say that X is a vector B-lattice if j(b) is a projection band for all b ∈ B. In this case we identify j(b) with the corresponding band projection [j(b)] and write B ⊂ P(X). A vector B-lattice X is said to be B-complete if for every family (xξ )ξ∈Ξ in X and every partition of unity (bξ )ξ∈Ξ in B there exist x ∈ X such that bξ x = bξ xξ for all ξ ∈ Ξ. This element x is called the mixture of (xξ )ξ∈Ξ by (bξ )ξ∈Ξ and we write x := mixξ∈Ξ bξ xξ (cp. 1.4.3). nDenote by St0 (B) the subspace of Orth(X) consisting of the operators k=1 λk πk where π1 , . . . , πn are pairwise disjoint members of B and λ1 , . . . , λn ∈ R. Clearly, A0 := St0 (B) is an f -subalgebra of Orth(X) and X is an f -module over A0 . If X is B-complete then for every partition of unity (πξ )ξ∈Ξ in B and every family of reals (λξ )ξ∈Ξ there exist an orthomorphism T := ξ∈Ξ λξ πξ ∈ Orth(X) such that πξ T = λξ πξ for all ξ ∈ Ξ. Let St(B) stand for the set of all orthomorphisms of this form. Then A := St(B) is an f -subalgebra of Orth(X) and X is an f -module over A. 98 Chapter 2. Boolean Valued Numbers 2.11.4. Theorem. If X ∈ V(B) is an internal vector lattice over the internal field R∧ , then X ↓ is a B-complete vector lattice over R and there exists a complete Boolean monomorphism j : B → P(X ↓) with b [[x y]] ⇐⇒ j(b)x j(b)y (x, y ∈ X ↓; b ∈ B). Moreover, there exists a Boolean isomorphism κ from P(X ↓) onto P(X )↓ such that the diagram commutes B❄ ⑧⑧ ❄❄❄ ⑧ ❄❄ı j ⑧⑧ ❄❄ ⑧⑧ ❄❄ ⑧ ⑧ ❄ ⑧⑧ / P(X )↓ P(X ↓) κ where ι is defined as in 1.10.2. ⊳ The proof can be given along the lines of the proof of the Gordon Theorem with obvious modiﬁcations. Alternatively we can use 1.8.6. In the latter case X is considered as an algebraic system with the universe |X |, the nullary operation 0, the 1-ary operations {+} ∪ R∧ , and the predicates {=, }. The symbol λ ∈ R∧ is identiﬁed with the operation x → λx (x ∈ X ). Then it should be observed that for each π ∈ P(X) the set cπ := {(x, y) ∈ X × X : πx = πy} is a congruence of the algebraic system X = (|X |, 0, +, (λ)λ∈R∧ , ) and the mapping π → cπ is an isomorphism of P(X) onto a complete Boolean algebra of congruences of X (cp. 1.7.5 and 1.7.9). ⊲ 2.11.5. For every vector B-lattice X there exists X , X δ ∈ V(B) such that the following hold: (1) [[X is a vector lattice over R∧ ]] = 1. (2) there is a lattice isomorphism h from X to X ↓ satisfying X ↓ = mix(h(X)) and b = h−1 ◦ j(b) ◦ h for all b ∈ B. (3) [[X δ is a Dedekind completion of X ]] = 1 and X δ ↓ is a Dedekind completion of X ↓; i.e., (X ↓)δ = (X δ )↓. ⊳ Let X be a vector B-lattice. Deﬁne d, P : X × X → B by putting d(x, y) := b ∈ B : b∗ x = b∗ y ; b ∈ B : b∗ x b∗ y P (x, y) := 2.11. Boolean Valued Vector Lattices 99 for x, y ∈ X. It is immediate from the deﬁnitions that d is a B-metric on X and for all x, y, u, v ∈ X and λ ∈ R the inequalities hold: d(x + u, y + v) d(x, y) ∨ d(u, v), d(λx, λy) d(x, y), P (x, y) △ P (u, v) d(x, u) ∨ d(y, v). It follows that A = (X, +, (λ)λ∈R , P ) is an algebraic B-system of signature {+, P } ∪ R. By 1.8.8 within V(B) there exists an algebraic system X := A of signature {+, P }∧ ∪ R∧ with the properties 1.8.8 (2–4). Direct calculation of truth values on using 1.7.6 shows that |ϕ|X = 1 with ϕ the formula of signature {+, P } ∪ R formalizing the sentence “X is a vector lattice over R.” It follows from 1.8.8 (4) that X is a vector lattice over R∧ within V(B) . Observe also that (2) follows from 1.8.8 (3). The claim (3) amounts to saying that [[X is an order dense majorizing sublattice of X δ ]] = 1 if and only if X ↓ is an order dense majorizing sublattice of X δ ↓ (cp. Luxemburg and Zaanen [297, Theorem 32.7]). An easy veriﬁcation of the latter is left to the reader. ⊲ 2.11.6. Theorem. If X is a universally complete vector lattice in V(B) , then X ↓ is a universally complete vector lattice and there exists a lattice isomorphism j from R↓ onto an order closed sublattice X0 ⊂ X with 1 := j(1∧ ) ∈ X0 a weak order unit of X. Moreover, there is a Boolean isomorphism κ from P(X )↓ onto P(X ↓) such that the diagram χ B −−−−→ P(R↓) ⏐ ⏐ ⏐j ⏐ ι κ P(X )↓ −−−−→ P(X ↓) commutes, where ι and χ are defined respectively as in 1.10.1 and 2.2.4 and κ is induced by j with j : π → [j(π1∧ )] (π ∈ P(R↓)). ⊳ Working within V(B) and using transfer, we put D := P(X ) and observe that X is lattice isomorphic to S(D) by 2.8.4. At the same time, by 1.10.1 D := D↓ is a complete Boolean algebra and there is a Boolean isomorphism from B onto an order closed subalgebra D0 ⊂ D. Moreover, according to 2.8.3 and 2.8.5 S(D) is a universally complete vector lattice and S(D0 ) is an order closed sublattice in S(D) isomorphic to R↓. Prove that S(D)↓ and S(D) are isomorphic algebraic systems. If ǫ ∈ S(D)↓ 100 Chapter 2. Boolean Valued Numbers then [[ǫ : R → D is a spectral function ]] = 1. Since spectral functions as well as operations and order on S(D) are uniquely determined by the values on a dense subﬁeld in R and since R∧ is a dense subﬁeld of R, we can replace ǫ by its restriction onto R∧ . Denote by e the modiﬁed descent of ǫ|R∧ ; i.e., e : R → D is deﬁned by [[e(t) = ǫ(t∧ )]] = 1 (t ∈ R). An easy calculation ensures that e is a spectral function. ⊲ 2.11.7. Let F be a dense subﬁeld of R and let X be a vector lattice over F. A Dedekind complete vector lattice X δ over R is said to be a Dedekind completion of X whenever X is lattice isomorphic to a majorizing order dense sublattice X δ (which is identiﬁed with X). If X is Archimedean then X has a Dedekind completion X δ unique up to lattice isomorphism. This fact can be proved by the method of cuts just as the classical result (cp. [297, Theorems 32.3 and 32.5] and [403, Theorems II.3.2 and IV.11.1]). It follows also that if X is Archimedean then X has a universal completion X u unique up to a lattice isomorphism; i.e., X δ is a universally complete vector lattice and X is lattice isomorphic to an order dense vector subspace of X. 2.11.8. Let X be a vector B-lattice. Then there exist X , X u ∈ V(B) such that (1) [[X is a vector lattice over R∧ and X u is a universal completion of X ]] = 1. (2) There is a lattice isomorphism h from X into X u ↓ such that (X u ↓, h) is a universal completion of X and b = h−1 ◦ j(b) ◦ h for all b ∈ B, where j is defined as in 2.11.4. ⊳ Put D := P(X) and B := P(X0 ) and let j be the embedding of B to D. Then S(D) and S(B) are universally complete vector lattices isomorphic to X and X0 , respectively. By 1.9.3 there exists a complete Boolean algebra D in V(B) and we have a Boolean isomorphism h from D onto D↓ and a Boolean isomorphism ι from B onto an order closed subalgebra in D↓ such that ι = j ◦ h. By transfer X := S(D) is a universally complete vector lattice in V(B) and P(X ) is isomorphic to D. It remains to appeal to 2.11.3. ⊲ 2.11.9. Let X be an f -module over Z (Y ) with Y a Dedekind complete vector lattice and B = P(Y ). Then there exist X , X δ , X u ∈ V(B) such that (1) [[X is a vector lattice over R]] = 1, X ↓ is an f -module over Au , 2.12. Variations on the Theme 101 and there is an f -module isomorphism h from X to X ↓ satisfying X ↓ = mix(h(X)); (2) [[X δ is a Dedekind completion of X ]] = 1, X δ ↓ is an f -module over Au , and X ↓ is f -module isomorphic to an order dense f -submodule in (X ↓)δ . (3) [[X u is a universal completion of X ]] = 1, X u ↓ is an f -module over Au , and X ↓ is f -module isomorphic to an order dense f -submodule in (X ↓)u . 2.12. Variations on the Theme In this section we raise the following question: Which uniformities are generated by the metrics that take values in some vector lattices? It is clear that if (X, ρ) is a metric space, Λ is a vector lattice and 0 < e ∈ Λ, then the Λ-valued metric (x, y) → ρ(x, y) · e (x, y ∈ X) determines the same uniformity as ρ. Consequently, the question raised becomes nontrivial only if we additionally require that the Λ-valued metric uses a substantial part of Λ rather than just its one-dimensional subspace spanned by e. This extra assumption, for instance, provides decomposability. Thus, we are to clarify necessary and suﬃcient conditions for a uniformity F on X to be generated by a decomposable metric with values in a universally complete vector lattice. 2.12.A. Vector Lattice Valued Metrics We introduce the main deﬁnitions and notation that are dealt with henceforth. 2.12.A.1. Consider a nonempty set X and a vector lattice Λ. A mapping ρ : X × X → Λ is called a (vector, or Λ-valued) semimetric on X if for all x, y, z ∈ X the axioms are valid: (1) ρ(x, y) 0, ρ(x, x) = 0, (2) ρ(x, y) = ρ(y, x), (3) ρ(x, y) ρ(x, z) + ρ(z, y). A semimetric ρ is said to be a metric if from ρ(x, y) = 0 it follows that x = y for all x and y in X. Much of the sequel is valid for general semimetrics, but we conﬁne exposition to the case of vector metrics. A pair (X, ρ) is said to be a Λmetric space, if X is a nonempty set and ρ is a metric on X with values in some vector lattice Λ. 102 Chapter 2. Boolean Valued Numbers 2.12.A.2. Given an arbitrary net (xα )α∈A in a Λ-metric space (X, ρ), we say that (a) (xα ) ρo-converges (ρr-converges) to an element x ∈ X if o-lim ρ(x, xα ) = 0 r-lim ρ(x, xα ) = 0 ; α∈A α∈A (b) (xα ) is ρo-fundamental (ρr-fundamental), if o-lim ρ(xα , xβ ) = 0 r-lim ρ(xα , xβ ) = 0 ; β,α∈A α,β∈A (c) the space (X, ρ) is ρo-complete (ρr-complete) if each ρofundamental (ρr-fundamental) net in X has a ρo-limit (ρr-limit); (d) a subspace X0 ⊂ X is ρo-closed (ρr-closed) if X0 contains the ρo-limits (ρr-limits) of all ρo-converging (ρr-converging) nets in X0 . A vector metric enables us to provide the underlying set both with a Boolean metric and a uniformity. 2.12.A.3. We assume henceforth that the vector lattice Λ under consideration is Dedekind complete. Take a Λ-metric space X with a Λvalued metric ρ. Let j be an isomorphism of a complete Boolean algebra B onto the base P(Λ) of Λ. Deﬁne the mapping d : X ×X → Λ by putting d(x, y) := j−1 ([ρ(x, y)]) for all x, y ∈ X. Recall that [u] stands for the band projection onto u⊥⊥ . We can easily check that d is a Boolean metric on X. Thus, each Λ-metric space transforms canonically into a B-set. This fact allows us to use Boolean valued models for the study of Λ-metric spaces. Take an arbitrary family (xξ )ξ∈Ξ in X and a partition of unity (bξ )ξ∈Ξ in B. An element x ∈ X coincides with the mixture mix(bξ xξ ) (relative to the canonical B-metric d) if and only if πξ d(xξ , x) = 0 for all ξ ∈ Ξ, where πξ = j(bξ ). For convenience, from now on assume that B = P(Λ). Recall that mixing in a B-set is not always possible. A Λ-metric space X as well as its metric ρ is called mix-complete or laterally complete if there exist mixtures of all families (xξ ) in X by all partitions of unity (bξ ) in B. We say that (X, ρ) is decomposable if there exist mixtures of all ﬁnite collections by all ﬁnite partitions of unity. 2.12.A.4. Let Λ be a universally complete vector lattice. Let E denote the ﬁlter of all order units in Λ; i.e., e ∈ E means that 0 e ∈ Λ and {e}⊥⊥ = Λ. The collection of sets {[−e, e], e ∈ E } constitutes a base 2.12. Variations on the Theme 103 of zero neighborhoods for the unique topology τ making (Λ, τ ) a complete separated topological group (but not a topological vector space). Given e ∈ E , put U (ρ, e) := {(x, y) ∈ X 2 : ρ(x, y) e} = ρ−1 ([−e, e]). Clearly, the sets U (ρ, e) (e ∈ E ) form a fundamental system of entourages for the unique uniformity on X. This uniformity will be referred to as the uniformity generated by the Λ-metric ρ. Below, while talking about the uniform structure of a Λ-metric space (X, ρ), we bear in mind the uniformity. 2.12.A.5. The inclusion-ordered set of all uniformities U (X) on a set X forms a complete lattice. The bottom 0 of the lattice is the uniformity having the single entourage X 2 . Consider the interval [0, F ] := {F ′ ∈ U (X) : 0 F ′ F }. The complete Boolean algebra of components of F is deﬁned as the complete Boolean algebra B ⊂ [0, F ] in which suprema are inherited from the lattice U (X) and F serves as the order unit of B. It is easily seen that if B is a complete Boolean algebra and there is an injective mapping χ : B → [0, F ] preserving the suprema of all sets and having χ(1) = F , then χ(B) is a complete Boolean algebra of components of F . 2.12.A.6. Now we give the main deﬁnition of the current section: Fix the complete Boolean algebra B of components of a uniformity F . Take a family (xξ )ξ∈Ξ in a space Xand a partition of unity (Fξ )ξ∈Ξ in B. We assume that there is x := ξ∈Ξ Fξ (xξ ), where the supremum is taken in the complete lattice of all ﬁlters on X and Fξ (xξ ) denotes the set of subsets V (xξ ) := {y ∈ X : (xξ , y) ∈ V } (V ∈ Fξ ). We call x the mixture of (xξ ) by (Fξ ) and denote x by B-mixξ∈Ξ (Fξ xξ ). The space X is called B-decomposable (B-complete) if there exist B-mixtures of all ﬁnite (arbitrary) families in X by all ﬁnite (arbitrary) partitions of unity in B. 2.12.A.7. A ﬁlterbase F0 is called a B-cyclic base of a uniformity F provided that (1) every entourage V0 ∈ F0 is closed under B-mixing; i.e., if ((xξ , yξ ))ξ∈Ξ lies in V0 , while (Fξ )ξ∈Ξ is a partition of unity in B such that there exist x := B-mixξ∈Ξ (Fξ xξ ) and y := B-mixξ∈Ξ Fξ yξ ), then (x, y) ∈ V0 ; (2) each set V ∈ F includes a subset of the form V0 := B- mix(Fξ Vξ ) := B- mix(Fξ xξ ) : xξ ∈ Vξ (ξ ∈ Ξ) , ξ∈Ξ ξ∈Ξ where (Vξ ) is a family in F0 and (Fξ ) is a partition of unity in B. 104 Chapter 2. Boolean Valued Numbers 2.12.A.8. We say that a uniform space is completely metrizable by a Dedekind complete vector lattice Λ if its uniformity is determined by a decomposable Λ-valued metric as in 2.12.A.4. Consider the three examples of mix-complete metric spaces. 2.12.A.9. Take a metric space (X , ρ) in the Boolean-valued model V(B) , with a ﬁxed complete Boolean algebra B. Let R be the reals within V(B) and Λ := R↓. If X = X ↓ and ρ̄ := ρ↓ then ρ̄ : X 2 → Λ is a metric and the space (X, ρ̄) is mix-complete. The latter space is ρo-complete if and only if the internal metric space (X , ρ) is complete within V(B) . 2.12.A.10. Let Q be an extremally disconnected compact space and let (X, ρ) be a metric space. Denote by C∞ (Q, X) the set of cosets of continuous mappings from comeager subsets of Q into X. To put it in more detail, an element z ∈ C∞ (Q, X) is uniquely determined by the conditions: (a) for every u ∈ z, there is a comeager subset Q(u) ⊂ Q (i.e. the complement of a meager subset) such that u is a continuous mapping from Q(u) into X; (b) if u, v ∈ z then u(t) = v(t) for all t ∈ Q(u) ∩ Q(v). Take arbitrary elements y, z ∈ C∞ (Q, X). Let u ∈ y and v ∈ z. Then the function t → ρ(u(t), v(t)) (t ∈ Q(u) ∩ Q(v)) is deﬁned on a comeager set and is continuous. Consequently, it determines the unique element w of C∞ (Q) := C∞ (Q, R); the element w is independent of the choice of u ∈ y and v ∈ z. We set ρ̄(y, z) := w by deﬁnition. Clearly, that ρ̄ is a vector metric on C∞ (Q, X) with values in C∞ (Q). If (X, ρ) is a metric space then (C∞ (Q, X), ρ̄) is a mix-complete Λmetric space with Λ := C∞ (Q). Moreover, C∞ (Q, X) is ρ̄o-complete if and only if X is complete. 2.12.A.11. Consider a metric space (X, ρ). Let τ be the topology on X determined by the metric ρ. A mapping ϕ : τ → B is called a Cauchy B-filter if ϕ satisﬁes the conditions: (1) ϕ(∅) = 0; (2) ϕ(U ∩ V ) = ϕ(U ) ∧ ϕ(V ) for all U, V ∈ τ ; (3) {ϕ(V ) : V × V ⊂ {ρ < ε}} = 1 for every 0 < ε ∈ R. Say that a Cauchy B-ﬁlter ϕ is minimal of ϕ has the uniform regularity property: ϕ(U ) = {ϕ(V ) : V ∈ τ, Uε,ρ (V ) ⊂ U, 0 < ε ∈ R} (U ∈ τ ), where Uε,ρ (V ) = {x ∈ X : (∃ v ∈ V ) ρ(v, x) < ε}. Denote the set 2.12. Variations on the Theme 105 of minimal Cauchy B-ﬁlters by BX . Let Λ be the universally complete vector lattice of all spectral systems in B (see Theorem 2.8.3). We will determine a Λ-valued metric on BX . Given ϕ, ψ ∈ BX we put eλ := {ϕ(U ) ∧ ψ(V ) : U, V ∈ τ ; U × V ⊂ {ρ < λ}}, if 0 < λ ∈ R and eλ := 0 if λ 0. It can be veriﬁed that the mapping e : λ → eλ (λ ∈ R) is a spectral system in B. We put r(ϕ, ψ) := e. The mapping r : BX × BX → Λ is a metric and (BX , r) is a mixcomplete Λ-metric space. 2.12.B. Metrization by Vector Lattices The aim of this section is to prove the following metrization result. 2.12.B.1. Theorem. A separated uniform space (X, F ) is completely metrizable by a Dedekind complete vector lattice Λ with unit if and only if X is B-decomposable and the uniformity F possesses a countable B-cyclic base with respect to some complete Boolean algebra B of components of F which is isomorphic to P(Λ). 2.12.B.2. Let ρ : X × X → Λ be a decomposable metric generating the uniformity F as in 2.12.A.4. Then F has a countable B-cyclic base with respect to some complete Boolean algebra B of components of F which is isomorphic to P(Λ). ⊳ Associate to each projection b ∈ B := B(Λ) the uniformity F b on X that is determined by the fundamental system of entourages U (bρ, e) := {(x, y) ∈ X 2 : bρ(x, y) < e}, where e ranges over the ﬁlter of order units E in Λ. Clearly, the mapping b → F b (b ∈ B) is injective, preserves suprema, and associates F with the unit of the algebra B by our assumption of metrizability. Consequently, the mapping gives an isomorphism of B onto the Boolean algebra of components of F . As for the conditions required, we, for example, will prove B-decomposability, where B is the image of B under the isomorphism indicated. Take x, y ∈ X and b ∈ B and put z := ρ-mix{bx, b∗ y}. This means that z ∈ X and bρ(x, z) = b∗ ρ(y, z) = 0. Let some neighborhoods of x ∗ and y in the uniform topology corresponding to F b and F b look like ∗ U := {u ∈ X : bρ(x, u) e} and V := {v ∈ X : b ρ(y, v) e}, where e ∈ E . If W := U ∩ V then for every w ∈ W we have bρ(z, w) b(ρ(z, x) + ρ(x, w)) = bρ(x, w) e, ∗ b ρ(z, w) b∗ (ρ(z, y) + ρ(y, w)) = b∗ ρ(y, w) e; 106 Chapter 2. Boolean Valued Numbers ∗ i.e., ρ(z, w) e. Therefore, W ∈ F (z). Thus, z = lim(F b (x) ∨ F b (y)). The similar reasoning demonstrates that the sequence {ρ n−1 1} (n ∈ N) forms a B-cyclic ﬁlterbase of the entourages of F . ⊲ 2.12.B.3. Let (X, F ) be a B-decomposable separated uniform space with B the complete Boolean algebra of components of F isomorphic to B. Then X is a decomposable B-set. ⊳ Assume that there is Boolean algebra B onto the Given a pair of elements x, y d(x, y) := an isomorphism b → F b of a complete Boolean algebra B of components of F . ∈ X, put ∗ b ∈ B : (x, y) ∈ Fb . It is obvious that d(x, y) = d(y, x) and d(x, x) = 0 for all x, y ∈ X. Assume that d(x, y) = 0. Then F = F1 = F b : (x, y) ∈ Fb ; 2 ) or x = y. Take b, c ∈ B such that consequently, b(x, y) ∈ F =∆(X c (x, y) ∈ F and (z, y) ∈ F . Since F b∧c is a uniformity on X, the set V := F c∧b is symmetric. Moreover, (x, z), (z, y) ∈ V ; hence, (x, y) ∈ V . From the deﬁnition of d we have d(x, y) (b ∧ c)∗ = b∗ ∨ c∗ . By taking inﬁma over b∗ and c∗ , we arrive to the triangle inequality for d. Thus, (X, d) is a B-set; let us prove that X is decomposable. To this end, take arbitrary x, y ∈ X and b ∈ B. By assumption z = B∗ mix{bx, b∗ y} exists. From the containment z ∈ (F b (x) ∨ F b (y)) it is ∗ clear that (x, z) ∈ F b and (y, z) ∈ F b . Hence, from the deﬁnition of d we obtain d(x, z) b∗ and d(x, y) b, or, which is the same, b ∧ d(x, z) = 0 and b∗ ∧ d(x, y) = 0. This means that z = d-mix(bx, b∗ y), where d-mix denotes the mixing operation on the B-set (X, d). So the decomposability of (X, d) is corroborated. The same reasoning shows that the mixtures B-mixξ∈Ξ (F b ξxξ ) and z = d-mixξ∈Ξ (bξ xξ ) coincide for all (xξ ) ⊂ X and (bξ ) ⊂ B. Therefore, in what follows we will simply write mix, while denoting the two mixing operations. ⊲ 2.12.B.4. We are able now to prove 2.12.B.1. ⊳ Let X ∈ V(B) be a Boolean valued representation of a B-set (X, d). Without loss of generality, we can assume that X ⊂ X ′ := X ↓ ⊂ V(B) and that [[x = y]] = d(x, y) for x, y ∈ X (cp. 1.7.2). Let F0 be a countable B-cyclic ﬁlterbase of F . Put F := {V ↑ : V ∈ F }↑, F0 := {V ↑ : V ∈ 107 2.12. Variations on the Theme F0 }↑. Show that (X , F) is a uniform space and that F0 is a countable ﬁlterbase of F. The fact that F is a ﬁlterbase within V(B) is plain from the calculation: [[(∀ A, B ∈ F) A ∩ B ∈ F]] = [[A ↑ ∩B ↑∈ F]] A,B∈F A,B∈F [[(A ∩ B) ↑∈ F]] = [[C ↑∈ F]] = 1. C∈F Assuming that A ∈ F , B ∈ V(B) , and [[A↑⊂ B ⊂ X 2 ]] = 1, and putting VB := B↓ ∩ X 2 we see that V B ∈ F and [[B ∈ F]] = 1, since A ⊂ A↑↓ ∩ X 2 ⊂ VB and [[B ∈ F]] = V ∈F [[V ↑ = B]] [[VB ↑ = B]] = 1. Now it is easy to estimate [[(∀ A ∈ F)(∀ B ⊂ X 2 )(A ⊂ B → B ∈ F]] = [[B ∈ F]] : [[A ↑⊂ B ⊂ X 2 ]] = 1 = 1, A∈F so that F is a ﬁlter within V(B) . Demonstrate that F0 is a ﬁlterbase of F. Take an arbitrary entourage A ∈ F . We will establish that there is an element B ∈ V(B) for which [[B ∈ F0 ]] = 1 and [[A↑ ⊂ B]] = 1. The last equalities are equivalent to B ∈ F0 ↓ = mix{V ↑ : V ∈ F0 } and mix(A) ⊂ B↓. The latter are fulﬁlled since F0 is a B-cyclic base (cp. 2.12.A.7 (2)). Observe that every set V ∈ F0 is cyclic (cp. 2.12.A.7 (1)); consequently, (V ◦ V ) ↑= V ↑ ◦V ↑. Also, the equality (V −1 ) ↑= (V ↑)−1 is true for every V ⊂ X. From here we see that [[(∀ U ∈ F)(∃ V ∈ F0 )V ◦ V ⊂ U ]] = 1, [[(∀ U ∈ F)(∃ V ∈ F)(V −1 = V ∧ V ⊂ U )]] = 1. If [[(x, y) ∈ (F0 )]] = 1 then (x, y) ∈ A ↑↓= mix(A) = A for every A ∈ F0 (cp. 2.12.A.7(1)). Since F is a Hausdorﬀ uniformity, it follows that x = y. Thus, [[ F0 = IX ]] = 1; i.e., F is a Hausdorﬀ uniformity within V(B) . Take a mapping ϕ from the naturals ω onto F0 . Put ψ(n) := ϕ(n)↑(n ∈ ω). Then ψ↑ is a mapping from ω ∧ onto F0 within V(B) . Since ψ↑(ω ∧ ) = ψ(ω)↑, we have im (ψ↑) = F0 by 1.6.8 and hence F0 is a countable set within V(B) . 108 Chapter 2. Boolean Valued Numbers Thus, V(B) |= [[(X, F) is a Hausdorﬀ uniform space with a countable base of the uniformity]]. By the well-known metrization theorem from general topology, the uniformity F is generated by some metric p. Put X := X ↓ and ρ′ := p↓. By 2.12.A.9, (X ′ , ρ′ ) is a Λ-metric space, where Λ = R↓. It is easy that U (p, ε)↓ = U (ρ′ , ε) for ε ∈ Λ+ , [[ε > 0]] = 1. If ρ is the restriction of the metric ρ′ to X then U (ρ, ε) = U (ρ′ , ε) ∩ X 2 ; consequently, ρ is the required metric on X. ⊲ 2.12.C. Boolean Compactness In this section we present the notion of a cyclically compact (or mixcompact) set arising as a Boolean valued interpretation of compactness. 2.12.C.1. Suppose that (X, ρ) is a Λ-metric space, (xn )n∈N ⊂ X, and x ∈ X. Say that a sequence (xn )n∈N approximates x if inf nk ρ(xn , x) = 0 for all k ∈ N. Call a set K ⊂ X mix-compact if K is mix-complete and for every sequence (xn )n∈N ⊂ K there is x ∈ K such that (xn )n∈N approximates x. It is clear that in case Λ = R mix-compactness is equivalent to compactness in the metric topology. 2.12.C.2. As is easily seen, mix-compactness is an absolute concept in the following sense: If X and Y are Λ-metric spaces, X is a decomposable subspace of Y , and K ⊂ X then the mix-compactness of K in X is equivalent to that in Y . 2.12.C.3. Suppose that X is a metric space within V(B) . (1) A subset K ⊂ X ↓ is mix-compact if and only if K is mixcomplete and V(B) |= “K↑ is a compact subset of X .” (2) V(B) |= “K is a compact subset of X ” if and only if K ↓ is a mix-compact subset of X ↓. ⊳ (1): The compactness of K↑ within V(B) is equivalent to V(B) |= (∀ σ : N∧ → K↑)(∃ x ∈ K↑)(∀ k ∈ N∧ ) inf p(σ(n), x) : k n ∈ N∧ = 0. Taking account of 1.6.8 and recalling the equality cyc(K) = K (cp. 1.6.6), we conclude that the above formula amounts to (∀ s : N → K) (∃ x ∈ K)(∀ k ∈ N)ϕ, where ϕ := (V(B) |= inf{p(s↑(n), x) : n k ∧ } = 0). 2.12. Variations on the Theme 109 It remains to observe that ϕ ⇐⇒ V(B) |= “ ∀ e ∈ R+ (∀ n k ∧ )(e p(s↑(n),x)) → e = 0 ” ⇐⇒ ∀ e ∈ Λ+ (∀ n k) e ρ(s(n), x) =⇒ e = 0 ⇐⇒ inf ρ(s(n), x) : n k = 0. (2): Put K := K ↓. If V(B) |= “K is a compact subset of X ” then, using the obvious mix-completeness of K and applying (1), we conclude that K is a mix-compact subset of X ↓. Conversely, if K is a mix-compact subset of X ↓ then, as K↑= K , we have V(B) |= “K is a compact subset of X ” due to (1). ⊲ 2.12.C.4. Denote by PrtN (B) the set of sequences ν : N → B that are partitions of unity of the Boolean algebra B. For ν1 , ν2 ∈ PrtN (B), the formula ν1 ≪ ν2 abbreviates the following: If m, n ∈ N and ν1 (m) ∧ ν2 (n) = 0B then m < n. Let (X, ρ) be a mix-complete Λ-metric space. Given a mix-complete subset K ⊂ X, a sequence s : N → K, and a partition ν ∈ PrtN (B), put sν := mixn∈N ν(n)s(n). A cyclic subsequence of s : N → K is each sequence of the form (sνk )k∈N , where (νk )k∈N ⊂ PrtN (B) and νk ≪ νk+1 for all k ∈ N. A subset K ⊂ X is called cyclically compact if K is mix-complete and each sequence of elements in K admits a cyclic subsequence convergent to an element of K in the metric ρ. 2.12.C.5. Let X be a mix-complete Λ-metric space. A subset K ⊂ X is cyclically compact if and only if K is mix-compact. ⊳ =⇒: Let K be a cyclically compact subset of X. Consider an arbitrary sequence s : N → K. By the deﬁnition of cyclic compactness there exist a sequence (νk )k∈N ⊂ PrtN (B) and an element x ∈ K such that (∀ k ∈ N)(νk ≪ νk+1 ) and o-limk→0 ρ(sνk , x) = 0. The inspection of the latter formulas shows that inf ρ(κ, x) : κ ∈ mix{s(n) : n k} = 0 for all k ∈ N and so the sequence s approximates x ∈ K, since for each κ = mixnk πn s(n), with (πn )nk ∈ PrtN (B), we have πm inf ρ(s(n), x) πm ρ(s(m), x) nk πm ρ(s(m), κ) + πm ρ(κ, x) = πm ρ(κ, x) ρ(κ, x) 110 Chapter 2. Boolean Valued Numbers for all m k and, consequently, inf ρ(s(n), x) = sup πm nk mk inf ρ(s(n), x) ρ(κ, x). nk ⇐=: Suppose now that K is a mix-compact subset of X and let s : N → K. According to 2.12.B.3 we can assume that X is a decomposable subset of X ↓, where V(B) |= “(X , p) is a metric space.” Put σ := s↑. Then V(B) |= σ : N → K↑. Moreover, 2.12.C.2 and 2.12.C.3 (1) imply V(B) |= “K↑ is a compact subset of X .” Applying the classical compactness criterion within V(B) , consider x ∈ K and N ∈ V(B) such that V(B) |= “N : N → N, N (k) < N (k + 1), 1 for each k ∈ N. p(σ(N (k)), x) k Put νk (n) := [[N (k ∧ ) = n∧ ]] for all k, n ∈ N. A routine veriﬁcation shows that νk ∈ PrtN (B) and (∀ k ∈ N) νk ≪ νk+1 . Moreover, for each k ∈ N we have V(B) |= sνk = σ(N (k ∧ )) and, consequently, ρ(sνk , x) k1 e. ⊲ 2.12.C.6. Let (X, ρ), Λ, and E be the same as in 2.11.A.4. Take a projection π ∈ B(X) and an order unit e ∈ E . The set Θ ⊂ X will be called a (ρ, e, π)-net in X if, for every x ∈ X, there is an element y ∈ Θ such that πρ(x, y) e. The next fact is an interpretation of the Hausdorﬀ compactness criterion in a Boolean valued model. 2.12.C.7. For a decomposable Λ-metric space (X, ρ), the following are equivalent: (1) X is cyclically compact. (2) X is ρo-complete and, for every e ∈ E , there exist a sequence (Θn )n∈N of finite subsets Θn ⊂ X and a countable partition of unity (πn )n∈N in P(Λ) such that mix(Θn ) is a (ρ, e, πn )-net in X for all n ∈ N. 2.13. Comments 2.13.1. (1) In the history of functional analysis, the rise of the theory of ordered vector spaces is commonly ascribed to the contributions of Birkhoﬀ, Freudenthal, Kantorovich, Nakano, Riesz, et al. At present, the theory of ordered vector spaces and its applications constitute a noble 2.13. Comments 111 branch of mathematics, representing one of the main sections of contemporary functional analysis. The various aspects of vector lattices and positive operators are presented in the monographs by Abramovich and Aliprantis [5]; Akilov and Kutateladze [22]; Aliprantis and Burkinshaw [26, 28]; Kusraev [222, 228]; Kantorovich, Vulikh, and Pinsker [196]; Lacey [275]; Lindenstrauss and Tzafriri [281]; Luxemburg and Zaanen [297]; Meyer-Nieberg [311]; Schaefer [356]; Schwarz [361]; Vulikh [403]; and Zaanen [427]). (2) The credit for ﬁnding the most important instance of ordered vector spaces, a Dedekind complete vector lattice or a Kantorovich space, belongs to Kantorovich. This notion appeared in Kantorovich’s ﬁrst article on this topic [191] where he stated an important methodological principle, the heuristic transfer principle for Kantorovich spaces: “the elements of a Dedekind complete vector lattice are generalized numbers.” (3) At the very beginning of the development of the theory, many attempts were made at formalizing the above heuristic principle. These led to the so-called theorems of relation preservation which claimed that if some proposition involving ﬁnitely many functional relations is proven for the reals then an analogous fact remains valid automatically for the elements of every Dedekind complete vector lattice (cp. [196, 403]). The depth and universality of Kantorovich’s principle were demonstrated within Boolean valued analysis. 2.13.2. (1) The Boolean valued status of the concept of Kantorovich space (= Dedekind complete vector lattice) is established by Gordon’s Theorem obtained in Gordon [133]. This fact can be interpreted as follows: A universally complete vector lattice is the interpretation of the reals in an appropriate Boolean valued model. Moreover, it turns out that each theorem on the reals (in the framework of ZFC) has an analog for the corresponding Dedekind complete vector lattice. The theorems are transferred by means of the precisely-deﬁned procedures: ascent, descent, and canonical embedding, that is, algorithmically as a matter of fact. Descending the basic scalar ﬁelds opens a turnpike to the intensive application of Boolean valued models in functional analysis. The technique of Boolean valued analysis demonstrates its eﬃciency in studying Banach spaces and algebras as well as lattice normed spaces and modules. The corresponding results are collected and elaborated in Kusraev and Kutateladze [249, Chapters 10–12]. (2) If if μ is a Maharam measure and B in Theorem 2.2.4 is the algebra of all μ-measurable sets modulo sets of μ-measure zero, then R↓ 112 Chapter 2. Boolean Valued Numbers is isomorphic to the universally complete vector lattice L0 (μ) of (cosets) measurable functions. This fact (for the Lebesgue measure on an interval) was already known to Scott and Solovay [368]. If B is a complete Boolean algebra of projections in a Hilbert space H then R↓ is isomorphic to the space . of all selfadjoint operators A on H admitting spectral resolution A = λ dEλ with Eλ ∈ B for all λ ∈ R. The two indicated particular cases of Gordon’s Theorem were intensively and fruitfully exploited by Takeuti [379, 380, 381]. (3) The object R↓ for general Boolean algebras was also studied by Jech [180, 181, 182] who in fact rediscovered Gordon’s Theorem. The diﬀerence is that in [184] a universally complete (complex) vector lattice with unit element is deﬁned by another system of axioms and is referred to as a complete Stone algebra. The contemporary forms of the above mentioned relation preservation theorems, basing on Boolean valued models, may be found in Gordon [135] and Jech [181] (cp. also [248]). (4) The forcing method splits naturally into the two parts: one is general and the other, special. The general part comprises the apparatus of Boolean valued models of set theory; i.e., the construction of a Boolean valued universe V(B) and interpretation of the set-theoretic propositions in V(B) . Here, a complete Boolean algebra B is arbitrary. The special part consists in constructing particular Boolean algebras B providing some special (usually, pathological and exotic) properties of the objects (for example, Dedekind complete vector lattices) obtained from V(B) . Both parts are of interest in their own right, but the most impressive results stem from their combination. In this book, like in the most part of research in Boolean valued analysis, we primarily use the general part of the forcing method, using in some places cardinal collapsing phenomena. The special part is widely employed for proving independence or consistency (cp. Bell [43], Dales and Woodin [101], Jech [184], Rosser [350], Takeuti and Zaring [388]). 2.13.3. (1) Theorems 2.3.2 and 2.3.4 are companions for Gordon’s Theorem from the very beginning of Boolean values analysis. The complex structure of C ↓ was intensively employed by Takeuti [380]–[384] and multiplication on R↓ was examined by Gordon [134, 136, 137]. Sometimes it is useful to consider another companions of the Gordon Theorem treating quaternions and octonions. (2) Let H be the quaternion algebra and let O be the Cayley algebra. Recall that the Cayley algebra is an 8-dimensional algebra over R which is noncommutative and nonassociative, and the elements of O are 2.13. Comments 113 Cayley numbers or octonions. Then [[ H∧ and O∧ are normed algebras over the ﬁeld R∧ ]] = 1. Let H and O stand for the norm completions respectively of H∧ and O∧ within V(B) . It is easy to show (using, for example, the Hurwitz Theorem) that [[ H is the quaternion algebra ]] = 1 and [[O is the Cayley algebra ]] = 1. If Q denotes the Stone space of B then the descents (restricted descents) of H and O can be described as C∞ (Q, H) and C∞ (Q, O) (C(Q, H) and C(Q, O)), respectively. These objects occur in classiﬁcation and representation theory of Jordan operator algebras; see Ajupov [15, 16]; Hanshe-Olsen and Störmer [165]. (3) Subsections 2.3.5–2.3.7 can be considered as analytical versions of Shoenﬁeld type absoluteness theorems (see Takeuti [380, Lemma 2.7]): Let B0 be a complete subalgebra of B and let R be the reals within V(B0 ) . If u1 , . . . , un ∈ R↓ and ϕ is a ZFC-formula of the class Σ12 or Π12 then [[ϕ(u1 , . . . , un )]]B = [[ϕ(u1 , . . . , un )]]B . 2.13.4. The interconnections between the properties of numerical objects in R and the corresponding objects in the universally complete vector lattice R↓, indicated in 2.4.3–2.4.7 were actually obtained by Gordon [135, Section 3] for countable sets and sequences. The general case in Section 2.3 is treated by repeating essentially the same argument. Proposition 2.4.9, allowing us to translate into the internal language of R the claims about traces and characteristics in R↓, was also established by Gordon; see [133, Theorem 3] and [135, Theorem 3]. These results underlie the technology of translating the knowledge about numbers to theorems on the elements, sequences, and subsets of universally complete vector lattices. 2.13.5. (1) The Representation Theorem 2.5.1 is due to Kusraev [223]. A close result (in other terms) is presented in Jech’s article [182], where the Boolean valued interpretation is developed for the theory of linearly ordered sets. Corollaries 2.5.7 and 2.5.8 are well known (cp. Kantorovich, Vulikh, and Pinsker [196] and Vulikh [403]). Some subsystems of the reals R appear not only as the Boolean valued representation of Archimedean vector lattices. (2) For instance, the following assertions were stated in Kusraev [223] and proved in Kusraev and Kutateladze [244, 248]: (a) a Boolean valued representation of an Archimedean lattice-ordered group is a subgroup of the additive group of R; (b) an Archimedean f -ring includes two complementary bands one of which has the zero multiplication and is realized as in (a) and the other, as a subring of R; (c) an Archimedean 114 Chapter 2. Boolean Valued Numbers f -algebra contains two complementary bands one of which is realized as in 2.5.1 and the other, as a sublattice and subalgebra of R considered as a lattice-ordered algebra over R∧ (also see Jech [182]). 2.13.6. The tests of 2.6.2 (2, 5) for o-convergence (in the case of sequences) were obtained by Kantorovich and Vulikh (cp. [196]). As is seen from 2.6.1, these tests are merely the interpretation of convergence properties of numerical nets (sequences). 2.13.7. (1) The Spectral Theorem 3.7.7 was proved by Freudenthal [127]. It remains true for vector lattices with the principal projection property (see Veksler [395]; Luxemburg and Zaanen [297]). Then Veksler introduced slightly diﬀerent concepts of weak and strong Freudenthal properties for general vector lattices and characterized them by the corresponding projection properties; see [395, Theorems 2.3, 2.5, 2.8–2.10]. (2) The weak and strong Freudenthal properties in the sense of 2.7.8 were introduced and studied by Lavrič [277]. To characterize the spaces with strong Freudenthal property we need the deﬁnition. Two elements of a vector lattice X are completely disjoint if they lie in two disjoint projection bands of X. The following was proved in Lavrič [277]: A vector lattice X has the strong Freudenthal property if and only if every two disjoint elements in X are completely disjoint. A vector lattice X has the weak Freudenthal property if and only if for every two elements e, d ∈ X there are disjoint elements e0 ∈ [0, e] and d0 ∈ [0, d] such that X(e + d) = X(e0 + d0 ). (3) The Boolean valued proofs of the Freudenthal Spectral Theorem, as well as the properties 2.7.2 and 2.7.3 (1) were given by Gordon [135], while 2.7.3 (2), 2.7.4, and 2.7.5 are collected in Kusraev and Kutateladze [248, 249]. Of course, these formulas as well as the estimates in 2.7.6 were mostly known and employed by various authors; see for example the papers of Nakano [318], Vulikh [403], Luxemburg and de Pagter [294]. (4) The formulas for ex similar to those of 5.7.3 (1, 2), 5.7.4 (2, 3), and 5.7.5 (1–3) are trivial: ex = e|x| = eαx (0 = α ∈ R); ex∧y = exy = ex ∧ ey and ex∨y = ex+y = ex ∨ ey (x 0, y 0); exy = ex ∧ ey (x, y ∈ X arbitrary). To ensure the latter we need only to interpret within V(B) the simple proposition (∀ s, t ∈ R)(st = 0 ↔ s = 0 ∧ t = 0) and apply 5.4.9. 2.13.8. The fact that for a complete Boolean algebra B the set S(B) of spectral functions is a universally complete vector lattice with the 2.13. Comments 115 Boolean algebra of bands isomorphic to B (cp. 2.8.3) is due to Kantorovich [196]. The claim of 2.8.4 was obtained by Pinsker (cp. [196]). The representation of an arbitrary Dedekind complete vector lattice as an order dense ideal in C∞ (Q) was established independently from one another by Vulikh and Ogasawara (cp. [196, 403]). 2.13.9. (1) The starting point of the theory of spectral measures was von Neumann’s classical theorem: Each normal operator on a Hilbert space admits a spectral resolution with commutable orthogonal projections. By the classical deﬁnition, a spectral measure is a Boolean homomorphism of a Boolean algebra of sets to the Boolean algebra of projections (cp. Dunford and Schwartz [112]). If need be, the countable additivity condition or some regularity requirements are added. Motivated by spectral theory, much eﬀort has been made to extend the spectral theory of hermitian operators on a Hilbert space to Banach spaces. The third part of the Dunford and Schwartz treatise [112] is devoted to the corresponding theory of spectral operators. Recall that an operator T is called spectral if there is a spectral measure P on the Borel sets of the complex plane such that P (A)T = T P (A) for all A ∈ Bor(C) and the spectrum of P (A)T |P (A) lies in the closure of A. (2) The Bade Reﬂexivity Theorem tells us that a bounded linear operator T on a Banach space X belongs to the strongly closed algebra generated by a σ-complete Boolean algebra B of projections on X if and only if T keeps invariant every B-invariant subspace of X (cp. Bade [39]). Schaefer [354] discovered the key role that is played by order in abstracting the method of spectral measures and Bade’s reﬂexivity results to locally convex spaces. This article has started the systematic study of the operator algebras generated by Boolean algebras of projections within the theory of Riesz spaces; see Dodds and de Pagter [106, 107]; Dodds and Ricker [109]; and Dodds, de Pagter, and Ricker [108]. 2.13.10. (1) The Borel functions of an element of an arbitrary Dedekind complete vector lattice with unit seem to be ﬁrst considered by Sobolev (see [367] and also [403]). Theorems 2.10.4 and 2.10.5 in the above generality were obtained by Kusraev and Malyugin [252, 254]. Some Boolean valued proof of Theorem 2.10.4 is also given by Jech in [180]. Further details are available in the books by Kusraev and Kutateladze [244, 248]. (2) Kusraev and Malyugin constructed in [254] the Borel functional calculus of countable and uncountable collections of elements of 116 Chapter 2. Boolean Valued Numbers Dedekind complete vector lattices. The following was proved in particular: Let X be a universally complete vector lattice with unit 1 and let x := (xk )k∈N be an arbitrary sequence in X. There exists a unique se from B(RN , R) quentially order continuous f -algebra homomorphism x to X such that x(dtk ) = xk for all k ∈ N. (3) From 2.10.2 it follows that: For every resolution of the identity (eα )α∈R with range in a σ-algebra B there is a unique spectral measure μ : Bor(R) → B satisfying μ((−∞, α)) = eα (α ∈ R). This fact was ﬁrstly revealed by Sobolev in [367]. But the extension method that led to 2.10.2 diﬀers signiﬁcantly from the Carathéodory extension and bases on the Loomis–Sikorski representation of Boolean σ-algebras (co. 2.10.1). (4) Under the assumptions of Theorem 2.10.5, for each e ∈ C(1) we have ef (x1 , . . . , xN ) = f (ex1 , . . . , exN ) + e∗ f (0, . . . , 0) (cp. Kantorovich, Vulikh, and Pinsker [196, Proposition 3.14]). Indeed, if ex := (ex1 , . . . , exN ), μ = μx , μ̄ := μex , and μ0 is the {0, 1}-valued measure on B(RN , R) with support {0} then eμ = μ̄ + e∗ μ0 by the deﬁnition of μx in 2.10.3 and 2.7.5 (4); therefore, f (x1 , . . . , xN ) = Iμ (f ) = Iμ̄ (f )+ Iμ0 (f ) = f (ex1 , . . . , exN ) + e∗ f (0, . . . , 0). (5) If the function f in Theorem 2.10.5 is positive homogeneous (f (λt) = λf (t) for λ ∈ R+ and t ∈ RN ) then f (x1 , . . . , xN ) do not depend on the choice of an order unit 1 ∈ X. This fact was ﬁrst observed by Riesz [347] and Vulikh [196, Theorema 3.54]. Homogeneous functional calculus in uniformly complete vector lattices stems from Lozanovskiı̆ [290] and Krivine [210]; see further development in Buskes, de Pagter, and van Rooij [79], Lindenstrauss and Tzafriri [281], Szulga [374]. Concerning various generalizations of the homogeneous functional calculus see Haydon, Levy, and Raynaud [170], Kusraev [237], Kusraev and Kutateladze [250], Tasoev [389]. 2.13.11. (1) Vector lattices within Boolean valued models were ﬁrst considered by Gordon [134]; Theorems 2.11.4 and 2.11.6 are essentially contained in [134, Theorem 1]. (2) The f -module structure is inevitable in the theory of order bounded operators, since L∼ (X, Y ) is always an f -module over the f algebra Orth(Y ). Nevertheless, f -modules were introduced and studied in Luxemburg and de Pagter [293] more than sixty years after Kantorovich had started a systematic study of order bounded operators [191, 193]. 2.13.12. (1) In 1934 Kurepa introduced the so-called espaces pseudodistanciès, i.e. the spaces metrized by means of an ordered vector space 2.13. Comments 117 [212]. Soon after that Kantorovich developed the theory of abstract normed spaces; i.e., the vector spaces with a norm that takes values in an order complete vector lattice [193]. These objects turned out very useful in the study of functional equations by successive approximations (cp. Kantorovich [192, 194]) and in the related areas of analysis (cp. Kantorovich, Vulikh, and Pinsker [196], Kollatz [203], and Kusraev [222]). Lattice normed spaces (Kusraev [228]) and randomly normed spaces (Haydon, Levy, and Raynaud [170]) are special examples of spaces with lattice valued metric. At the same time their structural properties never gained adequate research. Metrization by means of a semiﬁeld (= a kind of a vector lattice) was studied also by several authors in a series of articles by Uzbekistan mathematicians (see Antonovskiı̆, Boltjanskiı̆, and Sarymsakov [32]). (2) The claim of 2.12.B.1 was justiﬁed in Kusraev [225]. The proof employs the following simple idea: By Gordon’s Theorem, metrization by a Dedekind complete vector lattice is nothing else but the usual metrization (i.e. by means of the reals) in the corresponding Boolean valued model. Successive implementation of this idea results in the notion of Boolean algebra of components of uniformity which reﬂects the main structural peculiarity of the uniformities metrizable by order complete vector lattices. (3) The concept of cyclical compactness was ﬁrst studied by Kusraev [216, 222]. Section 8.5 in Kusraev [228] deals with the cyclically compact linear operators on B-cyclic Banach spaces and Kaplansky–Hilbert modules. Recently Gönüllü [146]–[148] undertook the study of Schatten type classes of operators (which are cyclically compact) on Kaplansky–Hilbert modules. (4) The notion of mix-compact subset of lattice normed space was introduced in Gutman and Lisovskaya [152]. Basing on Boolean valued analysis, they prove the analogs of the three classical theorems for arbitrary lattice normed spaces over universally complete Riesz spaces, namely, the boundedness principle, the Banach–Steinhaus Theorem, and the uniform boundedness principle for a compact convex set; see [152, Theorems 2.4, 2.6, 3.3]. These theorems generalize the analogous results by Ganiev and Kudajbergenov [128] which were established for Banach–Kantorovich spaces over the lattice of measurable functions by the speciﬁc technique of the theory of measurable Banach bundles with lifting (see Gutman [158, 160] and Kusraev [228]). 118 Chapter 2. Boolean Valued Numbers (5) Take an arbitrary metric space (X, p). Then, (X ∧ , p∧ ) is a metric space within V(B) . If τ is the topology on X generated by the metric p then [[τ ∧ is a base of the topology on X ∧ generated by the metric ρ∧ ]] = 1. Let (X , ρ) denote the completion of the metric space (X ∧ , ρ∧ ) within V(B) . The elements of X are the minimal Cauchy ﬁlters identiﬁable with the mappings τ ∧ : X → {0, 1}. Thus, with every ϕ ∈ X ↓ we uniquely associate the Cauchy B-ﬁlter ϕ̄ by the formula ϕ̄(V ) := [[V ∧ ∈ ϕ]] = [[ϕ(V ∧ ) = 1]] (V ∈ τ ). The mapping ϕ → ϕ̄ (ϕ ∈ X ↓) is an isometric bijection from X ↓ onto BX . If (X, ρ) is complete then the mapping that sends a B-filter ϕ ∈ BX to the function f : q → lim ϕ−1 (q) (q ∈ Q), ϕ−1 (q) := {ϕ−1 (V ) : V ∈ q} determines an isometric bijection of BX onto C∞ (Q, X). Isometry is understood in the sense of Λ-valued metrics (see 2.12.A.10 and 2.12.A.11). (6) The Boolean extensions BX of general uniform structures was studied in a series of articles by Gordon and Lyubetskiı̆ (see [134, 139, 140]). Boolean extensions of locally compact abelian groups as well as the corresponding harmonic analysis were developed by Takeuti [380, 381]. Other interesting results on the structure of Boolean extensions can also be found in the above articles. CHAPTER 3 Order Bounded Operators The aim of the three subsequent chapters is to apply Boolean valued analysis to order bounded operators and establish some variants of the Boolean valued transfer principle from functionals to operators. The presentation below is rather transparent as we use the well-developed technique of “nonstandard scalarization.” This technique implements the Kantorovich heuristic principle and reduces operator problems to the case of functionals. The principal scheme works as follows: First, we establish that some class of operators T admits a Boolean valued representation T which turns out to be a Boolean valued class of functionals. More precisely, we prove that each operator T ∈ T embeds into an appropriate Boolean valued model V(B) , becoming a functional τ ∈ T within V(B) . Then the Boolean valued transfer principle tells us that each theorem about τ within Zermelo–Fraenkel set theory has its counterpart for the original operator T interpreted as the Boolean valued functional τ . Translation of theorems from τ ∈ V(B) to T ∈ V is carried out by the Boolean valued ascending–descending machinery together with principles of Boolean valued analysis. This chapter focuses on the structure of disjointness preserving operators and some related concepts. To save room, using the facts of vector lattice theory we will accept the terminology and notation of Aliprantis and Burkinshow [28] and Meyer-Nieberg [311]. 3.1. Positive Operators This section collects some basic facts on positive operators that we need in what follows. 3.1.1. Let X and Y be vector lattices. A linear mapping T from X to Y is called a positive operator if T carries positive vectors to positive vectors; in symbols, 0 x ∈ X =⇒ 0 T x ∈ Y or T (X+ ) ⊂ Y+ . An 120 Chapter 3. Order Bounded Operators operator T is said to be regular if T can be written as a diﬀerence of two positive operators and order bounded or shortly o-bounded provided that T sends each order bounded subset of X to an order bounded subset of Y . We will often omit the indication to linearity if this is implied by context. Let L(X, Y ) stand for the space of all linear operators from X to Y . The sets of all regular, order bounded, and positive operators from X to Y are denoted by Lr (X, Y ), L∼ (X, Y ), and L+ (X, Y ) := L∼ (X, Y )+ , respectively. Clearly, Lr (X, Y ) and L∼ (X, Y ) are vector subspaces of L(X, Y ) and L+ (X, Y ) is a convex cone in L(X, Y ). The order on the spaces of regular and order bounded operators is induced from the cone of positive operators L+ (X, Y ); i.e., T 0 ⇐⇒ T ∈ L+ (X, Y ), S T ⇐⇒ S − T 0. 3.1.2. A linear operator T ∈ L(X, Y ) is said to be dominated by a positive operator S ∈ L(X, Y ) provided that |T x| S(|x|) for all x ∈ X. In this event S is called a dominant or majorant of T . A positive operator T is dominated by itself; i.e., |T x| T (|x|) for all x ∈ X. (1) A linear operator T is dominated if and only if T is regular. ⊳ Indeed, if S is a dominant of T then T = S − (S − T ), while (S − T ) and S are positive. If T = S − R for some positive S, R ∈ L(X, Y ) then T x |Sx| + |Rx| (S + R)(|x|); i.e., S + R is a dominant of T . ⊲ (2) Let T : X → Y be a regular operator and let S be a dominant of T . If a net (xα ) converges to x in X with regulator e ∈ X+ then (T xα ) converges to T x with regulator Se. In particular, every regular operator is r-continuous. ⊳ Assume that |xα − x| λn e for α α(n), where e ∈ X+ and limn→∞ λn = 0. Then for each dominant S of T we have |T xα − T x| S(|xα − x|) λn Se α α(n) , which implies the convergence of (T xα ) to T x with regulator Se. ⊲ (3) Kantorovich Lemma. Let X be a vector lattice, and let Y be an arbitrary real vector space. Assume that U is an additive and positive homogeneous mapping from X+ to Y ; i.e., U : X+ → Y satisfies the conditions: U (x + y) = U x + U y, U (λx) = λU x (0 λ ∈ R; x, y ∈ X+ ). 3.1. Positive Operators 121 Then U has the unique linear extension T on the whole lattice X. Moreover, if Y is a vector lattice and U (X+ ) ⊂ Y+ then T is positive. ⊳ Deﬁne T by diﬀerences: T x := U x+ − U x− (x ∈ X). Then T is a sought extension whose uniqueness is obvious from the representation x = x+ − x− (cp. 3.1.2 (1)). ⊲ We now formulate the celebrated Riesz–Kantorovich Theorem. 3.1.3. Riesz–Kantorovich Theorem. Let X and Y be vector lattices with Y Dedekind complete. The set L∼ (X, Y ) of all order bounded linear operators from X to Y , ordered by the cone of positive operators L∼ (X, Y )+ , is a Dedekind complete vector lattice. In particular, L∼ (X, Y ) = Lr (X, Y ). Deﬁnitions 3.1.1 make it clear that every positive operator is order bounded. Consequently, so is the diﬀerence of positive operators. In other words, every regular operator is order bounded. The converse fails in general, holding in case that Y is Dedekind complete, as follows from the Riesz–Kantorovich Theorem. 3.1.4. The proof of the Riesz–Kantorovich Theorem yields the formulas for presenting the lattice operations on L∼ (X, Y ) by pointwise calculations. The collection of these formulas is usually called the calculus of order bounded operators or shortly order calculus. We will exhibit the main formulas of order calculus below. Let X and Y be the same as above. For all S, T ∈ L∼ (X, Y ) and x ∈ X+ the following hold: (1) (S ∨ T )x = sup{Sx1 + T x2 : x1 , x2 0, x = x1 + x2 }. (2) (S ∧ T )x = inf{Sx1 + T x2 : x1 , x2 0, x = x1 + x2 }. (3) S + x = sup{Sy : 0 y x}. (4) S − x = sup Sy : −x y 0 = − inf{Sy : 0 y x}. (5) |S|x = sup{|Sy| : |y| x}. (6) |S|x = sup { nk=1 |Sxk | : x1 , . . . , xn 0, x = nk=1 xk , n ∈ N} . (7) |Sx| |S|(|x|) (x ∈ X). 3.1.5. An operator T : X → Y between vector lattices is said to be order continuous, provided that, for every net (xα ) in X order convergent to x ∈ X, the net (T xα ) order converges to T x in Y . Say that T is 122 Chapter 3. Order Bounded Operators sequentially order continuous, if for every sequence (xn ) in X with order limit x ∈ X, the sequence (T xn ) is order convergent to T x in Y . It is useful to note that a positive operator T is (sequential) order continuous if and only if inf(T (A)) = 0 in Y for an arbitrary (countable) downward directed set A ⊂ X with inf(A) = 0. The collections of all order bounded order continuous operators and order bounded sequentially order continuous operators from X to Y ∼ will be denoted by L∼ n (X, Y ) and Lc (X, Y ), respectively. (Note that every order continuous operator is order bounded; see Aliprantis and Burkinshaw [28, Lemma 1.54].) 3.1.6. Ogasawara Theorem. If X and Y are vector lattices with Y ∼ ∼ Dedekind complete then L∼ n (X, Y ) and Lc (X, Y ) are bands of L (X, Y ). ⊳ See Aliprantis and Burkinshaw [28, Theorem 1.57]. ⊲ The following results tells us that the classical Hahn–Banach Theorem remains valid for operators if we take Dedekind complete vector lattice as a range space. An operator p from a (real) vector space V to an ordered vector space Y is called sublinear whenever p(u+v) p(u)+p(v) and p(λv) = λp(v) for all u, v ∈ V and all 0 λ ∈ R. 3.1.7. Hahn–Banach–Kantorovich Theorem. Let V be a (real) vector space, let Y be a Dedekind complete vector lattice, and let p : V → Y be a sublinear operator. If U is a vector subspace of V and S : U → Y is a linear operator satisfying S(u) p(u) for all u ∈ U , then there exists some linear operator T : V → Y such that T u = Su (u ∈ U ) (i.e., T is a linear extension of S to all of V ) and T (v) p(v) for all v ∈ V . Now we present several extension results for positive operators which will be needed in what follows. 3.1.8. Theorem. Let X0 , X, and Y be vector lattices with Y Dedekind complete and X0 a vector sublattice in X. Assume that S0 : X0 → Y and T : X → Y are positive operators and S0 x T x for all 0 x ∈ X0 . Then there exists a positive operator S : X → Y extending S0 and satisfying S T . ⊳ If p(x) := T (x+ ) (x ∈ X) then p is a sublinear operator from X to Y and S0 x p(x) for all x ∈ X0 . By the Hahn–Banach–Kantorovich Theorem there exists a linear operator S : X → Y extending S0 and satisfying Sx p(x) for all x ∈ X. The latter implies that 0 S T . ⊲ 123 3.1. Positive Operators and Y be vector lattices with Y 3.1.9. Theorem. Let X0 , X, Assume that T0 : X0 → Dedekind complete and X0 an order ideal in X. for which Y is a positive operator and define X as the set of all x ∈ X T0 ([0, |x|] ∩ X0 ) is order bounded in Y . Then X is an order ideal in X including X0 and there exists a positive extension T of T0 to all of X such that T S for every positive extension S of T0 to X. Moreover, T is order continuous if so is T0 . ⊳ Deﬁne the operator T : X+ → Y as T x := sup T0 u : u ∈ X0 , 0 u x for all x ∈ X+ . Then T is additive and positive homogeneous, so it can be extended to X by diﬀerences (cp. 3.1.2 (3)). The resulting operator satisﬁes the desired conditions and is called the least extension of T0 (cp. Aliprantis and Burkinshaw [28, Theorem 1.30] and Kusraev [228, Propositions 3.1.3 (1, 2)]). ⊲ 3.1.10. Theorem. Let X0 be an order dense majorizing vector sublattice of a vector lattice X, and let Y be a Dedekind complete vector lattice. If T0 : X0 → Y is a positive order continuous operator then there exists a unique order continuous linear extension T : X → Y of T0 to all of X. ⊳ The required extension T : X → Y is deﬁned ﬁrst on X+ as T (x) := sup{T (x0 ) : x0 ∈ X0 and 0 x0 x} (x ∈ X+ ) and then T is extended to the whole of X by diﬀerences; see 3.1.2 (3). More details can be found in Aliprantis and Burkinshaw [28, Theorem 1.65]. ⊲ 3.1.11. Let X and Y be vector lattices. For a linear operator T from X to Y the following are equivalent: (1) T (x ∨ y) = T x ∨ T y (x, y ∈ X). (2) T (x ∧ y) = T x ∧ T y (x, y ∈ X). (3) x ∧ y = 0 =⇒ T x ∧ T y = 0 (x, y ∈ X). (4) T (x+ ) = (T x)+ (x ∈ X). (5) T (|x|) = |T x| (x ∈ X). (6) [0, T ] = [0, IY ] ◦ T . 3.1.12. Note that 3.1.11 (1) means by deﬁnition that T is a lattice homomorphism. So T is a lattice isomorphism whenever T enjoys one (and hence all) of the properties listed in 3.1.11. 124 Chapter 3. Order Bounded Operators An injective lattice homomorphism from X to Y is called a lattice (more exactly order) monomorphism, or an isomorphic embedding, or even a lattice isomorphism from X to Y . If a lattice homomorphism T : X → Y is one-to-one then X and Y are called lattice (or order) isomorphic. The same is worded also as follows: T is an order isomorphism between X and Y . The set of all lattice homomorphisms from X to Y is denoted by Hom(X, Y ). 3.1.13. Theorem. Let X and Y be vector lattices with Y Dedekind complete. If X0 is a majorizing vector sublattice of X and T0 : X0 → Y is a lattice homomorphism, then there exists a lattice homomorphism T : X → Y extending T0 . ⊳ See Aliprantis and Burkinshaw [28, Theorem 2.29] and Kusraev [228, Proposition 3.3.11 (2)]. ⊲ 3.1.14. Consider a vector lattice X. An order bounded linear operator π : X → X is an orthomorphism in X whenever for all x, y ∈ X from x ⊥ y it follows that T x ⊥ y. The set of all orthomorphisms in X is denoted by Orth(X). Clearly, Orth(X) is a vector subspace of L∼ (X) which we will consider with the order induced from L∼ (X). In case that X is a Dedekind complete vector lattice, Orth(X) coincides with the band of L∼ (X) which is generated by the identity operator IX . For more details on orthomorphisms see de Pagter [327] and Zaanen [427]. Some special properties of orthomorphisms will be addressed in Chapter 4. 3.2. Bilinear Operators In this section we introduce the classes of bilinear operators on products of vector lattices. The main purpose is to agree on notation and terminology and give a brief outline of some useful facts. For an extended discussion of this subject see the survey paper Bu, Buskes, and Kusraev [72]. 3.2.1. Let X, Y , and Z be vector lattices. A bilinear operator B from X × Y to Z is called positive if B(x, y) 0 for all 0 x ∈ X and 0 y ∈ Y . This amounts to saying that the linear operators B(u, ·) : y → B(u, y) (y ∈ Y ), B(·, v) : x → B(x, v) (x ∈ X) 125 3.2. Bilinear Operators are positive for all 0 u ∈ X and 0 v ∈ Y . Given a positive bilinear operator B, we have |B(x, y)| B(|x|, |y|) (x ∈ X, y ∈ Y ). A bilinear operator is called order bounded if it sends order bounded sets in X × Y to order bounded sets in Z, and regular if it can be represented as the diﬀerence of two positive bilinear operators. Denote by BLr (X, Y ; Z) and BL+ (X, Y ; Z) the sets of all regular and positive bilinear operators from X × Y to Z. 3.2.2. A bilinear operator B : X × Y → Z is said to be of order bounded variation if ΣB[x; y] := n m |B(xk , yl )| : 0 xk ∈ X (1 k n ∈ N), k=1 l=1 0 yl ∈ X (1 l m ∈ N), x = n xk , y = k=1 m yl l=1 is order bounded in Z for all 0 x ∈ X and 0 y ∈ Y . The set of all bilinear operators B from X × Y to Z of order bounded variation (order bounded) is denoted by BLbv (X, Y ; Z) BL∼ (X, Y ; Z) and forms an ordered vector space with the positive cone BL+ (X, Y ; Z). Obviously, BLr (X, Y ; Z) ⊂ BLbv (X, Y ; Z) ⊂ BL∼ (X, Y ; Z) and BLr (X, Y ; Z) has the induced order. The converse inclusion may be false. 3.2.3. If Z is Dedekind complete then BLr (X, Y ; Z) = BLbv (X, Y ; Z) and this space is a Dedekind complete vector lattice. In particular, every regular bilinear operator B ∈ BLr (X, Y ; Z) has the modulus |B| and |B|(x, y) = sup ΣB[x; y] (0 x ∈ X, 0 y ∈ Y ), |B(x, y)| |B|(|x|, |y|) (x ∈ X, y ∈ Y ). 3.2.4. For a bilinear operator B : X × Y → Z the following are equivalent: (1) B(u, ·) and B(·, v) are lattice homomorphisms for all u ∈ X+ and v ∈ Y+ . (2) |B(x, y)| = B(|x|, |y|) for all x ∈ X and y ∈ Y . (3) B(x, y)+ = B(x+ , y + ) + B(x− , y − ) for all x ∈ X and y ∈ Y . 126 Chapter 3. Order Bounded Operators (4) B(x, y) ∧ B(u, v) = 0, whenever x, u ∈ X+ and y, v ∈ Y+ are such that either x ∧ u = 0 or y ∧ v = 0. 3.2.5. A bilinear operator B : X × Y → Z is said to be a lattice bimorphism if B satisﬁes one of (and then all) the conditions of 3.2.4. The lattice bimorphisms are simple in structure modulo lattice homomorphisms, as will be shown below in 3.12.A.3: Each lattice bimorphism B : X × Y → Z admits the representation B(x, y) = S(x)T (y) (x ∈ X, y ∈ Y ), where S : X → Z u and T : Y → Z u are lattice homomorphisms with values in the universal completion Z u of Z and Z u is equipped with an f -algebra multiplication uniquely determined by a choice of an order unit in Z u . 3.2.6. Theorem. Let X and Y be vector lattices. Then there exist a unique up to isomorphism vector lattice X ⊗ Y and a bimorphism φ : X × Y → X ⊗ Y such that the following are satisfied: (1) Whenever Z is a vector lattice and ψ : X × Y → Z is a lattice bimorphism, there is a unique lattice homomorphism T : X ⊗ Y → Z with T ◦ φ = ψ. (2) The bimorphism φ induces an embedding of the algebraic tensor product X ⊗ Y into X ⊗ Y . (3) X ⊗ Y is dense in X ⊗ Y in the sense that for every v ∈ X ⊗ Y there exist x0 ∈ X and y0 ∈ Y such that for every ε > 0 there is an element u ∈ X ⊗ Y with |v − u| εx0 ⊗ y0 . (4) If 0 < v ∈ X ⊗ Y , then here exist x ∈ X+ and y ∈ Y+ with 0 < x ⊗ y v. ⊳ This fact was established in Fremlin [121, Theorem 4.2]. See another proof in Grobler and Labuschagne [152]. ⊲ 3.2.7. The vector lattice X ⊗ Y is called the Fremlin tensor product of vector lattices X and Y . The lattice bimorphism φ is conventionally denoted by ⊗ and the algebraic tensor product X ⊗ Y is regarded as already embedded into X ⊗ Y . Some additional remarks are in order here. (1) Let ψ and T be the same as in Theorem 3.2.6 (1). Suppose that for all x ∈ X+ and y ∈ X+ the equality ψ(x, y) = 0 implies x = 0 or 3.2. Bilinear Operators 127 y = 0. In this case T is injective and so sends X ⊗ Y onto a vector sublattice of X generated by im ψ := ψ(X × Y ). (2) In particular, if X0 and Y0 are vector sublattices in X and Y , respectively, then the tensor product X0 ⊗ Y0 is isomorphic to the vector sublattice in X ⊗Y generated by X0 ⊗Y0 . Therefore, X0 ⊗Y0 is regarded as a vector sublattice of X ⊗ Y ; see Fremlin [121, Corollaries 4.4 and 4.5]. 3.2.8. Theorem. Let X, Y , and Z be vector lattices with Z uniformly complete. Then for every positive bilinear operator B from X ×Y to Z there exists a unique positive linear operator T : X ⊗ Y → Z such that B = T ⊗. ⊳ See Fremlin [121, Theorem 4.2]. See another proof in Grobler and Labuschagne [152]. ⊲ 3.2.9. Thus, the Fremlin tensor product possesses the following universal property: the set of positive bilinear operators on the Cartesian product of two Archimedean vector lattices with values in a uniformly complete vector lattice is in a one-to-one correspondence with the set of positive linear operators on the Fremlin tensor product of given vector lattices. More precisely, if X, Y , and Z are vector lattices with Z relatively uniformly complete, then the mapping T → T ⊗ constitutes an isomorphism of the two pairs of ordered vector spaces: (1) Lr (X ⊗ Y, Z) and BLr (X, Y ; Z); (2) L∼ (X ⊗ Y, Z) and BLbv (X, Y ; Z). The ﬁrst assertion is immediate from Theorem 3.2.8 and the second was established in Buskes and van Rooij [83]. 3.2.10. A bilinear operator B : X × X → Z is called orthosymmetric if x ⊥ y implies B(x, y) = 0 for arbitrary x, y ∈ X. The diﬀerence of two positive orthosymmetric bilinear operators is called orthoregular. Denote by BLor (X, Z) the space of all orthoregular bilinear operators from X × X to Z and order BLor (X, Z) by the cone of positive orthosymmetric operators. Recall also that a bilinear operator B : X × X → G is said to be symmetric if B(x, y) = B(y, x) for all x, y ∈ X and positive semidefinite if B(x, x) 0 for every x ∈ X. It is not diﬃcult to see that a lattice bimorphism B : X × X → Y is orthosymmetric, symmetric, and positive semideﬁnite simultaneously (cp. Buskes and Kusraev [78, Proposition 1.7]). 128 Chapter 3. Order Bounded Operators 3.2.11. Let X be a vector lattice. A pair (X ⊙ , ⊙) is called a square of X provided that the following hold: (1) X ⊙ is a vector lattice; (2) ⊙ : X × X → X ⊙ is an orthosymmetric bimorphism; (3) for every vector lattice Y , whenever B is an orthosymmetric bimorphism from X ×X to Y , there exists a unique lattice homomorphism B̂ : X ⊙ → Y such that B = B̂⊙. 3.2.12. Theorem. The square of an Archimedean vector lattice exists and is essentially unique; i.e., if for some vector lattice X ⊚ and symmetric lattice bimorphism ⊚ : X × X → X ⊚ the pair (X ⊚ , ⊚) obeys the universal property 3.2.11 (3), then there exists a lattice isomorphism i from X ⊙ onto E ⊚ such that i⊙ = ⊚ (and, of course, i−1 ⊚ = ⊙). ⊳ Denote by J the least relatively uniformly closed order ideal in the Fremlin tensor product X ⊗ X which includes {x ⊗ y : x, y ∈ X, x ⊥ y}. Put X ⊙ := X ⊗ X/J. Let φ : X ⊗ X → E ⊙ be the quotient homomorphism and put ⊙ := φ ⊗. Then X ⊙ is an Archimedean vector lattice and ⊙ is a lattice bimorphism. Observe that ⊙ is orthosymmetric. Indeed, if x ⊥ y then x ⊗ y ∈ J = ker(φ), whence x ⊙ y = φ(x ⊗ y) = 0. Moreover, the pair (X ⊙ , ⊙) meets the universal property 3.2.11 (3). ⊲ 3.2.13. Theorem. Let X and Y be vector lattices with Y uniformly complete. Then for every bilinear orthoregular operator B : X × X → Y there exists a unique linear regular operator B̂ : X ⊙ → Y such that B(x, y) = B̂(x ⊙ y) (x, y ∈ X). The correspondence B → B̂ is an isomorphism of the ordered vector spaces BLor (X, Y ) and Lr (X ⊙ , Y ). Thus, the role of the square of vector lattices in the theory of positive orthosymmetric bilinear operators is similar to that of the Fremlin tensor product of vector lattices in the general theory of positive bilinear operators. 3.3. Boolean Valued Positive Functionals We will demonstrate in this section how Boolean valued analysis translates some results from order bounded functionals to operators. 3.3. Boolean Valued Positive Functionals 129 Below X and Y stand for vector lattices with Y an order dense sublattice of R↓, while R is the reals within VB and B = P(Y ). 3.3.1. The fact that X is a vector lattice over the ordered ﬁeld R may be rewritten as a restricted formula, say, ϕ(X, R). Hence, recalling the restricted transfer principle, we come to the identity [[ ϕ(X ∧ , R∧ ) ]] = 1 which amounts to saying that X ∧ is a vector lattice over the ordered ﬁeld R∧ within V(B) . The positive cone X+ is deﬁned by the restricted formula ϕ(X, X+ ) ≡ (∀ x ∈ X+ )(x ∈ X) ∧ (∀ x ∈ X)(x ∈ X+ ↔ x 0). Hence (X ∧ )+ = (X+ )∧ by restricted transfer. By the same reason |x∧ | = |x|∧ , (x ∨ y)∧ = x∧ ∨ y ∧ , (x ∧ y)∧ = x∧ ∧ y ∧ for all x, y ∈ X, since the lattice operations ∨, ∧, and |·| in X are deﬁned by restricted formulas. In particular, x ⊥ y ⇐⇒ [[x∧ ⊥ y ∧ ]] = 1 (x, y ∈ X). ∧ ∧ 3.3.2. Let X ∧∼ := L∼ R∧ (X , R) be the space of regular R -linear func∧ tionals from X to R. More precisely, R is considered as a vector space over the ﬁeld R∧ and by the maximum principle there exists X ∧∼ ∈ V(B) such that [[X ∧∼ is a vector space over R of order bounded R∧ -linear functionals from X ∧ to R which is ordered by the cone of positive functionals ]] = 1. A functional τ ∈ X ∧∼ is positive if [[(∀ x ∈ X ∧ )τ (x) 0]] = 1. It can easily be seen that the Riesz–Kantorovich Theorem remains true if X is a vector lattice over a dense subﬁeld P ⊂ R, while Y is a Dedekind complete vector lattice (over R), and L∼ (X, Y ) is replaced by L∼ P (X, Y ), the real vector space of all order bounded P-linear operators from X to Y which is ordered by the cone of positive operators; i.e., L∼ P (X, Y ) is a Dedekind complete vector lattice. Moreover, the formulas of order calculus of 3.1.4 are preserved. According to this observation X ∧∼ is a Dedekind complete vector lattice within V(B) and for all σ, τ ∈ X ∧∼ and x ∈ (X ∧ )+ we have ∧ , x = x1 + x2 }, (σ ∨ τ )x = sup{σx1 + τ x2 : x1 , x2 ∈ X+ ∧ , x = x1 + x2 }. (σ ∧ τ )x = inf{σx1 + τ x2 : x1 , x2 ∈ X+ Thus, the descent X ∧∼ ↓ of X ∧∼ is a Dedekind complete vector lattice. 3.3.3. Theorem. Let X and Y be vector lattices with Y universally complete and represented as Y = R↓. Given T ∈ L∼ (X, Y ), the modified ascent T ↑ is an order bounded R∧ -linear functional on X ∧ within 130 Chapter 3. Order Bounded Operators V(B) ; i.e., [[ T ↑ ∈ X ∧∼ ]] = 1. The mapping T → T ↑ is a lattice isomorphism between the Dedekind complete vector lattices L∼ (X, Y ) and X ∧∼ ↓. ∧ ⊳ Observe ﬁrst that T → T ↑ is a bijection from Y X to R X ↓. To this end, recall that for every T ∈ L∼ (X, Y ) the modiﬁed ascent T ↑ is deﬁned by the relation [[ T x = T ↑(x∧ ) ]] = 1 (x ∈ X), while for every τ ∈ X ∧∼ ↓ we have [[ τ : X ∧ → R ]] = 1 and so the modiﬁed descent τ ↓ : X → Y is available (cp. 1.6.8 and 1.5.8). Moreover, by the Escher rules 1.6.6 we have τ ↓↑ = τ and T ↑↓ = T . Assuming that T is linear and putting τ = T ↑, for all x, y ∈ X and λ ∈ R we deduce within V(B) : τ (x∧ + y ∧ ) = τ ((x + y)∧ ) = T (x + y) = T x + T y = τ (x∧ ) + τ (y ∧ ), τ (λ∧ x∧ ) = τ ((λx)∧ ) = T (λx) = λT x = λ∧ τ (x∧ ). The R∧ -linearity of τ within V(B) follows from the calculations: [[(∀ x ∈ X ∧ )(∀ y ∈ X ∧ )(τ (x + y) = τ (x) + τ (y))]] [[τ (x∧ + y ∧ ) = τ (x∧ ) + τ (y ∧ )]] = 1; = x,y∈X [[(∀ λ ∈ R∧ )(∀ x ∈ X ∧ )(τ (λx) = λτ (x))]] [[τ (λ∧ x∧ ) = λ∧ τ (x∧ )]] = 1. = λ∈R x∈X Suppose that T is order bounded and put ū = sup{|T x| : |x| u} for u ∈ X. Denote by ϕ(u, v) the formula (∀ x ∈ X ∧ )(|x| u → |τ (x)| v) and observe that [[ϕ(u∧ , ū)]] = [[|x∧ | u∧ ]] ⇒ [[|τ (x∧ )| ū]] x∈X = (1 ⇒ [[|T x| ū]]) = 1. x∈X It follows that τ is order bounded within V(B) : [[τ is order bounded]] = [[(∀ u ∈ X ∧ )(∃ v ∈ R)ϕ(u, v)]] [[(∃ v ∈ R)ϕ(u∧ , v)]] [[(∀ u ∈ X ∧ )ϕ(u∧ , ū)]] = 1. = u∈X u∈X 3.3. Boolean Valued Positive Functionals 131 Thus, T ∈ L∼ (X, Y ) implies [[τ ∈ X ∧∼ ]]. The converse can be handled in a similar way. Consequently, T → T ↑ is a linear bijection from L∼ (X, Y ) into [[X ∧∼ ]]. It follows from [[τ (x∧ ) 0]] = [[T x 0]] [[τ 0]] = x∈X+ x∈X+ that T is positive if and only if [[τ is positive]] = 1, so that T → T ↑ is the desired lattice isomorphism. ⊲ 3.3.4. We now formulate a few corollaries to 3.3.3. First, we introduce necessary deﬁnitions. An operator T ∈ L∼ (X, Y ) is said to be disjointness preserving if x ⊥ y implies T x ⊥ T y for all x, y ∈ X. Let L∼ dp (X, Y ) stand for the set of all order bounded disjointness preserving operators from X to Y . ∼ Let L∼ a (X, Y ) be the band generated by Hom(X, Y ) in L (X, Y ) and ∼ Ld (X, Y ) be the disjoint complement of Hom(X, Y ): ⊥⊥ , L∼ a (X, Y ) := Hom(X, Y ) ⊥ L∼ d (X, Y ) := Hom(X, Y ) . ∼ If Y is Dedekind complete then L∼ (X, Y ) = L∼ a (X, Y )⊕Ld (X, Y ) holds, ∼ and so every T ∈ L (X, Y ) has the unique decomposition T = Ta + Td , ∼ ∼ where Ta ∈ L∼ a (X, Y ) and Td ∈ Ld (X, Y ). The elements of Ld (X, Y ) are usually referred to as diﬀuse operators, while the elements of L∼ a (X, Y ) are called pseudoembedding operators or pseudoembeddings. Also, deﬁne within V(B) the band of order bounded R∧ -linear atomic, disjointness ∧ ⊥⊥ preserving, and diﬀuse functionals: namely (X ∧ )∼ , a := HomR∧ (X , R) ∧ ⊥ ∼ ∧ ∧ ∼ ∧ (X ∧ )∼ (X , R) . := L (X , R), and (X ) := Hom R dp dp d Recall that S ∈ L∼ (X, Y ) is a component or a fragment of 0 T ∈ L∼ (X, Y ) if S ∧ (T − S) = 0. Boolean valued representation of a band preserving operators obtained in Theorem 4.3.4 reduces some properties of band preserving operators to Boolean valued interpretations of the properties of solutions to Cauchy functional equation. 3.3.5. Corollary. Consider R, S, T ∈ L∼ (X, R↓) and b ∈ B. Put ρ := R↑, σ := S↑, τ := T ↑, and π := χ(b). The following are true: (1) b [[ σ τ ]] ⇐⇒ πS πT. (2) b [[ σ = |τ | ]] ⇐⇒ πS = π|T |. (3) b [[ ρ = σ ∨ τ ]] ⇐⇒ πR = πS ∨ πT. (4) b [[ ρ = σ ∧ τ ]] ⇐⇒ πR = πS ∧ πT. 132 Chapter 3. Order Bounded Operators (5) b [[ σ ⊥ τ ]] ⇐⇒ πS ⊥ πT. (6) b [[ σ ∈ C(τ ) ]] ⇐⇒ πS ∈ C(πT ). ⊳ According to 2.3.6 for each b ∈ B we have V([0,b]) |= b ∧ τ ∈ (X ∧ )∼ and (b ∧ τ )↓ = χ(b) ◦ T . Thus, it suﬃces to observe that the vector lattices L∼ (X, (b ∧ R)↓) and b ∧ (X ∧ )∼ + are lattice isomorphic in view of 1.3.7 and Theorem 3.3.3. ⊲ 3.3.6. Corollary. Consider S, T ∈ L∼ (X, Y ) and put τ := T ↑, σ := S↑. The following are true: (1) T ∈ Hom(X, Y ) ⇐⇒ [[ τ ∈ Hom(X ∧ , R) ]] = 1. ∧ ∼ (2) T ∈ L∼ dp (X, Y ) ⇐⇒ [[ τ ∈ (X )dp ]] = 1. ∧ ∼ (3) T ∈ L∼ a (X, Y ) ⇐⇒ [[ τ ∈ (X )a ]] = 1. ∧ ∼ (4) T ∈ L∼ d (X, Y ) ⇐⇒ [[ τ ∈ (X )d ]] = 1. ⊳ This is immediate from 3.3.5. ⊲ 3.3.7. Let X be a vector lattice and Y := R↓. Given T ∈ L∼ (X, Y ) and a family (Tξ )ξ∈Ξ in L∼ (X, Y ), put τ := T ↑ and τξ := Tξ ↑. Then for each partition of unity (bξ )ξ∈Ξ in B we have τ = mixξ∈Ξ bξ τξ if and only if the representation holds T x = oχ(bξ )Tξ x (x ∈ X). ξ∈Ξ ⊳ It follows from 3.3.5 (1) that bξ [[τ = τξ ]] if and only if χ(bξ )T = χ(bξ )Tξ . Summing the last identity over all ξ ∈ Ξ we see that the desired representation of T is equivalent to the relation τ = mixξ∈Ξ bξ τξ . ⊲ 3.4. Disjointness Preserving Operators We intend here to demonstrate that some properties of disjointness preserving operators are just Boolean valued interpretations of elementary properties of disjointness preserving functionals. 3.4.1. Theorem. For an order bounded linear functional f : X → R the following are equivalent: (1) f preserves disjointness. (2) |f | is a lattice homomorphism. 3.4. Disjointness Preserving Operators 133 (3) If g ∈ X ∼ and 0 g |f | then g = λ|f | for some λ ∈ [0, 1]. (4) If g is a component of |f | then either g = 0 or g = |f |. (5) Either f or −f is a lattice homomorphism. (6) |f |(|x|) = |f (|x|)| = |f (x)| for all x ∈ X. (7) ker(f ) := f −1 (0) is an order ideal in X. ⊳ (1) ⇐⇒ (2): Assume that f preserves disjointness and x, y ∈ X+ are disjoint, while |f |(x) ∧ |f |(y) > 0. Then by formula 3.1.4 (5) there exist x′ , y ′ ∈ X with |x′ | x, |y ′ | y, |f (x′ )| > 0, and |f (y ′ )| > 0. At the same time x′ ⊥ y ′ and we should have |f (x′ )| ∧ |f (y ′ )| = 0 by hypothesis; a contradiction. Thus, (1) =⇒ (2) and the converse follows from 3.1.4 (7). (2) =⇒ (3): Put h := |f | and observe that ker(h) ⊂ ker(g), since |g(x)| g(|x|) h(|x|) = |h(x)|. Thus g = λh h for some 0 λ 1. (3) =⇒ (4): According to (3), each component g of |f | is of the form g = λ|f |. It follows that 0 = g ∧ (|f | − g) = min{λ, (1 − λ)}|f |, so that either λ = 0 or λ = 1. (4) =⇒ (5): Since |f | is the sum of disjoint components f + and f − , either f − = 0, in which case |f | = f , or f + = 0, in which case |f | = −f . Moreover, |f | is a lattice homomorphism. Otherwise there is a pair of disjoint elements x, y ∈ X with |f |(x) > 0 and |f |(y) > 0. So, there exists a component g of f such that g(x) = f (x) and g(y) = 0. Thus neither g = 0 nor g = f , which is a contradiction. (5) =⇒ (6): In both cases of (5) the needed relation is trivial. (6) =⇒ (7): If |y| |x| and x ∈ ker(f ) then from (6) we have |f (y)| |f |(|y|) |f |(|x|) = |f (x)| = 0, and so y ∈ ker(f ). (7) =⇒ (1): We have only to note that for every pair of disjoint elements x, y ∈ X either x ∈ ker(f ) or y ∈ ker(f ). Assuming the contrary, we can choose nonzero s, t ∈ R with sx + ty ∈ ker(f ), since nonzero disjoint elements are linearly independent and ker(f ) is a hyperplane. It follows that |x| (|x| + (|t|/|s|)|y|) = |x + (t/s)y| ∈ ker(f ) and x ∈ ker(f ); a contradiction. ⊲ 3.4.2. Theorem. Assume that Y has the projection property. An order bounded linear operator T : X → Y is disjointness preserving if and only if ker(bT ) is an order ideal in X for every b ∈ P(Y ). ⊳ The necessity is obvious, so only the suﬃciency will be proved. Suppose that ker(bT ) is an order ideal in X for every b ∈ P(Y ). Let |y| |x| and b := [[T x = 0]]. Then bT x = 0 by 2.2.4 (G) and bT y = 0 by 134 Chapter 3. Order Bounded Operators the hypothesis. Recalling 2.2.4 (G) once again, we see that b [[T y = 0]]. Thus [[T x = 0]] [[T y = 0]] or, which is the same, [[T x = 0]] ⇒ [[T y = 0]] = 1. Put τ := T ↑ and ensure that ker(τ ) is an order ideal in X ∧ within V(B) . Since |x| |y| if and only if [[|x∧ | |y ∧ |]] = 1, we deduce: [[ker(τ ) is an order ideal in X ∧ ]] = [[(∀ x, y ∈ X ∧ ) (τ (x) = 0 ∧ |y| |x| → τ (y) = 0)]] [[(τ (x∧ ) = 0]] ∧ [[|y ∧ | |x∧ |]] ⇒ [[τ (y ∧ ) = 0]] = = x,y∈X [[T (x) = 0]] ⇒ [[T (y) = 0]] : x, y ∈ X, |y| |x| = 1. According to 3.4.1 (7) τ is a disjointness preserving functional within V(B) and so T is also disjointness preserving by 3.3.5 (1). ⊲ As is well known each order bounded disjointness preserving operator between vector lattices has a modulus. This is obvious in the special situation of functionals to which the general case is reduced by means of Boolean valued interpretation. 3.4.3. Meyer Theorem. For every order bounded disjointness preserving linear operator T : X → Y between vector lattices the modulus |T |, positive part T + , and negative part T − exist and are lattice homomorphisms. Moreover, |T |x = |T x|, T + x = (T x)+ , T − x = (T x)− (x ∈ X+ ). In particular, an order bounded disjointness preserving operator is regular. ⊳ By the Gordon Theorem we can assume that Y is an order dense sublattice in R↓. Again, put τ := T ↑ and note that τ ∈ (X ∧∼ )dp by 3.3.6 (2) and |τ | exists within V(B) . By 3.3.5 (2) |τ |↓ is the modulus of T in L∼ (X, R↓). Moreover, |τ | and |T | both are lattice homomorphisms in view of 3.4.1 (2) and 3.3.6 (1). But [[|τ |(x) = |τ x| for all x ∈ (X ∧ )+ ]] = 1 according to 3.4.1 (6). Putting this fact into V(B) and recalling 3.3.5 (2), we obtain |T |(x) = |T x| for all x ∈ X+ . It follows that |T |(u) = |T |(u+ ) − |T |(u− ) = |T (u+ )| − |T (u− )| ∈ Y for all u ∈ X, so that |T | exists in L∼ (X, Y ). Other properties of T + and T − can easily be deduced from above by using the formulas T + = (|T | + T )/2 and T − = (|T | − T )/2. ⊲ 3.4. Disjointness Preserving Operators 135 3.4.4. Theorem. Let Y have the projection property. For an order bounded disjointness preserving linear operator T ∈ L∼ (X, Y ) there exists a band projection π ∈ P(Y ) such that T + = π|T | and T − = π ⊥ |T |. In particular, T = (π − π ⊥ )|T | and |T | = (π − π ⊥ )T . ⊳ Once again we reduce the problem to the case of functionals by putting τ := T ↑. As before, τ ∈ (X ∧∼ )dp and by, 3.4.1 (5), either τ − = 0 or τ + = 0 within V(B) . Put π := [[τ − = 0]] and observe that π = [[|τ | = τ + ]] and π ⊥ = [[τ − = 0]] [[τ + = 0]], since [[τ − = 0 → τ + = 0]] = 1. By Corollary 3.3.5 (2, 3) we obtain π|T | = πT + and π ⊥ T + = 0. Putting together the last two relations we arrive at the ﬁrst of the desired identities π|T | = πT + + π ⊥ T + = T + . The second is immediate from the ﬁrst: π ⊥ |T | = |T | − π|T | = |T | − T + = T − . ⊲ 3.4.5. Corollary. Let X and Y be vector lattices and let T ∈ L∼ (X, Y ) be disjointness preserving. Then (T x)+ ⊥ (T y)− for all x, y ∈ X+ . ⊳ Given x, y ∈ X+ we can write (T x)+ = (T x) ∨ 0 T + x = π|T |x. Similarly, (T y)− π ⊥ |T |y, and so (T x)+ ∧ (T y)− = 0. ⊲ 3.4.6. Theorem. Let X and Y be vector lattices with Y Dedekind complete, and let S, T : X → Y be order bounded disjointness preserving operators. The following are equivalent: (1) T ∈ {S}⊥⊥. (2) T x ∈ {Sx}⊥⊥ for all x ∈ X. (3) πSx = 0 =⇒ πT x = 0 for all x ∈ X and π ∈ P(X). (4) There exists π ∈ Orth(J(T (X)), Y ) such that T = π ◦ S, where J(Y0 ) is an order ideal generated by Y0 in Y . ⊳ Assume that S and T are positive, since otherwise we can replace them by their modules. Again, put τ := T ↑ and σ := S↑ and observe that for k = 1, 2, 3, 4 we have (k) ⇐⇒ [[(k ◦ )]] = 1, where (1◦ ) := τ ∈ {σ}⊥⊥ , (2◦ ) := τ (x) ∈ {σ(x)}⊥⊥ for all x ∈ X ∧ , (3◦ ) := σ(x) = 0 → τ (x) = 0 for all x ∈ X ∧ , (4◦ ) := τ = ασ for some α ∈ R. Now working within V(B) we see that (2◦ ) → (3◦ ) and (4◦ ) → (1◦ ) are trivial, (3◦ ) implies that ker(σ) ⊂ ker(τ ) and so τ = ασ for some α ∈ R, 136 Chapter 3. Order Bounded Operators whence (4◦ ). Finally, if (1◦ ) holds then τ = supn∈N (nσ) ∧ τ and |τ (x)| = τ (|x|) ∈ {σ(|x|)}⊥⊥ , because ((nσ) ∧ τ )(|x|) nσ(|x|) ∈ {σ(|x|)}⊥⊥ . Thus, (1◦ ) → (2◦ ) and the proof is complete. ⊲ 3.4.7. Corollary. For a positive linear operator T : X → Y the following are equivalent: (1) T is a lattice homomorphism. (2) If S ∈ L∼ (X, Y ) and 0 S T then there exists an orthomorphism π ∈ Orth(Y ) such that 0 π IY and S = π ◦ T . (3) If S ∈ C(T ) then there exists π ∈ P(Y ) such that S = π ◦ T . ⊳ The proof goes along similar lines using 3.4.1 (3, 4) and 3.3.5 (1, 2, 6). ⊲ 3.4.8. Theorem. Let X and Y be vector lattices with Y Dedekind complete. For a pair of disjointness preserving operators T1 and T2 from X to Y there exist a band projection π ∈ P(Y ), a lattice homomorphism T ∈ Hom(X, Y ), and orthomorphisms S1 , S2 ∈ Orth(Y ) such that |S1 | + |S2 | = π, πT1 = S1 T, πT2 = S2 T, ⊥ im(π T1 )⊥⊥ = im(π ⊥ T2 )⊥⊥ = π ⊥ (Y ), π ⊥ T1 ⊥ π ⊥ T2 . ⊳ As usual, there is no loss of generality in assuming that Y = R↓. Put τ1 := T1 ↑ and τ2 := T2 ↑. The desired result is a Boolean valued interpretation of the following fact: If the disjointness preserving functionals τ1 and τ2 are not proportional then they are both nonzero and disjoint. Indeed, if τ := |τ1 | ∧ |τ2 | = 0 then both |τ1 | and |τ2 | are positive multiples of τ by 3.4.1 (3); therefore, τ1 and τ2 are proportional. Put b := [[τ1 and τ2 are proportional]] and π := χ(b). Then within V([0,b]) there exist a lattice homomorphism τ : X ∧ → R and reals σ1 , σ2 ∈ R such that τi = σi τ . If the function σ̄i is deﬁned as σ̄i : λ → σi λ (λ ∈ R), then the operators S1 := σ̄1 ↓, S2 := σ̄2 ↓, and T := τ ↓ (with the modiﬁed descents taken from V([0,b]) ; see 1.3.7) satisfy the ﬁrst line of required conditions. Moreover, π ⊥ = χ(b∗ ) and by transfer we have b∗ = [[τ1 = 0]] ∧ [[τ2 = 0]] ∧ [[|τ1 | ∧ |τ2 | = 0]], so that the second line of required conditions is also satisﬁed by 3.3.5 (5) and 3.8.4. ⊲ 3.4.9. Theorem. Let X and Y be vector lattices with Y Dedekind complete. The sum T1 + T2 of two disjointness preserving operators T1 , T2 : X → Y is disjointness preserving if and only if there exist pairwise disjoint band projections π, π1 , π2 ∈ P(Y ), orthomorphisms 3.5. Diﬀerences of Lattice Homomorphisms 137 S1 , S2 ∈ Orth(Y ) and a lattice homomorphism T ∈ Hom(X, Y ) such that the following system of relations is consistent π + π1 + π2 = IY , ⊥⊥ T (X) = π(Y ), πT1 = S1 T, |S1 | + |S2 | = π, π1 T2 = π2 T1 = 0, πT2 = S2 T. Consequently, in this case T1 + T2 = π1 T1 + π2 T2 + (S1 + S2 )T . ⊳ The suﬃciency is obvious. To prove the necessity we apply Theorem 3.4.8 and note that only the claim concerning π1 and π2 is needed to check. Using the same notation put b1 := [[τ2 = 0]], b2 := [[τ1 = 0]] and b := [[ both τ1 and τ2 are nonzero]]. Observe that the sum of two disjoint functionals that preserve disjointness is disjointness preserving if and only if at least one of them is zero, since otherwise each of them is proportional to their sum; a contradiction. Thus, in view of the transfer principle b∗ = b1 ∨ b2 or b0 ∨ b1 ∨ b2 = 1. Moreover, we can assume by replacing b1 with b1 ∧b∗2 , if necessary, that b0 , b1 , b2 are pairwise disjoint. Putting πi := χ(bi ) (i = 1, 2), we see that π1 π2 = π1 π = π2 π = 0 and π1 + π2 + π = IY . Using 2.2.4 (G), we conclude that b1 [[τ2 = 0]] implies π1 T2 = 0 and b2 = [[τ1 = 0]] implies π2 T1 = 0. ⊲ 3.4.10. Corollary. The sum T1 + T2 of two disjointness preserving operators T1 , T2 : X → Y is disjointness preserving if and only if T1 (x1 ) ⊥ T2 (x2 ) for all x1 , x2 ∈ X with x1 ⊥ x2 . ⊳ The necessity is immediate from Theorem 3.4.9, since T1 = π1 T1 + S1 T and T2 = π2 T2 + S2 T . To see the suﬃciency, observe that if T1 and T2 meet the above condition then Tk x1 ⊥ Tl x2 (k, l := 1, 2) and so (T1 + T2 )(x1 ) ⊥ (T1 + T2 )(x2 ) for every pair of disjoint elements x1 , x2 ∈ X. ⊲ 3.5. Differences of Lattice Homomorphisms This section answers the following question: Which closed hyperplane in a Banach lattice is a vector sublattice? It turns out that each hyperspace with this property is exactly the kernel of the diﬀerence of some lattice homomorphisms on the initial vector lattice. The starting point of this question is the celebrated Stone Theorem about the structure of vector sublattices in the Banach lattice C(Q, R) of continuous 138 Chapter 3. Order Bounded Operators real functions on a compact space Q. This theorem may be rephrased in the above terms as follows: 3.5.1. Stone Theorem. Each closed vector sublattice of C(Q, R) is the intersection of the kernels of some diﬀerences of lattice homomorphisms on C(Q, R). 3.5.2. In view of this theorem it is reasonable to refer to a diﬀerence of lattice homomorphisms on a vector lattice X as a two-point relation on X. We are not obliged to assume here that the lattice homomorphisms under study act into the reals R. Thus a linear operator T : X → Y between vector lattices is said to be a two-point relation on X whenever it is written as a diﬀerence of two lattice homomorphisms. An operator bT := b ◦ T with b ∈ B := P(Y ) is called a stratum of T . 3.5.3. The kernel ker(T ) of every two-point relation T := T1 −T2 with T1 , T2 ∈ Hom(X, Y ) is evidently a sublattice of X, since it is determined by an equation ker(T ) = {x ∈ X : T1 x = T2 x}. On using of the above terminology, the Meyer Theorem 3.4.3 reads as follows: Each order bounded disjointness preserving operator between vector lattices is a two-point relation. Thus, each stratum bT of an order bounded disjointness preserving operator T : X → Y is a two-point relation on X and so its kernel is a vector sublattice of X. In fact, the converse is valid too. 3.5.4. Theorem. An order bounded linear operator from a vector lattice to a Dedekind complete vector lattice is a two-point relation if and only if the kernel of its every stratum is a vector sublattice of the ambient vector lattice. ⊳ The proof presented below in 3.5.9 and 3.5.10 follows along the general lines: Using the canonical embedding and ascending to the Boolean valued universe V(B) , we reduce the matter to characterizing scalar twopoint relations on vector lattices over dense subﬁelds of the reals R. To solve the resulting scalar problem, we use one of the formulas of subdifferential calculus, namely the Moreau–Rockafellar Formula. ⊲ 3.5.5. We need some additional concepts. Recall that p : X → R ∪ {+∞} is called a sublinear functional if p(0) = 0, p(x + y) p(x) + p(y), and p(λx) = λp(x) for all x, y ∈ X and 0 < λ ∈ R. The subdiﬀerential (at zero) ∂p of a sublinear functional p is deﬁned as ∂p := {l : X → R : l is linear and lx p(x) for all x ∈ X}. 3.5. Diﬀerences of Lattice Homomorphisms 139 The eﬀective domain dom(p) := {x ∈ X : p(x) < +∞} of a sublinear functional p is a cone. By a cone K we always mean a convex cone which is a subset of X with the properties K + K = K and λK ⊂ K (λ ∈ R+ ). Evidently, p q implies ∂p ⊂ ∂q; the converse is also true whenever dom(p) = dom(q) = X. 3.5.6. Assume now that X is a vector lattice. If p is increasing (i.e., x1 x2 =⇒ p(x1 ) p(x2 )) then ∂p consists of positive functionals. It can easily be seen from the Hahn–Banach Theorem that the converse is also true whenever dom(p) = X. Indeed, we can pick a linear functional ∼ f ∈ ∂p with f (x1 ) = p(x1 ) and, assuming ∂p ⊂ X+ and x1 x2 , we get p(x1 ) = f (x1 ) f (x2 ) p(x2 ). Take a positive functional f on X. The representation f = f1 + · · · + fN will be called a positive (N -)decomposition of f whenever f1 , . . . , fN are positive functionals on X. In this event we say also that (f1 , . . . , fN ) is a positive decomposition of f . Given a positive functional f ∈ X ∼ deﬁne the function pf : X N → R as pf (x1 , . . . , xN ) = f (x1 ∨ · · · ∨ xN ). Then p is sublinear and increasing and ∂pf consists of all positive decompositions of f ; i.e., the representation N ∂pf = (f1 , . . . , fN ) : 0 fk ∈ X ∼ , f = fk k=1 holds. Indeed, (f1 , . . . , fN ) ∈ ∂pf means that for all x1 , . . . , xN we have f1 (x1 ) + · · · + fN (xN ) f (x1 ∨ · · · ∨ xN ). Taking xj to be zero for all j = i and xi 0 yields fi 0 and putting x1 = · · · = xN gives f = f1 + · · · + fN . This proves the inclusion ⊂, while the converse inclusion is trivial. 3.5.7. Moreau–Rockafellar Formula. Assume that X is a real vector space and p, q : X → R ∪ {+∞} are sublinear functionals. If dom(p) − dom(q) = dom(q) − dom(p) then ∂(p + q) = ∂p + ∂q. ⊳ The inclusion ∂p + ∂q ⊂ ∂(p + q) is trivial. To see the converse inclusion take l ∈ ∂r with r := p + q and construct a sublinear functional P : X × X → R such that (f, g) ∈ ∂P implies f ∈ ∂p, g ∈ ∂q, and l = f + g. By hypothesis X0 := dom(p) − dom(q) is a subspace of X and 140 Chapter 3. Order Bounded Operators therefore the set H(x, y) := {h ∈ X : x + h ∈ dom(p), y + h ∈ dom(q)} is nonempty for all x, y ∈ X0 . Deﬁne P0 : X0 × X0 → R as P0 (x, y) := inf{p(x + h) + q(y + h) − l(h) : h ∈ H(x, y)}. If π is a linear projection of X onto X0 then P = P0 ◦ π is the desired sublinear functional. It remains to observe that ∂P = ∅ by the Hahn– Banach Theorem. ⊲ 3.5.8. Decomposition Theorem. Assume that H1 , . . . , HN are cones in a vector lattice X. Assume further that f and g are positive functionals on X. The inequality f (h1 ∨ · · · ∨ hN ) g(h1 ∨ · · · ∨ hN ) holds for all hk ∈ Hk (k := 1, . . . , N ) if and only if to each positive decomposition (g1 , . . . , gN ) of g there is a positive decomposition (f1 , . . . , fN ) of f such that fk (hk ) gk (hk ) (hk ∈ Hk ; k := 1, . . . , N ). ⊳ Note that the left-hand side of the ﬁrst inequality of the claim is the sublinear functional pf of N variables. The right-hand side is the sublinear functional pg of the same variables. Put H := H1 × · · · × HN and deﬁne qH : X N → R ∪ {+∞} by letting qH (u) = 0 if u ∈ H and qH (u) = +∞ if u ∈ / H. Evidently, qH is sublinear and ∂qH consists of all N -tuples (g1 , . . . , gN ) such that gk : X → R is linear and gk |Hk 0 for all k := 1, . . . , N . Note that pg pf + qH . Using 3.5.6 and the Moreau–Rockafellar formula 3.5.7 we obtain ∂pg ⊂ ∂(pf + pH ) = ∂pf + ∂qH . Consequently for every positive decomposition g = g1 + · · · + gN we have (g1 , . . . , gN ) ∈ ∂pg and so (g1 , . . . , gN ) = (f1 , . . . , fN ) + (f¯1 , . . . , f¯N ) with (f1 , . . . , fN ) ∈ ∂pf and (f¯1 , . . . , f¯N ) ∈ ∂qH . It follows from the above remarks that (f1 , . . . , fN ) is a positive decomposition of f and gk |Hk = fk |Hk + f¯k |Hk fk |Hk . ⊲ 3.5.9. An order bounded functional on a vector lattice is a two-point relation if and only if its kernel is a vector sublattice of the ambient vector lattice. 3.5. Diﬀerences of Lattice Homomorphisms 141 ⊳ Let l be an order bounded functional on a vector lattice X. We may present l as the diﬀerence of the two disjoint functionals l = l+ − l− , where l+ and l− are the positive and negative parts of l. For convenience, we put f := l+ , g := l− , and H := ker(l). It suﬃces to demonstrate only that g is a lattice homomorphism; i.e., [0, g] = [0, 1]g (cp. 3.4.1 (3)). So, we take 0 g1 g and put g2 := g − g1 . We may assume that g1 = 0 and g1 = g, since otherwise there is nothing left to prove. By hypothesis, for all h1 , h2 ∈ ker(l) we have the inequality f (h1 ∨ h2 ) g(h1 ∨ h2 ). By the Decomposition Theorem there is a decomposition of f into the sum of some positive terms f = f1 + f2 such that f1 (h) − g1 (h) = 0 and f2 (h) − g2 (h) = 0 for all h ∈ H. Since H is the zero hyperplane of l = f −g, we see that there are reals α and β satisfying f1 −g1 = α(f −g) and f2 − g2 = β(f − g). Clearly, α + β = 1 (for otherwise f = g and l = 0). Therefore, one of the reals α and β is strictly positive. If α > 0 then we have g1 = αg for f and g are disjoint. Since g1 is not equal to zero, it follows that 0 α 1 and g1 ∈ [0, 1]g. If β > 0 then, arguing similarly, we see that g2 = βg. Since g1 = g; therefore, g2 = 0. Hence, 0 β 1 and we again see that g1 ∈ [0, 1]g. ⊲ 3.5.10. Proof of Theorem 3.5.4. We ought to demonstrate only the suﬃciency part of the claim. So, let T be an order bounded operator from X to Y and the kernel ker(bT ) := (bT )−1 (0) of each stratum of T is a vector sublattice of X. We reduce the problem to 3.5.9 by means of Boolean valued “scalarization.” Without loss of generality, we can assume that Y is a nonzero space embedded as an order dense ideal in the universally complete vector lattice R↓. Denote by τ := T ↑ the modiﬁed ascent of T to V(B) . Then by 1.6.8 [[τ : X ∧ → R]] = 1, (∀ x ∈ X) [[τ (x∧ ) = T x]] = 1. Straightforward calculations of truth values show that T + ↑ = τ + and T − ↑ = τ − within V(B) ; see 3.3.5. Moreover, [[ker(τ ) is a vector sublattice of X ∧ ]] = 1. Indeed, given x, y ∈ X, put b := [[T x = 0∧ ]] ∧ [[T y = 0∧ ]]. This means that x, y ∈ ker(bT ) by 2.2.4 (G). Hence, we see by hypothesis that bT (x ∨ y) = 0, whence b [[T (x ∨ y) = 0∧ ]] again by 2.2.4 (G). 142 Chapter 3. Order Bounded Operators Replacing T by τ yields [[τ (x∧ ) = 0∧ ∧ τ (y ∧ ) = 0∧ ]] [[τ (x ∨ y)∧ = 0∧ ]]. A straightforward calculation of Boolean truth values with use of the last estimate [[ker(τ ) is a vector sublattice of X ∧ ]] 3.6. Sums of Lattice Homomorphisms 143 = [[(∀ x, y ∈ X ∧ )(τ (x) = 0∧ ∧ τ (y) = 0∧ → τ (x ∨ y) = 0∧ )]] [[τ (x∧ ) = 0∧ ∧ τ (y ∧ ) = 0∧ ]] ⇒ [[τ ((x ∨ y)∧ ) = 0∧ ]] = 1. = x,y∈X completes the proof. ⊲ 3.6. Sums of Lattice Homomorphisms In this section we will give a description for an order bounded operator T whose modulus may be presented as the sum of two lattice homomorphisms in terms of the properties of the kernels of the strata of T . 3.6.1. Recall that a subspace H of a vector lattice is a G-space or Grothendieck subspace provided that H enjoys the property: (∀ x, y ∈ H) (x ∨ y ∨ 0 + x ∧ y ∧ 0 ∈ H). 3.6.2. Theorem. Let X and Y be vector lattices with Y Dedekind complete. The modulus of an order bounded operator T : X → Y is the sum of some pair of lattice homomorphisms if and only if the kernel of each stratum bT of T with b ∈ B := P(Y ) is a Grothendieck subspace of the ambient vector lattice X. We argue further as follows: Using the functors of canonical embedding and ascent to the Boolean valued universe V(B) , we reduce the matter to characterizing a Grothendieck hyperspace that serves as the kernel of an order bounded functional over a dense subring of the reals R. The scalar case is settled by the following four auxiliary propositions. 3.6.3. A functional l is the sum of some pair of lattice homomorphisms if and only if l is a positive functional with kernel a Grothendieck subspace. ⊳ Necessity is almost evident. Indeed, assume that l = f + g with f and g lattice homomorphisms. Take hk such that f (hk ) + g(hk ) = 0 for k := 1, 2. Then f (h1 ∨ h2 ∨ 0) = f (h1 ) ∨ f (h2 ) ∨ 0 = (−g(h1 )) ∨ (−g(h2 )) ∨ 0 = −g(h1 ) ∧ g(h2 ) ∧ 0. Similarly, g(h1 ∨ h2 ∨ 0) = −f (h1 ∧ h2 ∧ 0). Finally, these yield l(h1 ∨ h2 ∨ 0 + h1 ∧ h2 ∧ 0) = (f + g)(h1 ∨ h2 ∨ 0 + h1 ∧ h2 ∧ 0) = 0. Hence, ker(l) is a Grothendieck subspace. 144 Chapter 3. Order Bounded Operators Suﬃciency: Take l 0 and assume that ker(l) is a Grothendieck subspace. If l has no components other than 0 and l then l is a lattice homomorphism and we are done. Recall that component of f is an extreme point of the order interval [0, f ]. Let f be a component of l such that 0 = f and f = l. Denote by g the component of l disjoint from f ; i.e., g := l − f . Clearly, g = 0 and g = f . Check that [0, f ] = [0, 1]f . To this end, take a functional f1 such that 0 f1 f , f1 = 0, and f1 = f . Put f2 := f − f1 . Since H is a Grothendieck subspace; therefore, h1 ∨ h 2 ∨ h 3 + h 1 ∧ h 2 ∧ h 3 = (h1 − h3 ) ∨ (h2 − h3 ) ∨ 0 + (h1 − h3 ) ∧ (h2 − h3 ) ∧ 0 + 2h3 ∈ H for all h1 , h2 , h3 ∈ H. Consequently, (∀ h1 , h2 , h3 ∈ H) l(h1 ∨ h2 ∨ h3 ) l((−h1 ) ∨ (−h2 ) ∨ (−h3 )). The decomposition of f into the sum f = f1 + f2 yields the decomposition l = f1 + f2 + g of l into a sum of positive terms. By the Decomposition Theorem 3.5.8, there is a decomposition of l into a sum of positive terms l = l1 + l2 + l3 such that for all h ∈ H we have l1 (h) f1 (−h), l2 (h) f2 (−h), l3 (h) g(−h). Since H is the hyperplane of l, there are reals α1 , α2 , α3 ∈ R such that f1 + l1 = α1 (f + g), f2 + l2 = α2 (f + g), g + l3 = α3 (f + g). Summing up all these equalities and recalling that l = 0, we see that α1 + α2 + α3 = 2. Summing up the ﬁrst two equalities, we arrive at the following: f + l1 + l2 = (α1 + α2 )(f + g). We thus obtain (α1 + α2 − 1)f + (α1 + α2 )g 0 and α1 + α2 1 since f and g are disjoint. Similarly, (α3 − 1)g + α3 f 0 and (α3 − 1)g 0. Since g = 0; therefore, α3 1. Finally, α1 + α2 = 1 and α3 = 1. We thus conclude that l3 = f and l1 + l2 = g. Moreover, f1 − α1 f = α1 g − l1 . Since 0 f1 f and 0 l1 g; therefore, |f1 − α1 f | (1 + |α1 |)f and |α1 g − l1 | (1 + |α1 |)g. Since f and g are disjoint, we have f1 = α1 f . Since f f1 = 0, we see that 1 > α1 > 0 and the proof is complete. ⊲ 145 3.6. Sums of Lattice Homomorphisms 3.6.4. Given an order bounded functional l on a vector lattice X, assume that l+ = 0 and l− = 0. The kernel ker(l) is a Grothendieck subspace of X if and only if l+ and l− are lattice homomorphisms on X (or, which is the same, ker(l) is a vector sublattice). ⊳ Suﬃciency is obvious. Necessity: Put f := l+ = 0, g := l− = 0, and H := ker(f − g). Assume further that 0 f f , 0 g g, and h1 , h2 , h3 ∈ H. Clearly, f (h1 ∨ h2 ∨ h3 ) + g((−h1 ) ∨ (−h2 ) ∨ (−h3 )) f ((−h1 ) ∨ (−h2 ∨ (−h3 )) + g(h1 ∨ h2 ∨ h3 ) f (−h1 ) + g(h2 ). (1) Given x1 , x2 , x3 ∈ X, put ε3 (x, y, z) =: x1 ∨ x2 ∨ x3 ; σ(x) := −x; m(x1 , x2 , x3 ) := f(−x1 ) + g(x2 ); p := f ◦ ε3 ; q := g ◦ ε3 ◦ σ. Using subdiﬀerential calculus and the new notation, we can rephrase (1) as follows: m ∈ ∂(p + q + δ(H 3 )) = ∂(p) + ∂(q) + ∂(δ((H 3 )), where δ(U ) is the indicator of a set U (i.e., δ(U )x = 0 for x ∈ U and δ(U )x = +∞ for x ∈ U ) and ∂(s) is the subdiﬀerential of a sublinear functional s. The Decomposition Theorem yields some decompositions of f and g in the sums of positive terms f = f1 + f2 + f3 and g = g1 + g2 + g3 such that f1 + g1 ◦ σ − f ◦ σ ∈ ∂(δ(H)); f2 + g2 ◦ σ − g ∈ ∂(δ(H)); f3 + g3 ◦ σ ∈ ∂(δ(H)). Since H is the hyperplane of l; therefore, ∂(δ(H)) = {tl : t ∈ R}. Hence, there are reals α, β, γ ∈ R satisfying f1 − g1 + f = α(f − g); f2 − g2 − g = β(f − g); f3 − g3 = γ(f − g). Put t := α + β + γ − 1. Addition yields tf − f = tg − g. Hence, 0 |tf − f | = |tg − g| = |tf − f| ∧ |tg − g| (1 + |t|)(f ∧ g) = 0. Consequently, f = tf and g = tg. By hypotheses, 0 t 1. Therefore, [0, f ] = [0, 1]f and [0, g] = [0, 1]g, and so l is a diﬀerence of lattice 146 Chapter 3. Order Bounded Operators homomorphisms. We conclude that ker(l) is a vector sublattice and the proof is complete. ⊲ 3.6.5. The kernel of an order bounded functional is a Grothendieck subspace if and only if so is the kernel of the modulus of this functional. ⊳ Suﬃciency: Let l : X → R be an order bounded functional. If ker(|l|) is a Grothendieck subspace then |l| = l1 + l2 by 3.6.4, where l1 and l2 are lattice homomorphisms. Two lattice homomorphisms are either disjoint or proportional. Without loss of generality, we can assume that l1 and l2 are disjoint components of |l| each distinct from 0 and |l| (otherwise, l would be a lattice homomorphism). The order interval [0, |l|] lies in some plane since [0, 1]l1 + [0, 1]l2 = [0, |l|]. Consequently, every extreme point of [0, |l|] belongs to the set {0, l1 , l2 , |l|}. Since l+ and l− are also disjoint components of |l|, we see that l = l1 + l2 or l = l1 − l2 or l = l2 − l1 . In the ﬁrst case ker(l) is a Grothendieck subspace by 3.6.4; and in the remaining two cases it is a vector sublattice (thus, a Grothendieck subspace) of X. Necessity: Assume that ker(l) is a Grothendieck subspace. If either of the functionals l+ and l− equals zero then ker(|l|) = ker(l) and we are done. If l+ = 0 and l− = 0 then l+ and l− are lattice homomorphisms by 3.6.4. Thus, |l| is the sum of a pair of lattice homomorphisms. By 3.6.3, ker(|l|) is a Grothendieck subspace. ⊲ 3.6.6. The kernel of an order bounded functional is a Grothendieck subspace if and only if the modulus of this functional is the sum of a pair of lattice homomorphisms. ⊳ Let l be an order bounded functional. By 3.6.5 the kernel ker(l) of l is a Grothendieck subspace if and only if so is the subspace ker(|l|). Since |l| is a positive functional, we are done by 3.6.3. ⊲ 3.6.7. Proof of Theorem 3.6.2. We start with “scalarizing” the problem. Without loss of generality, we can assume that Y is a nonzero space embedded as an order dense ideal in the universally complete vector lattice R↓ which is the descent of the reals R within V (B) . We further let X ∧ stand for the standard name of X in V(B) . Clearly, ∧ X is a vector lattice over R∧ within the Boolean valued universe V(B) . Denote by l := T ↑ the modiﬁed ascent of T to V(B) . By Theorem 3.3.3 [[l ∈ (X ∧ )∼ ]] = 1 and [[l(x∧ ) = T x]] = 1 for all x ∈ X. Working 3.6. Sums of Lattice Homomorphisms 147 within V(B) , we see [[ker(l) is a Grothendieck subspace of X ∧ ]] = [[(∀ x, y ∈ X ∧ )(l(x) = 0∧ ∧l(y) = 0∧ → l(x∨y∨0+x∧y∧0) = 0∧ )]] (2) = [[l(x∧ ) = 0∧ ∧ l(y ∧ ) = 0∧ → l((x ∨ y ∨ 0 + x ∧ y ∧ 0)∧ ) = 0∧ ]]. x,y∈X Suﬃciency: Take x, y ∈ X and put b := [[T x = 0∧ ]] ∧ [[T y = 0∧ ]]. In view of 2.2.4 (G) this means that x, y ∈ ker(bT ). By hypothesis the kernel of each stratum bT is a Grothendieck subspace. Hence, bT (x ∨ y ∨ 0 + x ∧ y ∧ 0) = 0. In other words, [[T x = 0∧ ]] ∧ [[T y = 0∧ ]] [[T (x ∨ y ∨ 0 + x ∧ y ∧ 0) = 0∧ ]]. By (2) in follows that [[ker(l) is a Grothendieck subspace of X ∧ ]] = 1. Applying 3.6.6 to the order bounded functional l within V(B) and using the maximum principle, we see that l is the sum of two lattice homomorphisms l1 and l2 within V(B) . Deﬁne the operators T1 , T2 : X → R↓ as T1 := l1 ↓ and T2 := l2 ↓. According to 3.3.6 (1) T1 and T2 are lattice homomorphisms satisfying T1 + T2 = T . Since Y is an ideal of R↓, the ranges of T1 and T2 lie in Y . Necessity: Assume that |T | is the sum of a pair of lattice homomorphisms. By Theorem 3.3.3, the ascent of the sum of some summands is obviously the sum of the ascents of the summands, and so l is an order bounded functional within V(B) whose modulus is the sum of some pair of lattice homomorphisms. From 3.6.6 it follows that [[ker(l) is a Grothendieck subspace of X ∧ ]] = 1. Considering (2), we infer that the kernel of each stratum bT of T is a Grothendieck subspace of X. Indeed, for x, y ∈ X it follows by (2) that [[l(x∧ ) = 0∧ ∧ l(y ∧ ) = 0∧ → l((x ∨ y ∨ 0 + x ∧ y ∧ 0)∧ ) = 0∧ ]] = 1. Therefore, [[l(x∧ ) = 0∧ ]] ∧ [[l(y ∧ ) = 0∧ ]] [[l((x ∨ y ∨ 0 + x ∧ y ∧ 0)∧ ) = 0∧ ]]. 148 Chapter 3. Order Bounded Operators Consequently, if b ∈ B and bT x = bT y = 0 then b [[T x = 0]] ∧ [[T y = 0]] by 2.2.4 (G) or, taking into account the deﬁnition of l = T ↑, b [[lx∧ = 0∧ ]] ∧ [[ly ∧ = 0∧ ]] and we obtain [[l((x ∨ y ∨ 0 + x ∧ y ∧ 0)∧ ) = 0∧ ]] b. Finally, one more use of 2.2.4 (G) gives bT (x ∨ y ∨ 0 + x ∧ y ∧ 0) = 0. The proof of the theorem is complete. ⊲ 3.7. Polydisjoint Operators The aim of the present section is to describe the order ideal in the space of order bounded operators which is generated by the order bounded disjointness preserving operators in terms of n-disjoint operators. 3.7.1. Let X and Y be vector lattices and let n be a positive integer. A linear operator T : X → Y is n-disjoint if, for every disjoint collection of n + 1 elements x0 , . . . , xn ∈ X, the meet of |T xk | : k := 0, 1, . . . , n equals zero; symbolically: (∀ x0 , x1 . . . , xn ∈ X) xk ⊥ xl (k = l) =⇒ |T x0 | ∧ · · · ∧ |T xn | = 0. An operator is called polydisjoint if it is n-disjoint for some n ∈ N. Evidently, if an operator is n-disjoint then it is m-disjoint for all m n. A 1-disjoint operator is just a disjointness preserving operator. Theorem 3.6.2 tells us that 2-disjoint operators are just those satisfying the condition: the kernel of every stratum is a Grothendieck subspace. Consider some simple properties of n-disjoint operators. 3.7.2. Let X and Y be vector lattices with Y Dedekind complete. An operator T ∈ L∼ (X, Y ) is n-disjoint if and only if its modulus |T | is n-disjoint. ⊳ Suﬃciency is obvious from the inequality |T (x)| |T |(|x|) (x ∈ X). Suppose that the operator T is n-disjoint. Take pairwise disjoint elements e0 , . . . , en ∈ X+ . Observe that if |xk | ek then xk ⊥ xl (k = l); therefore, |T x0 | ∧ · · · ∧ |T xn | = 0. Passing to the supremum over x0 , . . . , xn in the last equality, we obtain |T |e0 ∧ · · · ∧ |T |en = 0 by 3.1.4 (5). ⊲ 3.7. Polydisjoint Operators 149 3.7.3. Let T1 , . . . , Tn be order bounded disjointness preserving operators from X to Y . Then T := T1 + · · · + Tn is n-disjoint. ⊳ Take x0 , x1 , . . . , xn ∈ X with xk ⊥ xl for all k = l. Then, from 2.1.6 (3) we have n k=1 |T xk | |Tι(0) x0 | ∧ · · · ∧ |Tι(n) xn |, ι∈I where I is the set of all mappings from {0, 1, . . . , n} to {1, . . . , n}. Evidently, each summand on the right-hand side contains at least two identical indices. Assuming that m := ι(k) = ι(l) for some k = l we deduce |Tι(0) x0 | ∧ |Tι(1) x1 | ∧ · · · ∧ |Tι(n) xn | |Tm xk | ∧ |Tm xl | |Tm |(|xk |) ∧ |Tm |(|xl |) = |Tm |(|xk | ∧ |xl |) = 0. n It follows k=1 |T xk | = 0 and the proof is complete. ⊲ In the next two propositions we will characterize n-disjoint order bounded functionals; i.e., “scalarize” the problem. 3.7.4. Assume that f ∈ C(Q)′ is n-disjoint for some n ∈ N. Then there exist q1 , . . . , qn ∈ Q and a1 , . . . , an ∈ R such that f= n ak δqk k=1 where δq ∈ C(Q)′ is the Dirac delta measure x → x(q) (x ∈ C(Q)) at q ∈ Q. ⊳ Assume that f is an n-disjoint functional. According to 3.7.2 there is no loss of generality in assuming that f is positive. Prove that the corresponding Radon measure μ is a linear combination of n Dirac delta measures. This is equivalent to saying that the support of μ contains at most n points. If there are n + 1 points q0 , q1 , . . . , qn ∈ Q in the support of μ then we can choose pairwise disjoint compact neighborhoods U0 , U1 , . . . , Un of these points and next take pairwise disjoint open sets Vk ⊂ Q with μ(Uk ) > 0 and Uk ⊂ Vk (k = 0, 1, . . . , n). Using the Tietze– Urysohn Theorem, construct a continuous function xk on Q which vanishes on Q\Vk and is identically one on Uk . Then x0 ∧x1 ∧· · ·∧xn = 0 but none of f (x0 ), f (x1 ), . . . , f (xn ) is equal to zero, since f (xk ) μ(Uk ) > 0 for all k := 0, 1, . . . , n. This contradiction shows that the support of μ contains at most n points. ⊲ 150 Chapter 3. Order Bounded Operators 3.7.5. An order bounded functional on a vector lattice is n-disjoin if and only if it is representable as a disjoint sum of n order bounded disjointness preserving functionals. This representation is unique up to permutation. ⊳ Let f be an n-disjoint functional on a vector lattice X and denote by m the least natural for which f is m-disjoint. Then there exists a disjoint collection x1 , . . . , xm ∈ X such that none of the reals f (x1 ), . . . , f (xm ) is equal to zero. Let J(e) stand for the order ideal in X generated by e := |x1 | + · · · + |xm |. By the Kakutani–Kreı̆ns Representation Theorem we can consider J(e) as a norm dense vector sublattice of C(Q) for some Hausdorﬀ compact topological space Q. Clearly, f |J(e) admits the unique extension f e by continuity to the whole of C(Q); moreover, f e is m-disjoint. By 3.7.4 f |J(e) = f e |J(e) is representable as a sum of m nonzero order bounded disjointness preserving e functionals f1e , . . . , fm . Given x ∈ X, we can choose m nonzero order e(x) e(x) bounded disjointness preserving functionals f1 , . . . , fm on J(e(x)) e(x) e(x) with e(x) = |x| + e such that f |J(e(x)) = f1 + · · · + fm . Finally, the e(x) functional fk deﬁned on X by letting fk (x) := fk (x) is order bounded and disjointness preserving, while f = f1 + · · · + fm . The functionals fk and fl with k = l are disjoint. Indeed, for every x ∈ X+ the functionals e(x) e(x) fk are disjoint and so and fl (fk ∧ fl )(x) = inf fk (x1 ) + fl (x2 ) : x1 , x2 ∈ X+ , x1 + x2 = x e(x) = inf fk e(x) (x1 ) + fl (x2 ) : x1 , x2 ∈ J(e(x))+ , x1 + x2 = x = 0. If m < n then some zero terms should be added. ⊲ 3.7.6. Let T : X → R↓ be an order bounded linear operator and τ := T ↑. Then τ is n∧ -disjoint if and only if T is n-disjoint. ⊳ Put b := [[τ is n∧ -disjoint]] and ensure that if T is n-disjoint then b = 1. Identifying n∧ with {0, . . . , n − 1}∧ and using 1.5.2, we deduce // b = (∀ ν : n∧ → X ∧ ) (∀ k, l n∧ )(k = l → ν(k) ⊥ ν(l)) 00 → inf∧ |τ (ν(k))| = 0 k∈n = inf{|τ (ν(ı))| : ı ∈ n∧ } = 0 : ν ∈ [[n∧ → X ∧ ]], (∀ k, l ∈ n∧ )(k = l → ν(k) ⊥ ν(l)) = 1 151 3.7. Polydisjoint Operators = inf{|τ (im(ν))| = 0 : ν↓ ∈ [n → X ∧ ↓], (∀ k = l) ν↓(k) ⊥ ν↓(l) = 1 . Since X ∧ ↓ = mix{x∧ : x ∈ X}, we can choose a partition of unity (bξ )ξ∈Ξ and a ﬁnite collection of families (xξ,k )ξ∈Ξ (k := 0, 1, . . . , n) in X such that ν↓(k) = mixξ∈Ξ (bξ x∧ξ,k ). It follows from [[ν↓(k) ⊥ ν↓(l)]] = 1 that xξ,k ⊥ xξ,l whenever bξ = 0 and k = l. Putting Aξ := {xξ,0 , . . . , xξ,n }, we can easily check that bξ [[im(ν) = A∧ξ ]]. Working within the relative universe V(Bξ ) with Bξ := [0, bξ ] and using 1.6.8 and 2.2.4 (G), we deduce τ (A∧ξ ) = T (Aξ )↑ and inf |T (Aξ )↑| = inf |T (Aξ )|, so that V(Bξ ) |= inf |τ (im(ν))| = inf |τ (A∧ξ )| = inf |T (Aξ )↑| = inf |T (Aξ )|. Thus, inf |T (Aξ )| = |T xξ,0 | ∧ · · · ∧ |T xξ,n | = 0 or, equivalently, [[inf |T (Aξ )| = 0]] = 1 for all ξ, since T is n-disjoint. Using 1.2.5 (3) we deduce bξ [[inf |τ (im(ν))| = inf |T (Aξ )|]] ∧ [[inf |T (Aξ )| = 0]] [[inf |τ (im(ν))| = 0]], so that b = 1. The converse is demonstrated similarly. ⊲ 3.7.7. Theorem. An order bounded operator from a vector lattice to a Dedekind complete vector lattice is n-disjoint for some n ∈ N if and only if it is representable as a disjoint sum of n order bounded disjointness preserving operators. ⊳ Assume that X and Y are vector lattices with Y Dedekind complete. Assume further that T ∈ L(X, Y ) is order bounded and n-disjoint. Let τ ∈ V(B) be an internal linear functional on X ∧ with [[T x = τ (x∧ )]] = 1 for all x ∈ X. Then τ is order bounded n-disjoint functional by 3.3.3 and 3.7.6. According to the transfer principle, applying 3.7.5 to τ yields some pairwise disjoint order bounded disjointness preserving functionals τ1 , . . . , τn on X ∧ with τ = τ1 + · · · + τn . It remains to observe that by Theorem 3.3.3 Tk = τk ↓ is an order bounded disjointness preserving operator from X to Y and T = T1 + · · · + Tn . Moreover, if k = l then Tk and Tl are disjoint by Corollary 3.3.5 (5). ⊲ 152 Chapter 3. Order Bounded Operators 3.7.8. The representation of an order bounded n-disjoint operator in Theorem 3.7.7 is unique up to mixing: if T = T1 +· · ·+Tn = S1 +· · ·+ Sm for two disjoint collections {T1 , . . . , Tn } and {S1 , . . . , Sm } of order bounded disjointness preserving operators then for every j = 1, . . . , m there exists a disjoint collection of band projections π1j , . . . , πnj ∈ P(Y ) such that Sj = π1j T1 + · · · + πnj Tn for all j := 1, . . . , m. ⊳ Let Tk , τ , and τk be the same as in the proof of 3.7.7 and σk := Sk ↑. Then [[τ = τ1 + · · · + τn∧ = σ1 + · · · + σm∧ ]] = 1. It follows from the uniqueness of the representation in 3.7.5 that [[(∀ j m∧ ) (∃ ı n∧ ) (σı = τj )]] = 1. Evaluating the n Boolean truth values for quantiﬁers according to 1.2.3 yields 1 = ı=1 [[σj = τı ]] for every j m. For every j m we can take a partition of unity (bıj )nı=1 such that [[σj = τı ]] bıj or, equivalently, σj = mixın bıj τı . It follows that Sj = π1j T1 + · · · + πnj Tn for all j := 1, . . . , m, where πıj = χ(bıj ) by 3.3.7. ⊲ 3.7.9. Corollary. A positive linear operator from a vector lattice to a Dedekind complete vector lattice is n-disjoint if and only if it is the disjoint sum of n lattice homomorphisms. 3.7.10. Corollary. The set of polydisjoint operators from a vector lattice to a Dedekind complete vector lattice coincides with the order ideal in the vector lattice of order bounded linear operators generated by lattice homomorphisms or, equivalently, by disjointness preserving operators. 3.8. Sums of Disjointness Preserving Operators In this section we examine the problem of ﬁnding conditions for the sum of a ﬁnite collection of order bounded disjointness preserving operators to be n-disjoint. We will start with the case of functionals. 3.8.1. For a finite collection of order bounded disjointness preserving functionals f1 , . . . , fN on X the following are equivalent: (1) fı + fj is disjointness preserving for all 1 ı, j N . (2) |f1 | + · · · + |fN | is a lattice homomorphism. (3) There exists a lattice homomorphism h : X → R such that fı = 3.8. Sums of Disjointness Preserving Operators 153 λı h (ı := 1, . . . , N ) for some λ1 , . . . , λN ∈ [0, 1] ⊂ R. (4) If fı = 0 and fj = 0, then |fı | ∧ |fj | = 0 for all 1 ı, j N . (5) fı = 0 and fj = 0 imply ker(fı ) = ker(fj ) for all 1 ı, j N . ⊳ (1) =⇒ (2): Assume that (1) holds, while h := |f1 | + · · · + |fN | is not a lattice homomorphism. Find naturals 1 ı, j N with fı = 0, fj = 0, and fı ⊥ fj . It follows that |fı + fj | = |fı | + |fj | is not a lattice homomorphism, whereas fı + fj is disjointness preserving; a contradiction. (2) =⇒ (3): This is immediate from 3.4.1 (3) with h := |f1 |+· · ·+|fN |. (5) =⇒ (3): If f1 = · · · = fN = 0, there is nothing to prove. Otherwise, choose a natural j N with fj = 0. Then for each nonzero fı we have ker(fj ) = ker(fı ) and so fı = λi fj for some nonzero λı ∈ R. Put λı = 0 whenever fı = 0. It remains to put h := |f1 | ∨ · · · ∨ |fN |. The remaining implications (3) =⇒ (4) =⇒ (5) and (3) =⇒ (1) are obvious. ⊲ 3.8.2. Assume that n, N ∈ N and n < N . For a finite collection of order bounded disjointness preserving functionals f1 , . . . , fN on X the following are equivalent: (1) The sum g1 +· · ·+gn+1 is n-disjoint for an arbitrary permutation (g1 , . . . , gN ) of (f1 , . . . , fN ). (2) |f1 | + · · · + |fN | is n-disjoint. (3) There is a permutation (g1 , . . . , gN ) of (f1 , . . . , fN ) such that g1 , . . . , gn are pairwise disjoint and, for ı := n + 1, . . . , N , the representation gı = λı gκ(ı) holds with some κ(ı) ∈ {1, . . . , n} and λ ∈ R, |λ| 1. ⊳ Simple arguments similar to those in 3.8.1 will do the trick. ⊲ 3.8.3. Having settled the scalar case, let us discuss the conditions under which the sum of order bounded linear operators is disjointness preserving or n-disjoint with n 1. The following deﬁnition is motivated by 3.7.8. Given two collections T := (T1 , . . . , TN ) and S := (S1 , . . . , SN ) of linear operators from X to Y , say that S is a P(Y )-permutation of T whenever there exists an N × N matrix (πı,l ) with entries from P(Y ), whose rows and columns are partitions of unity in P(Y ) such that Sı = N N l=1 πı,l Tl for all ı := 1, . . . , N (and so Tl = ı=1 πı,l Sı for all l := 1, . . . , N ). The range projection RT of an operator T : X → Y is the least band projection in Y with T = RT ◦T or, equivalently, RT is a band projection 154 Chapter 3. Order Bounded Operators onto the band T (X)⊥⊥ in Y . 3.8.4. Let T be an order bounded linear operator from X to Y := R↓ and τ := T ↑. Then χ([[τ = 0]]) coincides with the range projection RT . ⊳ It follows from the Gordon Theorem that, given y ∈ Y , the band projection [y] onto {y}⊥⊥ coincides with χ([[y = 0]]). Therefore, we can calculate b := [[τ = 0]] = [[(∃ x ∈ X ∧ )τ (x) = 0]] [[τ (x∧ ) = 0]] = [[T (x) = 0]]. = x∈X x∈X It remains to observe that χ(b) = RT . ⊲ x∈X χ([[T (x) = 0]]) = x∈X [T (x)] = 3.8.5. Given τ, σ ∈ V(B) with [[τ, σ : {1, . . . , N }∧ → (X ∧ )∼ ]] = 1. For l ∈ {1, . . . , N } put τl := τ ↓(l), σl := σ↓(l), Tl := τl ↓, and Sl := σl ↓. Denote (τ1 , . . . , τN ∧ ) := im(τ ) and (σ1 , . . . , σN ∧ ) := im(σ). Then (σ1 , . . . , σN ∧ ) is a permutation of (τ1 , . . . , τN ∧ ) within V(B) if and only if (S1 , . . . , SN ) is a P(Y )-permutation of (T1 , . . . , TN ). ⊳ Assume that (σ1 , . . . , σN ∧ ) is a permutation of (τ1 , . . . , τN ∧ ). Take some permutation ν : {1, . . . , N }∧ → {1, . . . , N }∧ such that σı = τν(ı) (ı ∈ {1, . . . , N }∧ ). By 1.5.8 ν↓ is a function from {1, . . . , N } to ({1, . . . , N }∧ )↓ = mix({1∧ , . . . , N ∧ }). Thus, for each ı ∈ {1, . . . , N } ∧ there exists a partition of unity (bı,l )N l=1 such that ν↓(ı) = mixlN (bı,l l ). Since ν is injective, we have 1 = [[(∀ ı, j ∈ {1, . . . , N }∧ )(ν(ı) = ν(j) → ı = j)]] = N [[ν(ı∧ ) = ν(j∧ ) → ı∧ = j∧ ]] ı,j=1 = N [[ν↓(ı) = ν↓(j)]] ⇒ [[ı∧ = j∧ ]], ı,j=1 and so [[ν↓(ı) = ν↓(j)]] [[ı∧ = j∧ ]] for all ı, j N . Taking this inequality and the deﬁnition of ν↓ into account yields bı,l ∧ bj,l [[ν↓(ı) = l∧ ]] ∧ [[ν↓(j) = l∧ ]] [[ν↓(ı) = ν↓(j)]] [[ı∧ = j∧ ]], so that ı = j implies bı,l ∧ bj,l = 0 (because ı = j ⇐⇒ [[ı∧ = j∧ ]] = O 3.8. Sums of Disjointness Preserving Operators 155 by 1.4.5 (2)). At the same time, the surjectivity of ν implies 1 = [[(∀ l ∈ {1, . . . , N }∧ )(∃ ı ∈ {1, . . . , N }∧ )l = ν(ı)]] = N N [[l∧ = ν↓(ı)]] = l=1 ı=1 N N bı,l . l=1 ı=1 Hence, (bı,l )N ı=1 is a partition of unity in B for all l = 1, . . . , N . By the choice of ν it follows that bı,l [[σı = τl ]], because of the estimations bı,l [[σ(ı∧ ) = τ (ν(ı∧ ))]] ∧ [[ν(ı∧ ) = l∧ ]] [[σ(ı∧ ) = τ (l∧ )]] = [[σı = τl ]]. Put πı,l := χ(bı,l ) and observe that bı,l [[σı (x∧ ) = τl (x∧ )]] [[Si x = Tl x]] for all x ∈ X and 1 ı, j N . Using 2.2.4 (G), we obtain πı,l Sı = πı,l Tl and so Sı = N l=1 πı,l Tl for all 1 ı N . Clearly, (πı,l ) is the N × N matrix as required in Deﬁnition 3.8.3. The suﬃciency can be seen by the same reasoning in the reverse direction. ⊲ 3.8.6. Theorem. For a finite collection of order bounded disjointness preserving linear operators T1 , . . . , TN from X to Y the following are equivalent: (1) Tı + Tj is disjointness preserving for all 1 ı, j N . (2) |T1 | + · · · + |TN | is a lattice homomorphism. (3) There exist a lattice homomorphism T : X → Y and orthomorphisms ̺1 , . . . , ̺j ∈ Z (Y ) such that Tı = ̺ı T (ı := 1, . . . , N ). (4) If RTı ◦ RTj R|Tı |∧|Tj | for all 1 ı, j N . (5) For π ∈ P(Y ) and 1 ı, j N the inequality π RTı ◦ RTj implies ker(πTı ) = ker(πTj ). ⊳ We may assume without loss of generality that Y = R↓. Put fı := Tı ↑ (ı := 1, . . . , N ). Note that 3.8.1 is valid within V(B) in view of the transfer principle, so that it suﬃces to ensure that 3.8.6 (k) is equivalent to the interpretation of 3.8.1 (k) within V(B) for all k = 1, . . . , 5. For k = 1, 2 the equivalences are obvious. Putting h := T ↑ and using 3.3.6 (1) yields [[3.8.1 (3)]] = 1 ⇐⇒ 3.8.6 (3). Furthermore, 3.8.6 (4) may be symbolized as Φ ≡ (∀ ı, j ∈ {1, . . . , N }∧ )(fı = 0 ∧ fj = 0 → |fı | ∧ |fj | = 0), so that [[Φ]] = [[fı = 0]] ∧ [[fj = 0]] ⇒ [[|fı | ∧ |fj | = 0]]. ı,jN 156 Chapter 3. Order Bounded Operators Thus, [[3.8.1 (4)]] = 1 if and only if [[fı = 0]]∧[[fj = 0]] [[|fı |∧|fj | = 0]] for all ı, j N . The latter is equivalent to 3.8.6 (4) by 3.8.4. The remaining equivalence of [[3.8.1 (5)]] = 1 and 3.8.6 (5) is veriﬁed by combining the above arguments with the proof of 3.4.2. ⊲ 3.8.7. Theorem. Let n, N ∈ N and n < N . For a collection of order bounded disjointness preserving linear operators T1 , . . . , TN from X to Y the following are equivalent: (1) For an arbitrary P(Y )-permutation S1 , . . . , SN of T1 , . . . , TN the sum S1 + · · · + Sn+1 is n-disjoint. (2) |T1 | + · · · + |TN | is n-disjoint. (3) There exists a P(Y )-permutation S1 , . . . , SN of T1 , . . . , TN such that S1 , . . . , Sn arepairwise disjoint and each of Sn+1 , . . . , SN is repren sentable as Sj = ı=1 αı,j Sı for some pairwise disjoint α1,j , . . . , αn ∈ Z (Y ) (j := n + 1, . . . , N ). ⊳ We can assume that Y = R↓ and put τl := Tl ↑. The equivalence (1) ⇐⇒ (2) is immediate and we need only check (2) ⇐⇒ (3). Moreover, there is no loss of generality in assuming that T1 , . . . , TN are lattice homomorphisms so that τ1 , . . . , τN are also assumed to be lattice homomorphisms within V(B) . (2) =⇒ (3): Supposing (2) and working within V(B) , observe that τ1 + · · · + τN is n∧ -disjoint and so there exists a permutation ν : {1, . . . , N }∧ → {1, . . . , N }∧ such that τν(1) , . . . , τν(n) are pairwise disjoint lattice homomorphisms, while each of the homomorphisms τν(n+1) , . . . , τν(N ) is proportional to some of τν(1) , . . . , τν(n) with a constant of modulus 1. The latter is formalized as follows: Φ ≡ (∀ ı ∈ {n + 1, . . . , N }∧ )(∃ j ∈ {1, . . . , n}∧ ) (∃ β ∈ R)(|β| 1 ∧ τν(ı) = βτν(j) ). Put Sı := τν(ı∧ ) ↓ (ı := 1, . . . , N ). Then (S1 , . . . , SN ) is a P(Y )permutation of (T1 , . . . , TN ) and (S1 , . . . , Sn ) are pairwise disjoint by 3.3.5 (5). Moreover, [[Φ]] = 1 by transfer. Hence, 1= N n [[(∃ β)(β ∈ R)(|β| 1 ∧ τν(ı∧ ) = βτν(j∧ ) )]]. ı=n+1 j=1 It follows that for each n + 1 ı N there is a partition of unity {bı,1 , . . . , bı,n } in B such that bı,j [[(∃ β)(β ∈ R)(|β| 1 ∧ τν(ı∧ ) = 3.9. Representation of Disjointness Preserving Operators 157 βτν(j∧ ) )]]. According to the maximum principle there exists βı,j ∈ R↓ with bı,j [[|βı,j | 1]] ∧ [[τν(ı∧ ) = βı,j τν(j∧ ) ]]. Observe that for each x ∈ X we have bı,j [[τν(ı∧ ) = βı,j τν(j∧ ) )]] [[τν(ı∧ ) (x∧ ) = βı,j τν(j∧ ) (x∧ )]] ∧ [[τν(ı∧ ) (x∧ ) = Si x]] ∧ [[τν(j∧ ) (x∧ ) = Sj x]] [[Si x = βı,j Sj x]]. using the Gordon Theorem, Putting πı,j := χ(bı,j ) and αı,j := πı,j βı,j and we see that πı,j Si x = αı,j Sj x, whence Sı = nj=1 αı,j Sj as required. (3) =⇒ (2): This is demonstrated along the above lines making use of 3.7.6, 3.8.2, and 3.8.5. ⊲ 3.9. Representation of Disjointness Preserving Operators The main result of the present section is representation of an arbitrary order bounded disjointness preserving operator as a strongly disjoint sum of operators admitting some weight-shift-weight factorization. 3.9.1. Let B be a complete Boolean algebra and let ϕ be a 2-valued Boolean homomorphism on B with 2 := {0, 1} ⊂ R. Deﬁne D(ϕ) as the set of all spectral systems x ∈ S(B) satisfying ϕ(x(s)) = 0 and ϕ(x(t)) = 1 for some s, t ∈ R. For x ∈ D(ϕ) we can choose t = −s > 0, since the function t → ϕ(x(t)) is increasing, so that D(ϕ) := x ∈ S(B) : (∃ s ∈ R+ ) ϕ(x(s)) = 1, ϕ(x(−s)) = 0 . Moreover, for every ﬁnite collection x1 , . . . , xn ∈ S(B) there is 0 < s ∈ R such that ϕ(xı (s)) = 1 and ϕ(xı (−s)) = 0 for all ı = 1, . . . , n. Recall that S(B) is a universally complete vector lattice with zero element 0̄ and weak order unit 1̄ deﬁned as 0̄(t) := 1 if t > 0 and 0̄(t) := 0 if t 0, 1̄(t) := 1 if t > 1 and 1̄(t) := 0 if t 1 (cp. 2.8.2 and 2.8.3). Moreover, b → b̄ in 2.8.2 is a Boolean isomorphism of B onto C(1̄) and we will identify these two Boolean algebras. Denote by J(1) the order ideal in S(B) generated by 1̄. 3.9.2. D(ϕ) is simultaneously an order dense ideal in S(B) and an f -subalgebra with unit 1̄. ⊳ Take arbitrary x, y ∈ D(ϕ), 0 < α ∈ R, and z ∈ S(B) with 0 z |x|, and choose s ∈ R+ such that ϕ(x(s)) = ϕ(y(s)) = 1 and 158 Chapter 3. Order Bounded Operators ϕ(x(ε − s)) = ϕ(x(−s)) = ϕ(y(−s)) = 0 for some 0 < ε ∈ R. Then ϕ((x + y)(2s)) = 1 and ϕ((x + y)(−2s)) = 0 by 2.7.6 (5), ϕ((αx)(αs)) = 1 and ϕ((αx)(−αs)) = 0 by 2.7.5 (1), ϕ(|x|(s)) = 1 by 2.7.6 (3) and ϕ(|x|(−s)) = 0 by 2.7.4 (1), since |x| 0̄. Moreover, ϕ(z(s)) = 1 and ϕ(z(−s)) = 0 by 2.7.4 (1) and the above proved property of |x|. It follows that x + y, αx, |x|, and z lie in D(ϕ) and so D(ϕ) is an order ideal. It remains to observe that for the same x and y we have ϕ(|xy|(s2 )) = 1 and ϕ(|xy|(−s2 )) = 0 by 2.7.6 (6), so that xy ∈ D(ϕ) and hence D(ϕ) is an f -subalgebra of S(B) containing 1̄. ⊲ 3.9.3. A spectral system x ∈ S(B) is contained in D(ϕ) if and only if there exists a countable partition of unity (bn ) in B such that ϕ(bm ) = 1 for some m ∈ N and bn x ∈ J(1) for all n ∈ N. ⊳ By 3.9.2 we can assume that x is positive. Take a partition (tn )∞ n=0 of the real half-line R+ and put bn := x(tn ) − x(tn−1 ) for all n ∈ N. Clearly, (bn ) is a partition of unity in B. If x ∈ D(ϕ) then ϕ(x(tm )) = 1 for some m ∈ N and we can choose the ﬁrst natural m with this property. Then ϕ(bm ) = ϕ(x(tm )) − ϕ(x(tm−1 )) = 1. At the same time x(t) x(tn ) bn whenever t tn and so (bn x)(t) = bn ∧ x(t) + b∗n = 1 by 2.7.5 (4). It follows from 2.7.4 (1) that 0 bn x tn 1. Conversely, if a partition of unity (bn ) satisfy the above condition then ϕ(b∗m ) = 0 and ϕ((bm x)(t0 )) = 1 for some t0 tm . In view of 2.7.5 (4) we have ϕ(x(t0 )) = ϕ(bm ∧ x(t0 ) + b∗m ) = ϕ((bm x)(t0 )) = 1, so that x ∈ D(ϕ). ⊲ 3.9.4. The order ideal J(1) is uniformly dense in D(ϕ). ∞ ⊳ By 3.9.3 x ∈ D(ϕ) can be written as x = o- n=1 bn xn , where xn ∈ J(1) for all n ∈ N and (bn ) is a partition of unity in B with ϕ(bm ) = 1 for some m ∈ N. Put yn = nk=1 bk xk and e = o- ∞ n=1 nbn xn . Clearly, e exists in S(B) and e ∈ D(ϕ) by 3.9.3. Moreover, yn ∈ J(1), and |x − yn | (1/n)e (n ∈ N). ⊲ 3.9.5. Let B be a complete Boolean algebra and let ϕ be a 2-valued Boolean homomorphism on B. Then there exists a unique lattice homomorphism ϕ : D(ϕ) → R with ϕ| B = ϕ. Moreover, ϕ(x) = sup{t ∈ R : ϕ(x(t)) = 0} = inf{t ∈ R : ϕ(x(t)) = 1} (x ∈ D(ϕ)). ⊳ The above formula correctly deﬁnes some function ϕ : D(ϕ) → R, since for every x ∈ D(ϕ) the two sets A = (ϕ ◦ x)−1 (0) and B = (ϕ ◦ 3.9. Representation of Disjointness Preserving Operators 159 x)−1 (1) form a disjoint partition of the real line with s < t for all s ∈ A and t ∈ B. It is immediate from 2.7.5 (1) and the deﬁnition of ϕ that ϕ(αx) = αϕ(x) for all α ∈ R+ . From 2.7.6 (2) we see that ϕ(−x(t)) = 1 implies ϕ(x(ε − t)) = 0 (ε > 0) and ϕ(x(−t)) = 1 implies ϕ(−x(t)) = 0. Consequently, for every 0 < ε ∈ R, making use of 2.7.6 (2) we deduce ϕ(−x) = inf{t ∈ R : ϕ((−x)(t)) = 1} inf{t ∈ R : ϕ(x(ε − t)) = 0} = − sup{t − ε ∈ R : ϕ(x(t)) = 0} = −ϕ(x) + ε = − inf{t ∈ R : ϕ(x(t)) = 1} + ε = sup{t ∈ R : ϕ(x(−t)) = 1} + ε sup{t ∈ R : ϕ(−x(t)) = 0} + ε = ϕ(−x) + ε. It follows that ϕ(−x) = −ϕ(x). Observe now that if ϕ((x + y)(r)) = 0 and r = s + t, then either ϕ(x(s)) = 0 or ϕ(y(t)) = 0. Using this fact we deduce ϕ(x + y) = sup{r ∈ R : ϕ((x + y)(r)) = 0} inf{t ∈ R : ϕ(x(s)) = 0 or ϕ(y(t)) = 0} = sup{s ∈ R : ϕ(x(t)) = 0} + sup{t ∈ R : ϕ(y(t)) = 0} = ϕ(x) + ϕ(y). Replacing x by −x and y by −y and applying the identity ϕ(−x) = Thus, ϕ̂ is a linear −ϕ(x) just proved we obtain ϕ(x + y) = ϕ(x) + ϕ(y). functional. Moreover ϕ is a lattice homomorphism, since the identity ϕ(x ∨ y) = ϕ(x) ∨ ϕ(y) is immediate from 2.7.4 (2). The uniqueness of ϕ follows from 3.9.4 in view of 3.1.2 (2). ⊲ 3.9.6. Let X be a vector lattice with the projection property and B := B(X). If f : X → R is a nonzero disjointness preserving functional then there exists a unique Boolean homomorphism ϕ : B → {0, 1} such that im(f |K ) = ϕ(K)R (K ∈ B). ⊳ Deﬁne ϕ : B → {0, 1} by putting ϕ(K) := 0 if K ⊂ ker(f ) and ϕ(K) := 1 otherwise. Assume that K ∈ B is not contained in ker(f ). Then f (x) = 0 for some x ∈ K and K ⊥ ⊂ {x}⊥ ⊂ ker(f ). Thus, for every K ∈ B either K ⊂ ker(f ) or K ⊥ ⊂ ker(f ). Using this simple 160 Chapter 3. Order Bounded Operators properties one can easily ensure that ϕ(K ∧ L) = ϕ(K) ∧ ϕ(L) for all K, L ∈ B. Now use the projection property and observe that either ϕ(K) = 1 or ϕ(K ⊥ ) = 1, since X = K + K ⊥ and ϕ(X) = 1. It follows that ϕ(K)⊥ = ϕ(K ⊥ ) and ϕ is a Boolean homomorphism from B to {0, 1}. Moreover, by the deﬁnition of ϕ we have im(f |K ) = {0} = ϕ(K)R whenever K ⊂ ker(f ) and im(f |K ) = R = ϕ(K)R otherwise. ⊲ The Boolean homomorphism ϕ constructed from f is called the shadow of f . It is immediate from the deﬁnition that f and |f | have the same shadow. The Boolean homomorphism ϕ induces a homomorphism from P(X) to 2 deﬁned as π → ϕ(π(X)) and denoted by the same letter ϕ. From the deﬁnition of the shadow it is also clear that ϕ(π)f ◦ π = f ◦ π and ϕ(π)f ◦ π ⊥ = 0. Therefore, f ◦ π = ϕ(π)f for all π ∈ P(X). 3.9.7. In the rest of this section, X and Y are vector lattices considered as order dense sublattices in their universal completions X u and Y u . Moreover, we assume that Y is Dedekind complete. We ﬁx the weak order units 1 and 1̂ in X u and Y u , respectively, so that X u and Y u are also f -algebras with the multiplicative units 1 and 1̂. Recall that orthomorphisms in X u and Y u are multiplication operators and we identify them with the corresponding multipliers. Note that some notions in this section depend on a speciﬁc choice of the unities 1 and 1̂. For every e ∈ X u , there exists a unique element 1/e ∈ X u such that e(1/e) = [e]1. The product x(1/e) is denoted by x/e for brevity. Put X/e := {x/e ∈ X u : x ∈ X}. Then X/e is a vector sublattice of X u and h : x → x/e is an order bounded band preserving operator from X onto X/e with h(e) = [e]1. If e is invertible in the f -algebra X u then h is a lattice isomorphism of X onto X/e and h(e) = 1. 3.9.8. Theorem. Let X be a vector lattice over an ordered field P with Q ⊂ P ⊂ R and B := B(X). Let f : X → R be a nonzero order bounded band preserving P-linear functional. Then there exist a 2valued Boolean homomorphism ϕ on B and α ∈ R such that X/e ⊂ D(ϕ) and f (x) = αϕ(x/e) (x ∈ X). (1.1) ⊳ Because f is nonzero, there exists e ∈ X+ with 0 < p := |f |(e) ∈ P. There is no loss of generality in assuming that e is a weak order unit, since f preserves disjointness and {e}⊥ ⊂ ker(f ). Denote by g : X → R a lattice homomorphism which is an extension of |f | to the Dedekind completion X δ . (Assume further that X ⊂ X δ and X δ is an order 3.9. Representation of Disjointness Preserving Operators 161 dense ideal in S(B).) Such extension exists by Theorem 3.1.13. We can also identify B(X) and B(X δ ), since B → B ∩ X is an isomorphism of B(X δ ) onto B(X). As was mentioned in 3.9.7, h : x → x/e is a lattice isomorphism from X δ onto X δ /e. By 3.9.6 there exists a unique Boolean homomorphism ϕ : B → {0, 1} such that g([K]x) = ϕ(K)g(x) (x ∈ X δ , K ∈ B). Observe now that if X δ /e ⊂ D(ϕ) then g1 := g ◦h−1 and g2 := ϕ| X δ /e are real lattice homomorphisms on X δ /e with g1 (1) = p and g2 (1) = 1. As can be seen using 3.4.1, two lattice homomorphisms are either disjoint or proportional. But the ﬁrst case is impossible, since (g1 ∧ g2 )(1) = inf{g(πe) + ϕ(π ⊥ 1) : π ∈ P(X δ )} = inf{ϕ(K)p + ϕ(K ⊥ ) : K ∈ B(X δ )} min{p, 1} > 0, where K = π(X δ ). Thus, g1 and g2 are proportional and, since g|X = ±f , we have g = αϕ ◦ h for some α ∈ R. It remains to show that X δ /e ⊂ D(ϕ). Observe ﬁrst that e is a weak order unit in X δ , whence h(e) = 1 = 1̄. Moreover, b(t) = ebt (t ∈ R) for all b ∈ B (cp. 2.8.4). Take arbitrary 0 x ∈ X δ /e and n ∈ N and 1−ex x/n note that eλ = exnλ exn = eλ n , whenever 0 λ < 1. If λ 1 then 1−ex 1−ex x/n x/n x/n eλ 1 = eλ n and eλ = 0 for λ < 0. Thus, eλ eλ n for all ˙ λ ∈ R, whence 1 − exn x/n by 2.7.4(1). Now we can estimate |g1 (1)|(1 − ϕ(exn )) = |g1 (1) − ϕ(exn )g1 (1)| = |g1 (1 − exn )| |g1 (1)|/n → 0. It follows that ϕ(exn ) = 1 for some n ∈ N, so that x ∈ D(ϕ). ⊲ 3.9.9. Theorem. Let X and Y be vector lattices and let T : X → Y be an order bounded disjointness preserving operator such that {T (e)}⊥⊥ = Y for some e ∈ X+ . Then there exit an order dense sublattice Y0 in Y u , an order dense ideal D(Φ) in X u , a lattice homomorphism Φ : D(Φ) → Y0 , and an orthomorphism W from Y0 to Y such that x → x/e is an orthomorphism from X into D(Φ), 1 ∈ D(Φ), Φ(1) = 1̂, and T (x) = W Φ(x/e) (x ∈ X). (1.2) ⊳ There is no loss of generality in assuming that e and |T |e are weak order units in X and Y , respectively, since T vanishes on {e}⊥. 162 Chapter 3. Order Bounded Operators In accordance with the Gordon Theorem, we can assume that Y is an order dense sublattice of R↓ and T = τ ↓ for an internal order bounded disjointness preserving functional τ : X ∧ → R with R, τ ∈ V(B) and B = B(Y ). Then [[τ (e∧ ) = T e = 0]] = 1 and so [[τ (e∧ ) = 0]] = 1. Working within V(B) we can apply Theorem 3.9.8 and pick a Boolean homomorphism ϕ : B(X ∧ ) → {0, 1} and α ∈ R such that X ∧ /e∧ ⊂ D(ϕ) ⊂ (X ∧ )u , ϕ : D(ϕ) → R is a lattice homomorphism with ϕ(1 ∧) = ∧ 1 and τ (x) = αϕ(x/e) for all x ∈ X . Clarify some details of such a representation. Recall that X ∧ is an order dense sublattice in (X u )∧ and (X u )∧ is an order dense sublattice in (X ∧ )u . It follows that 1∧ ∈ (X u )∧ is a weak order unit in (X ∧ )u . Moreover (X ∧ )u is an f -algebra with the multiplicative unit 1∧ and D(ϕ) is an order dense ideal and an f -subalgebra in (X ∧ )u containing 1∧ . Thus, the multiplication operator μe : x → x/e∧ on (X ∧ )u induces a lattice isomorphism of X ∧ into D(ϕ). If mα denotes a linear function y → αy on R, then τ = mα ◦ ϕ ◦ μe . Now we examine the descent of this representation. By Theorem 2.11.6 Z := (X ∧ )u ↓ is a universally complete vector lattice. Moreover, Z0 := D(ϕ)↓ is an order dense ideal in Z containing 1∧ and ϕ↓ is a lattice homomorphism from Z0 to R↓. Note that x → x∧ is a lattice homomorphism from X u into (X ∧ )u ↓, so that we can identify X u with a sublattice in (X ∧ )u ↓ (and so 1 with 1∧ ). Keeping this in mind, denote by Φ and w the restrictions of ϕ↓ onto Z0 ∩ X u and μ↓ onto X, respectively. Since the multiplication on (X ∧ )u ↓ is the descent of the multiplication on (X ∧ )u , we conclude that μ↓ is the multiplication operator with the same multiplier e, that is μ↓ : x → x/e. By the same reasons mα ↓ is the multiplication operator on R↓ with the multiplier α ∈ R↓. Put Y0 = Φ(X/e) and denote by W the restriction of mα ↓ onto Y0 . Then Y0 is a sublattice of Y u , and W is an orthomorphism from Y0 to Y . From 1.5.5 (1) we have ◦ μe ↓)x = (W ◦ Φ ◦ w)x T x = τ ↓(x) = (mα ↓ ◦ ϕ↓ for all x ∈ X and the proof is complete. ⊲ 3.9.10. The above representation is called the weight-shift-weight factorization of T . The operator Φ, whose existence is asserted in 3.9.9, is called the shift of T . We say that an operator S : D(S) → Y0 is a shift operator, if D(S) and Y0 are order dense ideals in X u and Y u , respectively, and S is the shift of some order bounded disjointness preserving 3.9. Representation of Disjointness Preserving Operators 163 operator T : X → Y . The operator Φ of the representation can be deﬁned for an arbitrary order bounded disjointness preserving operator T just as in the proof of Theorem 3.9.9, but there is not enough room in X u to provide the weight-shift-weight factorization of T . A weight system is a family of pairs w := (bξ , eξ ) ξ∈Ξ such that (bξ )ξ∈Ξ is a partition of unity in P(Y u ) and (eξ )ξ∈Ξ is a family of positive elements in X and the representation bξ T = Wξ ◦ Φ ◦ (·/eξ ) holds for all ξ ∈ Ξ. In this case, ob Φ(1/e ξ ξ ) = 1̂ and, putting ξ∈Ξ W := o- ξ∈Ξ bξ T eξ , we obtain the representation T = obξ W ◦ Φ ◦ (·/eξ ), ξ∈Ξ which will be written shortened as follows: T = W ◦ Φ ◦ w. 3.9.11. Theorem. Let X and Y be vector lattices and let T : X → Y be an order bounded disjointness preserving operator. Then there exist Φ and W as in Theorem 3.9.9 and a weight system w ∈ W (X, B) such that ◦ w. Tx = W ◦ Φ ⊳ Let τ be the same as in 3.9.10. If [[τ = 0]] = 1 then T = 0 and there is nothing to prove. If [[τ = 0]] = 1 then, in view of the ZFC-theorem ψ := τ = 0 → (∃p ∈ R)(∃ e ∈ X ∧ )(0 < p ∧ p = τ (e)), we have by transfer [[ψ]] = 1 and according to maximum principle there exists p ∈ Y such that 1= [[p = T (e)]]. [[p = τ (e∧ ) ∧ p > 0]] = [[0 < p]] ∧ e∈X e∈X Thus, p is a weak order unit in Y and there exists a partition of the unit (bξ )ξ∈Ξ in B, and a family (eξ )ξ∈Ξ in X such that bξ [[T eξ = p]] for all ξ. It follows that bξ T eξ = bξ p. By Theorem 3.9.9, for every ξ ∈ Ξ, we have X/eξ ⊂ D(Φ), and there exist a vector sublattice Yξ ⊂ bξ Y u and an orthomorphism Wξ : Yξ → Y such that (·/eξ ) is an orthomorphism from X to Xξ and πξ ◦ T = Wξ ◦ πξ ◦ Φ◦ (·/eξ ). Clearly, w = (bξ , e ξ )ξ∈Ξ is u a weight system for Φ. If Y0 is a sublattice in Y generated by ξ∈Ξ Yξ and W is the restriction of the sum o- ξ∈Ξ bξ ◦ Wξ to Y0 , then T x = obξ ◦ T x = o- (Wξ ◦ bξ ◦ Φ)(x/e ξ ) = (W ◦ Φ ◦ w)x ξ∈Ξ ξ∈Ξ for all x ∈ X. The proof is complete. ⊲ 164 Chapter 3. Order Bounded Operators 3.10. Pseudoembedding Operators In this section we will give a description of the band generated by disjointness preserving operators in the vector lattice of order bounded operators. First we examine the scalar case. 3.10.1. For an arbitrary vector lattice X there exist a unique cardinal γ and a disjoint family (ϕα )α<γ of nonzero lattice homomorphisms ϕα : X → R such that every f ∈ X ∼ admits the unique representation f = fd + oλα ϕα α<γ where fd ∈ Xd∼ and (λα )α<γ ⊂ R. The family (ϕα )α<γ is unique up to permutation and positive scalar multiplication. ⊳ The Dedekind complete vector lattice X ∼ splits into the direct sum of the atomic band Xa∼ and the diﬀuse band Xd∼ ; therefore, each functional f ∈ E ∼ admits the unique representation f = fa + fd with fa ∈ Xa∼ and fd ∈ Xd∼ . Let γ be the cardinality of the set K of onedimensional bands in Xa∼ (= atoms in B(X ∼ )). Then there exists a family of lattice homomorphisms (ϕα : X → R)α<γ such that K = {ϕ⊥⊥ : α α < γ}. It remains to observe that the mapping sending a family of reals (λα )α<γ to the functional x → α<γ λα ϕα (x) implements a lattice isomorphism between Xa∼ and some ideal in the vector lattice Rγ . If (ψα )α<γ is a disjoint family of nonzero real lattice homomorphisms on X with Xa∼ = {ψα : α < γ}⊥⊥ , then for all α, β < γ the functionals ϕα and ψβ are either disjoint or proportional with a strictly positive coeﬃcient, so that there exist a permutation (ωβ )β<γ of (ϕα )α<γ and a unique family (μβ )β<γ in R+ such that ψβ = μβ ωβ for all β < γ. ⊲ 3.10.2. Given two families (Sα )α∈A and (Tβ )β∈B in L∼ (X, Y ), say that (Sα )α∈A is a P(Y )-permutation of (Tβ )β∈B wheneverthere exists a double family (πα,β )α∈A, β∈B in P(Y ) such that Sα = β∈B πα,β Tβ for all α ∈ A, while (πα,β̄ )α∈A and (πᾱ,β )β∈B are partitions of unity in B(Y ) for all ᾱ ∈ A and β̄ ∈ B. It is easily seem that in case Y = R this amounts to saying that there is a bijection ν : A → B with Sα = Tν(α) for all α ∈ A; i.e., (Sα )α∈A is a permutation of (Tβ )β∈B . We also say that (Sα )α∈A is Orth(Y )-multiple of (Tα )α∈A whenever there exists a family of orthomorphisms (πα )α∈A in Orth(Y ) such that Sα = πα Tα for all α ∈ A. In case Y = R we evidently get that Sα is a scalar multiple of Tα for all α ∈ A. 3.10. Pseudoembedding Operators 165 Using above notation deﬁne the two mappings S : A → X ∧∼ ↓ and T : B → X ∧∼ ↓ within V(B) by putting S (α) := Sα ↑ (α ∈ A) and T (β) := Tβ ↑ (β ∈ B). 3.10.3. Define the internal mappings τ, σ ∈ V(B) as σ := S ↑ and τ := T ↑. Then (σ(α))α∈A∧ is a permutation of (τ (β))β∈B∧ within V(B) if and only if (Sα )α∈A is a P(Y )-permutation of (Tβ )β∈B . ⊳ Assume that (σ(α))α∈A∧ is a permutation of (τ (β))β∈B∧ within V(B) . Then there is a bijection ν : B∧ → A∧ such that σ(α) = τ (ν(α)) for all α ∈ A∧ . By 1.5.8 ν↓ is a function from A to (B∧ )↓ = mix({β ∧ : β ∈ B}). Thus, for each α ∈ A there exists a partition of unity (bα,β )β∈B such that ν↓(α) = mixβ∈B (bα,β β ∧ ). Since ν↓ is injective, we have 1 = [[(∀α1 , α2 ∈ A∧ )(ν(α1 ) = ν(α2 ) → α1 = α2 )]] [[ν(α∧1 ) = ν(α∧2 ) → α∧1 = α∧2 ]] = α1 ,α2 ∈A = [[ν↓(α1 ) = ν↓(α2 )]] ⇒ [[α∧1 = α∧2 ]], α1 ,α2 and so [[ν↓(α1 ) = ν↓(α2 )]] [[α∧1 = α∧2 ]] for all α1 , α2 ∈ A. Taking this inequality and the deﬁnition of ν↓ into account yields bα1 ,β ∧ bα2 ,β [[ν↓(α1 ) = β ∧ ]] ∧ [[ν↓(α2 ) = β ∧ ]] [[ν↓(α1 ) = ν↓(α2 )]] [[α∧1 = α∧2 ]], so that α1 = α2 implies bα1 ,β ∧ bα2 ,β = 0 (because α1 = α2 ⇐⇒ [[α∧1 = α∧2 ]] = O by 1.4.5 (2)). At the same time, surjectivity of ν implies 1 = [[(∀ β ∈ B∧ )(∃ α ∈ A∧ )β = ν(α)]] bα,β . [[β ∧ = ν↓(α)]] = = β∈B α∈A β∈B α∈A It follows that (bα,β )α∈A is a partition of unity in B for all β ∈ B. By the choice of ν it follows that bα,β [[σ(α∧ ) = τ (β ∧ )]], because of the estimations bα,β [[σ(α∧ ) = τ (ν(α∧ ))]] ∧ [[ν(α∧ ) = β ∧ ]] [[σ(α∧ ) = τ (β ∧ )]] = [[S (α) = T (β)]]. Put now πα,β := χ(bα,β ) and observe that bα,β [[S (α)x∧ = T (β)x∧ ]] [[Sα x = Tβ x]] for all α ∈ A, β ∈ B, and x ∈ X. Using 2.2.4 (G), we obtain 166 Chapter 3. Order Bounded Operators πα,β Sα = πα,β Tβ and so Sα = β∈B πα,β Tβ for all α ∈ A. Clearly, (πα,β ) is the family as required in Deﬁnition 3.10.2. The suﬃciency is shown by the same reasoning in the reverse direction. ⊲ 3.10.4. Recall that the elements of the band L∼ d (X, Y ) := Hom(X, Y )⊥ are referred to as diﬀuse operators; see 3.3.4. An order bounded operator T : X → Y is said to be pseudoembedding if T be⊥⊥ longs to the complimentary band L∼ , the band a (X, Y ) := Hom(X, Y ) generated by all disjointness preserving operators. A nonempty set D of positive operators from X to Y is called strongly generating if D is disjoint and S(X)⊥⊥ = Y for all S ∈ D. If, in addition, D ⊥⊥ = B, then we say also that D strongly generates the band B ⊂ L∼ (X, Y ) or B is strongly generated by D. In case Y = R, the strongly generating sets in X ∼ = L∼ (X, R) are precisely those that consist of pairwise disjoint nonzero positive functionals. Given a cardinal γ and a universally complete vector lattice Y , say that a vector lattice X is (γ, Y )-homogeneous if the band L∼ a (X, Y ) is strongly generated by a set of lattice homomorphisms of cardinality γ and for every nonzero projection π ∈ P(Y ) and every strongly generating set D in L∼ a (X, πY ) we have card(D) γ. We say also that X is (γ, π)homogeneous if π ∈ P(Y ) and X is (γ, πY )-homogeneous. Evidently, the (γ, R)-homogeneity of a vector lattice X amounts just to saying that the band Xa∼ is generated in X ∼ by a disjoint set of nonzero lattice homomorphisms of cardinality γ or, equivalently, the cardinality of the set of atoms in P(X ∼ ) equals γ. Take D ⊂ L∼ (X, R↓) and ∆ ∈ V(B) with [[∆ ⊂ (X ∧ )∼ ]] = 1. Put D↑ := {T ↑ : T ∈ D}↑ and ∆↓ := {τ ↓ : τ ∈ ∆↓}. Let mix(D) stand for the set of all T ∈ L∼ (X, R↓) representable as T x = o- ξ∈Ξ πξ Tξ x (x ∈ X) with arbitrary partition of unity (πξ )ξ∈Ξ in P(R↓) and family (Tξ )ξ∈Ξ in D. 3.10.5. Let ∆ ⊂ (X ∧ )∼ is a disjoint set of nonzero positive functionals of cardinality γ ∧ within V(B) . Then there exists a strongly generating set of positive operators D from X to R↓ of cardinality γ such that ∆ = D↑ and ∆↓ = mix(D). ⊳ If ∆ obeys the conditions then there is φ ∈ V(B) such that [[φ : γ → ∆ is a bijection]] = 1. Note that φ↓ sends γ into ∆↓ ⊂ (X ∧ )∼ ↓ by 1.5.8. By Theorem 3.3.3, we can deﬁne the mapping α → Φ(α) from γ to L∼ (X, R↓) by putting Φ(α) := (φ↓(α))↓. Put D := {Φ(α) : α ∈ γ} and note that D ⊂ ∆↓ . By 1.6.6 and surjectivity of φ we have ∆↓ = ∧ 3.10. Pseudoembedding Operators 167 ϕ(γ ∧ )↓ = mix{φ↓(α)) : α ∈ γ} and combining this with 3.3.7 we get ∆ = D↑ and ∆↓ = mix(D). The injectivity of φ implies that to [[(∀α, β ∈ γ ∧ )(α = β → φ(α) = φ(β)]] = 1. Replacing the universal quantiﬁer by the supremum over α, β ∈ γ ∧ , from 1.4.5 (1) and 1.4.5 (2) we deduce that 1= [[α∧ = β ∧ ]] ⇒ [[ϕ(α∧ ) = φ(β)∧ ]] = [[Φ(α) = Φ(β)]], α,β∈γ α,β∈γ α=β and so α = β implies Φ(α) = Φ(β) for all α, β ∈ γ. Thus Φ is injective and the cardinality of D is γ. The fact that D is strongly generating follows from 3.3.5 (5) and 3.8.4. ⊲ 3.10.6. If D is a strongly generating set of positive operators from X to R↓ of cardinality γ then ∆ = D↑ ⊂ (X ∧ )∼ is a disjoint set of nonzero positive functionals of cardinality |γ ∧ | within V(B) . ⊳ Assume that D ⊂ L(X, R↓) is a strongly generating set of cardinality γ. Then there is a bijection f : γ → D↑. Moreover, α = β implies [[f (α) ⊥ f (β)]] = 1 by 3.3.5 (5) and [[f (α) = 0]] = 1 by 3.8.4. Interpreting in V(B) the ZFC-theorem (∀f, g ∈ X ∼ )(f = 0 ∧ g = 0 ∧ f ⊥ g → f = g) yields [[f (α) = f (β)]] = 1 for all α, β ∈ γ, α = β. It follows that φ := f ↑ is a bijection from γ ∧ onto ∆ = (D↑)↑, so that the cardinality of ∆ is |γ ∧ |. The proof is completed by the arguments similar to those in 3.10.5. ⊲ 3.10.7. A vector lattice X is (γ, R↓)-homogeneous for some cardinal γ if and only if [[ γ ∧ is a cardinal and X ∧ is (γ ∧ , R)-homogeneous ]] = 1. ⊳ Suﬃciency: Assume that γ ∧ is a cardinal and X ∧ is (γ ∧ , R)-homogeneous within V(B) . The latter means that (X ∧ )∼ a is generated by a disjoint set of nonzero lattice homomorphisms ∆ ⊂ (X ∧ )∼ of cardinality γ ∧ within V(B) . By 3.10.5 there exists a strongly generating set D in ∧ L∼ a (X, R↓) of cardinality γ such that ∆ = D↑ . Take nonzero π ∈ P(R↓) −1 and put b := χ (π). Recall that we can identify L∼ (X, π(R↓)) and L∼ (X, (b ∧ R)↓). If D ′ is a strongly generating set in L∼ a (X, π(R↓)) of cardinality β then D↑′ strongly generates (X ∧ )∼ and has cardinality |β ∧ | a within the relative universe V([0,b]) . By 1.3.7 γ ∧ = |β ∧ | β ∧ and so γ β. Necessity: Assume now that X is (γ, R↓)-homogeneous and a set lattice homomorphisms D of cardinality γ generates strongly the band 168 Chapter 3. Order Bounded Operators ∧ ∼ L∼ a (X, R↓). Then ∆ = D↑ generates the band (X )a and the cardinal∧ ∧ ities of ∆ and γ coincide; i.e., |∆| = |γ |. By 1.9.11 the cardinal |γ ∧ | has the representation |γ ∧ | = mixαγ bα α∧ , where (bα )αγ is a partition of unity in B. It follows that bα [[∆ is a generating set in (X ∧ )∼ a of cardinality α∧ ]] = 1. If bα = 0 then bα ∧ ∆ is a generating set in ∧ ∧ ∧ [0,bα ] (X ∧ )∼ . Put a of cardinality |γ | = α γ in the relative universe V πα = χ(bα ) and πα ◦ D := {πα ◦ T : T ∈ D}. Clearly, bα ∧ ∆ = (πα D)↑ and so πα ◦ D strongly generates the band L∼ a (X, R↓). By hypothesis D is (γ, R↓)-homogeneous, consequently, α γ, so that α = γ, since α γ if and only if α∧ γ ∧ . Thus, |γ ∧ | = γ ∧ whenever bα = 0 and γ ∧ is a cardinal within V(B) . ⊲ 3.10.8. Let X be a (γ, Y )-homogeneous vector lattice for some universally complete vector lattice Y and a nonzero cardinal γ. Then there exists a strongly generating family of lattice homomorphisms (Φα )α<γ from X to Y such that each operator T ∈ L∼ a (X, Y ) admits the unique representation T = o- α<γ σα ◦ Φγ,α , where (σα )α<γ is a family of orthomorphisms in Orth(Y ). ⊳ This is immediate from the deﬁnitions in 3.10.4. ⊲ 3.10.9. Theorem. Let X and Y be vector lattices with Y universally complete. Then there are a nonempty set of cardinals Γ and a partition of unity (Yγ )γ∈Γ in B(Y ) such that X is (γ, Yγ )-homogeneous for every γ ∈ Γ. ⊳ We may assume without loss of generality that Y = R↓. The transfer principle tells us that according to 3.10.1 there exists a cardinal κ within V(B) such that (X ∧ )∼ a is generated by a disjoint set H of nonzero R∧ -linear lattice homomorphisms of cardinality κ or, equivalently, [[X ∧ is (κ, R)-homogeneous ]] = 1. By 1.9.11 there is a nonempty set of cardinals Γ and a partition of unity (bγ )γ∈Γ in B such that κ = mixγ∈Γ bγ γ ∧ . It follows that bγ [[X ∧ is (γ ∧ , R)-homogeneous ]] for all γ ∈ Γ. Passing to the relative subalgebra Bγ := [0, bγ ] and considering 1.3.7 we conclude that V(Bγ ) |= “X ∧ is (γ ∧ , bγ ∧ R)-homogeneous”, so that X is (γ, (bγ ∧ R)↓)-homogeneous by 3.10.7. In view of 2.3.6 (bγ ∧ R)↓ is lattice isomorphic to Yγ , so the desired result follows. ⊲ 3.10.10. Theorem. Let X and Y be vector lattices with Y universally complete. Then there is a nonempty set of cardinals Γ, a partition of unity (Yγ )γ∈Γ in B(Y ), and to each cardinal γ ∈ Γ there is a disjoint family of lattice homomorphisms (Φγ,α )α<γ from X to Yγ such that (1) Φγ,α (X)⊥⊥ = Yγ = {0} for all γ ∈ Γ and α < γ. 3.11. Diﬀuse operators 169 (2) X is (γ, Yγ )-homogeneous for all γ ∈ Γ. (3) For each order dense sublattice Y0 ⊂ Y each T ∈ L∼ (X, Y0 ) admits the unique representation T = Td + ooσγ,α ◦ Φγ,α , γ∈Γ α<γ with Td ∈ L∼ d (X, Y ) and σγ,α ∈ Orth(Φγ,α , Y0 ). For every γ ∈ Γ the family (Φγ,α )α<γ is unique up to P(Y )permutation and Orth(Yγ )+ -multiplication. ⊳ The existence of (Yγ )γ∈Γ and (Φγ,α )γ∈Γ, α<γ with the required properties is immediate from 3.10.8 and 3.10.9. The uniqueness follows from 3.10.1 and 3.10.3. ⊲ 3.11. Diffuse operators In this section, we give necessary and suﬃcient conditions under which an order bounded linear operator is diﬀuse. We ﬁrst handle the case of functionals and then obtain a general result by means of Boolean valued interpretation of the scalar result. 3.11.1. We need a property of additive measures on Boolean algebras. Consider a Boolean algebra B. A function μ : B → R is called additive if μ(a ∨ b) = μ(a) + μ(b) a, b ∈ B with a ∧ b = 0 and for all completely additive whenever μ( D) = d∈D μ(d) for every disjoint subset D ⊂ B. A positive (that is, (∀ b ∈ B)μ(b) 0) additive function μ is completely additive if and only if it is order continuous; i.e., limα μ(bα ) = 0 for every decreasing net (bα ) in B with 0 = inf α bα . Say that b0 ∈ B is a μ-atom if μ(b0 ) = 0 and for every b ∈ B with b b0 either μ(b) = 0 or μ(b − b0 ) = 0. An additive function μ is said to be nonatomic on B if there are no μ-atoms in B or, equivalently, for each b ∈ B the relation μ(b) = 0 implies the existence of b0 ∈ B such that b0 b, μ(b0 ) = 0 and μ(b − b0 ) = 0. 3.11.2. Assume that B is a complete Boolean algebra and μ : B → R is a nonatomic order continuous additive function. Then for all b ∈ B and 0 α μ(b) there exists bα ∈ B with bα b and α = μ(bα ). ⊳ Let b ∈ B and 0 α μ(b) be given. Put D := {d ∈ B : d b, μ(d) α} and, given c, d ∈ B, put c d whenever μ(c−d) = 0. It can 170 Chapter 3. Order Bounded Operators easily be checked involving order continuity of μ that every chain in an ordered set (D, ) has an upper bound in D. Thus, by the Kuratowski– Zorn Lemma, D has a maximal element, say bα ; i.e., if c ∈ D and bα c, then μ(c − bα ) = 0. We claim that μ(bα ) = α. Indeed, otherwise μ(b − bα ) = μ(b) − μ(bα ) α − μ(bα ) > 0 and, since μ is nonatomic, there is c1 b0 := b − bα with 0 < μ(c1 ) < μ(b0 ). Moreover, with the choice b1 := c1 or b1 := b0 − c1 this yields 0 < μ(b1 ) (1/2)μ(b0 ). Repeating the same argument, we obtain a sequence (bn ) in B such that b0 bn bn−1 and 0 < μ(bn ) (1/2)μ(bn−1 ) (1/2n )μ(b0 ) for all n ∈ N. Choose n with (1/2n )μ(b0 ) α − μ(bα ) and put d := bn ∨ bα . Then d b, bα d, and μ(d − bα ) = μ(bn ) > 0. This contradicts the maximality of bα . ⊲ 3.11.3. Theorem. An order bounded functional f on a vector lattice X is diﬀuse if and only if for all 0 x ∈ X and 0 < ε ∈ R there is a finite disjoint collection of positive functionals f1 , . . . , fN ∈ X ∼ such that |f | = f1 + · · · + fN , fk (x) ε (k := 1, . . . , N ). ⊳ Assume that h is a nonzero lattice homomorphism with h |f | and choose x ∈ X+ such that h(x) = 1. If |f | = f1 + · · · + fN for a collection of pairwise disjoint positive functionals f1 , . . . , fN ∈ X ∼ , then h fk for some 1 k n. Thus 1 = h(x) fk (x) and the above necessary condition cannot be fulﬁlled. To prove the suﬃciency, take a diﬀuse f ∈ X ∼ and ﬁx x ∈ X+ and 0 < ε ∈ R. There is no loss of generality in assuming that f is positive and f0 (x) > 0 for every component f0 ∈ C(f ). Put B := C(f ) and deﬁne μ : B → R by μ(b) := b(x). Clearly, μ is order continuous nonatomic additive function on B. Pick 0 = α0 < α1 < · · · < αN = μ(f ) with αı − αı−1 < ε (ı := 1, . . . , N ). By 3.11.2 we can choose a ﬁnite sequence b1 · · · bN in B such that μ(bı ) = αı for all ı := 1, . . . , N . If b0 := b∗N then μ(b0 ) = μ(f ) − μ(bN ) = 0. It remains to put fı := bı − bı−1 (ı := 1, . . . , N ) and observe that f1 + · · · + fN = f and fı (x) = μ(fı ) = μ(bı ) − μ(bı−1 ) = αı − αı−1 < ε for all ı := 1, . . . , N . ⊲ 3.11.4. Theorem. Let X and Y be vector lattices with Y Dedekind complete. For T ∈ L∼ (X, Y ), the following are equivalent: (1) T is diﬀuse. 171 3.11. Diﬀuse operators (2) For every x ∈ X we have n ı=1 Tı |x| : |T | = n ı=1 Tı , Tı ⊥ Tj (ı = j), n ∈ N = 0. (3) Given x ∈ X+ , 0 < ε ∈ R, and π ∈ P(Y ) with π|T |x = 0, there exist a nonzero projection ρ π and pairwise disjoint positive operators T1 , . . . , TN from X to ρY such that ρ|T | = T1 +· · ·+TN and Tk x ε|T |x for all k := 1, . . . , N . (4) Given x ∈ X+ , 0 < ε ∈ R, and π ∈ P(Y ) with π|T |x = 0, there exists a countable partition (πn ) of π such that for every n ∈ N the operator πn |T | decomposes into the sum of pairwise disjoint positive operators T1,n , . . . , Tn,n from X to πn Y satisfying Tk,n x ε|T |x for all k := 1, . . . , n. ⊳ It is an easy exercise to check the equivalences (2) ⇐⇒ (3) ⇐⇒ (4). The proof of (1) ⇐⇒ (4) is obtained by interpreting Theorem 3.11.3 within V(B) where B := P(Y ). By the Gordon Theorem we can take Y = R↓ without loss of generality. Moreover, the problem reduces easily to the case of positive T not involving Boolean valued arguments. Put τ := T ↑ and note that, according to Corollary 3.3.6 (4), T is diﬀuse if and only if [[τ is diﬀuse ]] = 1. Theorem 3.11.3 is valid within V(B) by transfer, and so the sentence “τ is diﬀuse” is equivalent to the formula (∀ x ∈ X ∧ )(∀ 0 < ε ∈ R∧ )(∃ n ∈ N∧ )(∃ ν)ϕ(x, ε, n, ν, τ, X) ∼ where ϕ(x, ε, n, ν, τ, X) stands for the assertion: ν : {1, . . . , n} → X+ , τ = ν(1) + · · · + ν(n) and ν(ı) ⊥ ν(j), (ı = j), ν(ı)x ετ (x) for all ı, j := 1, . . . , n. By 1.5.2 quantiﬁcations over X ∧ , R∧ , and N∧ can be replaced by order operations in B over X, R, and N: 1= [[(∃ ν)ϕ(x∧ , ε∧ , n∧ , ν, τ, X ∧ )]]. x∈X 0<ε∈R n∈N This amounts to saying that for all x ∈ X and 0 < ε ∈ R we can choose a countable partition of unity (bn ) in B with bn [[(∃ ν)ϕ(. . .)]]. In view of the maximum principle, for each n ∈ N there exists νn ∈ V(B) such that bn [[ϕ(x∧ , ε∧ , n∧ , νn , τ, X ∧ )]]. Passing to relative subalgebra Bn := [0, bn ] and taking 1.3.7 and 1.4.6 into account we see that the last inequality is fulﬁlled if and only if ϕ(x∧ , ε∧ , n∧ , ν̄n , τ̄ , X ∧ ) 172 Chapter 3. Order Bounded Operators with ν̄n := bn ∧ νn ∈ V(Bn ) and τ̄ := bn ∧ τ ∈ V(Bn ) is true within V(Bn ) or, in more details (with n := {1, . . . , n} and e := τ̄ (x∧ ) for short), V(Bn ) |= ν̄n : n∧ → (X ∧ )∼ + ∧ τ̄ = ν̄n (1) + · · · + ν̄n (n), V(Bn ) |= (∀ ı, j ∈ n∧ )(ı = j → ν̄n (ı) ⊥ ν̄n (j)) ∧ (ν̄n (ı)x∧ ε∧ e). By 1.5.8 the modiﬁed descent T := ν̄n ↓ maps {1, . . . , n} into (X ∧ )∼ + ↓. Put πn := χ−1 (bn ) and Tı := T (ı)↓ and observe that, by Theorem 3.3.5 (5), {T1 , . . . , Tn } is a disjoint collection of positive operators from X to πn Y with πn T = T1 + · · · + Tn . Moreover, for each x ∈ X we have bn [[Ti x = T (ı)x∧ = ν̄n (ı∧ )x∧ ]] ∧ [[ν̄n (ı∧ )x∧ ε∧ e]] [[Ti x ε∧ e]], so that Ti x εT x. Thus we arrived at the conclusion that (4) holds if and only if [[τ is diﬀuse ]] = 1, which completes the proof. ⊲ 3.11.5. Theorem. Let X, Y , and Z be vector lattices with Z Dedekind complete. For B ∈ BL∼ (X, Y ; Z), the following are equivalent: (1) B is diﬀuse. (2) For all x ∈ X and y ∈ Y the identity holds n Bı (|x|, |y|) : |B| = ı=1 n ı=1 Bı , Bı ⊥ Bj (ı = j), n ∈ N = 0. (3) Given x ∈ X+ , y ∈ Y+ , 0 < ε ∈ R, and π ∈ P(Z) with π|B|(x, y) = 0, there exist a nonzero projection ρ π and pairwise disjoint positive bilinear operators B1 , . . . , BN from X × Y to ρZ such that ρ|B| = B1 + · · · + BN ; Bk (x, y) ε|B|(x, y) for all k := 1, . . . , N . (4) Given x ∈ X+ , y ∈ Y+ , 0 < ε ∈ R, and π ∈ P(Z) with π|B|(x, y) = 0, there exists a countable partition (πn ) of π such that for every n ∈ N the operator πn |B| decomposes into the sum of pairwise disjoint positive operators B1,n , . . . , Bn,n from X × Y to πn Z satisfying Bk,n (x, y) ε|B|(x, y) for all k := 1, . . . , n. 3.12. Variations on the Theme 173 ⊳ The proof can be obtained by reasoning along the lines of Theorem 3.11.4. Alternatively, it can be reduced to the case of linear operators by applying Fremlin’s Theorem 3.2.8. ⊲ 3.12. Variations on the Theme In this section we apply the Boolean value approach to the three types of problems: the multiplicative representation of a lattice multimorphism, the characterization of disjointness preserving sets of operators, and the Sobczyk–Hammer type decomposition for measures with values in Dedekind complete vector lattices. 3.12.A. Representation of Lattice Multimorphisms 3.12.A.1. Let X and Y be vector lattices. Recall that a bilinear operator B : X × X → Y is said to be orthosymmetric if x ⊥ y implies B(x, y) = 0 for all x, y ∈ X, symmetric if B(x, y) = B(y, x) for all x, y ∈ X, and positive semidefinite if B(x, x) 0 for every x ∈ X. A bilinear operator B is said to be a lattice bimorphism if the partial mappings B(x, ·) : y ′ → b(x, y ′ ) (y ′ ∈ Y ) and B(·, y) : x′ → B(x′ , y) (x′ ∈ X) are lattice homomorphisms for all 0 x ∈ X and 0 y ∈ Y . 3.12.A.2. Given a lattice bimorphism β : X × Y → R, there are two lattice homomorphisms σ : X → R and τ : Y → R such that β(x, y) = σ(x)τ (y) (x ∈ X, y ∈ Y ). If, in addition, X = Y and β is symmetric then we can take σ = τ . ⊳ We assume that the lattice bimorphism β : X × Y → R is nonzero, since otherwise we have nothing to prove. Choose 0 x0 ∈ X so that τ := β(x0 , ·) be a nonzero lattice homomorphism. Take u ∈ X+ and put e = x0 + u. It is clear that the three lattice homomorphisms βx0 , βu , and βe are connected by the equality βe = βx0 + βu . By the Kutateladze Theorem 3.1.11 ((1) ⇐⇒ (6)), for appropriate r, s ∈ R+ we have τ = βx0 = rβe and βu = sβe . Since τ = 0, we have r > 0 and so βu = γτ , where γ := s/r. So, for every u ∈ X+ there exists a number γ(u) 0 such that βu = γ(u)τ ; i.e., β(u, y) = γ(u)τ (y) for all u ∈ X+ and y ∈ Y . Hence, in particular, γ(x0 ) = 1. 174 Chapter 3. Order Bounded Operators Since, for y ∈ Y+ , the functional β(·, y) is a lattice homomorphism; therefore, for all u, u′ ∈ X+ and λ ∈ R+ we have γ(u + u′ )τ (y) = γ(u)τ (y) + γ(u′ )τ (y), γ(u ∨ u′ )τ (y) = γ(u)τ (y) ∨ γ(u′ )τ (y), γ(λu)τ (y) = λγ(u)τ (y) (y ∈ Y+ ). Hence, γ is additive, positive homogeneous, and join preserving. Put σ(x) := γ(x+ ) − γ(x− ) (x ∈ X). Then the functional σ extends γ onto the whole lattice X and σ is a lattice homomorphism. Moreover, for x ∈ X and y ∈ Y we have β(x, y) = β(x+ , y) − β(x− , y) = γ(x+ )τ (y) − γ(x− )τ (y) = σ(x)τ (y). If X = Y and β is symmetric then σ(x)τ (y) = τ (x)σ(y) 1 for all x ∈ X and y ∈ Y ; hence, τ (y) = τ (x0 )σ(y). By putting ρ := τ (x0 )σ we obtain the representation β(x, y) = ρ(x)ρ(y). ⊲ 3.12.A.3. Theorem. Given an arbitrary lattice bimorphism B : X × Y → Z, there are two lattice homomorphisms S : X → Z u and T : Y → Z u such that B(x, y) = S(x)T (y) (x ∈ X, y ∈ Y ). If, in addition, X = Y and B is symmetric then we can take S = T . ⊳ The reduction of the general case to the scalar case is carried out by means of Boolean valued analysis. To apply the latter, it is important to observe that 3.12.A.2 remains valid on replacing X and Y with vector lattices over an ordered ﬁeld F satisfying the inclusions Q ⊂ F ⊂ R. Furthermore, it is worth taking account of the fact that the functionals β, σ, and τ act into R and are F-linear. The Kutateladze Theorem, which is a key tool of the above proof, remains valid for those functionals. Turning to the general case and recalling the Gordon Theorem, we can assume that the universally complete vector lattice Z u is the descent R↓ of the reals R from the Boolean valued model V(B) with B := P(Z u ). Take the ordered ﬁeld F to be the standard name R∧ of R. Then X ∧ and 3.12. Variations on the Theme 175 Y ∧ are vector lattices over F within V(B) . The technique of ascending and descending (cp. 1.6.8) yields existence of β ∈ V(B) such that [[ β : X ∧ × Y ∧ → R ]] = 1, [[ β is a lattice F-bimorphism ]] = 1, [[ β(x∧ , y ∧ ) = B(x, y) ]] = 1 (x ∈ X, y ∈ Y ). The above fact on the structure of real lattice homomorphisms is valid within V(B) according to the transfer principle 1.4.1. Applying the maximum principle 1.4.2, we ﬁnd elements σ, τ ∈ V(B) such that [[ σ : X ∧ → R ]] = [[ τ : Y ∧ → R ]] = 1, [[σ and τ are lattice F-homomorphisms]] = 1, [[(∀ x ∈ X ∧ ) (∀ y ∈ Y ∧ ) β(x, y) = σ(x)τ (y)]] = 1. Let S and T denote the modiﬁed descents of σ and τ as deﬁned in 1.5.8: S : X → R↓, T : Y → R↓, ∧ [[S(x) = σ(x )]] = [[T (y) = τ (y ∧ )]] = 1 (x ∈ X, y ∈ Y ). By Corollary 3.3.6 (1), S and T are lattice homomorphisms. Moreover, the representation B(x, y) = S(x)T (y) is valid, since for x ∈ X and y ∈ Y we have [[B(x, y) = β(x∧ , y ∧ ) = σ(x∧ )τ (y ∧ ) = S(x)T (y)]] = 1. The fact that B is symmetric amounts to the fact that β is symmetric within V(B) ; therefore, we can take σ and τ coincident, which is equivalent to the equality S = T . ⊲ 3.12.A.4. Say that a bilinear operator B : X ×Y → G is disjointness preserving if for all x ∈ X and y ∈ Y we have x1 ⊥ x2 =⇒ B(x1 , y) ⊥ B(x2 , y) (x1 , x2 ∈ X), y1 ⊥ y2 =⇒ B(x, y1 ) ⊥ B(x, y2 ) (y1 , y2 ∈ Y ). It is clear that a bilinear operator B is disjointness preserving if and only if B(x, ·) : Y → G and B(·, y) : X → G are disjointness preserving for all x ∈ X and y ∈ Y . A positive disjointness preserving bilinear operator is a lattice bimorphism, since B(x, ·) and B(·, y) are lattice homomorphisms for x 0 and y 0. 176 Chapter 3. Order Bounded Operators 3.12.A.5. Corollary. Each order bounded disjointness preserving bilinear operator B : X × Y → Z is representable as the product B(x, y) = S(x)T (y) (x ∈ X, y ∈ Y ), where one of the two operators S : X → Z u and T : Y → Z u can be chosen to be a lattice homomorphism, while the other be bounded and disjointness preserving. If, in addition, X = Y and B is symmetric, then we can take T = πS − π ⊥ S, where π is a band projection in Z u . ⊳ Let a bilinear operator B be order bounded and disjointness preserving. By Theorems 3.2.8 and 3.4.4 there exists a projection π in Z u such that πB = B + and π ⊥ B = −B − ; moreover, B + and B − are lattice bimorphisms. According to Theorem 3.12.A.3 we have the representations B + (x, y) = S1 (x)T1 (y), B − (x, y) = S2 (x)T2 (y) (x ∈ X, y ∈ Y ), where S1 , S2 : X → Z u and T1 , T2 : Y → Z u are lattice homomorphisms. Denote the bilinear operator (x, y) → S(x)T (y) by S ⊙ T . We have B = B + − B − = πB + π ⊥ B = πS1 ⊙ T1 − π ⊥ S2 ⊙ T2 = πS1 ⊙ πT1 − π ⊥ S2 ⊙ π ⊥ T2 . Put S = πS1 − π ⊥ S2 and T = πT1 + π ⊥ T2 . Then S is order bounded and disjointness preserving, T is a lattice homomorphism, and S ⊙ T = (πS1 − π ⊥ S2 ) ⊙ (πT1 + π ⊥ T2 ) = πS1 ⊙ πT1 + πS1 ⊙ π ⊥ T2 − π ⊥ S2 ⊙ πT1 − π ⊥ S2 ⊙ π ⊥ T2 = πS1 ⊙ πT1 − π ⊥ S2 ⊙ π ⊥ T2 = B as required. The rest is obvious. ⊲ 3.12.A.6. Corollary. Let X and Y be vector lattices, T an order bounded disjointness preserving operator from X to Y , and Y0 a vector sublattice of Y generated by T (X). Then there exists a unique algebra and lattice homomorphism T2 from Orth(X) to Orth(Y0 ) such that T2(π)(T x) = T (πx) (π ∈ Orth(X), x ∈ X). ⊳ Assume without loss of generality that Y = Y0 . Moreover, the proof can be reduced to the case of positive B according to Theorem 3.4.3. It suﬃces to note that if T : X → Y is a lattice homomorphism, then the bilinear operator B : Orth(X) × X → Y deﬁned 3.12. Variations on the Theme 177 as (π, x) → T (πx) is a lattice bimorphism and apply the above result. Indeed, by Theorem 3.12.A.3 there are lattice homomorphisms S̄ : Orth(X) → Y u and T̄ : X → Y u such that T (πx) = S̄(π)T̄ (x) for all x ∈ X and π ∈ Orth(X). It follows that the element u := S̄(IX ) is an order unit in Y u and T̄ (X)⊥⊥ = Y u , since T x = uT̄ x for all x ∈ X, and hence u−1 exists in the f -algebra Y u . Denote now by T2(π) the multiplication operator y → u−1 S̄(π)y (y ∈ Y ) and observe that T2(π) is an orthomorphism on Y . Indeed, if y = T x then T2(π)y = S̄(π)u−1 T x = S̄(π)T̄ x = T (πx) ∈ Y , so that T2(π) sends T (X) and hence the whole Y into Y . Clearly, T2(IX ) = IY , since by the same reason T2(IX )T x = S̄(IX )u−1 T x = S̄(IX )T̄ x = T x. Moreover, T2 is a lattice homomorphism, since so is S̄. Given π, ρ ∈ P(Y ) and x ∈ X we have u−1 S̄(πρ)T̄ x = u−1 T (πρx) = u−1 S̄(π)T̄ (ρx) = u−1 S̄(π)u−1 T (ρx) = u−1 S̄(π)u−1 S̄(ρ)T̄ x, whence T2(πρ) = T2(π)T2(ρ) and T2 is an f -algebra homomorphism. Observe ﬁnally that T (πx) = S̄(π)T̄ x = u−1 S̄(π)T x = T2(π)T x. ⊲ 3.12.A.7. Theorem. Assume that X, Y , and Z are vector lattices with Z having the projection property. An order bounded bilinear operator B : X × Y → Z is disjointness preserving if and only if for every π ∈ P(Z) the subspaces Xπ := {ker(πB(·, y)) : y ∈ Y } and Yπ := {ker(πB(x, ·)) : x ∈ X} are order ideals respectively in X and Y , and the kernel of every stratum πB of B with π ∈ P(Z) is representable as Xσ × Yτ : σ, τ ∈ P(Z); σ ∨ τ = π . ker(πB) = ⊳ The necessity is immediate from 3.12.A.5. The proof of the sufﬁciency can be deduced from a corresponding scalar result by means of interpreting it within the appropriate Boolean valued model similar to that in Theorem 3.4.2. As to the scalar case, the following is true: An order bounded bilinear functional β : X × Y → R is disjointness preserving if and only if ker(β) = (X0 × Y ) ∪ (X × Y0 ) for some order ideals X0 ⊂ X and Y0 ⊂ Y . Indeed, assume that the latter is fulﬁlled and take y ∈ Y . If y ∈ Y0 then β(·, y) ≡ 0, otherwise ker(β(·, y)) = X0 and β(·, y) is disjointness preserving by 3.4.1 (7). Similarly, β(x, ·) is disjointness preserving for all x ∈ X. The converse follows from 3.12.A.2. ⊲ 178 Chapter 3. Order Bounded Operators 3.12.B. Disjointness Preserving Sets of Operators. 3.12.B.1. A nonempty subset D of L∼ (X, Y ) is called n-disjoint in L∼ (X, Y ) if |T0 x0 | ∧ · · · ∧ |Tn xn | = 0 for all T0 , . . . , Tn ∈ D and pairwise disjoint x0 , . . . , xn ∈ X. An n-disjoint set M in L∼ (X, Y ) is said to be maximal if every n-disjoint set in L∼ (X, Y ) including M coincides with M . A 1-disjoint set of operators is also called disjointness preserving. More precisely, a nonempty subset D of L∼ (X, Y ) is disjointness preserving in L∼ (X, Y ) if S(u) ⊥ T (v) for all S, T ∈ D and u, v ∈ X with u ⊥ v. Observe some immediate consequences of the deﬁnition. An order bounded operator T from X into Y is n-disjoint if and only if the singleton {T } is an n-disjoint set in L∼ (X, Y ). Therefore, each member of an n-disjoint set in L∼ (X, Y ) is an order bounded n-disjoint operator. Moreover, the nonempty subset D of L∼ (X, Y ) is n-disjoint in L∼ (X, Y ) if and only if each collection of n + 1 elements {T0 , . . . , Tn } of the members of D is n-disjoint. 3.12.B.2. Suppose that X is a vector sublattice of Y . A mapping T : X → Y is called an orthomorphism from X to Y if T is order bounded and x ⊥ y implies T x ⊥ y for all x ∈ X and all y ∈ Y . The set of all orthomorphisms from X to Y is denoted by Orth(X, Y ). It is easily seen that if T is an orthomorphism from X to Y then T (X) ⊂ X ⊥⊥ . Moreover, the representation holds: Orth(X, Y ) = {T |X : T ∈ Orth(Y u ), T (X) ⊂ Y }. Indeed, the universal completion Y u of a vector lattice Y is an f algebra with a multiplicative unit. Each orthomorphism T from X to Y extends uniquely to an orthomorphism T̂ on Y u . Each orthomorphism on Y u is a multiplication operator. Therefore, if T ∈ Orth(X, Y ), then there exists some y ∈ X u such that T (x) = yx holds for all x ∈ X, so that y · X ⊂ Y (cp. Aliprantis and Burkinshaw [28, Theorem 2.63]). 3.12.B.3. Consider an example. For D ⊂ L∼ (Y, Z) and T : X → Y put D ◦T := {S ◦T : S ∈ D}. If T, T1 , . . . , Tn are lattice homomorphisms from X to Y then Orth(T (X), Y ) ◦ T is a disjointness preserving set and the set Orth(T1 (X), Y ) ◦ T1 + · · · + Orth(Tn (X), Y ) ◦ Tn is n-disjoint. The next aim is to demonstrate that this example is typical. 3.12.B.4. Given n pairwise disjoint nonzero real lattice homomorphisms h1 , . . . , hn on a vector lattice X, there exist pairwise disjoint 3.12. Variations on the Theme 179 elements x1 , . . . , xn ∈ X such that hı (xj ) = δij for all ı, j := 1, . . . , n (with δı,j the Kronecker symbol). ⊳ Pick uı ∈ X+ with hı (uı ) > 0 and put u := u1 + · · · + un . By the Kakutani–Kreı̆ns Representation Theorem the order ideal Xu in X generated by u can be identiﬁed with a norm dense vector sublattice of C(Q) containing constants and separating points, where C(Q) is the Banach lattice of continuous functions on a Hausdorﬀ compact topological space Q. Moreover, u corresponds under this identiﬁcation to the identically one function 1 ∈ C(Q). Then the restrictions h1 |Xu , . . . , hn |Xu are pairwise disjoint lattice homomorphisms. Let ĥı stand for the extension of hı |Xu to C(Q) by norm continuity. Clearly, ĥ1 , . . . , ĥn are also pairwise disjoint nonzero lattice homomorphisms and so there exist distinct points q1 , . . . , qn ∈ Q such that ĥı coincides with the Dirac measure δqı : x → x(qı ) (x ∈ C(Q)). By the Tietze–Urysohn Theorem we can ﬁnd pairwise disjoint continuous functions y1 , . . . , yn ∈ C(Q) such that yı (qı ) = 1 and 0 yı (q) 1 for all q ∈ Q and ı := 1, . . . , n. Take ȳı ∈ Xu so that &yı − ȳı & < ε < 1/2 and note that hı (ȳı ) − ε > 1 − 2ε > 0 and ȳı − ε1 yı . Put xı := (hı (ȳı ) − ε)−1 (ȳı − ε1) ∨ 0 and observe that {x1 , . . . , xn } ⊂ X is the required collection. ⊲ 3.12.B.5. A nonempty set D in X ∼ is n-disjoint if and only if there exist pairwise disjoint lattice homomorphisms h1 , . . . , hn : X → R such that D ⊂ R · h1 + . . . + R · hn . Moreover, D is maximal if and only if either D = Hom(X, R) = {0} or D = R · h1 + · · · + R · hm with nonzero h1 , . . . , hm and m := min{n, cat(X ∼ )}, where cat(X ∼ ) stands for the cardinality of atoms in P(X ∼ ) (see 3.10.4). In this event the collection {h1 , . . . , hm } is unique up to permutation. ⊳ The suﬃciency is obvious, so only the necessity will be proved. Suppose that D = {0}. There is no loss of generality in assuming that f ∈ D implies |f | ∈ D. According to Theorem 3.7.7 each functional in D is decomposable into a sum of disjointness preserving components. Let D0 stand for the set of all such components of all functionals in D. We claim that, assuming n-disjointness of D, there is at most n nonzero pairwise disjoint members in D0 . Let {h1 , . . . , hm } be a disjoint collection of nonzero lattice homomorphisms in D0 . By the above we can pick m nonzero pairwise disjoint elements x0 , . . . , xm ∈ X+ such that hı (xj ) = δı,j for all 1 ı, j m. By construction, for each ı m we can choose 0 fı ∈ D with fı = hı + · · · , so that fı (xı ) = hı (xı ) + · · · hı (xı ) = 1. It follows that |fı (xı )| ∧ · · · ∧ |fm (xm )| 1 and so m n by 180 Chapter 3. Order Bounded Operators assumption. Evidently, R · h1 + · · · + R · hm is a maximal n-disjoint set in X ∼ including D. ⊲ 3.12.B.6. Given a nonempty set D in L∼ (X, R↓), put D↑ := {T ↑ : T ∈ L∼ (X, R↓)} and ∆ := (D↑)↑. If D is n-disjoint in L∼ (X, R↓) for some natural n ∈ N, then [[∆ is n∧ -disjoint in (X ∧ )∼ ]] = 1. Moreover, D is maximal if and only if [[∆ is maximal ]] = 1. ⊳ Assuming that D is n-disjoint in L∼ (X, R↓), prove that ∆ is n∧ disjoint in (X ∧ )∼ within V(B) . The sentence “∆ is n∧ -disjoint in (X ∧ )∼ ” can be written as Φ ≡ ∀ τ : {0, . . . , n}∧ → ∆ ∀ κ : {0, . . . , n}∧ → X ∧ (∀ ı, j n∧ )(ı = j → κ(ı) ⊥ κ(j)) → |τ (ı)κ(ı)| = 0 . ın∧ We have to prove that [[Φ]] = 1. Calculating the Boolean truth values for the universal quantiﬁers and taking 1.5.9 into account, we see that [[Φ]] = 1 if and only if [[|T (0)k(0)| ∧ · · · ∧ |T (n)k(n)| = 0]] = 1 for all mappings T = τ ↓ : {0, . . . , n} → ∆↓ and k = κ↓ : {0, . . . , n} → X ∧ ↓ with [[κ(ı∧ ) ⊥ κ(j∧ )]] = 1 for all ı = j. Since ∆↓ = mix(D↑) and X ∧ = mix({x∧ ↓ : x ∈ X}), there exists a partition of unity (bξ )ξ∈Ξ in B and for each ı = 0, . . . , n there are families (Tξ,ı )ξ∈Ξ in L∼ (X, R↓) and (xξ,ı )ξ∈Ξ in X such that T (ı) = mixξ∈Ξ (bξ Tξ,ı ↑) and k(ı) = mixξ∈Ξ (bξ x∧ξ,ı )ξ∈Ξ . Note that bξ [[κ(ı∧ ) = k(ı) = x∧ξ,ı ]], so that xξ,ı ⊥ xξ,j whenever bξ = 0 and ı = j. Thus, from the n-disjointness of D we deduce that bξ [[ |Tξ,0 xξ,0 | ∧ · · · ∧ |Tξ,n xξ,n | = 0]] [[T (ı)x∧ξ,ı = Tξ,ı (xξ,ı )]] ∧ [[k(ı) = x∧ξ,ı ]] ∧ ın [[|T (0)k(0)| ∧ · · · ∧ |T (n)k(n)| = 0]]. This yields the required relation, since ξ∈Ξ bξ = 1. ⊲ 3.12.B.7. Given a nonempty D ⊂ L(X, R↓), put RD := T ∈ D}. Then RD = χ([[∆ = {0}]]). {RT : ⊳ This is immediate from 3.8.4 and Deﬁnition of ∆ in 3.12.B.6. ⊲ We have gathered now all of the ingredients for proving the main result of this section. 181 3.12. Variations on the Theme 3.12.B.8. Theorem. Let X and Y be vector lattices with Y having the projection property. A nonempty set D in L∼ (X, Y ) is ndisjoint if and only if there exist pairwise disjoint lattice homomorphisms T1 , . . . , Tn from X to Y u such that D is contained in Orth(T1 (X), Y ) ◦ T1 + · · · + Orth(Tn (X), Y ) ◦ Tn . Moreover, D is maximal if and only if, additionally, there is a partition of unity π0 , . . . , πn in P(Y u ) such that π0 ◦ D = Hom(X, π0 Y ) = {0}, D = Orth(T1 (X), Y ) ◦ T1 + · · · + Orth(Tn (X), Y ) ◦ Tn , πm + · · · + πn = RTm (m := 1, . . . , n). The collection T1 , . . . , Tn in this representation is unique up to P(Y )permutation. ⊳ The claim reduces to the case of Y universally complete, since by the Gordon Theorem we can assume that Y = R↓ without loss of generality. Let D be an n-disjoint set in L∼ (X, R↓) and ∆ is deﬁned as in 3.12.B.6. Working within V(B) and using the transfer principle we conclude that ∆ is n∧ -disjoint in (X ∧ )∼ and, by 3.12.B.5, ∆ ⊂ R · τ (1∧ ) + · · · + R · τ (n∧ ) for some τ : {1, . . . , n}∧ → Hom(X ∧ , R). Just as in 3.12.B.6 put T := τ ↓ and note that T sends {1, . . . , n} to Hom(X ∧ , R)↓. If T ∈ D then [[T ↑ ∈ ∆]] = 1, so that there exists α ∈ V(B) with // [[α : {1, . . . , n}∧ → R]] = T ↑ = 00 α(ı)T (ı) = 1. ∧ ın Put αı := α↓(ı) and Tı := T (ı)↓ for all ı := 1, . . . , n. Then α1 , . . . , αn ∈ R↓, T1 , . . . , Tn ∈ Hom(X, R↓), so that the chain of internal identities T x = T ↑x∧ = α(ı)T (ı)x∧ = ∧ ın ın αı T (ı)↓x = ın αi Ti x with arbitrary x ∈ X yields the required representation T = nı=1 αi Tı . Actually we have proved more: It is clear from the above argument that ∧ the double descent (Λ↓)↓ := {τ ↓ : τ ∈ Λ↓} of Λ := R · τ (1 ) + · · · + ∧ R · τ (n ) consists of all operators representable as ın αi Tı for some α1 , . . . , αn ∈ R↓. Assume now that D is maximal. Then [[∆ is maximal]] = 1 by 3.12.B.6. The maximality condition in 3.12.B.5 can be symbolized as 182 Chapter 3. Order Bounded Operators follows: Ψ ≡ ∆ = Hom(X ∧ , R) = {0} ∨ ∃ m ∈ {1, . . . , n}∧ (∀ ı m)(τ (ı) = 0) ∧(∆ = R · τ (1) + · · · + R · τ (m)) . Put b0 := [[∆ = Hom(X ∧ , R) = {0}]]. By transfer [[Ψ]] = 1, and the calculation of Boolean valued truth values yields b∗0 = [[∆ = 0]] = n [[∆ = R · T (1) + · · · + R · T (m)]] ∧ m=1 m [[T (ı) = 0]]. ı=1 It follows that there exists a ﬁnite partition of unity {b0 , b1 , . . . , bn } in B such that bm [[∆ = R · T (1) + · · · + R · T (m)]] and bm [[T (ı) = 0]] (ı m) for all m := 1, . . . , n. Put πm := χ(bm ) and observe that {π0 , π1 , . . . , πn } is a partition of unity in P(R↓). Note also that (Λ↓)↓ = mix(D), so that (Λ↓)↓ = D, since D is maximal. Combing the above and using 3.12.B.6, we see that RD = π0⊥ = π1 + · · · + πn , πm RT1 ◦ . . . ◦ RTm , πm ◦ D = πm ◦ (R↓ · T1 + · · · + R↓ · Tm ) (m := 1, . . . , n). The ﬁrst identity gives π0 ◦ D = Hom(X, π0 Y ) = {0}. The second yields πm +· · ·+πn RTm (m := 1, . . . , n). Replacing Tm by (πm +· · ·+πn ) Tm , if need be, and summing the third identities over m brings about the required maximality conditions. ⊲ 3.12.C. Atomic Decomposition of Vector Measures. 3.12.C.1. Let A be a Boolean algebra and let Y be a vector lattice. By a vector measure we mean an arbitrary mapping μ : A → Y which is finitely additive, i.e., μ(a1 ∨a2 ) = μ(a1 )+μ(a2 ) for all disjoint a1 , a2 ∈ A . A measure μ is bounded if μ(A ) is an order bounded subset of Y . Denote by ba(A , Y ) the space of all bounded Y -valued measures and put ba(A ) := ba(A , R). A measure μ ∈ ba(A , Y ) is positive if μ(a) 0 for all a ∈ A . It is well known that ba(A , Y ) is a Dedekind complete vector lattice whose positive cone coincides with the set of positive measures. Moreover, |μ|(a) = sup{μ(b) : b ∈ A , b a} for all a ∈ A . A measure μ ∈ ba(A , Y ) is said to be disjointness preserving if a1 ∧ a2 = 0 implies |μ(a1 )| ∧ |μ(a2 )| = 0 for all a1 , a2 ∈ A . We say that μ is diﬀuse if μ is disjoint from all disjointness preserving measures 3.12. Variations on the Theme 183 and atomic if μ lies in the band generated by disjointness preserving measures. 3.12.C.2. Theorem. Let A be a Boolean algebra and let Y be a universally complete vector lattice represented as Y = R↓. Given μ ∈ ba(A , Y ), the modified ascent m := μ↑ is an order bounded finitely additive real measure on A ∧ within V(B) ; i.e., [[m ∈ ba(A ∧ , R)]] = 1. The mapping μ → μ↑ is a lattice isomorphism between the Dedekind complete vector lattices ba(A , Y ) and ba(A ∧ , R)↓. ⊳ The proof can be extracted from Theorem 3.3.3. ⊲ Henceforth m will denote the bounded measure from A ∧ into R within V(B) corresponding to μ ∈ ba(A , Y ) by the above theorem. Observe some immediate consequences: (1) μ is disjointness preserving if and only if m is disjointness preserving within V(B) ; (2) μ is atomic if and only if m is atomic within V(B) ; (3) μ is diﬀuse if and only if m is diﬀuse within V(B) . 3.12.C.3. Hammer–Sobczyk Decomposition Theorem. Let μ be a finitely additive real measure on a Boolean algebra A . Then there exist a sequence (νn )n∈N of pairwise disjoint {0, 1}-valued measures on A , a sequence (rn )n∈N of reals, and a diﬀuse measure μ0 on A , such ∞ ∞ that |r | < ∞ and, μ = μ + n 0 n=1 n=1 rn νn . Furthermore, this decomposition is unique. ⊳ See Rao K. P. S. B. and Rao M. B. [342, Theorem 5.2.7]. ⊲ 3.12.C.4. Take a measure μ ∈ ba(A , Y ) and a nonzero element π ∈ P(Y ). The symbol [e] stands for the projection onto the band {e}⊥⊥ generated by the element e ∈ Y . An element a ∈ A is called a π-atom of the measure μ if π [|μ|(a)] and for all a0 ∈ A , a0 a, the elements πμ(a0 ) and πμ(a \ a0 ) are disjoint. In case Y = R we speak of atoms instead of π-atoms. More precisely, an atom of a measure μ ∈ ba(A ) is an element a0 ∈ A such that μ(a0 ) = 0 and for every a ∈ A , a a0 , either μ(a) = 0, or μ(a0 \ a) = 0. 3.12.C.5. Fix b ∈ B and put π := χ(b). An element a ∈ A is a π-atom of the measure μ if and only if b [[a∧ is an atom of m]] = 1. ⊳ The sentence “a∧ is an atom of m” can be formalized as Φ(m, a∧ , A ∧ ) ≡ |m|(a∧ ) = 0 ∧ (∀a0 ∈ A ∧ )(a0 a∧ → (m(a0 ) = 0 ∨ m(a∧ \ a0 ) = 0). 184 Chapter 3. Order Bounded Operators Thus, the estimate b [[Φ(m, a∧ , A )]] amounts to the system of inequalities b [[|m|(a) = 0]] and b 1 ⇒ [[m(a∧0 ) = 0 ∨ m(a∧ \ a∧0 ) = 0]] for all a0 ∈ A , a0 a or, equivalently, b [[μ(a0 ) = 0]] ∨ [[μ(a \ a0 ) = 0]] (a0 a). b [[|μ|(a) = 0]], The ﬁrst inequality, means that b [|μ|(a)], and the remaining one is satisﬁed if and only if b = b1 ∨b2 for some b1 , b2 ∈ B with χ(b1 )μ(a0 ) = 0, χ(b2 )μ(a \ a0 ) = 0, and b1 ∧ b2 = 0. To ensure this, we need only to put b1 := [[μ(a0 ) = 0]] ∧ b, and b2 := [[μ(a \ a0 ) = 0]] ∧ b. The identities χ(b1 )μ(a0 ) = 0 and χ(b2 )μ(a \ a0 ) = 0 are equivalent to the inequalities χ(b2 ) [μ(a0 )] and χ(b1 ) [μ(a \ a0 )], which in turn mean that πμ(a0 ) and πμ(a \ a0 ) are disjoint. ⊲ 3.12.C.6. Let Y be a Dedekind complete vector lattice. Boolean homomorphisms μ1 , μ2 : A → P(Y ) are disjoint in the vector lattice ba(A , Orth(Y )) if and only if there exist a partition of unity (πξ )ξ∈Ξ in P(Y ) and a family (aξ )ξ∈Ξ in A , such that πξ μ1 (aξ ) = 0 and πξ μ2 (a∗ξ ) = 0 for all ξ ∈ Ξ. ⊳ Assume that Y u = R↓ and put mı := μı ↑ (ı = 1, 2). Since B is the descent of the two-element Boolean algebra {0, 1}B ∈ V(B) by 1.8.1, V(B) |= “m1 and m2 are {0, 1}-valued measures.” Clearly, μ1 and μ2 are disjoint in ba(A , Orth(Y )) if and only if V(B) |=“m1 and m2 are disjoint elements of the vector lattice ba(A ∧ )” by Theorem 3.12.C.2 (cp. 3.3.5 (5)). At the same time, the disjointness of m1 and m2 is equivalent to 1 = [[(∃ a ∈ A )(m1 (a) = 0 ∧ m2 (a∗ ) = 0)]] [[m1 (a∧ ) = 0]] ∧ [[m2 ((a∗ )∧ ) = 0]] = a∈A = [[μ1 (a) = 0]] ∧ [[μ2 (a∗ ) = 0]]. a∈A This amounts to saying that there exist a partition of unity (bξ )ξ∈Ξ in B and a family (aξ )ξ∈Ξ in A , such that bξ [[μ1 (aξ ) = 0]] and bξ [[μ2 (a∗ξ ) = 0]] for all ξ ∈ Ξ. This is equivalent to the desired condition with πξ := χ(bξ ) by 2.2.4 (G). ⊲ 3.12.C.7. For every measure μ ∈ ba(A , Y ) the following are equivalent: (1) μ is disjointness preserving. 185 3.12. Variations on the Theme (2) There exists a Boolean homomorphism h : A → B such that μ(a) = h(a)μ(1) for all a ∈ A . (3) If b [|μ|(a)] for some a ∈ A and b ∈ B, then a is a χ(b)-atom of μ. ⊳ Recall 3.12.C.5 and use the arguments similar to 3.12.C.6. ⊲ We are now in a position to state the B-atomic decomposition theorem and the fact that the B-atomic component of a vector measure is the sum of a disjoint sequence of “spectral measures.” 3.12.C.8. Theorem. Assume that Y is a Dedekind complete vector lattice. For every measure μ ∈ ba(A , Y ) there exist a diﬀuse measure μ0 ∈ ba(A , Y ), a sequence (νn )n∈N of pairwise disjoint Boolean homomorphisms from A into P(Y ), and ∞a sequence (yn )n∈N in Y , such that |yn+1 | |yn | (n ∈ N), the series k=1 |yn | is o-convergent, and μ(a) = μ0 (a) + ∞ νn (a)yn (a ∈ A ). n=1 This representation is unique in the following sense: If μ0 ∈ ba(A , Y ), (ν̄n )n∈N and (ȳn )n∈N obey the above conditions, then μ̄0 = μ0 and there exists an N × N matrix (πk,n ) whose rows and columns are partitions on unity in P(Y ) such that for all a ∈ A and n ∈ N we have ν̄n (a) = ∞ πk,n νk (a), ȳn = k=1 ∞ πk,n yk . k=1 ⊳ Theorem 3.12.C.2 enables us to “scalarize” the problem. By transfer we can apply the Hammer–Sobczyk Decomposition Theorem to the measure m := μ↑ ∈ ba(A ∧ , R) within V(B) . Using the maximum principle, we can pick m0 ∈ V(B) , (hn ) ⊂ V(B) , and (yn ) ⊂ R such that the following hold within V(B) : m0 : A ∧ → Y is a diﬀuse measure; hn : A ∧ → {0, 1} is a sequence of pairwise disjoint measures; ∞ |yn+1 | |yn | (n ∈ N) and the series |yn | converges; n=1 m(a) = m0 (a) + ∞ n=1 hn (a)yn (a ∈ A ∧ ). 186 Chapter 3. Order Bounded Operators Let μ0 and νn be the modiﬁed descents of the measures m0 and hn , ˙ respectively. Clearly, μ0 ∈ ba(A , Y ) is diﬀuse by 3.12.C.2(3). From 3.12.C.2 and 3.12.C.7 it follows that (νn ) is a sequence of pairwise disjoint Boolean homomorphisms. The o-convergence of the series ∞ n=1 |yn | and the required representation of μ are deduced using 2.4.7. The proof of uniqueness is based on the argument similar to that of 3.10.3. ⊲ Let us conclude the section with the characterization of diﬀuse measures which is similar to that for diﬀuse operators (cp. 3.11.4). 3.12.C.9. Theorem. Let Y be a Dedekind complete vector lattice. For every measure μ ∈ ba(A , Y ) the following are equivalent: (1) μ is diﬀuse. (2) μ has no summands of the form a → h(a)y (a ∈ A ) with 0 = y ∈ Y and h : A → B a Boolean homomorphism. (3) For all 0 e ∈ Y and π ∈ P(Y ) with πe = 0 there exist a nonzero projection π0 π and a finite disjoint family measures μ1 , . . . , μn ∈ ba(A , Y ) such that μ = μ1 + · · · + μn and π0 |μk |(1) e (k := 1, . . . , n). 3.13. Comments 3.13.1. (1) In 1936, Kantorovich [193] laid grounds for the theory of regular operators in vector lattices. Also, the Riesz–Kantorovich Theorem (3.1.2) appeared in this article for the ﬁrst time. Riesz [346] formulated an analogous assertion for the space of continuous linear functionals over the lattice C[a, b] in his famous talk at the International Mathematical Congress in Bologna in 1928 and thereby became enlisted in the cohort of the founders of the theory of ordered vector spaces. (2) Abramovich in [1] developed a version of the calculus of 3.1.4 in which suprema and inﬁma can be taken over partitions of the argument into disjoint parts. For the modulus of a regular operator, this fact was independently established by Luxemburg and Zaanen in [298]. (3) The problem of dominated extension of linear operators originates with the Hahn–Banach Theorem (see a nice survey by Buskes [77] in which the history, interconnections, and numerous generalizations of the Hahn–Banach Theorem are collected). Theorem 3.1.7 was discovered 3.13. Comments 187 by Kantorovich [191] in 1935. In fact, the converse is also true: A preordered vector space Y has dominated extension property (i.e., Theorem 3.1.7 holds true for all U , V , p, and S) if and only if Y has the least upper bound property. This fact was ﬁrst established by Bonnice and Silvermann [62] and To [392]; an elegant proof with decisive simpliﬁcations is due to Ioﬀe [177]; for more details also see Kusraev and Kutateladze [247]. (4) Theorem 3.1.7 can be considered as an exemplar application of the heuristic transfer principle for Dedekind complete vector lattices (see 2.13.1 (2, 3)). It claims that the Kantorovich principle is valid in relation to the classical Dominated Extension Theorem; i.e., we may replace the reals in the standard Hahn–Banach Theorem by an arbitrary Dedekind complete vector lattice and a linear functional by a linear operator with values in this lattice. (5) Theorem 3.1.9 determines the least (or minimal) extension operator E from L∼ (X0 , Y )+ to L∼ (X, Y )+ which is additive and positively homogeneous, so that E can be extended by diﬀerences to a positive operator to the whole L∼ (X0 , Y ) (for the properties of E see, for instance, Aliprantis and Burkinshaw [28] and Kusraev [228]). The extension theorem for positive order continuous operators (Theorem 2.1.10) is due to Veksler [396]. Theorem 3.1.13 was proved independently by Kutateladze [263, 265], Lipecki [284], and Luxemburg and Schep [296]. Various approaches to Hahn–Banach type theorems for lattice homomorphisms are discussed in Bernau [52]. (6) The theory of positive operators with a vast ﬁeld of applications is thoroughly covered in many books. The following (incomplete) list of monographs that deal with vector lattices and positive (order bounded) operators provides an impression of the subject: Abramovich and Aliprantis [5]; Abramovich, Arenson, and Kitover [6]; Abramovich and Kitover [8]; Akilov and Kutateladze [22]; Aliprantis and Burkinshow [28]; Fremlin [124, 126]; Jameson [178]; Kantorovich, Vulikh, and Pinsker [196]; Krasnosel′skiı̆ [206]; Krasnosel′skiı̆, Zabreı̆ko, Pustylnik, and Sobolevskiı̆ [208]; Krasnosel′skiı̆, Lifshits, and Sobolev [207]; Kriger [209]; Kusraev [222, 228]; Kutateladze (ed.) [272]; Lacey [275]; Lindenstrauss and Tzafriri [280, 281]; Luxemburg and Zaanen [297]; MeyerNieberg [311]; Nagel (ed.) [316]; Nakano [319, 318]; Peressini [335]; Schaefer [355, 356]; Schwarz [361]; Vulikh [403]; and Zaanen [427, 428]. 3.13.2. (1) The study of the order properties of bilinear operators in vector lattices was started more than a half-century ago. The ﬁrst 188 Chapter 3. Order Bounded Operators publication on the topic by Nakano was in 1953. But this article had not attracted specialists and the new achievements had appeared only in the 1970s in the articles by Fremlin [121, 123], Wittstock [416, 417], Cristescu [97], and Kusraev [215]. In two decades after that the bilinear operators were not topical within the theory of operators in vector lattices. But it stands to reason to mention that several particular cases (like bilinear functionals, multiplication on a lattice ordered algebra, and various tensor products) were studied from time to time by various authors; see Schaefer [357, 358]. A more detailed history and the state of the art in the area are reﬂected in the surveys by Boulabiar, Buskes, and Triki [68, 69]; Bu, Buskes, and Kusraev [72]; and Kusraev [234, 235]. (2) In spite of the nice universal property (Theorem 3.2.8), Fremlin’s tensor product has the essential disadvantage: The isomorphism of 3.2.9 does not preserve order continuity. For an order continuous T ∈ Lr (X ⊗ Y, Z) the bilinear operator T ⊗ ∈ BLr (X, Y ; Z) is also order continuous but the converse may fail. An example can be extracted from Fremlin [123] in which he introduced also a construction for the “projective” △ tensor product X ⊗ Y of Banach lattices X and Y as the completion of Fremlin’s tensor product X ⊗ Y under “positive-projective” norm & · &|π| (see [123, Theorem 1 E]. If X = L2 ([0, 1]), then X ⊗ X is order dense in △ △ X ⊗ X but the norm of X ⊗ X is not order continuous (cp. [123, 4 B and 4 C]). Thus, there exists a (norm continuous) positive linear functional △ l ∈ (X ⊗ X)′ which is not order continuous. Clearly, the restriction l0 of l to X ⊗ X is not order continuous, too. At the same time the positive bilinear functional b = l0 ⊗ is separately order continuous, since X has an order continuous norm. (3) The class of orthosymmetric bilinear operators on vector lattices was introduced in Buskes and van Rooij [81] and received much attention in the succeeding years. An inseparable companion of the orthosymmetric bilinear operators turns out to be the concept of square of vector lattice, developed by the same authors in another paper by Buskes and Rooij [82]; Deﬁnition 3.2.11 and Theorems 3.2.12 and 3.2.13 are taken from this paper. (4) For α, s, t ∈ R, α > 0, we denote tα := |t|α sgn(t) and σα (s, t) := 1/α (s + t1/α )α . In a uniformly complete vector lattice X, we introduce new vector operations ⊕ and ∗, while the original order remain unchanged: x ⊕ y := σ(x, y), λ ∗ x := λα x (x, y ∈ X; λ ∈ R). Then X (α) := (X, ⊕, ∗, ) is again a vector lattice. If X is uniformly complete 3.13. Comments 189 then X ⊙ = X (1/2) and x ⊙ y := (xy)1/2 for all x, y ∈ X. If (X, & · &) is a Banach lattice then we can deﬁne also a homogeneous function & · &α : X (α) → R by &x&α := &x&1/α . The pair (X (α) , & · &α ) is called an α-convexification of X if α > 1 and α-concavification if α < 1; cp. Lindenstrauss and Tzafriri [282, pp. 53, 54] and Szulga [374, Deﬁnition 4.4.1]. For the homogeneous functional calculus in uniformly complete vector lattices and Banach lattices see Lindenstrauss and Tzafriri [282, Theorem 1.d.1] and Buskes, de Pagter, and van Rooij [79]. (5) A Banach lattice E is called p-concave (1 p < ∞) if there is a constant C < ∞ such that for every ﬁnite collection {x1 , . . . , xn } ⊂ X the inequality holds (Lindenstrauss and Tzafriri [282, Deﬁnition 1.d.3]): 1 13 3 n n p p3 3 p p 3. 3 C3 &xk & |xk | 3 k=1 k=1 If X is a p-convex Banach lattice for some p 1 then (X (α) , &·&α ) is also a Banach lattice provided that αp 1; see Szulga [374, Proposition 4.8]. In particular, X ⊙ equipped with the norm &x ⊙ x&⊙ := &x&2 becomes a Banach lattice if q > 2 (also see Bu, Buskes, Popov, Tcacius, and Troitsky [73]). (6) The theory of positive bilinear operators partially presented in this book can be developed for positive multilinear operators. In particular, Fremlin’s tensor product of two vector lattices and Buskes–van Roij square of a vector lattice together with their universal properties for the classes of positive bilinear operators (see Theorems 3.2.6 and 3.2.8) and positive orthosymmetric bilinear operators (Theorems 3.2.12 and 3.2.13) were extended to multilinear case by Schep [359] and Boulabiar and Buskes [65], respectively. 3.13.3. In 1935 Kantorovich’s in his ﬁrst deﬁnitive article on vector lattices [191] wrote: “In this note, I deﬁne the new type of space that I call a semiordered linear space. The introduction of such a space allows us to study linear operations of one abstract class (those with values in such a space) as linear functionals.” Theorem 3.3.3 with technical Corollaries 3.3.5 and 3.3.6 presents one of the mathematical realization of this heuristic principle. The two other mathematical forms of Kantorovich’s heuristic principle will be presented in Chapters 4 and 5. 3.13.4. (1) The ﬁrst appearance of disjointness preserving operators in the literature occurred in 1943 in the article [402] by Vulikh implicitly under the name multiplicative linear operations. A systematic study 190 Chapter 3. Order Bounded Operators of disjointness preserving operators began from the articles [11, 12] by Abramovich, Veksler, and Koldunov. Various aspects of disjointness preserving operators have been studied by many authors over the years, and we indicate only a portion of these publications: on multiplicative representations on function spaces (Abramovich, Veksler, and Koldunov [11, 12], Araujo, Beckenstein, and Narici [33], Abramovich, Arenson, and Kitover [6], Henriksen and Smith [171]); on weight-shift factorization (Gutman [157]–[160], [162]); on spectral theory (Arendt [34], Arendt and Hart [36], Huijsmans and de Pagter [174], Meyer-Nieberg [311]); on the inverses of disjointness preserving operators (Abramovich and Kitover [8]); on the various properties of the band generated by disjointness preserving operators and the corresponding band projections (Huijsmans and de Pagter [174], Kolesnikov [200, 201], Tabuev [376]); on polar decomposition (Abramovich, Arenson, and Kitover [8], Boulabiar and Buskes [64], Grobler and Huijsmans [151]). Sometimes order bounded disjointness preserving operators are called Lamperti operators [34, 66] or separating mappings [33, 171]. For more historical remarks and references we refer to the survey [63] by Boulabiar. (2) Theorem 3.4.2 is taken from Kusraev and Kutatetladze [251]. A bilinear version of this result which can proved by using similar arguments see below in 3.12.A.7. The ﬁrst proof of the Theorem 3.4.3 was given in [310] by Meyer. This proof is based upon the Kuratowski–Zorn Lemma (i.e., the Axiom of Choice). Later, the two proofs of Theorem 3.4.3, free of the Kuratowski–Zorn Lemma, were obtained by Bernau in [51] and de Pagter in [329], respectively. Theorem 3.4.4 is immediate from Abramovich, Arenson, and Kitover [6, Theorem 3.3]. In Section 3.4 we show that Boolean valued approach provides a new insight into this circle of problems. Theorems 3.4.8 and 3.4.9 were obtained in Kusraev and Kutateladze [251]. (3) It was proved by Kusraev and Tabuev in [257] that a bilinear version of Meyer’s Theorem is also true: Let X, Y , and G be vector lattices and let B : X × Y → G be an order bounded disjointness preserving bilinear operator. Then b possesses the positive part B + , the negative part B − , and the modulus |B| which are lattice bimorphisms. Moreover, B + (x, y) = B(x, y)+ and B − (x, y) = B(x, y)− for 0 x ∈ X, 0 y ∈ Y , and |B|(|x|, |y|) = |B(x, y)| for all x ∈ X and y ∈ Y . In particular, B is regular. (4) Combining Theorem 3.9.11 with Corollary 3.12.A.5 yields the following result due to Kusraev and Tabuev [258]. 3.13. Comments 191 Theorem. Let X, Y , and Z be vector lattices with Z Dedekind complete and B : X × Y → Z an order bounded disjointness preserving bilinear operator. Then there exist a partition of unity (ρξ )ξ∈Ξ in the Boolean algebra P(Z) and families of positive elements (eξ )ξ∈Ξ in X and (fξ )ξ∈Ξ in Y such that the representation holds ρξ W · σ(x/eξ ) · τ (y/fξ ) (x ∈ X, y ∈ Y ), B(x, y) = oξ∈Ξ where σ and τ are shift operators into Z u with D(σ) and D(τ ) being order dense ideals in X u and Y u , respectively, and W : Z u → Z u is the operator of multiplication by o- ξ∈Ξ ρξ B(eξ , fξ ) (see 3.9.10). (5) The noncommutative analogs of disjointness preserving operators have been studied as well. We present only one result of Wolﬀ [415]. Let A and B be two C ∗ -algebras. A bounded linear operator T : A → B is called disjointness preserving if T (x∗ ) = (T x)∗ for all x ∈ A and yz = 0 implies T (y)T (z) = 0 for all hermitian x, y ∈ A. Let Ih and M (Ih ) stand for the principal ideal in the commutant {h}′ of h generated by h and the multiplier algebra of Ih , respectively. Assume now that A is unital and T is a disjointness preserving operator from A to B which sends the unity of A to h ∈ B. Then T (A) ⊂ Ih and there exists a Jordan ∗-homomorphism S from A into M (Ih ) such that S(1A ) = 1M(Ih ) and T x = hS(x) for all x ∈ A. 3.13.5. (1) Theorem 3.5.4 was established by Kutateladze in [273]. The Moreau–Rockafellar Formula is one of the key tools in subdiﬀerential calculus; various aspects and applications can be found in Kusraev and Kutateladze [247]. Theorem 3.5.8 was stated and proved in this form in Kutateladze [261]. Obviously, the Riesz space in this theorem may be viewed over an arbitrary dense subﬁeld of the reals R. (2) Descending Theorem 3.5.4 from an appropriate Boolean valued universe or, which is equivalent, using the characterization of the modules admitting convex analysis, we can arrive to an analogous description for a dominated module homomorphism with kernel a Riesz subspace in modules over an almost rational subring of the orthomorphism ring of the range (cp. Abramovich, Arenson, and Kitover [6, Theorem 3.3] and Kutateladze [266]). (3) From Theorem 3.5.4 it is immediate that the Stone Theorem cannot be abstracted far beyond the limits of AM -spaces. Indeed, if each closed Riesz subspace of a Banach lattice is an intersection of twopoint relations then there are suﬃciently many Riesz homomorphisms 192 Chapter 3. Order Bounded Operators to separate the points of the Banach lattice under consideration (cp. Schaefer [356, Chapter 3, Section 9]). 3.13.6. Theorem 3.6.2 was obtained by Kutateladze in [274]. The history of the property 3.6.1 is as follows: In 1955 Grothendieck [156] distinguished the subspaces that satisfy 3.6.1 in the space C(Q, R) of continuous real functions on a compact space Q. He determined such a subspace as the set of functions f satisfying some family of relations of the form f (qα1 ) = λα f (qα2 ) (qα1 , qα2 ∈ Q; λα ∈ R, α ∈ A). These subspaces were discovered by Grothendieck and gave the examples of the L1 -predual Banach spaces other than AM -spaces. In 1969 Lindenstrauss and Wulpert characterized these subspaces by using 2.6.1. They also introduced the concept of G-space (cp. [283]). 3.13.7. (1) The notion of n-disjoint operator between vector lattices (Deﬁnition 3.7.1) as well as the main results of Section 3.7 (Theorem 3.7.7 and Corollaries 3.7.9 and 3.7.10 without assuming that the summands are pairwise disjoint) are due to Bernau, Huijsmans, and de Pagter [53] (see also Bernau [52]). Radnaev [341, 340] noticed that, ﬁrst, the disjointness preserving operators T1 , . . . , Tn in the decomposition T = T1 + · · · + Tn can be chosen pairwise disjoint and, second, this decomposition is unique up to “mixture permutation” (cp. 3.7.8). Similar results for dominated operators on lattice normed spaces are collected in Kusraev [228, 5.2.1 and Theorem 5.2.7]. (2) Radnaev [341, 340] found a purely algebraic approach to the proof of Theorem 3.7.7 (see [228, 2.1.10, 5.2.6, 5.2.7]). In the same article he gave various characterizations of n-disjoint operators (see [228, 5.2.1 (2) and 5.2.5]) employing Kutateladze’s canonical sublinear operator method [262, 264] (also see Rubinov [352]). 3.13.8. (1) In [331] de Pagter and Schep raised the problem of ﬁnding conditions for the sum of two order bounded disjointness preserving operators to be disjointness preserving. In Section 3.8 the problem is examined for arbitrary ﬁnite sums in the more general setting of n-disjoint operators. The Boolean valued approach to the problem and the main results are taken from Kusraev and Kutateladze [251]. The equivalence (1) ⇐⇒ (4) in Theorem 3.8.6 is essentially (a) ⇐⇒ (c) in de Pagter and Schep [331, Proposition 2.13 (5)]. (2) Let X, Y , and Z be vector lattices with Z Dedekind complete. Say that the ﬁnite collections {x0 , x1 , . . . , xn } ⊂ X and {y0 , y1 , . . . , yn } ⊂ Y are bidisjoint, if for every two naturals 0 i, j n, 3.13. Comments 193 i = j, either xi ⊥ xj , or yi ⊥ yj . A bilinear operator B from X × Y to Z is called n-disjoint, if |B(x0 , y0 )| ∧ |B(x1 , y1 )| ∧ · · · ∧ |B(xn , yn )| = 0, for all bidisjoint collections {x0 , x1 , . . . , xn } in X and {y0 , y1 , . . . , yn } in Y . For a regular bilinear operator B : X × Y → Z there exists a linear regular operator T : X ⊗ Y → Z such that B = T ⊗, where X ⊗ Y is the Fremlin tensor product of X and Y (see Theorem 3.2.6). It was proved in Kusraev and Tabuev [257] that B is n-disjoint if and only if T is n-disjoint. These facts enable us to transfer some results on regular n-disjoint linear operators to regular n-disjoint bilinear operators. In particular, some versions of Theorems 3.8.6 and 3.8.7 hold for bilinear operators. 3.13.9. (1) In Section 3.9 we present the Boolean valued approach to Gutman’s representation theory for disjointness preserving operators [160, 162]. The main idea of [160, 162] can be worded as follows: The shadow of an operator T : X → Y between vector lattices is the mapping sh := shT : B(X) → B(YT ) deﬁned as sh(B) = T (B)⊥⊥ (B ∈ B(X)). If X and Y have the projection property then we can also deﬁne sh : P(X) → P(YT ) by sh(π) = [T π(X)]; i.e., sh(π) is the band projection onto (T π(X))⊥⊥ . In the latter case a linear operator T : X → Y is disjointness preserving if and only if its shadow shT is a Boolean homomorphism. The shadow generates the so-called shift operator which is a lattice homomorphism on a certain order dense ideal of the universal completion of the departure vector lattice. Both are closely related with the initial disjointness preserving operator and concentrate, in a sense, its multiplicative properties. Using these simplest types of operators, we can construct weighted shift operators; i.e., the composites W ◦ S ◦ w of two orthomorphisms w and W and a shift operator S. Moreover, an arbitrary disjointness preserving operator is representable as the strongly disjoint sum of weighted shift operators (see [228, Subsections 5.3.2, 5.3.6, and 5.3.10]). (2) The shadow of an operator (without introducing the term) was ﬁrst considered by Luxemburg in [291] for lattices homomorphisms and then by Kusraev in [224] for a disjointness preserving operators in lattice normed spaces. The shadow of an operator may fail to be a Boolean homomorphism unless X has the projection property or T is order continuous. It was proved by de Pagter and Schep in [331, Proposition 2.8] that the shadow shT of a lattice homomorphism T is a Boolean homomorphism if and only if T has the unique positive linear extension to the Dedekind completion of X. 194 Chapter 3. Order Bounded Operators 3.13.10. (1) Theorem 3.10.10, the main result of Section 3.10, was proved by Tabuev in [376, Theorem 2.2] with standard tools. The pseudoembedding operators are closely connected with the so-called order narrow operators. A linear operator T : X → Y is order narrow if for every x ∈ X+ there exists a net (xα ) in X such that |xα | = x for all α and (T xα ) is order convergent to zero in Y ; see [308, Deﬁnition 3.1]. The main result by Maslyuchenko, Mykhaylyuk, and Popov in [308, Theorem 11.7 (ii)] states that if X and Y are Dedekind complete vector lattices with X atomless and Y an order ideal of some order continuous Banach lattice then an order bounded order continuous operator is order narrow if and only if it is pseudoembedding. (2) The term pseudoembedding operator stems from a result by Rosenthal [349] which asserts that a nonzero bounded linear operator in L1 is a pseudoembedding if and only if it is a near isometric embedding when restricted to a suitable L1 (A)-subspace. Systematic study of narrow operators was started by Plichko and Popov in [337]. For a detailed presentation of the theory of narrow operators see the recent book by Popov and Randrianantoanina [338] and the references therein. 3.13.11. (1) Theorems 3.11.4 and 3.11.5 are due to Tabuev [375, 376]. The characterization of diﬀuse functionals in 3.11.3 belongs to H. Gordon [141]. For positive operators in Banach function spaces over separable metric space the diﬀuse decomposition was ﬁrst investigated by Weis [406, 407]. The general decomposition and projection results were obtained by Huijsmans and de Pagter in [174] and by Tabuev in [375]. The projection onto the band of diﬀuse and pseudoembedding operators was studied by Kolesnikov [201]. (2) The main tools in Huijsmans and de Pagter [174] and Tabuev [375] are sublinear operators pT and qT respectively deﬁned as pT (x) := inf T x1 ∨ · · · ∨ T xn : |x| x1 ∨ · · · ∨ xn ; x1 , . . . , xn ∈ E+ , n ∈ N , n qT (x) := inf T1 |x| ∨ · · · ∨ Tn |x| : T = Tk , k=1 ∼ T1 , . . . , Tn ∈ L (E, F )+ , Tk ⊥ Tl (k = l) , 3.13. Comments 195 where x ∈ X and T is a positive operator from X to Y . Then T is diﬀuse if and only if pT (x) = 0 for all x ∈ X [174] if and only if qT (x) = 0 for all x ∈ X [375]. In Maslyuchenko, Mykhaylyuk, and Popov [308] the operator pT is also called the Enflo–Starbird function in the wake of Enﬂo and Starbird [113]. 3.13.12. (1) Theorem 3.12.A.3 and Corollary 3.12.A.5 were obtained in Kusraev and Tabuev [258]. Corollary 3.12.A.6, a special case of Theorem 4.12.A.3, is due to Hart [167]. Theorem 3.12.A.7 is a recent result by Kusraev and Kutateladze [251]. (2) Theorem 3.12.B.8, the main result of 3.12.B, was proved in Kusraev and Kutateladze [251]. This result gives a complete description of n-disjoint sets of operators. In the particular setting of disjointness preserving operators this problem was motivated by the research of Benamor and Boulabiar [44, 46, 45] and stated explicitly in Boulabiar [63, Problem 5.8]: Given a lattice homomorphism T from X into Y , under what conditions is D := Orth(Y ) ◦ T maximal? (3) The decomposition theorem in 3.12.C.3 is due to Sobczyk and Hammer [366]. Theorem 3.12.C.8 tells us that every ﬁnitely additive measures with values in a Dedekind complete vector lattice can be written as the sum of two measures, one of which is diﬀuse, the other is a countable sum of ﬁnitely additive “spectral measure.” This is a special case of a more general Sobczyk–Hammer type decomposition theorem for ﬁnitely additive measures with values in a Banach–Kantorovich space which was obtained by Kusraev and Malyugin in [252, 253]. CHAPTER 4 BAND PRESERVING OPERATORS WP: When are we so happy in a vector lattice that all band preserving linear operators turn out to be order bounded? This question raised by Wickstead in [408] is often referred to as the Wickstead problem. The answer depends on the vector lattice in which the operator in question acts. There are several results that guarantee automatic order boundedness for a band preserving operator between concrete classes of vector lattices. The goal of this chapter is to examine the Wickstead problem in universally complete vector lattices for various classes of band preserving operators: projections, algebraic operators, derivations, algebra homomorphisms, etc. Boolean valued representation of band preserving operators reduces this task to examining the classical Cauchy functional equation. 4.1. Orthomorphisms In this section we introduce the class of band preserving operators and brieﬂy overview some properties of orthomorphisms. 4.1.1. Let X and Y be vector sublattices of a vector lattice Z. For a linear operator T from X into Y the following are equivalent: (1) x ⊥ z implies z ⊥ T x for all x ∈ X and z ∈ Z. (2) T x ∈ {x}⊥⊥ for all x ∈ X (with the disjoint complements taken in Z). (3) T (K ∩ X) ⊂ K for all bands K ∈ B(Z). ⊳ The implications (1) =⇒ (2) =⇒ (3) are immediate from the definitions. To ensure that (3) =⇒ (1), put K := {z}⊥ and note that for an arbitrary z ∈ Z the relation x ⊥ z and x ∈ K are equivalent, so that T x ∈ T (K ∩ X) ⊂ K by (3), whence z ⊥ T x. ⊲ 4.1. Orthomorphisms 197 4.1.2. A linear operator T from X to Y is called band preserving 1 provided that one (and hence all) of the conditions 4.1.1 (1–3) holds. If T is band preserving and T x ⊥ X for some x ∈ X, then T x ∈ X ⊥ ⊂ {x}⊥ and, in view of 4.1.1 (2), T x ∈ {x}⊥⊥ ∩ {x}⊥ = {0}. Thus, T (X) ⊂ Y ∩ X ⊥⊥ and we will only deal in the sequel with the case Z = X ⊥⊥ . A band preserving operator T need not be order bounded (cp. Theorems 4.4.9, 4.6.4, and 4.7.6 below). An order bounded band preserving operator π : X → Y is called an orthomorphism from X to Y . Let Orth(X, Y ) signify the set of all orthomorphisms from X to Y . An orthomorphism T ∈ Orth(X, Y ) is called an extended orthomorphism of Y whenever X is an order dense ideal of Y and a weak orthomorphism of Y provided that X is an order dense sublattice of Y . In the case Y = X we speak of the orthomorphisms of X and put Orth(X) := Orth(X, X). Let Z (X) stand for the set of regular operators T ∈ L(X) satisfying −cIX T cIX for some c ∈ R+ . By 3.1.11 (6) Z (X) ⊂ Orth(X). The space Z (X) is often called the ideal center of X, since Z (X) coincides with the order ideal in Orth(X) generated by the identity operator IX . 4.1.3. Let X be a vector lattice. Then Orth(X) is a semiprime commutative f -algebra with the composite as ring multiplication and with the identity operator as weak order unit. ⊳ Since an orthomorphism is disjointness preserving, it is also regular by the Meyer Theorem 3.4.3. Moreover, the collection of orthomorphisms Orth(X) is a vector lattice under the induced order from the space of regular operators. The vector lattice Orth(X) has some natural multiplicative structure: given π and ρ in Orth(X), put πρ := π ◦ ρ. It follows easily from the deﬁnition that πρ ∈ Orth(X). Obviously, Orth(X) is a lattice ordered algebra (cp. 2.3.1) and the identity operator is the ring unity. It remains to check that if π ⊥ ρ then σπ ⊥ ρ and πσ ⊥ ρ for all π, ρ, σ ∈ Orth(X)+ . Indeed, if π ∧ ρ = 0 then πx ∧ ρx = (π ∧ ρ)x = 0 for all x ∈ X+ . Since σ is band preserving, the relation πx ⊥ ρx implies σ(πx) ⊥ ρx or (σπ)x ∧ ̺x = 0. Hence (σπ) ⊥ ̺. At the same time, putting x0 := x ∨ σx we deduce 0 (πσ)x ∧ ρx πx0 ∧ ρx0 = 0, which implies that (πσ) ⊥ ρ. Commutativity and semiprimeness of Orth(X) can be seen from 4.3.8. ⊲ 1 In the Russian literature the term nonextending is also in use. 198 Chapter 4. Band Preserving Operators 4.1.4. A vector lattice X is said to have a cofinal family of band projections (or a cofinal family of projection bands) if for every nonzero band B ⊂ X there exists a nonzero projection band B0 ⊂ B. Equivalently, X has a coﬁnal family of band projections if for each nonzero band B in X there exists a nonzero band projection π on X such that π(X) ⊂ B. The space of continuous functions C(K) is a vector lattice with a coﬁnal family of band projections whenever K is a zero-dimensional compact space. 4.1.5. Let Z be a vector lattice with a cofinal family of band projections and let X and Y be vector sublattices of Z. For a linear operator T : X → Y the following are equivalent: (1) T is band preserving. (2) πx = 0 implies πT x = 0 for all x ∈ X and π ∈ P(Z). (3) πx = πy implies πT x = πT y for all x, y ∈ X and π ∈ P(Z). ⊳ If T is band preserving and πx = 0 for a band projection π ∈ P(Z), then T x ∈ {x}⊥⊥ ⊂ ker(π) by 4.1.1 (2), so that πT x = 0. Conversely, assume that T is not band preserving, while (2) holds. Then, according to 4.1.1 (2) there is x ∈ X such that T x ∈ / {x}⊥⊥ and so the band ⊥⊥ ⊥ K := {T x} ∩{x} is nonzero. By hypothesis there is a projection band π ∈ P(Z) with π(Z) ⊂ K, so that πT x = 0 and πx = 0; a contradiction. Thus (1) and (2) are equivalent. The equivalence of (2) and (3) follows trivially from the linearity of T . ⊲ 4.1.6. Let Z be a vector lattice with a cofinal family of band projections, X an order ideal of Z, and Y a vector sublattice of Z. For a linear operator T : X → Y the following are equivalent: (1) T is band preserving. (2) πT x = T πx for all x ∈ X and π ∈ P(Z). (3) πT π ⊥ x = 0 for all x ∈ X and π ∈ P(Z) with π ⊥ := IX − π. ⊳ (1) =⇒ (2): If T is band preserving then T (K∩X) ⊂ K and T (K ⊥ ∩ X) ⊂ K ⊥ for every band K ∈ B(Z). Assume that K is a projection band. Put π := [K] and π ′ := [K ⊥ ]. Then π|X ∈ P(X), π ′ |X ∈ P(X), and π ′ |X = IX − π|X . Consequently, T x = T πx + T π ′ x with T πx ∈ K and T π ′ x ∈ K ⊥ , so that πT x = πT πx = T πx. (2) =⇒ (3): Replace x by π ⊥ x in (2) to get (3). 199 4.1. Orthomorphisms (3) =⇒ (1): If (3) holds then πT x = πT πx for all x ∈ X and π ∈ P(Z). Therefore, πx = 0 implies trivially that πT x = 0 and we are done by 4.1.5 (2). ⊲ 4.1.7. Let X and Y be vector sublattices of a laterally complete vector lattice Z with Z = X ⊥⊥ . Each band preserving linear operator T from X into Y extends uniquely to a band preserving linear operator T λ from λ(X) to λ(Y ). Moreover, T is order bounded if and only if so is T λ and in this case |T λ | = |T |λ . ⊳ An arbitrary x ∈ λ(X) may be presented as x = o- ξ∈Ξ πξ xξ where (πξ )ξ∈Ξ is a partition of unity in P(Z) and (xξ )ξ∈Ξ is a family in X; see 2.5.3. Deﬁne T λ : λ(X) → λ(Y ) by putting λ T x := ox = oπξ xξ , πξ T xξ ξ∈Ξ ξ∈Ξ Ξ. The deﬁnition is sound. or equivalently, πξ T λ x := πξ T xξ for all ξ ∈ Indeed, given another representation x = o- η∈H ρη yη with a partition of unity (ρη )η∈H in P(Z) and a family (yη )η∈H in X, we have πξ ρη xξ = πξ ρη x = πξ ρη yη and so πξ ρη T xξ = πξ ρη T yη by 4.1.5 (3). Consequently, o- ξ∈Ξ πξ T xξ = o- ξ∈Ξ o- πξ ρη T xξ η∈H = o- η∈H o- ξ∈Ξ πξ ρη T yη = o- ρη T y η . η∈H To show the linearity of T λ , take x, y ∈ λ(X) and observe that there are families (xξ ) and (yξ ) in X and a common partition of unity (πξ ) in P(Z) such that πξ x = πξ xξ and πξ y = πξ yξ for all ξ. It follows that πξ (αx + βy) = πξ (αxξ + βyξ ) with α, β ∈ R. From the deﬁnition of T λ we deduce πξ T λ (αx + βy) = πξ T (αxξ + βyξ ) = απξ T xξ + βπξ T yξ = πξ (αT λ x + βT λ y), whence T λ (αx + βy) = αT λ x + βT λ y. To see that T λ is band preserving, observe that if πx = 0 then ππξ xξ = πξ πx = 0 and so πξ (πT λ x) = ππξ T xξ = 0 by 4.1.5 (2). As ξ is arbitrary, πT λ x = 0, as required. 200 Chapter 4. Band Preserving Operators Suppose that T̂ is another band preserving linear operator from λ(X) to λ(Y ) with T̂ |X = T . If x ∈ λ(X) is as above, then πξ x = πξ xξ with xξ ∈ X and, according to 4.1.5 (3), πξ T̂ x = πξ T̂ xξ = πξ T x = πξ T λ xξ = πξ T λ x. Since ξ is arbitrary, it follows that T̂ x = T λ x. If T is order bounded then by the Meyer Theorem |T | exists and |T x| = |T |(|x|) for all x ∈ X. By what we have proved |T | has a unique extension |T |λ to λ(X) and |T |λ is positive. Given x ∈ λ(X) as above, we have |T λ (x)| = oπξ |T xξ | = oπξ |T |(|xξ |) = |T |λ (|x|). ξ∈Ξ ξ∈Ξ From this we see that T λ is order bounded and |T λ | = |T |λ . Conversely, if T λ is order bounded then |T λ | exists and |T x| = |T λ x| = |T λ |(|x|) for all x ∈ D. Thus, T is order bounded. ⊲ 4.1.8. Let X, Y , Z, T , and T λ be the same as in 4.1.7 and X is order dense in Z. Then λ(X) = X λ and T is injective if and only if so is T λ . ⊳ We need only to check that T λ is injective whenever T is injective. It follows from the deﬁnition of T λ that T λ is injective if and only if πT x = 0 implies πx = 0 for all x ∈ X and π ∈ P(Z). Suppose the contrary to our claim that there are x ∈ X and π ∈ P(Z) such that πT x = 0, while πx = 0. We can assume further that x > 0 because T x+ ⊥ T x− and so πT x+ = πT x− = 0, while either πx+ = 0 or πx− = 0. Choose x0 ∈ X with 0 < x0 πx, put ρ := [x0 ], and note that ρ⊥ T x0 = 0 because T x0 ∈ {x0 }⊥⊥ . At the same time ρ π and so ρT x0 = 0. Thus, T x0 = 0, while x0 = 0; a contradiction. ⊲ 4.2. The Cauchy Functional Equation Here we shortly address the celebrated Cauchy equation of the classical calculus. In the next section we demonstrate that the contracting operators in universally complete vector lattices are solutions in disguise of the Cauchy equation and the Wickstead problem amounts to that of regularity of all solutions to the equation under some extra condition of regularity type. 4.2.1. By F we denote either R or C. The Cauchy functional equation with f : F → F unknown has the form f (x + y) = f (x) + f (y) (x, y ∈ F). 4.2. The Cauchy Functional Equation 201 Clearly, every solution to the equation is automatically Q-homogeneous; i.e., it satisﬁes another functional equation: f (qx) = qf (x) (q ∈ Q, x ∈ F). In the sequel we will be interested in a more general situation. Namely, we will consider the simultaneous functional equations 4 f (x + y) = f (x) + f (y) (x, y ∈ F), f (px) = pf (x) (p ∈ P, x ∈ F), (L) where P is a subﬁeld of F (which includes Q). In case F = C we assume that i ∈ P, so that Q ⊕ iQ ⊂ P. Denote by FP the ﬁeld F which is considered as a vector space over P. Clearly, solutions to the simultaneous equations (L) are precisely P-linear functions from FP to FP . 4.2.2. Let E be a Hamel basis for a vector space FP , and let F (E , F) be the space of all functions from E to F. The solution set of (L) is a vector space over F isomorphic with F (E , F). Such an isomorphism can be implemented by sending a solution f to the restriction f |E of f to E . ⊳ The solution set of (L) coincides with the space LP (F) of all Plinear operators in FP . Suﬃce it to mention that LP (F) and F (E , F) are isomorphic vector spaces. Let F0 (E , P) be the set of ﬁnitely supported functions ϕ from E to P; i.e., each ϕ : E → P is such that the support {e ∈ E : ϕ(e) = 0} of ϕ is ﬁnite. Then F0 (E , P) is a vector space over P isomorphic with FP . Such an isomorphism can be implemented by sending ϕ ∈ F0 (E , P) to the sum xϕ := e∈E ϕ(e)e. The inverse isomorphism x → ϕ is determined by expansion of x ∈ X in E . Given ψ ∈ F (E , F), put fψ (xϕ ) := ϕ(e)ψ(e) (ϕ ∈ F0 (E , F)). e∈E This yields an isomorphism of F (E , F) to LP (F). The inverse isomorphism takes the form f → f |E . The deﬁnitions of the isomorphisms ϕ → xϕ and ψ → fψ are meaningful, since there are only ﬁnitely many nonzero terms under the sign of summation. ⊲ 202 Chapter 4. Band Preserving Operators 4.2.3. Corollary. Let P be a subfield of R. The general form of a P-linear function f : R → R is given as f (x) = xe φ(e) if x = xe e, e∈E e∈E where φ : E → R is an arbitrary function. ⊳ This is immediate from 4.2.2. ⊲ 4.2.4. Theorem. Each solution of (L) is either F-linear or everywhere dense in F2 := F × F. ⊳ A solution f of (L) is F-linear if and only if f has presentation f (x) = cx (x ∈ F), with c := f (1). If f is not F-linear, then there are x1 , x2 ∈ F satisfying f (x1 )/x1 = f (x2 )/x2 . This yields the linear independence of v1 := (x1 , f (x1 )) and v2 := (x2 , f (x2 )) in the vector space F2 over F. Indeed, if α1 v1 + α2 v2 = 0 for some α1 , α2 ∈ F, then α1 x1 + α2 x2 = 0 and α1 f (x1 ) + α2 f (x2 ) = 0, while the two simultaneous equations has only the trivial solution α1 = α2 = 0, since the relevant determinant x1 f (x2 ) − x2 f (x1 ) is other than zero by hypothesis. Thus, each pair (x, y) ∈ F2 admits the presentation (x, y) = α1 v1 + α2 v2 for some α1 , α2 ∈ F. Since P is dense in F, each neighborhood of (x, y) contains a vector of the form p1 v1 + p2 v2 , with p1 , p2 ∈ P. (Recall that Q ⊕ iQ ⊂ P whenever F = C.) Therefore, {p1 v1 + p2 v2 : p1 , p2 ∈ P} is dense in F2 . At the same time this set lies in f , since the P-linearity of f implies that p1 v1 + p2 v2 = (p1 x1 + p2 x2 , p1 f (x1 ) + p2 f (x2 )) = (p1 x1 + p2 x2 , f (p1 x1 + p2 x2 )) for all p1 , p2 ∈ P. ⊲ 4.2.5. Let E be a Hamel basis of the space FP and let φ : E → F be an arbitrary function. The unique P-linear extension f : F → F of φ is continuous if and only if φ(e)/e = c = const for all e ∈ E . Moreover, in this event f admits the representation f (x) = cx for all x ∈ F. ⊳ By the P-linearity of f we have f (p) = pf (1) (p ∈ P). If f is continuous, then using the denseness of P in F, we arrive at the desired presentation with c := f (1). If g(e) = ce for all e ∈ E then the function x → cx is a P-linear extension of g, and by uniqueness of such extension f (x) = cx (x ∈ F), whence the continuity of f follows. ⊲ 4.2. The Cauchy Functional Equation 203 Thus, for a solution f to (L) to admit the presentation f (x) = cx (x ∈ R) we must impose some condition of regularity and continuity exempliﬁes such a condition. Let us list some other regularity conditions. First, agree to call an additive function f : R → R order bounded if f is bounded when restricted to every interval [a, b] ⊂ R. 4.2.6. Each solution f of (L) in case F = R admits the representation f (x) = cx (x ∈ R) with some c ∈ R if and only if one of the following is fulfilled: (1) f is continuous at some point. (2) f is increasing or decreasing. (3) f is order bounded. (4) f is bounded above or below on some interval ]a, b[ ⊂ R with a < b. (5) f is bounded above or below on some measurable subset of positive Lebesgue measure. (6) f is Lebesgue measurable. ⊳ We start with demonstrating (4) by way of example. Necessity is obvious. To prove suﬃciency, assume that f is bounded above by a real M on ]a, b[. Then the open set {(s, t) ∈ R2 : a < s < b, M < t} is disjoint from f , and so f cannot be dense in R2 . But if f fails to admit the desired representation then f is dense in R2 . Clearly (1), (2), and (3) follow from (4). The proofs of (5) and (6) are available in [14, Ch. 2, Theorem 8] and [211, Theorem 9.4.3]. ⊲ 4.2.7. Turn now to F = C and let P := P0 + iP0 for some subﬁeld of a subﬁeld P0 ⊂ R. In this event the solution set of (L) is the complexiﬁcation of the solution set of (L) in the case that P := P0 . In more detail, if g : R → R is a P0 -linear function, then g extends uniquely to the P-linear function 2 g : C → C deﬁned as g̃(z) = g(x)+ig(y) (z = x + iy ∈ C). Conversely, if f : C → C is a P-linear function, then there exists a unique pair of P0 -linear functions f1 , f2 : R → R satisfying f (z) = f˜1 (z) + if˜2 (z) (z ∈ C). Therefore, each solution f to (L) has the form f = f˜1 + if˜2 , where f1 , f2 : R → R are P0 -linear and fi (R) ⊂ R (i = 1, 2). Say that f is monotone or bounded provided that so are f1 and f2 . It is now easy to see that the following are true: 204 Chapter 4. Band Preserving Operators (1) The function f is dense in C2 if and only if both f1 and f2 are dense in R2 .2 (2) A solution f of (L) in case F = C, P = P0 + iP0 , and P0 ⊂ R admits the representation f (x) = cx (x ∈ C) with some c ∈ C if and only if one of the conditions 4.2.4 (1–6) is fulﬁlled. 4.2.8. Theorem. Let P be a subfield of F, while P := P0 + iP0 for some subfield P0 ⊂ R, in case F = C. The following are equivalent: (1) F = P. (2) Every solution to (L) is order bounded. ⊳ It is trivial that (1) =⇒ (2). Prove the converse by way of contradiction. The assumption F = P implies that each Hamel basis E for the vector space FP contains at least two nonzero distinct elements e1 , e2 ∈ E . Deﬁne the function ψ : E → F so that ψ(e1 )/e1 = ψ(e2 )/e2 . Then the P-linear function f = fψ : F → F, coinciding with ψ on E , would exist by 4.2.2 and be dense in F2 by 4.2.4. Therefore, fψ could not be order bounded (cp. 4.2.6 and 4.2.7). ⊲ 4.2.9. Consider the two more collections of simultaneous functional equations: ⎧ ⎪f (x + y) = f (x) + f (y) (x, y ∈ F), ⎨ (A) f (px) = pf (x) (p ∈ P, x ∈ F), ⎪ ⎩ f (xy) = f (x)f (y) (x, y ∈ F), ⎧ ⎪ ⎨f (x + y) = f (x) + f (y) (x, y ∈ F), (D) f (px) = pf (x) (p ∈ P, x ∈ F), ⎪ ⎩ f (xy) = f (x)y + xf (y) (x, y ∈ F). The nonzero solutions to (A) are called P-automorphisms of the ﬁeld F, while the solutions of D are called P-derivatives of F. The identity automorphism and the zero derivation are called trivial. The problem of existence of nontrivial solutions to (A) and (D) needs more delicate results from ﬁeld theory which will be presented in Section 4.11. 4.2.10. Sometimes it is important to deal with f satisfying the equation f (x + y) = f (x) + f (y) only for (x, y) ∈ G, where G is a subset of R × R. In this case the term restricted Cauchy functional equation is in common parlance (cp. [211, § 13.6]). 2 Recall that each function f : X → Y from X to Y is a subset of X × Y . 4.3. Representation of Band Preserving Operators 205 4.3. Representation of Band Preserving Operators Here we will demonstrate that a band preserving operator can be represented in an appropriate Boolean valued model as a solution of the restricted Cauchy functional equation. This fact enables us to study the properties of band preserving operators with the help of the theory of functional equations. 4.3.1. We consider a pair of vector lattices X and Y with Y a nonzero vector sublattice of the universal completion X u . Let Lbp (X, Y ) be the set of all band preserving linear operators from X to Y . Clearly, Lbp (X, Y ) is a vector space. Moreover, Lbp (X, Y ) becomes a faithful unitary module over the f -algebra A := Orth(Y ) on letting πT := π ◦ T , since the multiplication by an element of A is band preserving and the composite of band preserving operators is band preserving too. The part of Lbp (X, Y ) comprising all order bounded operators ∼ is denoted by L∼ bp (X, Y ). Clearly, Lbp (X, Y ) is an A-submodule of Lbp (X, Y ). Moreover, according to the Meyer Theorem, L∼ bp (X, Y ) is a vector sublattice of L∼ (X, Y ). Denote Lbp (X) := Lbp (X, X) and ∼ L∼ bp (X) := Lbp (X, X). 4.3.2. Let RR stand for the reals R within V(B) considered as a vector space over the ﬁeld R∧ . Thus, the expression “X is a subspace of RR ” means that X is an R∧ -subspace of R. Actually, in this case X is a totally ordered vector space over the ordered ﬁeld R∧ or, trivially, a vector sublattice of RR . Let X and Y be nonzero vector sublattices of RR . By LR∧ (X , Y ) we denote the element of V(B) that represents the space of all R∧ -linear operators from X to Y . Then LR∧ (X , Y ) is a vector space over R∧ within V(B) , and LR∧ (X , Y )↓ is a unitary semiprime module over the f -algebra R∧ ↓. Just as in 4.3.1, denote by L∼ R∧ (X , Y ) the part of LR∧ (X , Y ) consisting of order bounded functions. Recall that R↓ is a universally complete vector lattice and a semiprime f -algebra with unity 1 := 1∧ , while X := X ↓ and Y := Y ↓ are laterally complete vector sublattices in R↓. Moreover, X u = Y u = R↓, so that we can deﬁne Lbp (X, Y ). The main purpose of this section is to demonstrate that Lbp (X, Y ) is isomorphic to LR∧ (X , Y )↓. 4.3.3. Let X be a vector lattice, X u = R↓, and let Y be a vector sublattice of X u . A linear operator T : X → Y is band preserving if and 206 Chapter 4. Band Preserving Operators only if T is extensional; i.e., [[x1 = x2 ]] [[T x1 = T x2 ]] for all x1 , x2 ∈ X. ⊳ Let B ≃ B(X) and χ : B → P(R↓) is the same as in the Gordon Theorem. Then, in view of the properties of χ in 2.2.4 (G), T is extensional if and if χ(b)x1 = χ(b)x2 implies χ(b)T x1 = χ(b)T x2 for all x1 , x2 ∈ X and b ∈ B. But the latter means that T is band preserving by 4.1.5. ⊲ 4.3.4. Theorem. Let X and Y be vector sublattices of R↓ and R↓ = X u . The mapping T → T ↑ defines the isomorphisms of Lbp (X, Y ) ∼ to LR∧ (X , Y )↓ and L∼ bp (X, Y ) to LR∧ (X , Y )↓. Both isomorphisms are onto whenever X and Y are laterally complete and in this case the inverse isomorphisms are defined by the mapping τ → τ ↓. ⊳ Put X̄ := X ↓ and Ȳ := Y ↓ and observe that X̄ = mix(X) = X λ and Ȳ = mix(Y ) = λ(Y ); see 2.5.3. By 4.1.7 each T ∈ Lbp (X, Y ) admits the unique band preserving extension T̄ ∈ Lbp (X̄, Ȳ ). Therefore, Lbp (X, Y ) can naturally be identiﬁed with a subspace of Lbp (X̄, Ȳ ). Each T ∈ Lbp (X, Y ) is extensional by 4.3.3, and so T has the ascent τ := T ↑ presenting the unique mapping from X into Y such that [[τ (x) = T x]] = 1 for all x ∈ X (see 1.6.5). Using this identity and the deﬁnition of the ring structure on R↓ (cp. 2.2.2), we see that τ (x ⊕ y) = T (x + y) = T x + T y = τ (x) ⊕ τ (y), τ (λ∧ ⊙ x) = T (λ · x) = λ · T x = λ∧ ⊙ τ (x) hold within V(B) for all x, y ∈ X and λ ∈ R. Hence, [[ τ ∈ LR∧ (X , Y ) ]] = 1; i.e., [[ τ : X → Y is an R∧ -linear function ]] = 1, where ⊕ and ⊙ stand for the operations on X and Y , while + and · are the operations on X and Y . Conversely, if τ ∈ LR∧ (X , Y )↓ then the descent τ ↓ : X̄ → Ȳ is extensional by 1.5.6. A similar argument as above shows that if τ is R∧ linear within V(B) then τ ↓ is a linear operator. Now it is clear from 1.6.7 that the ascent functor as well as the descent functor deﬁnes a one-to-one correspondence between Lbp (X̄, Ȳ ) and LR∧ (X , Y )↓. Show that the above one-to-one correspondences preserve addition and scalar multiplication. This can be done by simple calculations, revealing isomorphisms, which is immediate from the identities (S + T )↑x = (S + T )x = Sx + T x = S↑x ⊕ T ↑x = (S↑ ⊕ T ↑)x (x ∈ R↓); 4.3. Representation of Band Preserving Operators 207 (α·S)↑x = (α·S)x = α·(Sx) = α⊙(S↑x) = (α⊙S↑)x (α, x ∈ R↓), where ⊕ and ⊙ stand for the operations on Y and LR∧ (X , Y )↓, while + and · are the operations on Ȳ and Lbp (X̄, Ȳ ). Thus, Lbp (X̄, Ȳ ) and LR∧ (X , Y )↓ are isomorphic. It remains to show that the ascent and descent preserve order boundedness. Take τ ∈ LR∧ (X , Y )↓. The sentence “τ is order bounded within V(B) ” can be written as (with [c, d]⋄ stand for a order interval within V(B) ) 1 = [[(∀ a ∈ X+ )(∃ b ∈ Y+ )τ ([−a, a]⋄ ) ⊂ [−b, b]⋄ ]] [[(∃ b ∈ Y+ )τ ([−a, a]⋄ ) ⊂ [−b, b]⋄ ]]. = a∈X̄ By the maximum principle for every a ∈ X̄+ there exists ba ∈ Ȳ such that [[τ ([−a, a]⋄ ) ⊂ [−ba , ba ]⋄ ]] = 1. The last identity is equivalent to τ ↓[−a, a] ⊂ [−ba , ba ] because of the three relations: [c, d]⋄ ↓ = [c, d], [[A ⊂ B]] = 1 ⇐⇒ A↓ ⊂ B↓, τ (A)↓ = τ ↓(A↓). The ﬁrst relation is immediate from the deﬁnition of the descent of an order in 2.2.3, the second is easily deduced with the help of 1.5.2, and the third follows from 1.5.3. ⊲ 4.3.5. Put Lbp (X) := Lbp (X, X) and End(RR ) := LR∧ (R, R). The modules Lbp (R↓) and End(RR )↓ are isomorphic, and such an isomorphism can be implemented by sending a band preserving operator to its ascent. Moreover, the isomorphism preserves order boundedness. ⊳ This is immediate from 4.3.4. ⊲ We now formulate a few corollaries to Boolean valued representation of an order bounded operators obtained in Theorem 3.3.3. An operator T ∈ L∼ (X, Y ) is said to be disjointness preserving if x ⊥ y implies T x ⊥ T y for all x, y ∈ X. Let L∼ dp (X, Y ) stand for the set of all order bounded disjointness preserving operators from X to Y . 4.3.6. Theorem. If T : X → X is an injective band preserving operator on a vector lattice X, then its inverse T −1 : T (X) → X is also band preserving. ⊳ This is immediate from 4.1.7 and 4.1.8 and Theorem 4.3.4. If X λ and T λ are the same as in 4.1.8 then we can assume that X λ = X ↓ 208 Chapter 4. Band Preserving Operators and T λ = τ ↓ for some X , τ ∈ V(B) with [[X is a subspace of RR ]] = 1 and [[τ : X → X is an R∧ -linear function]] = 1. Observe that τ is injective within V(B) if and only if for every x ∈ X λ we have [[τ x = 0 ↔ x = 0]] = 1 or, equivalently, [[τ ↓(x) = 0]] = [[x = 0]]. By 2.2.4 (G) and 1.5.6 this amounts to saying that T λ (x) = 0 ⇐⇒ x = 0 for all x ∈ X λ or, which is the same, T λ is injective. By 4.1.8 and our hypothesis T λ is injective. Consequently, [[τ is injective]] = 1 and by the transfer and maximum principles there exists τ −1 ∈ V(B) such that [[τ (X ) is a subspace of RR and τ −1 : τ (X ) → X is R∧ -linear function]] = 1. By 2.5.1 (1) Xτ := τ (X )↓ is a vector sublattice of X λ . It follows from 1.5.5 (1) that (T λ )−1 = (τ ↓)−1 = (τ −1 )↓, so that T λ is a linear operator from Xτ to X λ . By 4.3.4 (T λ )−1 is band preserving. We arrive at the desired conclusion by appealing again to 4.1.7, since T (X) ⊂ Xτ and the restriction of (T λ )−1 to T (X) is band preserving. ⊲ 4.3.7. Assume that X and Y are P-linear subspaces of R. A Plinear function τ : X → Y is order bounded if and only if there exists p ∈ P+ such that |τ (x)| p|x| for all x ∈ X . In this case τ admits the representation τ (x) = cx (x ∈ X ) for some c ∈ R. ⊳ Suﬃciency is obvious, so only the necessity should be proved. If τ is order bonded then there are 0 < q ∈ P and 0 < e ∈ X such that τ ([−e, e]) ⊂ [−q, q]. Given x ∈ X , take an arbitrary α ∈ P with α |x| and choose p ∈ P+ such that q/e p. Since ex/α ∈ [−e, e], we have τ (ex/α) ∈ [−q, q] or |τ (x)| (q/e)α pα. Thus |τ (x)| p|x| as α |x| is arbitrary. In particular, τ is uniformly continuous and admits the unique continuous extension τ̄ : R → R. From the continuity and P-linearity of τ̄ we see that τ̄ (x) = τ (x) = cx (x ∈ X ) where c := τ̄ (1). ⊲ 4.3.8. Let X be vector lattice and let Y be a sublattice of X u . A band preserving operator T : X → Y is order bounded if and only if T can be presented as T x = c · x (x ∈ X) for some fixed c ∈ X u . ⊳ If T admits the above representation then T ([−a, a]) ⊂ [−|c|a, |ca|] for all a ∈ X+ , so that T is order bounded. To prove the converse assume that X u = R↓ and put X := X↑ and Y := Y ↑. Working within V(B) and using transfer, apply 4.3.7 to the function τ := T ↑ from X to Y which is R∧ -linear and order bounded. Thus, [[(∃ c ∈ R)(∀ x ∈ X )τ (x) = c ⊙ x]] = 1. By the maximum principle there exists c ∈ R↓ such that [[τ (x) = c ⊙ x]] = 1 for all x ∈ X ↓. It follows that for all x ∈ X ⊂ X ↓ 4.3. Representation of Band Preserving Operators 209 we have 1 = [[T x = τ (x)]] ∧ [[τ (x) = c ⊙ x]] [[T x = c · x]] and so T x = cx. ⊲ 4.3.9. Let X be a vector lattice with X u = R↓ and let Y be a nonzero vector sublattice of X u . Every band preserving operator from X to Y is order bounded if and only if X := X↑ is a one-dimensional subspace of R over R∧ within V(B) , with B := B(X). In symbols, (B) Lbp (X, Y ) = L∼ |= (∃ e ∈ X )X = R∧ e. bp (X, Y ) ⇐⇒ V ⊳ ⇐=: If X = R∧ e for some e ∈ X then every R∧ -linear function τ : X → Y within V(B) evidently admits the representation τ (x) = cx for all x ∈ X with c = τ (e) and, by 4.3.7, τ is order bounded. By 4.3.4 every band preserving operator from X to Y is order bounded. =⇒: If there is no e ∈ X with R∧ e = X , then each Hamel basis E for the vector space X over R∧ has at least two distinct elements e1 = e2 . Deﬁning some function φ0 : E → Y so that φ0 (e1 )/e1 = φ0 (e2 )/e2 , we can extend φ0 to an R∧ -linear function φ : X → Y as in 4.2.2 which could not be order bounded by 4.2.4 and 4.3.7. Therefore, the descent of φ would be a band preserving linear operator that fails to be order bounded by 4.3.4. ⊲ 4.3.10. Let X be a vector sublattice of a vector lattice Y and let T : X → Y be a band preserving linear operator. Then there is a band B of Y λ such that the restriction of T to X ∩ B is order bounded and the restriction of T to X ∩ B ⊥ has the property ⊥ ⊥ (∀ x ∈ X ∩ B+ ) (∀ n ∈ N) (∃ un ∈ λ(X) ∩ B+ ) such that un x and |T λ un | nx. (∗) In particular, the restriction T |J of T to every nonzero order ideal J of X ∩ B ⊥ is not order bounded. ⊳ There is no loss of generality in assuming that X = Y u = R↓. Let an operator T λ from λ(X) to Y λ is deﬁned as in 4.1.7. Put τ := T λ ↑, b = [[τ : X → Y is order bounded]] and πb := χ(b). Then [[b∧τ : b∧X → b ∧ Y is order bounded]] = 1b := b by 1.3.7. Moreover, by 2.3.6, (b ∧ Y )↓ can naturally be identiﬁed with the band B := πb (Y ↓) = πb (Y λ ), while (b ∧ τ )↓ can be identiﬁed with the restriction of T λ to λ(X) ∩ B. Thus, 210 Chapter 4. Band Preserving Operators the restriction is order bounded together with its restriction to X ∩ B which coincides with T |X∩B . Observe further that b∗ = [[τ : X → Y is not order bounded]] and by 1.3.7 we again have [[b∗ ∧ τ : b∗ ∧ X → b∗ ∧ Y is not order bounded]] = 1b∗ := b∗ . Moreover, πb′ := IY λ − πb = χ(b∗ ) and, by 2.3.6 (b∗ ∧Y )↓ can be identiﬁed with the band B ⊥ := πb′ (Y λ ) and (b∗ ∧τ )↓ can be identiﬁed with the restriction of T λ to λ(X) ∩ B ⊥ . For brevity, put τ0 := b∗ ∧τ , X0 := b∗ ∧X , and Y0 := b∗ ∧Y . Since X0 is linearly ordered and Archimedean, the fact that τ0 is not bounded can be formalized as follows: (∀ 0 x ∈ X0 ) (∀ n ∈ N∧ ) (∃ un ∈ X0 ) 0 un x ∧ |τ (yn )| nx . By transfer this sentence is true within V(B) . Calculation of the Boolean truth values of the two universal quantiﬁers and application of the maximum principle to the existential quantiﬁer leads to the assertion: for all 0 x ∈ X ∩ B ⊥ and n ∈ N there exists un ∈ λ(X) ∩ B ⊥ such that |T λ (un )| nx, which is precisely (∗). Assume that there are 0 < x ∈ X ∩B ⊥ and y ∈ Y+ such that |T u| y for all u ∈ [0, x]. Then |T λ (v)| y for all v ∈ λ(X) with 0 v x. Indeed, if v ∈ [0, x] and πξ v = πξ vξ (ξ ∈ Ξ) for a family (vξ )ξ∈Ξ in X and a partition of unity (πξ )ξ∈Ξ in P(Y λ ), then we have also πξ v = πξ uξ with uξ = x ∧ vξ ∈ X ∩ [0, x] (ξ ∈ Ξ). It follows that |πξ T λ (v)| = |πξ T uξ | πξ y and so |T λ (v)| y. If a sequence (un ) is chosen in accordance with (∗), then nx |T λ (un )| y (n ∈ N); a contradiction. Consequently, the restriction of T to any nonzero order ideal in X ∩ B ⊥ is not order bounded. ⊲ 4.4. Dedekind Cuts and Continued Fractions The behavior of Dedekind cuts and continued fractions in a Boolean valued model clariﬁes that R∧ coincides with the internal reals R ∈ V(B) if and only if the complete Boolean algebra B is σ-distributive. 4.4.1. Consider an ordered set L. A Dedekind cut in L is a pair (a, b) of nonempty subsets a ⊂ L and b ⊂ L such that a consists of all lower bounds of b and b consists of all upper bounds of a (in symbols, 4.4. Dedekind Cuts and Continued Fractions 211 the set of all Dedekind cuts in L and a = b and b = a). Denote by L introduce the order on L by putting (a, b) (a′ , b′ ) for (a, b), (a′ , b′ ) ∈ L ′ ′ if and only if a ⊂ a or, equivalently, b ⊂ b. Assign to each u ∈ L the Dedekind cut û := (←, u], [u, →) , where (←, u] := {v ∈ L : v u} and is an order complete lattice; i.e., each [u, →) := {v ∈ L : v u}. Then L nonempty upper bounded subset has supremum, and each nonempty bounded below subset has inﬁmum. Moreover, u → û is a one-to-one preserving suprema and inﬁma and for every cut mapping of L to L (a, b) ∈ L̂ we have sup{û : u ∈ a} = (a, b) = inf{v̂ : v ∈ b}. is called a Dedekind completion of L. The order complete lattice L is 4.4.2. In particular, if L := Q then the Dedekind completion Q isomorphic to R. If (α1 , β1 ) and (α2 , β2 ) are Dedekind cuts in Q then (α1 , β1 ) + (α2 , β2 ) = (α0 , β0 ) with α0 := β1 + β2 and β0 := α1 + α2 ; if, in addition, (αi , βi ) 0̂ (i := 1, 2) then (α1 , β1 ) · (α2 , β2 ) = (α, β) with α := β1 · β2 and β := α1 · α2 . Here and below we put u + v := {x + y : x ∈ u, y ∈ v} and u · v := {x · y : x ∈ u, y ∈ v}. Assume now that L is a vector lattice. Introduce the addition and with (a, b), (a1 , b1 ), (a2 , b2 ) ∈ L and t ∈ R scalar multiplication on L as follows: (a1 , b1 ) + (a2 , b2 ) := b1 + b2 , a1 + a2 , ⎧ ⎪ if t > 0, ⎨(ta, tb), t(a, b) := t(a, b) := (tb, ta), if t < 0, ⎪ ⎩ t(a, b) := (L− , L+ ), if t = 0. becomes a Dedekind complete vector lattice With these operators, L Moreover, and the mapping ι : u → û is a lattice isomorphism of L to L. (L, ι) is a Dedekind completion of the vector lattice L. 4.4.3. If Q is the rationals within V(B) , then V(B) |= Q = Q∧ . ⊳ By transfer and the maximum principle there are Z and Q ∈ V(B) such that [[Z is the ring of integers]] = [[Q is the ring of rationals]] = 1. We have to show that [[Z = Z∧ ]] = [[Q = Q∧ ]] = 1. 212 Chapter 4. Band Preserving Operators We knew already that [[ℵ0 = (ω0 )∧ ]] = 1 (cp. 1.9.9 (1)). So, using the fact that the deﬁnition N := ω \ {0} is a bounded ZF-formula, we can write within V(B) that ℵ0 \ {0} = ω ∧ \ {0} = (ω \ {0})∧ = N∧ . Hence, N∧ is the set of naturals within V(B) . Let ω̄ = {. . . , −n, . . . , −1, 0} be an isomorphic copy of ω with the reverse order: −n −m ⇐⇒ m n. Then the set of integers can be deﬁned as the direct sum (= disjoint union) Z := ω̄+N. Since the direct sum as well as ω̄ is given by a bounded formula, we can write within V(B) as follows: Z = ℵ0 + N∧ = ω ∧ + (ω̄ + N)∧ = Z∧ . Recall that the set of rationals is deﬁned as the factor set Q := Z × N/R, where the coset of (m, n) stands for the rational m/n, and the equivalence of the pairs (m, n)R(m′ , n′ ) means that mn′ = nm′ . This deﬁnition is also written as a bounded formula and so within V(B) we have Q = Z × N∧ /R∧ = Z∧ × N∧ /R∧ = (Z × N/R)∧ = Q∧ . By analogy we see that the equality Q = Q∧ within V(B) may be viewed as the coincidence of the respective algebraic systems, since the ring and ﬁeld axioms are given by bounded formulas. ⊲ 4.4.4. For all a ⊂ Q and b ⊂ Q, the following holds: (a, b) is a Dedekind cut in Q ⇐⇒ [[ (a∧ , b∧ ) is a Dedekind cut in Q∧ ]] = 1. ⊳ The formula ϕ(a, b, Q) := (a ⊂ Q) ∧ (b ⊂ Q) ∧ (a = b) ∧ (b = a) stating that a and b comprise a Dedekind cut in Q, is bounded. Indeed, the formula a ⊂ Q is bounded (see 1.1.4) and a = b can be written as (∀ r ∈ a)(∀ s ∈ b)(r s) ∧ (∀ r ∈ Q)((∀ s ∈ b)(r s → r ∈ a) which is also a bounded formula. Similarly, b = a is a bounded formula. So we are done by restricted transfer (cp. 1.4.7). ⊲ 4.4.5. If A ∼ B and P(B ∧ ) = P(B)∧ then P(A∧ ) = P(A)∧ . ⊳ Given a mapping β : A → B deﬁne β̃ : P(A) → P(B) as β̃ : C → β(C). If β is a bijection then β̃ is also a bijection. Moreover, 213 4.4. Dedekind Cuts and Continued Fractions ∧ by restricted transfer, the mappings β ∧ : A∧ → B ∧ and β̃ := (β̃)∧ : P(A)∧ → P(B)∧ are one-to-one within V(B) . By transfer the mapping β5∧ : P(A∧ ) → P(B ∧ ) is one-to-one too. Clearly, P(A)∧ is a subset of P(A∧ ). It remains to show that the restriction of β5∧ to P(A)∧ coincides ∧ with β̃ : ∧ ∧ [[(∀ u ∈ P(A)∧ )β5∧ (u) = β̃ (u)]] = [[β5∧ (u∧ ) = β̃ (u∧ )]] = ∧ [[β5∧ (C ) = β̃ (C )]] = ∧ C⊂A ∧ u∈P(A) [[β ∧ (C ∧ ) = β̃(C)∧ ]] = 1. ⊲ C⊂A 4.4.6. If B is σ-distributive then V(B) |= R ⊂ R∧ . ⊳ Assume that B is σ-distributive. By 1.9.13 (3) P(ω ∧ ) = P(ω)∧ and P(Q∧ ) = P(Q)∧ by 4.4.5. To demonstrate the desired inclusion we are to show only that [[t ∈ R]] = 1 implies [[t ∈ R∧ ]] = 1. Assume that [[t ∈ R]] = 1; i.e., t is a Dedekind cut within V(B) . We then see within V(B) that ∃ a ∈ P(Q) ∃ ã ∈ P(Q) ϕ(a, ã, Q) ∧ t = (a, ã), where ϕ is the same as in 4.4.4. Considering that P(Q) = P(Q)∧ in view of 4.4.3 and calculating the truth value of the above formula, we infer 1= [[ϕ(a∧ , ã∧ , Q∧ )]] ∧ [[t = (a, ã)∧ ]]. a⊂Q ã⊂Q Choose a partition of unity (bξ ) ⊂ B and two families (aξ ) and (ãξ ) in P(Q) so that bξ [[ϕ(a∧ξ , ã∧ξ , Q∧ )]] ∧ [[t = (aξ , ãξ )∧ ]]. It follows that t = mixξ bξ (aξ , ãξ )∧ , and bξ [[ϕ(a∧ξ , ã∧ξ , Q∧ )]]. If bξ = 0 then [[ϕ(a∧ξ , ã∧ξ , Q∧ )]] = 1, since ϕ(x1 , x2 , x3 ) is a bounded formula and the truth value [[ϕ(x∧1 , x∧2 , x∧3 )]] of a bounded formula may be either 0 or 1 by the deﬁnitions and rules of calculating truth values. By 4.4.4 ϕ(aξ , ãξ , Q); i.e., (aξ , ãξ ) is a Dedekind cut. Evidently, bξ [[t = (aξ , āξ )∧ ∈ R∧ ]]. Hence, [[t ∈ R∧ ]] = 1. ⊲ 4.4.7. To prove the converse implication in 4.4.6 we use continued fractions. Put I := {t ∈ R : 0 < t < 1, t is irrational}, I := {t ∈ R : 0 < t < 1, t is irrational} within V(B) . 214 Chapter 4. Band Preserving Operators It is well known that there is a bijection λ : I → NN sending a real t to the sequence λ(t) = a : N → N of partial continued fractions of the continued fraction expansion of t: t= 1 a(1) + 1 1 a(2)+ a(3)+··· . Given the two sequences a : N → N and s : N → I, consider the bounded formula ϕ0 (a, s, t, N) stating that s(1) = t−1 and 1 1 a(n) = − a(n), , s(n + 1) = s(n) s(n) for all n ∈ N, where [α] is the integer part of 0 < α ∈ R which is expressed by the bounded formula ψ(α, [α], N): [α] ∈ N ∧ [α] α ∧ (∀ n ∈ N)(n α → n [α]). The equality λ(t) = a means the existence of a sequence s : N → I such that ϕ0 (a, s, t, N). Call the bijection λ the continued fraction expansion. ∧ By transfer, the continued fraction expansion λ̃ : I → (ℵ0 )ℵ0 = (N∧ )N (B) exists within V . 4.4.8. Within V(B) , the restriction of λ̃ to I∧ coincides with λ∧ ; i.e., V(B) |= (∀ t ∈ I∧ ) λ̃(t) = λ∧ (t). ⊳ The desired is true if λ̃(t∧ ) = λ(t)∧ for all t ∈ I. By the deﬁnition of λ̃ we have to demonstrate the validity within V(B) of the formula: ∧ (∃ s ∈ I N ) ϕ0 (λ(t)∧ , s, t∧ , N∧ ). By the deﬁnition of λ there is a sequence σ : N → I satisfying ϕ0 (λ(t), σ, t, N). Since ϕ0 is bounded, 1 = [[ϕ0 (λ(t)∧ , σ ∧ , t∧ , N∧ )]]. Note that σ ∧ : N∧ → I∧ ⊂ I ; i.e., ∧ [[σ ∧ ∈ I N ]] = 1. Summarizing the above, we can write ∧ [[ (∃ s ∈ I N ) ϕ0 (λ(t)∧ , s, t∧ , N∧ ) ]] [[ϕ0 (λ(t)∧ , σ ∧ , t∧ , N∧ )]] = 1. ⊲ 4.4.9. Theorem. Assume that X is a universally complete vector lattice, B := P(X), and R stands for the reals within V(B) . Then the following are equivalent: (1) B is σ-distributive. 4.5. Hamel Bases in Boolean Valued Models 215 (2) R = R∧ within V(B) . (3) Every band preserving linear operator in X is order bounded. ⊳ The implication (1) =⇒ (2) amounts to 4.4.6. Prove that V(B) |= R = R∧ implies σ-distributivity of B. By hypothesis im(λ̃) = I = I∧ = im(λ∧ ) within V(B) . Hence, λ̃ and λ∧ are bijections, λ̃ extends λ∧ by 4.4.8, and their images coincide. Clearly, the domains coincide in this event too (and, moreover, λ̃ = λ∧ ). ∧ Therefore, (NN )∧ = (N∧ )N . By 1.9.13 (2) B is σ-distributive. The equivalence (2) ⇐⇒ (3) follows from 4.3.7. ⊲ 4.5. Hamel Bases in Boolean Valued Models As can be seen from 4.3.9, the important feature of a vector lattice is the internal dimension of its Boolean valued representation considered as a vector lattice over R∧ . It stands to reason to ﬁnd out what construction in a vector lattice corresponds to a Hamel basis for within the Boolean valued representation. 4.5.1. Let X be a vector lattice with a coﬁnal family of band projections. We will say that x, y ∈ X are distinct at π ∈ P(X) provided that π|x − y| is a weak order unit in π(X) or, equivalently, if π(X) ⊂ |x − y|⊥⊥ . Clearly, x and y diﬀer at π whenever ρx = ρy implies πρ = 0 for all ρ ∈ P(X). A subset E of X is said to be locally linearly independent provided that, for an arbitrary nonzero band projection π in X and each collection of the elements e1 , . . . , en ∈ E that are pairwise distinct at π, and each collection of reals λ1 , . . . , λn ∈ R, the condition π(λ1 e1 + · · · + λn en ) = 0 implies that λk = 0 for all k := 1, . . . , n. In other words, E is locally linearly independent if for all π ∈ P(X) every subset of π(E ), consisting of nonzero members pairwise distinct at π, is linearly independent. An inclusion maximal locally linearly independent subset of X is called a local Hamel basis for X. 4.5.2. Each vector lattice X with a cofinal family of band projections has a local Hamel basis for X. ⊳ It suﬃces to apply the Kuratowski–Zorn Lemma to the inclusion ordered set of all locally linearly independent subsets of X. ⊲ 4.5.3. A locally linearly independent set E in X is a local Hamel basis for X if and only if for every x ∈ X there exists a partition of 216 Chapter 4. Band Preserving Operators unity (πξ )ξ∈Ξ in P(X) such that for every ξ ∈ Ξ the projection πξ x is a finite linear combination of nonzero elements of πξ E pairwise distinct at π. This representation of πξ x is unique in the band πξ (X). ⊳ ⇐=: Assume that E ⊂ X is locally linearly independent but fails to be a Hamel basis. Then we can ﬁnd x ∈ X such that E ∪ {x} is locally linearly independent. Therefore, there is no nonzero band projection π for which πx is a linear combination of nonzero elements from πE pairwise distinct at π. This contradicts the existence of a partition of unity with the above mentioned properties. =⇒: If E is a local Hamel basis for X then E ∪ {x} is not locally linearly independent for an arbitrary x ∈ X. Thus, there exist a nonzero band projection π and e1 , . . . , en ∈ E such that either ρx = 0 for some nonzero band projection ρ π, or ρx = ρek for some k ∈ {1, . . . , n} and nonzero band projection ρ π, or π(λ0 x + λ1 e1 + · · · + λn en ) = 0 for some λ0 , λ1 , . . . , λn ∈ R, while πe1 , . . . , πen , πx are nonzero and pairwise distinct at π and not all λ0 , λ1 , . . . , λn are equal to zero. In the latter case the equality λ0 = 0 contradicts the local linear independence of E , so that λ0 = 0. In all cases there is a nonzero band projection π such that πx is representable as a linear combination of πe1 , . . . , πen . The set of such band projections π is minorizing in P(X), since in above reasoning we can replace x by σx with an arbitrary band projection σ ∈ P(X). The existence of the required partition of unity follows from the fact that every minorizing subset of a complete Boolean algebra admits a disjoint reﬁnement (the exhaustion principle). ⊲ 4.5.4. The claim of 4.5.3 admits the reformulation: A locally linearly independent set E in X is a local Hamel basis if and only if for every x ∈ X there exist a partition of unity (πξ )ξ∈Ξ in P(X) and a family of reals (λξ,e )ξ∈Ξ,e∈E such that λξ,e πξ e x = oξ∈Ξ e∈E and for every ξ ∈ Ξ the set {e ∈ E : λξ,e = 0} is ﬁnite and consists of nonzero elements pairwise distinct at πξ . Moreover, the representation is unique in the sense that if x admits another representation x = oκω,e ρω e , ω∈Ω e∈E 4.5. Hamel Bases in Boolean Valued Models 217 and for every ω ∈ Ω the set {e ∈ E : κω,e = 0} is ﬁnite and consists of nonzero elements pairwise distinct at ρω , then for all ξ ∈ Ξ, ω ∈ Ω, and e ∈ E the relation πξ ρω e = 0 implies λξ,e = κω,e . 4.5.5. Assume that E , X ∈ V(B) , [[ X is a vector subspace of the vector space RR ]] = 1, [[ E ⊂ X ]] = 1, and X := X ↓. Then [[ E is a linearly independent subset of the vector space X (over R∧ ) ]] = 1 if and only if E ↓ is a locally linearly independent subset of X. ⊳ ⇐=: Put E ′ := E ↓ and assume that E ′ is locally linearly independent. Given a natural n, let the formula ϕ(n, τ, σ) expresses the following: τ and σ are mappings from n := {0, 1, . . . , n − 1} into R∧ and E respectively, σ(k) = σ(l) for diﬀerent k and l in n, and k∈n τ (k)σ(k) = 0. Denote the formula (∀ τ )(∀ σ) ϕ(n, τ, σ) → (∀ k ∈ n) τ (k) = 0 by ψ(n). Then the linear independence of E within V(B) amounts to the equality 1 = [[ (∀ n ∈ N∧ ) ψ(n) ]] = [[ ψ(n∧ ) ]]. n∈N Hence, we are left with proving that [[ ψ(n∧ ) ]] = 1 for all n ∈ N. Calculate the truth values, using the construction of the formula ψ and the rules of Boolean valued analysis (cp. 1.5.2). The result is as follows: [[ (∀ k ∈ n∧ ) τ (k) = 0 ]] : τ, σ ∈ V(B) ; [[ ϕ(n∧ , τ, σ) ]] = 1 . Take some τ, σ ∈ V(B) and n ∈ N such that [[ϕ(n∧ , τ, σ)]] = 1. Then [[τ : ∧ n∧ → R∧ ]] = 1 and [[ σ : n → E ]] = 1. Moreover, [[ σ(k) = σ(l) for ∧ distinct k and l in n , and k∈n∧ τ (k)σ(k) = 0 ]] = 1. Let t : n → R∧ ↓ and let s : n → E ′ stand for the modiﬁed descents of τ and σ (cp. 1.5.8). Then 1 = [[(∀ k, l ∈ n∧ ) k = l → σ(k) = σ(l) ]] [[σ(k ∧ ) = σ(l∧ )]] = [[s(k) = s(l)]], = k,l∈n k=l k,l∈n k=l and so s(k) and s(l) diﬀer at the identity projection for k and l distinct. Furthermore, n−1 t(k)s(k) = 0 = τ (k)σ(k) = 0 = 1. k=0 k∈n∧ 218 Chapter 4. Band Preserving Operators n−1 Hence, k=0 t(k)s(k) = 0. Since t(k) ∈ R∧ ↓ for all k ∈ n, there is a partition of unity (bξ )ξ∈Ξ in B and, to each k ∈ n, there is a numerical family (λξ,k )ξ∈Ξ such that λξ,k χ(bξ )1 (k := 0, 1, . . . , n − 1). t(k) = oξ∈Ξ Inserting these expressions into the equality n−1 k=0 t(k)s(k) = 0, we obtain ⎞ ⎛ n−1 n−1 ⎝o0= χ(bξ ) λξ,k s(k). λξ,k χ(bξ )1⎠s(k) = ok=0 ξ∈Ξ ξ∈Ξ k=0 n−1 Consequently, χ(bξ ) k=0 λξ,k s(k) = 0 and, since s(k) and s(l) diﬀer at χ(bξ ) for distinct k, l ∈ n, by the deﬁnition of local linear independence we have λξ,k = 0 (k = 0, 1, . . . , n − 1). Thus t(k) = 0 (k = 0, 1, . . . , n − 1), and so 1= [[τ (k ∧ ) = 0]] = [[(∀ k ∈ n∧ ) τ (k) = 0]], [[t(k) = 0]] = k∈n k∈n which was required. =⇒: Assume that [[ E is an R∧ -linearly independent set in X ]] = 1. Consider π ∈ P(X), n ∈ N, t : n → R, and s : n → E ′ such that π = 0, s(k) and s(l) are distinct at π for diﬀerent k, l ∈ n, n−1 and π k=0 t(k)s(k) = 0. Our goal is now to prove that t(k) = 0 (k := 0, . . . , n − 1). Let τ, σ ∈ V(B) be the modiﬁed ascents of t and s (cp. 1.6.8). Then, within V(B) , we have τ : n∧ → R∧ , σ : n∧ → E , and (∀ k, l ∈ n∧ ) k = l → σ(k) = σ(l) ∧ τ (k ∧ )σ(k ∧ ) = 0 k∈n∧ → (∀ k ∈ n∧ ) τ (k) = 0. Calculating the truth value of the latter formula, we obtain b := k,l∈n k=l [[s(k) = s(l)]] ∧ n−1 k=0 t(k)s(k) = 0 n−1 [[t(k)∧ = 0]]. k=0 4.6. Locally One-Dimensional Vector Lattices 219 According to the initial properties of π, s, and t, by 2.2.4 (G) we have π χ(b) implying that πt(k)∧ = 0 for all k ∈ n again by 2.2.4 (G). Since π = 0; therefore, t(k) = 0 for all k := 0, . . . , n − 1. ⊲ 4.5.6. Let E0 be a locally linearly independent subset of X and E := E0 ↑. Then [[ E is R∧ -linearly independent in X ]] = 1. In particular, mix(E0 ) is locally linearly independent. ⊳ By 4.5.5 it suﬃces to show that E0′ := mix(E0 ) = E ↓ = E0 ↑↓ is locally linearly independent. Take some nonzero band projection π in X, elements e1 , . . . , en ∈ E0′ that diﬀer at π, and reals λ1 , . . . , λn ∈ R satisfying π(λ1 e1 + · · · + λn en ) = 0. There are a partition of unity (bξ ) in B and families (gξ,k ) ⊂ E0 such that ek = o- ξ χ(bξ )gξ,k . Clearly, ρ := πχ(bη ) = 0 for some index η. The elements gη,1 , . . . , gη,n diﬀer pairwise at ρ and ρ(λ1 gη,1 + · · · + λn gη,n ) = 0. Since E0 is locally linearly independent, λ1 = · · · = λn = 0. ⊲ 4.5.7. Theorem. Assume that E , X ∈ V(B) , [[ E ⊂ X ]] = 1, [[ X is a vector subspace of RR ]] = 1, and X := X ↓. Then [[ E is a Hamel basis for the vector space X (over R∧ )]] = 1 if and only if E ↓ is a local Hamel basis for X. ⊳ This is immediate from 4.5.5 and 4.5.6. ⊲ 4.6. Locally One-Dimensional Vector Lattices In this section we examine locally one-dimensional vector lattices and show that a universally complete vector lattice is locally one-dimensional if and only if all band preserving operators in it are automatically order bounded. 4.6.1. A vector lattice X is said to be locally one-dimensional if for every two nondisjoint x1 , x2 ∈ X there exist nonzero components u1 and u2 of x1 and x2 respectively such that u1 and u2 are proportional. Every atomic vector lattice is evidently locally one-dimensional, but the converse is not true. Below in 4.7.7–4.7.10 we will demonstrate that there exists a purely nonatomic locally one-dimensional universally complete vector lattice. An element x ∈ X is locally constant with respect to u ∈ X+ if there exist a numerical family (λξ )ξ∈Ξ and partition (πξ )ξ∈Ξ of [x] in P(X) such that πξ x = λξ πξ u for all ξ ∈ Ξ. In this event x = o- ξ∈Ξ λξ πξ u. 220 Chapter 4. Band Preserving Operators 4.6.2. Let X be a vector lattice with a cofinal family of band projections, let X λ be a lateral completion of X, and let X ∈ V(B) be a Boolean valued representation of X with B := P(X). The following are equivalent: (1) X is locally one-dimensional. (2) V(B) |= “X is a one-dimensional vector lattice over R∧ .” (3) There is a singleton local Hamel basis for X λ . (4) Every pair of locally independent members in X is disjoint. ⊳ We can assume without loss of generality that X ⊂ X λ = X ↓ and X = {0}. (1) =⇒ (2): Given x, y ∈ X, put b0 := [[|x| ∧ |y| = 0]] and X0 := {|x| ∧ |y|}⊥⊥ . Since X has a coﬁnal family of projection bands, it follows from (1) that there exists a partition (Xξ )ξ∈Ξ in B(X) of X0 such that [Xξ ]x = αξ [Xξ ]y with some 0 = αξ ∈ R for all ξ ∈ Ξ. Put bξ := χ−1 ([Xξ ]) ∧ and α0 := mixξ∈Ξ bξ αξ and observe that b0 = ξ∈Ξ bξ , bξ [[x = α∧ξ y]] (ξ ∈ Ξ), and [[α0 ∈ R∧ ]] = 1. From this we deduce bξ [[x = α∧ξ y]] ∧ [[α0 = α∧ξ ]] ∧ [[α0 ∈ R∧ ]] [[x = α0 y]] ∧ [[α0 ∈ R∧ ]] [[(∃ α ∈ R∧ )x = αy]] = [[x and y are proportional]]. Thus, we have proved that b0 [[x and y are proportional]] or, what is the same, [[|x| ∧ |y| = 0]] ⇒ [[x and y are proportional]] = 1 for all x, y ∈ X. A simple calculation completes the proof: [[X is a one-dimensional vector lattice over R∧ ]] = [[(∀ x ∈ X )(∀ y ∈ X )|x| ∧ |y| = 0 → x and y are proportional]] [[|x| ∧ |y| = 0]] ⇒ [[x and y are proportional]] = 1. = x,y∈X (2) =⇒ (3): Working within V(B) choose a nonzero e ∈ X so that X ≃ R∧ e. Then e ∈ X λ and {e} is a local Hamel basis for X λ by 4.5.6, since [[{e} is a Hamel basis for X ]] = 1. (3) =⇒ (4): Let {e} be a singleton local Hamel basis for X λ and consider a pair of locally independent members x, y ∈ X. It follows that there exist a partition of unity (πξ )ξ∈Ξ in P(X λ ) and numerical families (αξ )ξ∈Ξ and (βξ )ξ∈Ξ such that πξ x = αξ πξ e and πξ y = βξ πξ e for all 4.6. Locally One-Dimensional Vector Lattices 221 ξ ∈ Ξ. If x and y are not disjoint then there exists η ∈ Ξ with αη βη = 0. Choose nonzero band projection π ∈ P(X) with π πη . Then πx and πy are proportional; a contradiction. (4) =⇒ (1): Take a pair of nondisjoint elements x1 , x2 ∈ X. By (4) the set {x1 , x2 } is locally linearly dependent. Thus, there exists a nonzero band projection π ∈ P(X) such that {πx1 , πx2 } is a linearly dependent pair of distinct elements. It follows that u1 := πx1 and u2 := πx2 are proportional components of x1 and x2 . ⊲ 4.6.3. For each laterally complete vector lattice X with a weak order unit 1 the following are equivalent: (1) X is locally one-dimensional. (2) All elements of X+ are locally constant with respect to 1. (3) All elements of X+ are locally constant with respect to an arbitrary weak order unit e ∈ X. (4) {1} is a local Hamel basis for X. (5) Every local Hamel basis for X consists of pairwise disjoint members. ⊳ The equivalence (1) ⇐⇒ (4) and the implication (1) =⇒ (5) are immediate from 4.6.2. Obviously, (3) =⇒ (2). To prove the converse, note that, given x ∈ X, we can choose a partition of unity (πξ )ξ∈Ξ in P(X) such that for each ξ ∈ Ξ both πξ x and πξ e are multiples of πξ 1. So, πξ x is a multiple of πξ e and (2) =⇒ (3). A similar argument shows that {1} is a local Hamel basis if and only if so is {e} for every order unit e ∈ X. Thus, if (5) holds and E is a local Hamel basis for X then e := sup P exists and {e} is a local Hamel basis for X. It follows that (5) =⇒ (4). Clearly, (4) =⇒ (2) by 4.5.3. To complete the proof, we have to show (2) =⇒ (5). If (5) fails then we can choose a nonzero band projection π and a local Hamel basis containing two members e1 and e2 such that both πe1 and πe2 are nonzero multiples of π1. Consequently, π(λ1 e1 +λ2 e2 ) = 0 for some λ1 , λ2 ∈ R and we arrive at the contradictory conclusion that {e1 , e2 } is not locally linearly independent. ⊲ 4.6.4. Theorem. Let X be a universally complete vector lattice. Then the following are equivalent: (1) X is locally one-dimensional. (2) Every band preserving operator on X is order bounded. 222 Chapter 4. Band Preserving Operators ⊳ By the Gordon Theorem we can assume that X = R↓ with R ∈ V(B) and B ≃ P(X). Thus, the problem reduces to existence of a discontinuous solution to the Cauchy functional equation in 4.2.1. From 4.3.5 we see that 4.6.4 (i) ⇐⇒ 4.2.8 (i) (i = 1, 2) if in 4.2.8 replace R by R and P by R∧ . Thus the claim follows from 4.2.8 by transfer. ⊲ 4.6.5. It is worth comparing the above proof of 4.6.4 with the standard proof that does not involve Boolean valued representation. ⊳ (1) =⇒ (2): Recall that a linear operator T : X → X is band preserving if and only if πT = T π for every band projection π in X cp. 4.1.1 (4) . Assume that T is band preserving and put ρ := T 1. Since an arbitrary e ∈ X+ can be expressed as e = supξ∈Ξ λξ πξ 1, we deduce πξ T e = T (πξ e) = T (λξ πξ 1) = λξ πξ T (1) = πξ (e)T (1) = πξ eρ, whence T e = ρe. It follows that T is a multiplication operator in X which is obviously order bounded. (2) =⇒ (1): Assume that (1) is false. According to 4.6.4 (4) there is a local Hamel basis E for X containing two members e1 and e2 that are not disjoint. Then the band projection π := [e1 ] ∧ [e2 ] is nonzero. (Here and below [e] is the band projection onto {e}⊥⊥ .) For an arbitrary x ∈ X there exists a partition of unity (πξ )ξ∈Ξ such that πξ x is a ﬁnite linear combination of elements of E . Assume the elements of E have been labelled so that πξ x = λ1 πξ e1 + λ2 πξ e2 + · · · . Deﬁne T x to be the unique element in X with πξ T x := λ1 ππξ e2 . It is easy to check that T is a well deﬁned linear operator from X into itself. Take x, y ∈ X with x ⊥ y and let (πξ )ξ∈Ξ be a partition of unity such that both πξ x and πξ y are ﬁnite linear combinations of elements from E . Reﬁning the partition of unity if need be, we can also require that at least one of the elements πξ x and πξ y equals zero for all ξ ∈ Ξ. If πξ y = 0 then πξ x = 0, and so the corresponding λ1 e1 is equal to zero. If ππξ = 0 then λ1 = 0, and in any case πξ T x = 0. It follows that T x ⊥ y and T is band preserving. If T were order bounded then T would be presentable as T x = ax (x ∈ X) for some a ∈ X (cp. 4.1.6 (4)). In particular, T e2 = ae2 and, since T e2 = 0 by deﬁnition, we have 0 = [e2 ]|a| π|a|. Thus πe2 = T (πe1 ) = aπe1 = 0, contradicting the deﬁnition of π. ⊲ 4.6.6. Let P is a proper subfield of R. There exists an P-linear subspace X in R such that X and R are isomorphic vector spaces over P but they are not isomorphic as ordered vector spaces over P. 4.6. Locally One-Dimensional Vector Lattices 223 ⊳ Recall that the real ﬁeld R has no proper subﬁeld of which it is a ﬁnite extension; see, for example, Coppel [96, Lemma 17]. It follows that R is an inﬁnite dimensional vector space over the ﬁeld P. Let E be a Hamel basis of a P-vector space R. Since E is inﬁnite, we can choose a proper subset E0 E of the same cardinality: |E0 | = |E |. If X denotes the P-subspace of R generated by E0 , then X0 R and X and R are isomorphic as vector spaces over P. If X and R were isomorphic as ordered vector spaces over P, then X would be order complete and, as a consequence, we would have X = R; a contradiction. ⊲ 4.6.7. Theorem. Let X be a nonlocally one-dimensional universally complete vector lattice. Then there exist a vector sublattice X0 ⊂ X and a band preserving linear bijection T : X0 → X such that T −1 is also band preserving but X0 and X are not lattice isomorphic. ⊳ We can assume without loss of generality that X = R↓ and [[R = R∧ ]] = 1. By 4.6.6 there exist an R∧ -linear subspace X in R and R∧ linear isomorphism τ from X onto R, while X and R are not isomorphic as ordered vector spaces over R∧ . Put X0 := X ↓, T := τ ↓ and S := τ −1 ↓. The mappings S and T and are band preserving and linear by 4.3.4. Moreover, S = (τ ↓)−1 = T −1 by 1.5.5 (2). It remains to observe that X0 and X are lattice isomorphic if and only if X and R are isomorphic as ordered vector spaces. ⊲ 4.6.8. Let γ be a cardinal. A vector lattice X is said to be Hamel γ-homogeneous whenever there exists a local Hamel basis of cardinality γ in X consisting of weak order units pairwise distinct at IX . (Two elements x, y ∈ X are distinct at IX if |x − y| is a weak order unit in X; see 4.5.1.) For π ∈ P(X) denote by κ(π) the least cardinal γ for which πX is Hamel γ-homogeneous. Say that X is strictly Hamel γ-homogeneous whenever X is Hamel γ-homogeneous and κ(π) = γ for all nonzero π ∈ P(X). 4.6.9. Theorem. Let X be a universally complete vector lattice. There is a band X0 in X such that X0⊥ is locally one-dimensional and there exists a partition of unity (πγ )γ∈Γ in P(X0 ) with Γ a set of infinite cardinals such that πγ X0 is strictly Hamel γ-homogeneous for all γ ∈ Γ. ⊳ Assume that X = Y ↓ with B = P(X). Put b0 := [[R = R∧ ]] and X0 := (b0 ∧ R)↓; see 2.3.6. Then b∗0 = [[R = R∧ ]] and X0⊥ = (b∗0 ∧ R)↓, so that the band X0⊥ is locally one-dimensional by 4.6.2 (1, 2). Next we can assume by passing to the model V([0,b0 ]) that b0 = 1 and X0 = X. Thus by 1.3.7 we have [[R = R∧ ]] = 1 and therefore 224 Chapter 4. Band Preserving Operators [[R is an inﬁnite dimensional vector space over R∧ ]] = 1; i.e., the algebraic dimension of R is an inﬁnite cardinal, say α, within V(B) . By 1.9.11 there exists a set Γ of inﬁnite cardinals and a partition of unity (bγ )γ∈Γ such that bγ [[α = γ ∧ ]] for all γ ∈ Γ. It follows that bγ [[γ ∧ is the algebraic dimension dimR∧ (R) of R over R∧ ]]. Put πγ := χ(bγ ). Again, passing to the model V([0,bγ ]) and making use of 1.3.7 and 2.3.6, we ﬁnd that bγ X = (bγ ∧ R)↓ and [[dimR∧ (bγ ∧ R) = γ ∧ ]] = 1, so that we can assume X = πγ X and [[dimR∧ (R) = γ ∧ ]] = 1. Let E be a Hamel basis for R over R∧ and let σ : γ ∧ → E be a bijection within V(B) . Then the modiﬁed descent s := σ↓ : γ → E ↓ is an injection. Put E0 := σ↓(γ) and by s := σ↓|γ . Clearly, s : γ → E0 is a bijection and it remains to ensure that E0 is a local Hamel basis in X consisting of weak order units pairwise distinct at IX . Since no Hamel basis contains the zero element, we have [[σ(β ∧ ) = 0∧ ]] = [[s(β) = 0]]. 1 = [[(∀ β ∈ γ ∧ )(σ(β) = 0)]] = β∈γ β∈γ It follows that es(β) = [[s(β) = 0]] = 1 and so s(β) is a weak order unit for all β < γ. Similarly, interpreting in V(B) the fact that σ is one-to-one and using the equivalence β1 = β2 ⇐⇒ [[β1∧ = β2∧ ]] = 1, we deduce 1 =[[(∀ β1 , β2 ∈ γ ∧ )(β1 = β2 ↔ σ(β1 ) = σ(β2 ))]] [[β1∧ = β2∧ ]] ⇔ [[σ(β1∧ ) = σ(β2∧ )]] = β1 ,β2 ∈γ = [[|s(β1 ) − s(β2 )| = 0]] : β1 , β2 ∈ γ, β1 = β2 . Thus, |s(β1 ) − s(β2 )| is a weak order unit in X, since e|s(β1 )−s(β2 )| = [[|s(β1 ) − s(β2 )| = 0]] = 1 for all β1 , β2 ∈ γ, β1 = β2 . Thus, πγ X0 is Hamel γ-homogeneous. To complete the proof we have to ensure that πγ X0 is strictly Hamel γ-homogeneous. This is immediate from the following: πγ X0 is strictly γ-homogeneous if and only if bγ [[dim(X ) = γ ∧ ]]. The latter can be proved as in [228, Theorem 8.3.11]. ⊲ 4.7. σ-Distributive Boolean Algebras 4.7. σ-Distributive Boolean Algebras 225 In this section we demonstrate that a universally complete vector lattice is locally one-dimensional if and only if the Boolean algebra of its band projections is σ-distributive; moreover, such vector lattice may be chosen purely nonatomic. 226 Chapter 4. Band Preserving Operators 4.7.1. Let B be an arbitrary Boolean algebra. A subset of B with supremum unit is called a cover of B. The partitions of unity in B are referred to as partitions of B for brevity. Let C be a cover of B. A subset C0 of B is said to be refined from C if, for each c0 ∈ C0 , there exists c ∈ C such that c0 c. An element b ∈ B is refined from C provided that {b} is reﬁned from C; i.e., b c for some c ∈ C. If (Cn )n∈N is a sequence of covers of B and b ∈ B is reﬁned from each of the covers Cn (n ∈ N), then we say that b is reﬁned from (Cn )n∈N . We also refer to a cover whose all elements are reﬁned from (Cn )n∈N as reﬁned from the sequence. 4.7.2. Let B be a σ-complete Boolean algebra. The following are equivalent: (1) B is σ-distributive. (2) There is a (possibly, uncountable) cover refined from each sequence of countable covers of B. (3) There is a (possibly, infinite) cover refined from each sequence of finite covers of B. (4) There is a cover refined from each sequence of two-element partitions of B. ⊳ A proof of (1) ⇐⇒ (2) can be found in Sikorski [365, 19.3]). Item (4) is a paraphrase of 1.9.12 (3) in the deﬁnition of σ-distributivity. The implications (2) =⇒ (3) =⇒ (4) are obvious. ⊲ 4.7.3. Let B be a complete Boolean algebra. The following are equivalent: (1) B is σ-distributive. (2) There is a (possibly, uncountable) partition refined from each sequence of countable partitions of B. (3) There is a (possibly, infinite) partition refined from each sequence of finite partitions of B. (4) There is a partition refined from each sequence of two-element partitions of B. ⊳ The claim follows from 4.7.2 in view of the exhaustion principle. ⊲ 4.7.4. Let Q stand for the Stone space of B and denote by Clop(Q) the Boolean algebra of all clopen sets in Q. We say that a function g ∈ C∞ (Q) is refined from a cover C of the Boolean algebra Clop(Q) if, 4.7. σ-Distributive Boolean Algebras 227 for every two points q ′ , q ′′ ∈ Q satisfying the equality g(q ′ ) = g(q ′′ ), there exists an element U ∈ C such that q ′ , q ′′ ∈ U . If (Cn )n∈N is a sequence of covers of Clop(Q) and a function g is reﬁned from each of the covers Cn (n ∈ N), then we say that g is refined from (Cn )n∈N . 4.7.5. There is a function of C(Q) refined from each sequence of finite covers of Clop(Q). ⊳ Let (Cn )n∈N be a sequence of ﬁnite covers of Clop(Q). By induction, it is easy to construct a sequence of partitions Pm = {U1m , U2m , . . . , U2mm } of Clop(Q) with the following properties: (1) for every n ∈ N, there is m ∈ N such that the partition Pm is reﬁned from Cn ; m+1 m+1 ∨ U2j for all m ∈ N and j ∈ {1, 2, . . . , 2m }. (2) Ujm = U2j−1 Given m ∈ N, deﬁne the 2-valued function χm ∈ C(Q) as follows: χm := m−1 2 m χ(U2i ), i=1 where χ(U ) is the characteristic function of U ⊂ Q also denoted by 1U ∞ in the sequel. Since the series m=1 31m χm is uniformly convergent, its sum g belongs to C(Q). We will show that g is reﬁned from (Cn )n∈N . By property (1) of the sequence (Pm )m∈N , it suﬃces to establish that g is reﬁned from (Pm )m∈N . Assume the contrary and consider the least m ∈ N such that g is not reﬁned from Pm . In this case, there are two points q ′ , q ′′ ∈ Q satisfying the equality g(q ′ ) = g(q ′′ ) and belonging to distinct elements of Pm . Since g is reﬁned from Pm−1 (for m > 1), from property (2) of the sequence (Pm )m∈N it follows that q ′ and q ′′ belong to m some adjacent elements of Pm , i.e. elements of the form Ujm and Uj+1 , m ′ with j ∈ {1, . . . , 2 − 1}. For deﬁniteness, suppose that q belongs to an element with an even index and q ′′ , to that with an odd index; i.e., χm (q ′ ) = 1 and χm (q ′′ ) = 0. Since χi (q ′ ) = χi (q ′′ ) for all i ∈ {1, . . . , m − 1}; therefore, g(q ′ ) − g(q ′′ ) = ∞ 1 1 χi (q ′ ) − χi (q ′′ ) + m i 3 3 i=m+1 ∞ 1 1 1 − = > 0, 3m i=m+1 3i 2 · 3m 228 Chapter 4. Band Preserving Operators which contradicts the equality g(q ′ ) = g(q ′′ ). ⊲ 4.7.6. Theorem. A universally complete vector lattice X is locally one-dimensional if and only if the complete Boolean algebra P(X) is σ-distributive. ⊳ Let Q be the Stone space of the Boolean algebra P(X). Suppose that X is locally one-dimensional and consider an arbitrary sequence (Pn )n∈N of ﬁnite partitions of Clop(Q). By 4.7.3, to prove the σdistributivity of X, it suﬃces to reﬁne a cover of Clop(Q) from (Pn )n∈N . By 4.7.5, we can reﬁne g ∈ C∞ (Q) from the sequence (Pn )n∈N . Since X is locally one-dimensional, there exists a partition (Uξ )ξ∈Ξ of Clop(Q) such that g is constant on each of the sets Uξ . Show that (Uξ )ξ∈Ξ is reﬁned from (Pn )n∈N . To this end, ﬁx arbitrary indices ξ ∈ Ξ and n ∈ N and establish that Uξ is reﬁned from Pn . We may assume that Uξ = ∅. Let q0 be an element of Uξ . Finiteness of Pn allows us to ﬁnd an element U of Pn such that q0 ∈ U . It remains to observe that Uξ ⊂ U . Indeed, if q ∈ Uξ then g(q) = g(q0 ) and, since g is reﬁned from Pn , the points q and q0 belong to the same element of Pn ; i.e., q ∈ U . Assume now that the Boolean algebra P(X) is σ-distributive and consider an arbitrary g ∈ C∞ (Q). By the deﬁnition of locally onedimensional vector lattice, it suﬃces to construct a partition (Uξ )ξ∈Ξ of Clop(Q) such that g is constant on each of the sets Uξ . Given n a natural n and integer m, denote by Um the interior of the closure m+1 of the set of all points q ∈ Q for which m and put n g(q) < n n Pn := Um : m ∈ Z . By 4.7.3, from the sequence (Pn )n∈N of countable partitions of Clop(Q), we can reﬁne some partition (Uξ )ξ∈Ξ . Clearly, this is a desired partition. ⊲ 4.7.7. Theorem. There exists a purely nonatomic σ-distributive complete Boolean algebra. There exists a purely nonatomic locally onedimensional universally complete vector lattice. ⊳ According to 4.7.6 we have only to prove the existence of a purely nonatomic σ-distributive complete Boolean algebra. An algebra of this kind is constructed below in 4.7.9 and 4.7.10. ⊲ 4.7.8. A Boolean algebra B is σ-inductive provided that each decreasing sequence of nonzero elements of B has a nonzero lower bound. A subalgebra B0 of B is dense if, for every nonzero b ∈ B, there exists a nonzero element b0 ∈ B0 such that b0 b. As is well known, to every Boolean algebra B there is a complete including B as a dense subalgebra (cp. Sikorski [365, Boolean algebra B 4.7. σ-Distributive Boolean Algebras 229 is unique up to isomorphism and called a comSection 35]). This B pletion of B. Obviously, a completion of a purely nonatomic Boolean algebra is purely nonatomic. Moreover, the following lemma tells us that a completion of a σ-inductive algebra is σ-distributive. 4.7.9. If a σ-complete Boolean algebra B has a σ-inductive dense subalgebra then B is σ-distributive. ⊳ Let B0 be a σ-inductive dense subalgebra of B. Consider an arbitrary sequence (Cn )n∈N of countable covers of B, denote by C the set of all elements in B that are reﬁned from (Cn )n∈N , and assume by way of contradiction that C is not a cover of B. Then there is a nonzero element b ∈ B disjoint from all elements of C. By induction, we construct the sequences (bn )n∈N and (cn )n∈N as follows: Let c1 be an element of C1 such that b ∧ c1 = 0. Since B0 is dense, there is an element b1 ∈ B0 such that 0 < b1 b ∧ c1 . Suppose that bn and cn are already constructed. Let cn+1 be an element of Cn+1 such that bn ∧ cn+1 = 0. As bn+1 we take an arbitrary element of B0 such that 0 < bn+1 bn ∧ cn+1 . Thus, we have constructed sequences (bn )n∈N and (cn )n∈N such that bn ∈ B0 , bn cn ∈ Cn and 0 < bn+1 bn b for all n ∈ N. Since B0 is σ-inductive, B0 contains a nonzero element b0 that satisﬁes b0 bn for all n ∈ N. By the inequalities b0 cn , we see that b0 is reﬁned from (Cn )n∈N ; i.e., b0 belongs to C. On the other hand, b0 b, which contradicts the disjointness of b from all elements of C. ⊲ 4.7.10. Let B be the quotient Boolean algebra P(N)/I where I is the ideal of P(N) comprising all finite subsets of N. Then the com of B is purely nonatomic and σ-inductive. pletion B ⊳ In view of 4.7.9 we have to prove that B is σ-distributive. The pure nonatomicity of B is obvious. In order to prove that B is σ-inductive, it suﬃces to consider an arbitrary decreasing sequence (bn )n∈N of inﬁnite subsets of N and construct an inﬁnite subset b ⊂ N such that the diﬀerence b\bn is ﬁnite for each n ∈ N. We can easily obtain the desired set b := {mn : n ∈ N} by induction, letting m1 := min b1 and mn+1 := min{m ∈ bn+1 : m > mn }. ⊲ 4.7.11. Let (Ω, Σ, μ) be a Maharam measure space. The Boolean algebra B := B(Ω, Σ, μ) := Σ/μ−1 (0) of measurable sets modulo negligible sets is σ-distributive if and only if Bis atomic and so isomorphic to the boolean P(A) of a nonempty set A . 230 Chapter 4. Band Preserving Operators ⊳ Indeed, suppose that B is not atomic. By choosing a nonzero atomless coset b0 ∈ B of ﬁnite measure, taking an instance B0 ∈ b0 , and replacing (Ω, Σ, μ) with (B0 , Σ0 , μ|Σ0 ), where Σ0 = {B ∩ B0 : B ∈ Σ}, we can assume that μ is ﬁnite and B is atomless. Deﬁne a strictly positive countably additive function ν : B → R by ν(b) = μ(B), where b ∈ B is the coset of B ∈ Σ. Since every ﬁnite atomless measure admits halving, by induction it is easy to construct a sequence of ﬁnite partitions m m 1 1 1 1 Pm := {bm 1 , b2 , . . . , b2m } of 1 ∈ B with 1 = b1 ∨ b2 , ν(b1 ) = ν(b2 ), m+1 m+1 m+1 m+1 m and bj = b2j−1 ∨ b2j , ν(b2j−1 ) = ν(b2j ), for all m ∈ N and j ∈ {1, 2, . . . , 2m }. Since ν(bm j ) → 0 as m → ∞ for each j, there is no partition reﬁned from (Pm )m∈N . It remains to refer to 4.7.3 (1, 3). ⊲ 4.8. Band Preserving Projections In this section we describe the band preserving projection operators on a Dedekind complete vector lattice. First we expatiate on the concept of component (see 2.1.8). 4.8.1. Let X be a vector lattice and u ∈ X. An element v ∈ X is said to be a component or fragment of u if |v| ∧ |u − v| = 0. The collection of all components of u is denoted by C(u). This notation agrees with that in 2.1.8, since C(u) ⊂ X+ whenever u 0. A subset X0 ⊂ X is called componentwise closed in X if C(u) is contained in X0 for each u ∈ X0 . If X has the principal projection property then C(u) = {πu : π ∈ P(X)}. Thus, in this event, X0 is componentwise closed in X if and all if X is invariant under each band projection, i.e., if π(X0 ) ⊂ X0 for all π ∈ P(X). Let X be a vector lattice with the principal projection property. A projection P on X is band preserving if and only if ker(P ) and im(P ) are componentwise closed sublattices of X. ⊳ By 4.1.6 P is band preserving precisely when ker(P ) and im(P ) are invariant under all band projections. But the latter is equivalent to saying that ker(P ) and im(P ) are componentwise closed. Thus, the claim is true if the componentwise closed sublattices are replaced by componentwise closed subspaces. To complete the proof observe that if a vector lattice has the projection property then for all x, y ∈ X the representations x ∨ y = πx + (IX − π)y and x ∧ y = πy + (IX − π)x hold with π := sup{ρ ∈ P(X) : ρx ρy}. Thus, every pair of elements 4.8. Band Preserving Projections 231 x, y ∈ X lies in a subspace together with x ∨ y and x ∧ y, as the latter are the sums of components of x and y. ⊲ 4.8.2. Let P be a band preserving linear operator on a vector lattice X. Assume that X λ = X ↓ for a vector subspace X of RR and p = P λ ↑. Then P is a projection if and only if so is p within V(B) . ⊳ Observe that (P ◦ P )λ = P λ ◦ P λ . Indeed, given a family (xξ ) in X and a partition of unity (πξ ) in P(X λ ) with πξ x = πξ xξ for all ξ, we have πξ P λ x = πξ P xξ by deﬁnition of P λ . Considering that P λ commutes with all band projections in X λ , we can write πξ (P λ ◦ P λ )x = P λ (πξ P λ x) = P λ (πξ P xξ ) = πξ P λ (P xξ ) = πξ (P ◦ P )xξ , so that the required relation follows from the deﬁnition of P λ . It remains to note that the relations P λ ◦P λ = P λ and [[p◦p = p]] = 1 are equivalent, since P λ = p↓, P λ ◦ P λ = (p ◦ p)↓, and [[(P λ ◦ P λ )↑ = p ◦ p]] = 1 according to 1.6.4, 1.5.5 (1), and 1.6.6. ⊲ 4.8.3. Let X be a laterally complete vector lattice, and let X ∈ V(B) be the Boolean valued representation of X with B := P(X). Assume that Vec(X ) stands for the collection of all X0 ∈ V(B) such that [[X0 is a vector subspace of X (over R∧ )]] = 1 and Lat(X) stands for the set of vector sublattices of X which are componentwise closed and laterally complete. Then the mapping X0 → X0 ↓ is a one-to-one correspondence from Vec(X )↓ onto Lat(X). ⊳ This is immediate from 2.5.3 and 1.6.6. ⊲ 4.8.4. Let P be a subfield of R and let X be a subspace of RP . The following are equivalent: (1) X = Pe for some 0 = e ∈ X ; i.e., X is one-dimensional. (2) There are no P-subspaces in X other than {0} and X . (3) There are no P-linear projection on X other then 0 and IX . (4) All P-linear projections on X commute. (5) The composite of two P-linear projections on X is a P-linear projection as well. ⊳ The implications (1) =⇒ (2) =⇒ (3) =⇒ (4) =⇒ (5) are trivial. To ensure the remaining implication (5) =⇒ (1), assume that X is not one-dimensional; i.e., a Hamel basis E for X contains at least two 232 Chapter 4. Band Preserving Operators members e1 , e2 ∈ E . Deﬁne the two projections p and q in X by putting p(e1 ) = p(e2 ) = (e1 + e2 )/2, q(e1 ) = e1 , q(e2 ) = 0, and p(e) = q(e) = 0 for all e ∈ E \ {e1 , e2 }. Then p and q do not commute, since p(q(e1 )) = (e1 + e2 )/2 and q(p(e1 )) = e1 /2. ⊲ 4.8.5. Theorem. For a laterally complete vector lattice X the following are equivalent: (1) X is locally one-dimensional. (2) Each laterally complete componentwise closed sublattice in X is a band. (3) Each band preserving projection on X is a band projection. (4) All band preserving projections on X commute. (5) The composite of two band preserving projections on X is a projection. ⊳ There is no loss of generality in assuming that X = X ↓ with X a subspace of RR within V(B) , B := P(X). By transfer we can apply 4.8.4 within V(B) on replacing P by R∧ and R by R. The rest follows from 4.6.2 (1, 2), 4.8.2, and 4.8.3. ⊲ 4.8.6. Corollary. Let X be a universally complete vector lattice which is not locally one-dimensional. Then there exists a projection operator P on X such that P commutes with all band projections but, nevertheless, P is not a band projection. ⊳ This follows from the equivalence (1) ⇐⇒ (2) in 4.8.5, since P is band preserving if and only if P commutes with all band projections; see 4.1.6. ⊲ 4.8.7. Let X be a vector lattice with the principal projection property and let T : X → X be a band preserving operator. For a disjoint family (yξ )ξ∈Ξ in (im(T ))+ there exists a disjoint family (xξ )ξ∈Ξ in X+ such that yξ = T xξ for all ξ ∈ Ξ. ⊳ Observe that if T u 0 for some u ∈ X then T u− = 0. Indeed, a band preserving operator is disjointness preserving, so that T u+ ⊥ T u− and so T u+ − T u− = T u 0 implies T u− = 0. Now, given a disjoint family (yξ )ξ∈Ξ in (im(T ))+ , for every ξ ∈ Ξ choose uξ ∈ X with yξ = T uξ and put xξ := πξ u+ ξ with π := [yξ ]. Then (xξ )ξ∈Ξ is a disjoint family in X+ and, by 4.1.6, yξ = πP uξ = P πξ u+ ξ = P xξ for all ξ ∈ Ξ. ⊲ 4.8. Band Preserving Projections 233 4.8.8. Let X be a laterally complete vector lattice. A subspace X0 of X is the range of a band preserving projection operator if and only if X0 is componentwise closed and laterally complete sublattice. Moreover, in this event there exists a componentwise closed and laterally complete sublattice X1 ⊂ X such that X = X0 ⊕ X1 . ⊳ If X0 is a componentwise closed and laterally complete sublattice of X then, in view of 4.8.3, X0 = X0 ↓ for some vector subspace X0 ⊂ X within V(B) . Working within V(B) choose some complementary subspace X1 ⊂ X and let p be a projection on X with im(p) = X0 and ker(p) = X1 . By 4.8.2 P := p↓ is a band preserving projection and im(P ) = im(p)↓ = X0 ↓ = X0 . Conversely, assume that X0 = P (X) for some band preserving projection P on X. By 4.8.1 X0 is componentwise closed. To show that X0 is laterally complete take a disjoint family (yξ )ξ∈Ξ in (X0 )+ and, using 4.8.7, choose a disjoin family (xξ )ξ∈Ξ in X+ such that yξ = P xξ for all ξ ∈ Ξ. As X is laterally complete, there exists x := supξ∈Ξ xξ . Clearly, y = P x is the least upper bound of the family (yξ )ξ∈Ξ , since πξ y = P πξ x = P πξ xξ = yξ for all ξ ∈ Ξ. It remains to observe that X = X0 ⊕ X1 , whenever X1 := X1 ↓ and X1 is an (arbitrary) complementary subspace of X within V(B) . ⊲ 4.8.9. Let X be a Dedekind complete vector lattice. The following are equivalent: (1) Each principal band in X is universally complete. (2) For each x ∈ X+ , for each disjoint sequence (xn ) in C(x), and for each sequence (λn ) in R+ there exists in X the element ∞ n=1 λn xn = sup m m∈N n=1 λn xn . ⊳ Only the implication (2) ⇐⇒ (1) is nontrivial. Assume that (2) is fulﬁlled and verify that for an arbitrary e ∈ X+ the band B := {e}⊥⊥ is universally complete. Take 0 x ∈ B u and let (exλ )λ∈R stands for the spectral system of x with respect to e (considered as a unit element in B u ). Fix a partition of the real line β := (tn )n∈Z ; i.e., tn < tn+1 (n ∈ N) and limn→±∞ tn = ±∞. Observe that x(β) x x(β) where x(β) := tn (extn+1 − extn ), x(β) := tn+1 (extn+1 − extn ). n∈Z n∈Z 234 Chapter 4. Band Preserving Operators By (2) we have x(β), x(β) ∈ X ∩ B u = B and hence B u ⊂ B. ⊲ A Dedekind complete vector lattice X satisfying any of the equivalent conditions in 4.8.9 is called principally universally complete. 4.8.10. A projection P on a principally universally complete vector lattice X is band preserving if and only the following hold: (1) ker(P ) and im(P ) are componentwise closed. (2) For every principal band B in X the intersections B ∩ker(P ) and B ∩ im(P ) are laterally complete. ⊳ According to 4.8.8 the above conditions (1) and (2) are equivalent to saying that the restriction of P to every principal band is band preserving. In particular, P x ∈ {x}⊥⊥ for all x ∈ X. From this it is immediate that P (B) ⊂ B for every band B ∈ B(X), because x ∈ B implies P x ∈ {x}⊥⊥ ⊂ B. ⊲ 4.8.11. Theorem. Let X be a Dedekind complete vector lattice and let P be a band preserving projection operator on X. Then there exists a unique pair of complimentary bands X1 and X2 such that the following hold: (1) X1 is the maximal band such that the restriction of P to X1 is order bounded and, in particular, P |X1 is a band projection. (2) X2 principally universally complete and the restriction P |X2 is described as in 4.8.10. ⊳ Take X = Y in 4.3.10 and put X1 := B and X2 := B ⊥ . In view of 4.8.10 we have only to prove that X2 is principally universally complete. Take 0 x ∈ X2 , a disjoint sequence (xn ) of components of x, and a sequence (λn ) of positive scalars. According to 4.3.10 for each n ∈ N we can mﬁnd yn ∈ X such that 0 yn xn and |P yn | nλn xn . Obviously, n=k (1/n)yn (1/m)x for all k m ∈ N, and so the series ∞ n=1 (1/n)yn converges uniformly to some y ∈ Y . Since the terms of the series are pairwise disjoint, we have |P y| |P ((1/n)yn )| λn xn for all n ∈ N, whence ∞ n=1 λn xn exists in X. Appealing to 4.8.9 completes the proof. ⊲ 4.9. Algebraic Band Preserving Operators In this section a description of algebraic orthomorphisms on a vector lattice is given and the Wickstead problem for algebraic operators is examined. 4.9. Algebraic Band Preserving Operators 235 4.9.1. Let P[x] be a ring of polynomials in variable x over a ﬁeld P. An operator T on a vector space X over a ﬁeld P is said to be algebraic if there exists a nonzero ϕ ∈ P[x], a polynomials with coeﬃcients in P, for which ϕ(T ) = 0. For an algebraic operator T there exists a unique polynomial ϕT such that ϕT (T ) = 0, the leading coeﬃcient of ϕT equals to 1, and ϕT divides each polynomial ψ with ψ(T ) = 0. The polynomial ϕT is called the minimal polynomial of T . The simple examples of algebraic operators yield a projection P (an idempotent operator, P 2 = P ) in X with ϕP (λ) = λ2 − λ whenever P = 0, IX , and a nilpotent operator S (S m = 0 for some m ∈ N) in X with ϕS (λ) = λk , k m. For an operator T on X, the set of all eigenvalues of T will be denoted throughout by σp (T ). A real λ is a root of ϕT if and only if λ ∈ σp (T ). In particular, σp (T ) is ﬁnite. 4.9.2. Let X be a vector lattice and b − a2 > 0 for some a, b ∈ R. Then T 2 + 2aT + bI is a weak order unit in Orth(X) for every T ∈ Orth(X). ⊳ Since I := IX is a weak order unit in Orth(X), so is (b − a2 )I. Moreover, in Orth(X) the inequalities hold: 0 < (b − a2 )I (b − a2 )I + (T + aI)2 = T 2 + 2aT + bI. Consequently, T 2 + 2aT + bI is a weak order unit in Orth(X) as well. ⊲ 4.9.3. Let X be a vector lattice and let T in Orth(X) be algebraic. Then , ϕT (x) = (x − λ). λ∈σp (T ) ⊳ We claim that there are no quadratic polynomials in the factorization of T into irreducible elements in R[X]. Otherwise, there would exist a, b ∈ R with b − a2 > 0 and a nonzero polynomial ψ ∈ R[X] such that ϕT (x) = (x2 + 2ax + b)ψ(x). This would entail that (T 2 + 2aT + bI)ψ(T ) = ϕT (T ) = 0. But ψ(T ) ∈ Orth(X) and so ψ(T ) = 0 by 4.9.2, which contradicts the minimality of ϕT . Accordingly, , ϕT (x) = (x − λ)nλ λ∈σp (T ) for some nλ ∈ N (λ ∈ σp (T )). Choose n a common multiple of the collection {nλ : λ ∈ σp (T )}. Obviously, ϕT divides the polynomial 236 - Chapter 4. Band Preserving Operators n n = 0 in Orth(X). and therefore λ∈σp (T ) (T −λI) λ∈σp (T ) (x−λ) Since the f -algebra Orth(X) is semiprime by 4.1.3, we ﬁnd λ∈σp (T ) (T − λI) = 0, whence the desired identity follows. ⊲ 4.9.4. Consider the universally complete vector lattice X = R↓. Let T be a band preserving linear operator on X and let τ be an R∧ linear function on R. For ϕ ∈ R[x], ϕ(x) = a0 + a1 x + · · · + an xn define ∧ ϕ̂ ∈ R∧ [x] by ϕ̂(x) = a∧0 + a∧1 x + · · · + a∧n xn . Then ϕ̂(τ )↓ = ϕ(τ ↓), ϕ(T )↑ = ϕ̂(T ↑). ∧ ⊳ It follows from 1.5.5 (1) and 1.6.4 that (τ n )↓ = (τ ↓)n and (T n )↑ = ∧ (T ↑)n . It remains to apply 4.3.5. ⊲ 4.9.5. A linear operator T on a vector lattice X is said to be diagonal if T = λ1 P1 + · · · + λm Pm for some collections of reals λ1 , . . . , λm and projection operators P1 , . . . , Pm on X with Pı ◦ Pj = 0 (ı = j). In the equality above, we can and will assume that P1 + · · · + Pn = IX and that λ1 , . . . , λm are pairwise diﬀerent. An algebraic operator T is diagonal if and only if the minimal polynomial of T have the form ϕT (x) = (x − λ1 ) · · · (x − λm ) with pairwise distinct λ1 , . . . , λm ∈ R. We call an operator T on X strongly diagonal if there exist pairwise disjoint band projections P1 , . . . , Pm and reals λ1 , . . . , λm such that T = λ1 P1 + · · · + λm Pm . In particular, each strongly diagonal operator on X is an orthomorphism. It is easily seen that the set of all strongly diagonal operators on X is an f -subalgebra of Orth(X). 4.9.6. Let T = λ1 P1 + · · · + λm Pm be a diagonal operator on a vector lattice X. Then T is band preserving if and only if the projection operators P1 , . . . , Pm are band preserving. ⊳ The suﬃciency is obvious. To prove the necessity, observe ﬁrst that if T is band preserving then so is T n for all n ∈ N and so ϕ(T ) is band preserving for every polynomial ϕ ∈ R[x]. Next, make use of the representation Pj = ϕj (T ) (j := 1, . . . , m), where ϕj ∈ R[x] is an interpolation polynomial deﬁned by ϕj (λk ) = δjk with δjk the Kronecker symbol. ⊲ 4.9.7. Let X be a vector lattice. A linear operator T on X is strongly diagonal if and only if T is an algebraic orthomorphism on X. ⊳ The necessity follows from 4.9.5. Let T be an orthomorphism in X and ϕ(T ) = 0, where ϕ is a minimal polynomial of T , so that ϕ(λ) = (λ − λ1 ) · · · (λ − λm ) with λ1 , . . . , λm ∈ R. Since T admits the unique 4.9. Algebraic Band Preserving Operators 237 extension to an orthomorphism on X u , we can assume without loss of generality that X = X u = R↓ and τ = T ↑. Then [[τ (x) = λ0 x (x ∈ R)]] = 1 for some λ0 ∈ R. It is seen from 4.9.4 that ϕ̂(λ0 ) = 0 and so (λ0 − λ∧1 ) · · · (λ0 − λ∧m ) = 0 or λ0 ∈ {λ∧1 , . . . , λ∧m } within V(B) . Put Pl := χ(bl ) with bl := [[λ0 = λ∧l ]] and observe that {P1 , . . . , Pm } is a partition of unity in P(X). Moreover, given x ∈ X, we can estimate bl [[T x = τ x = λ0 x]]∧[[λ0 = λ∧l ]] [[T x = λ∧l x]], so that Pl T x = Pl (λl x) = λl Pl (x). Summing up over l = 1, . . . , m, we get T x = λ1 P1 x + · · · + λm Pm . ⊲ 4.9.8. Theorem. Let X be a universally complete vector lattice. The following assertions are equivalent: (1) The Boolean algebra P(X) is σ-distributive. (2) Every algebraic operator in Lbp (X) is order bounded. (3) Every algebraic operator in Lbp (X) is strongly diagonal. (4) Every diagonal operator in Lbp (X) is strongly diagonal. (5) Every projection operator in Lbp (X) is a band projection. (6) Every nilpotent operator in Lbp (X) is order bounded. (7) Every nilpotent operator in Lbp (X) is trivial. ⊳ (1) =⇒ (2): Follows from 4.6.4 and 4.7.7. (2) =⇒ (3): Follows from 4.9.7. (3) =⇒ (4): A diagonal operator is algebraic by deﬁnition (cp. 4.9.5). (4) =⇒ (5): This is evident. (5) ⇐⇒ (1): Follows from 4.8.5 ((1) ⇐⇒ (3)). (2) =⇒ (6): A nilpotent operator is algebraic by deﬁnition. (6) =⇒ (7): A nilpotent orthomorphism is trivial; i.e., the f -algebra Orth(X) is semiprime (cp. 4.1.3). (7) =⇒ (1): Arguing for a contradiction, assume that P(X) is not σdistributive and construct a nonzero band preserving nilpotent operator in X. By 4.4.9 ((1) ⇐⇒ (2)) V(B) |= R = R∧ and in this case R is an inﬁnite-dimensional vector space over R∧ within V(B) ; see 4.6.6. Let E ⊂ R be a Hamel basis and choose an inﬁnite sequence (en )n∈N of pairwise distinct elements in E . Fix a natural m > 1 and deﬁne an R∧ linear function τ : R → R within V(B) by letting τ (ekm+i ) = ekm+i−1 if 2 i m, τ (ekm+1 ) = 0 for all k := 0, 1, . . ., and τ (e) = 0 if e = en for all n ∈ N. In other words, if R0 is the R∧ -linear subspace of R generated by the sequence (en )n∈N , then R0 is an invariant subspace for τ and τ is the linear operators associated to the inﬁnite block matrix 238 Chapter 4. Band Preserving Operators diag(A, . . . , A, . . .) with equal blocks a square matrix of dimension m, ⎛ 0 1 0 ⎜0 0 1 ⎜ ⎜ A = ⎜ ... ... ... ⎜ ⎝0 0 0 0 0 0 in the principal diagonal and A ⎞ ... 0 . . . 0⎟ ⎟ .⎟ . .. . .. ⎟ ⎟ . . . 1⎠ ... 0 It follows that τ is discontinuous and τ m = 0 by construction. Consequently, T := τ ↓ is a band preserving linear operator in X and T m = 0 by 4.9.4, but T is not order bounded; a contradiction. ⊲ 4.10. Band Preserving Operators on Complex Vector Lattices Consider some properties of band preserving operators in a complex vector lattice. 4.10.1. A vector lattice X is called square-mean closed if for all x, y ∈ X the set {(cos θ)x + (sin θ)y : 0 θ < 2π} has a supremum s(x, y) in X. Every uniformly complete vector lattice is square-mean closed. But a square-mean closed Archimedean vector lattice need not be relatively uniformly complete. Recall that a complex vector lattice is the complexiﬁcation XC := X ⊕ iX := {x + iy : x, y ∈ X} of a real square-mean closed vector lattice X; see 2.3.3. Thus, each element z ∈ XC in a complex vector lattice has the absolute value |z| deﬁned as |z| := s(x, y) (z := x + iy ∈ XC ). 1 Clearly, |z| = x2 + y 2 in the sense of homogeneous functional calculus and so |x| ∨ |y| |z| |x| + |y|. The mapping z → |z| of XC to X satisﬁes the relations (λ ∈ C; z, z1 , z2 ∈ XC ; z̄ := x − iy): (1) |z| 0; |z| = 0 ⇐⇒ z = 0; (2) |λz| = |λ||z|; |z| = |z̄|; (3) |z1 + z2 | |z1 | + |z2 |; 6 6 (4) 6|z1 | − |z2 |6 |z1 − z2 |. 4.10. Band Preserving Operators on Complex Vector Lattices 239 A subset A ⊂ XC is order bounded if the set {|z| : z ∈ XC } is order bounded in X. As in the real case, the notion of disjointness of elements z := x + iy and w := u + iv in XC is deﬁned by the formula z ⊥ w ⇐⇒ |z| ∧ |w| = 0 and is equivalent to the relation {x, y} ⊥ {u, v}. The disjoint complement A⊥ of a nonempty set A ⊂ XC is deﬁned by A⊥ := {z ∈ XC : z ⊥ w for all w ∈ A}. Say that XC is Dedekind complete (σ-complete) if X is Dedekind complete (σ-complete). 4.10.2. A vector sublattice of XC is a vector subspace Y ⊂ XC such that z ∈ Y implies z̄ ∈ Y and |z| ∈ Y . An ideal J in XC is deﬁned as the linear subspace which is solid: |w| |z| with w ∈ XC and z ∈ J implies w ∈ J. As in the real case, a band in XC can be deﬁned as {z ∈ XC : (∀ w ∈ V ) z ⊥ w}, where V is a nonempty subset of XC . The sublattices, ideals, and bands of XC are precisely the complexiﬁcations of sublattices, ideals, and bands of X (cp. Schaefer [356, Chapter II, § 11] and Zaanen [427, Section 91] for more detail). A band B is a projection band if XC = B ⊕ B ⊥ . Each projection band B is the range of a projection P on XC with kernel B ⊥ called a band projection. As in the real case B(XC ) and P(XC ) stand respectively for the Boolean algebras of all band and all band projections in XC . 4.10.3. Let X and Y be real vector spaces considered as real subspaces of XC and YC , respectively. Each R-linear operator T : X → Y admits the unique extension to the C-linear operator TC : XC → YC deﬁned as TC (x + iy) := T x + iT y (x + iy ∈ XC ). The operator TC is usually identiﬁed with T , so that the vector space L(X, Y ) of R-linear operators from X to Y is viewed as a real vector subspace of L(XC , YC ) comprising the operators satisfying T (X) ⊂ Y . With this agreement in mind it is easily seen that an operator T ∈ L(XC , YC ) is uniquely representable as T = T1 + iT2 , where T1 , T2 ∈ L(X, Y ), that is, T z = T1 x − T2 y + i(T2 x + T1 y) (z = x + iy ∈ XC ). Thus, the space L(XC , YC ) of C-linear operators is isomorphic to the complexiﬁcation of the real space L(X, Y ) of R-linear operators; i.e., L(XC , YC ) = L(X, Y )C . 4.10.4. Assume now that X and Y are real vector lattices. An operator T = T1 + iT2 is positive provided that T1 0 and T2 = 0 and 240 Chapter 4. Band Preserving Operators order bounded provided that for every e ∈ X+ there is f ∈ Y+ satisfying |T z| f whenever z ∈ XC , |z| e. The space L∼ (XC , YC ) of all order bounded linear operators from XC into YC is the complexiﬁcation of the space of all order bounded linear operators from X into Y : L∼ (XC , YC ) = L∼ (X, Y )C = L∼ (X, Y ) ⊕ iL∼ (X, Y ). An operator T = T1 + iT2 ∈ L(XC , YC ) is said to be regular if T1 and T2 are regular. If Y is Dedekind complete then L∼ (XC , YC ) is also a Dedekind complete complex vector lattice. In particular, every operator T = T1 + iT2 has the modulus |T | and the Riesz–Kantorovich formula holds true; i.e., for every u ∈ X+ we have |T |u = |T1 + iT2 |u = sup |T z| = |z|u sup |(T1 + iT2 )(x + iy)|. |x+iy|u A lattice homomorphism is an operator T = T1 + iT2 ∈ L(XC , YC ) with T2 = 0 and T1 a lattice homomorphism from X to Y . Clearly, T is a lattice homomorphism if and only if |T z| = |T |(|z|) for all x ∈ XC . It is also worth mentioning that if P = P1 + iP2 is a projection onto the band B = B1 + iB2 then P2 = 0 and P1 is a projection onto the band B. More details can be found in Abramovich and Aliprantis [5, Chapter 3], Schaefer [356, Chapter II] and Zaanen [427, Section 92]. Suppose that Y is a sublattice of a vector lattice X. A linear operator T from YC to XC is band preserving provided that z ⊥ w =⇒ T z ⊥ w (z ∈ YC , w ∈ XC ), where the disjointness relations are understood in XC (cp. 4.1.1). 4.10.5. A linear operator T := T1 + iT2 from YC to XC is band preserving if and only if such are the real linear operators T1 and T2 from Y to X. ⊳ Assume that T1 and T2 are band preserving. If z := x + iy and w := u + iv are disjoint then {x, y} ⊥ {u, v}. Therefore, {x, y} ⊥ {T1 u − T2 v, T1 v + T2 u}. Hence, z ⊥ T w, since T w = (T1 u − T2 v) + i(T1 v + T2 u). Conversely, if T is band preserving and x ∈ X and y ∈ Y are disjoint then x ⊥ T y = T1 y + iT2 y hence, x ⊥ {T1 y, T2 y}, so that T1 and T2 are band preserving. ⊲ 4.10.6. In particular, if X is a vector lattice enjoying the principal projection property and Y is an order dense ideal of X then a linear 4.10. Band Preserving Operators on Complex Vector Lattices 241 operator T = T1 + iT2 : YC → XC is band preserving if and only if πTk z = Tk πz (z ∈ YC , k = 1, 2) for all π ∈ P(XC ). An order bounded band preserving operator on XC is called an orthomorphism and the set of all orthomorphisms on XC is denoted by Orth(XC ). Clearly, Orth(XC ) is the complexiﬁcation of Orth(X); i.e., Orth(XC ) = Orth(X)C . 4.10.7. Deﬁne a complex f -algebra to be the complexiﬁcation AC of a real square-mean closed f -algebra A (cp. 4.10.1). The multiplication on A extends naturally to AC by the formula (x + iy)(u + iv) = (xu − yv) + i(xv + yu), and so AC becomes a commutative complex algebra. Moreover, |z1 z2 | = |z1 ||z2 | (z1 , z2 ∈ AC ). In this situation AC is called a complex f -algebra (cp. Beukers, Huijsmans, and de Pagter [53]; Zaanen [427]). A complex f -algebra AC is semiprime whenever z ⊥ w is equivalent to zw = 0 for all z, w ∈ AC . If Z is a universally complete vector lattice with a ﬁxed order unit 1 ∈ Z then there is a unique multiplication on Z which makes Z into an f -algebra and 1 into the multiplicative unity. Thus, ZC is an example of a complex f -algebra. We will always keep this circumstance in mind while considering a universally complete vector lattice as an f -algebra. 4.10.8. Given an algebra A over a ﬁeld P and a subalgebra A0 of A, we call a P-linear operator D : A0 → A a P-derivation (or simply a derivation if P is meant) provided that D(uv) = D(u)v + uD(v) (u, v ∈ A0 ). A P-endomorphism of an algebra A is a P-linear multiplicative operator M : A → A; i.e., M is P-linear and satisfy the equation M (uv) = M (u)M (v) (u, v ∈ A). A bijective P-endomorphism is a P-automorphism. We simply speak of endomorphisms and automorphisms whenever P is meant. The kernel of a derivation is a subalgebra and the kernel of an automorphism is a ring ideal. A nonzero derivation is called nontrivial. The identical automorphism is commonly referred to as the trivial automorphism. If P = R or P = C in the above deﬁnitions of a P-derivation then we speak of real derivation and complex derivation, respectively. Let Z stand for a real universally complete vector lattice with a ﬁxed f -algebra multiplication and X be an f -subalgebra of Z. 242 Chapter 4. Band Preserving Operators 4.10.9. Let D ∈ L(XC , ZC ) and D = D1 + iD2 . The operator D is a complex derivation if and only if D1 and D2 are real derivations from X into Z. If X is minorizing in Z and X ⊥⊥ = Z then each derivation from XC into ZC is a band preserving operator. ⊳ To ensure that the ﬁrst assertion holds we only have to insert D := D1 + iD2 in the equality D(uv) = D(u)v + uD(v), take u := x ∈ X and v := y ∈ X, and then equate the real and imaginary parts of the resulting relation. According to this fact and 4.10.5, it remains only to establish that every real derivation is a band preserving operator. Let D : X → Z be a real derivation. Take disjoint x, y ∈ X. Since the relation x ⊥ y in an f -algebra implies xy = 0, we have 0 = D(xy) = D(x)y + xD(y). But the elements D(x)y and xD(y) are disjoint as well by the deﬁnition of f -algebra; therefore, D(x)y = 0 and xD(y) = 0. Hence, since the f -algebra X is semiprime, we obtain D(x) ⊥ y and x ⊥ D(y). Now, consider disjoint x ∈ X and z ∈ Z. By hypothesis, the order ideal I generated by (X ∩ {x}⊥ ) ∪ {x} is order dense in Z. Therefore, without loss of generality we may assume |z| = supα yα for some family (yα ) in X+ . We have yα ⊥ D(x) as just proved and consequently, z ⊥ D(x). ⊲ 4.10.10. Put X := R↓ and let Lbp (XC ) be the set of all band preserving linear operators in XC . Denote by End(CC ) the member of V(B) that depicts the C∧ -vector space of all C∧ -linear mappings from C into C . Then the faithful unitary XC -modules Lbp (XC ) and End(CC )↓ are put into isomorphy by sending a band preserving operator to its ascent. ⊳ Recall that C ∈ V(B) is deﬁned as C := R ⊕ iR and by the Gordon Theorem the descent C ↓ = R↓ ⊕ iR↓ is a universally complete complex vector lattice and a complex f -algebra simultaneously. Moreover, [[ C∧ = R∧ ⊕ iR∧ is a dense subﬁeld of C ]] = 1. (We write i instead of i∧ .) It is easy to observe that Lbp (XC ) = Lbp (X)C , [[End(CC ) = End(R)C ]] = 1. The claim follows from 4.3.5 and 4.10.5. ⊲ 4.11. Automorphisms and Derivations on the Complexes Here we recall the information on ﬁeld theory which we need for further analysis of the two collections of simultaneous functional equations (A) and (D) in Section 4.2. 4.11. Automorphisms and Derivations on the Complexes 243 4.11.1. Consider some ﬁelds K and L. If K is a subﬁeld of L, then L is an extension of K. An extension L of a ﬁeld K is called algebraic provided that each element of L is a root of some nonzero polynomial (in a sole variable) with coeﬃcients in K. In other words, an extension L of K is algebraic in case every x ∈ L is algebraic over K; i.e., to each x ∈ L there are ﬁnitely many a0 , . . . , an ∈ K, n 1, some of them nonzero, such that a0 + a1 x + · · · + an xn = 0. An extension L of K is transcendental over K if L is not algebraic. Recall that a ﬁeld K is algebraically closed provided that each nonconstant polynomial with coeﬃcients in K has at least one root in K. In other words, K is algebraically closed if and only if every algebraic extension of K is K. The algebraic closure of a ﬁeld K is an extension of K that is algebraic over K and algebraically closed. It is proved in ﬁeld theory that each ﬁeld K has some algebraic closure that is unique up to K-isomorphism (cp. Bourbaki [70] and Van der Waerden [405]). 4.11.2. Let L be an extension of a ﬁeld K. The pairwise distinct x1 , . . . , xn ∈ L are called algebraically independent over K provided that for each polynomial P in n variables with coeﬃcients in K from P (x1 , . . . , xn ) = 0 it follows that P ≡ 0; i.e., all coeﬃcients of P are equal to zero. The deﬁnition prompts us to say that the algebraic independence of x1 , . . . , xn amounts to the linear independence over K of the set of al monomials of the form xi11 xi22 · · · xinn , where n ∈ N and i1 , . . . , in ∈ N. A subset E of L is called algebraically independent provided that every ﬁnite subset of E is algebraically independent. So, the empty set is algebraically independent. An inclusion maximal subset E of L algebraically independent over K is called a transcendence basis for L. Let K(E ) stand for the inclusion least subﬁeld of L which includes K and E ⊂ L. In this event we say that K(E ) results from K by adjunction of E . In case L = K(E ) and E is algebraically independent, L is called a pure extension of K, while E is a pure transcendence basis of L over K. 4.11.3. Steinitz Theorem. Each extension L of a field K has a transcendence basis E over K. In this event L is an algebraic extension of the pure extension K(E ). ⊳ See Bourbaki [70, Chapter 5, Section 5, Theorem 1]. ⊲ 244 Chapter 4. Band Preserving Operators 4.11.4. Isomorphism Extension Theorem. Assume that L is an extension of a field K and E is a transcendence basis for L over K. Assume further that ı is an isomorphism of K to some field K ′ and L′ is an algebraically closed extension of K ′ . Then to each algebraically independent family (le )e∈E of elements of L′ there is an isomorphism ı′ of L to L′ extending ı and satisfying the condition ı′ (e) = le for all e ∈ E . ⊳ See Bourbaki [70, Chapter 5, Section 4, Proposition 1]. ⊲ 4.11.5. A mapping d : K → L is a derivation of K ⊂ L to L provided that d(x + y) = d(x) + d(y) and d(xy) = d(x)y + xd(y) for all x, y ∈ K. The general result on extension of derivations to be formulated in the next subsection uses the concept of separable extension. We will not expatiate upon the formal deﬁnition of separable extension and relevant information, but the interested reader can ﬁnd all details in Zariski and Samuel [429]. For our ends, it suﬃces to mention that if K is algebraically closed or has characteristic zero, then every extension of K is separable. 4.11.6. Derivation Extension Theorem. Let k be a subfield of L, while K is an extension of k lying in L. For a derivation d from k to L the following hold: (1) If K is a pure transcendental extension of k with a pure transcendence basis E ⊂ K over k, then to each family (le )e∈E of elements of L there corresponds the unique derivation D from K to L extending d such that De = le for all e ∈ E . (2) If K is a separable algebraic extension of k, then to d there corresponds the unique derivation D from K to L extending d. ⊳ See Bourbaki [70, Chapter 5, Section 9, Propositions 4 and 5]. ⊲ 4.11.7. Let C be a transcendental extension of a field P. Then there is a nontrivial P-automorphism of C. ⊳ Let E be a transcendence basis for the extension C over P. Since C is an algebraically closed extension of P(E ), each P-automorphism φ of the ﬁeld P(E ) extends to a P-automorphism Φ of the ﬁeld C by Theorem 4.11.4 (see Bourbaki [70, Chapter 5, § 5, Theorem 1]. It is clear that if φ is nontrivial then so is Φ. To construct a nontrivial P-automorphism in P(E ), we ﬁrstly consider the case when E contains only one element e; i.e., when C is an algebraic extension of a simple transcendental extension P(e). Take a, b, c, d ∈ P such that ad − bc = 0. Then e′ = (ae + b)/(ce + d) is a generator of the 4.11. Automorphisms and Derivations on the Complexes 245 ﬁeld P(e) which diﬀes from e. The ﬁeld P(e) = P(e′ ) is isomorphic to the ﬁeld of rational fractions in one variable t; consequently, the linearfractional substitution t → (at+b)/(ct+d) deﬁnes a P-automorphism φ of the ﬁeld P(e) which sends e to e′ (cp. Van der Waerden [405, Section 39]). Assume now that E contains at least two distinct elements e1 and e2 and take an arbitrary one-to-one mapping φ0 : E → E for which φ0 (e1 ) = e2 . Again, using the circumstance that C is an algebraically closed extension of P(E ), we can construct a P-automorphism φ of C such that φ0 (e) = φ(e) for all e ∈ E (see Theorem 4.11.4). Clearly, φ is nontrivial. ⊲ 4.11.8. Let C be a transcendental extension of a field P. Then there is a nontrivial P-derivation on C. ⊳ We again use a transcendence basis E for the extension C over P. It is well known that each derivation of P extends onto a purely transcendental extension; moreover, this extension is deﬁned uniquely by prescribing arbitrary values at the elements of a transcendence basis (see Theorem 4.11.6 (1)). Thus, for every mapping d : E → C, there is a unique derivation D : P(E ) → C such that D(e) = d(e) for all e ∈ E and D(x) = 0 for x ∈ P. Now, C is a separable algebraic extension of P(E ); consequently, D admits the unique extension to some derivation D : C → C by Theorem 4.11.6 (2). It is obvious that the freedom in the choice of d guarantees that D is nontrivial. ⊲ 4.11.9. Theorem. Let C be an extension of an algebraically closed subfield P. Then the following are equivalent: (1) P = C. (2) Every P-linear function in C is order bounded. (3) There is no nontrivial P-derivation on C. (4) There is no nontrivial P-automorphism of C. (5) Every P-endomorphism of C is the zero or the identity function. ⊳ If P = C then every P-linear function f : C → C is of the form f (z) = cz (z ∈ C) for some c ∈ C; therefore (1) =⇒ (2) and (1) =⇒ (3) trivially. If f is multiplicative then c2 = c and hence c = 0 or c = 1, whence (1) =⇒ (4) and (1) =⇒ (5). The converse implications follows from 4.11.7, 4.11.8, and Theorem 4.2.8. ⊲ 246 Chapter 4. Band Preserving Operators 4.12. Automorphisms and Derivations on Complex f -Algebras In this section we characterize the universally complete complex f -algebras admitting nontrivial automorphisms and derivations. The results are obtained by means of Boolean valued interpretation of some properties of the complexes that appeared in the previous section. 4.12.1. Theorem. The field C∧ is algebraically closed in C within V(B) . In particular, the following dichotomy holds within V(B) : either C∧ = C or C is a transcendental extension of C∧ . ⊳ The second part is obvious from the ﬁrst. Prove that the ﬁeld C∧ is algebraically closed in C . Working within V(B) , assume that z0 ∈ C is a root of a nonzero polynomial with coeﬃcients in C∧ . We can formalize this assertion as follows: ϕ(z0 ) ≡ (∃ n ∈ ω)(∃ κ : n → C∧ ) κ(l)z0l = 0 ∧ ((∃ l ∈ n)κ(l) = 0), l∈n where n := {0, 1, . . . , n − 1}. Thus, [[ϕ(z0 )]] = 1, and eliminating the Boolean estimates for quantiﬁers by means of the maximum principle 1.4.2, we ﬁnd a countable partition of unity (bn ) ⊂ B and a sequence (κn ) ⊂ V(B) for which [[κn : n∧ → C∧ ]] bn , [[(∃ l ∈ n∧ )κn (l) = 0)]] bn , (n−1)∧ [[κn (0∧ ) + κn (1∧ )z0 + · · · + κn ((n − 1)∧ )z0 = 0]] bn (n ∈ ω). It suﬃces to establish the inequality [[z0 ∈ C∧ ]] bn for a ﬁxed n ∈ ω. In the arguments below, without loss of generality we can assume that bn = 1, since otherwise we can replace B with the Boolean algebra Bn := [0, bn ] with unity bn and V(B) with V(Bn ) with application of 1.3.7 to the complete Boolean homomorphism π : b → b ∧ bn from B to Bn . Note that X := C∧ ↓ is an f -subalgebra in C ↓ and consists of piecewise constant elements. More exactly, an element z ∈ C belongs to X if and only if z has the representation z = o- ξ λξ πξ (1), where (πξ ) is a partition of unity in B = P(C ↓) and (λξ ) is a family of complex numbers with the same set of indices. 4.12. Automorphisms and Derivations on Complex f -Algebras 247 Let kn : {0, 1, . . . , n − 1} → X be the modiﬁed descent of κn ; see 1.5.8. Since kn (0), kn (1), . . . , kn (n − 1) ∈ X, we can choose a par tition of unity (πξ ) ⊂ B, πξ = 0, such that kn (l) = o- ξ λl,ξ πξ (1), l := 0, . . . , n − 1. If λ0,ξ = λ1,ξ = · · · = λn−1,ξ = 0 for some ξ then [[kn (l) = 0]] [[kn (l) = λ∧l,ξ ]] ∧ [[λ∧l,ξ = 0∧ ]] πξ for all l; consequently, n−1 ∧ [[κn (l ) = 0]] = l=0 n−1 [[kn (l) = 0]] = [[kn (l) = 0]] n−1 l=0 ∗ πξ∗ < 1. l=0 But this contradicts the relation 1 = [[(∃ l ∈ n∧ )κn (l) = 0)]] = n−1 [[κn (l∧ ) = 0]]. l=0 The relation (n−1)∧ [[κn (0∧ ) + κn (1∧ )z0 + · · · + κn ((n − 1)∧ )z0 = 0]] = 1 kn (0) + kn (1)z0 + · · · + kn (n − 1)z0n−1 = 0; therefore, implies the equality using the above representation for kn , we obtain a family of equations with constant complex coeﬃcients λ0,ξ + λ1,ξ πξ z0 + · · · + λn−1,ξ πξ z0n−1 = 0; moreover, for each ξ, not all of λ0,ξ , . . . , λn−1,ξ are zero. Let Q be a clopen set in the Stone space of the Boolean algebra B which corresponds to the projection πξ . Then the Dedekind complete vector lattice πξ C ↓ is isomorphic to C∞ (Q, C); moreover, the element πξ (1) goes into the identically one function on Q. If f ∈ C∞ (Q, C) is the image of an element πξ z0 under the indicated isomorphism then we arrive at the relation λ0,ξ + λ1,ξ f (q) + · · · + λn−1,ξ f (q)n−1 = 0 (q ∈ Q). By the Fundamental Theorem of Algebra, the continuous function f has at most n values; consequently, f is a step-function. But then the element πξ z0 is piecewise constant and so it belongs to X. Clearly, z0 ∈ X and hence 1 = [[z ∈ X↑]] = [[z ∈ C∧ ]]. ⊲ 4.12.2. Thus, under the canonical embedding of the complexes into the Boolean valued model, either C∧ = C or the ﬁeld of complexes is a transcendental extension of some subﬁeld of C . The same is true for the reals. To analyze this situation, we need the notion of an algebraic or transcendence basis of a ﬁeld over some subﬁeld. 248 Chapter 4. Band Preserving Operators Let P be a subﬁeld of C such that C is a transcendental extension of P. By the Steinitz Theorem, there is a transcendence basis E ⊂ C. This means that E is algebraically independent over P and C is an algebraic extension of the ﬁeld P(E ) obtained by addition of the elements of E to P. The ﬁeld P(E ) is a pure extension of P. 4.12.3. Let D(C ↓) be the set of all complex derivations on the f algebra C ↓ and let MN (C ↓) be the set of all complex band preserving automorphisms of C ↓. Let DC∧ (C ) and MC∧ (C ) be the elements of V(B) that depict the sets of all C∧ -derivations and all C∧ -automorphisms in C . Clearly, D(C ↓) is a module over C ↓ and [[ DC∧ (C ) is a complex vector space]] = 1. The descent and ascent produce isomorphisms between DC∧ (C )↓ and D(C ↓) as well as bijections between MC∧ (C )↓ and MN (C ↓). ⊳ The proof follows from 4.10.10. We only have to note that an operator T ∈ EndN (C ↓) is a complex derivation (automorphism) if and only if [[ τ := T ↑ is a C∧ -derivation (C∧ -automorphism) ]] = 1. ⊲ 4.12.4. An order bounded derivation and an order bounded band preserving automorphism of a universally complete f -algebra XC are trivial. ⊳ We may assume that XC = C ↓. If T is a derivation (a band preserving automorphism) of the f -algebra XC then [[ τ := T ↑ is a C∧ derivation (C∧ -automorphism) of C ]] = 1. Moreover, T is order bounded if and only if [[ τ is order bounded in C ]] = 1. But every order bounded C∧ -derivation on the ﬁeld C is zero and every order bounded C∧ -automorphism is the identity mapping. In the ﬁrst case we have T = 0 and in the second, T = I. ⊲ 4.12.5. If V(B) |= C∧ = C then there exist a nontrivial derivation and a nontrivial band preserving automorphism on the universally complete complex f -algebra C ↓. ⊳ It follows from the condition C∧ = C that C is a transcendental extension of C∧ within V(B) (cp. 3.12.1). By 4.11.9, there exist a nontrivial C∧ -derivation δ : C → C and a nontrivial C∧ -automorphism α : C → C . If D := δ↓ and A := α↓ then, according to 4.12.3, D is a nontrivial derivation and A is a nontrivial band preserving automorphism of the f -algebra C ↓. ⊲ 4.12.6. Theorem. Let B be a complete Boolean algebra, C the complexes within V(B) , and X := C ↓ a universally complete complex 4.12. Automorphisms and Derivations on Complex f -Algebras 249 f -algebra, the descent of C . Then the following are equivalent: (1) B is σ-distributive. (2) V(B) |= C = C∧ . (3) Every band preserving linear operators on X is order bounded. (4) There is no nontrivial derivation on X. (5) There is no nontrivial band preserving automorphism on X. (6) Every band preserving endomorphism of X is a band projection. ⊳ By Theorem 4.4.9 a Boolean algebra B is σ-distributive if and only if V(B) |= R = R∧ . At the same time, by restricted transfer 1.4.7 we have V(B) |= R∧ ⊕ iR∧ = C∧ . Thus V(B) |= C = C∧ if and only if V(B) |= R = R∧ . It follows that (1) ⇐⇒ (2). Observe that the assertion 4.12.6 (k + 1) is the interpretation of 4.11.9 (k) within V(B) for k = 1, . . . , 5. We now get the other equivalences by appealing to 4.10.10 and 4.12.5. ⊲ 4.12.7. Corollary. Let X be a universally complete real vector lattice with a fixed structure of an f -algebra. Then for the complex f algebra XC the following are equivalent: (1) B := P(X) is a σ-distributive Boolean algebra. (2) There is no nontrivial complex derivation on XC . (3) There is no nontrivial band preserving complex automorphisms of XC . 4.12.8. Using the same arguments as above, we can show that some analogs of 4.12.1 and 4.11.8 hold for the reals. More precisely, the following are valid: (1) [[ R∧ is algebraically closed in R ]] = 1; (2) If V(B) |= R∧ = R, then V(B) |= “ R is a transcendental extension of R∧ ”; (3) If R is a transcendental extension of a ﬁeld P then there is a nontrivial P-derivation on R. But 4.11.7 is not valid for the reals: there is no nontrivial automorphism on R. This is connected with the fact that R is not an algebraically closed ﬁeld. 250 Chapter 4. Band Preserving Operators 4.12.9. A derivation (an automorphism) S on X is called essentially nontrivial provided that πS = 0 (πS = πIX ) imply π = 0 for every band projection π ∈ P(X). A complete Boolean algebra B is said to be purely non-σ-distributive if none of its relative Boolean algebras [0, b] with nonzero b ∈ B is σ-distributive. Assume that Z is a universally complete real vector lattice and P(X) is purely non-σ-distributive. Then, by Theorem 4.12.7, for every band projection π ∈ P(X) there exist a nontrivial complex derivation and a nontrivial band preserving complex automorphisms on πZC . Therefore, we can ﬁnd also an essentially nontrivial complex derivation and a an essentially nontrivial band preserving complex automorphisms on ZC making use of the exhausting principle (= every minorizing set in a complete Boolean algebra admit a disjoint reﬁnement). 4.12.10. For each complete Boolean algebra B there exists an element b ∈ B such that the relative Boolean algebra B0 := [0, b] is σdistributive, while the relative Boolean algebra [0, b∗ ] is purely non-σdistributive. ⊳ Put b = [[R = R∧ ]] and note that V(B0 ) |= R = R∧ (we use the same symbols R and R∧ within V(B) and V(B0 ) for reals and standard reals). By Theorem 4.4.9 B0 := [0, b] is σ-distributive. If d ∈ B, d b∗ , and [0, d] is σ-distributive then again by Theorem 4.4.9 d [[R = R∧ ]] ∧ b∗ = b ∧ b∗ = 0. ⊲ 4.12.11. Corollary. If (Ω, Σ, μ) is an atomless Maharam measure space then the following hold: (1) There exists an essentially nontrivial R-derivation on L0R (Ω, Σ, μ). (2) There exists an essentially nontrivial C-derivation on L0C (Ω, Σ, μ). (3) The identity operator is the only automorphism of L0R (Ω, Σ, μ). (4) There exists an essentially nontrivial band preserving automorphism of L0C (Ω, Σ, μ). ⊳ This is immediate from 4.12.10, Corollary 4.12.7, and Remarks in 4.12.8 and 4.12.9 in view of 4.7.11. ⊲ 4.13. Involutions and Complex Structures The main result of this section tells us that in a real non-locally-onedimensional universally complete vector lattice there are band preserving complex structures and nontrivial band preserving involutions. 4.13. Involutions and Complex Structures 251 4.13.1. A linear operator T on a vector lattice X is called involutory or an involution if T ◦ T = IX (or, equivalently, T −1 = T ) and is called a complex structure if T ◦ T = −IX (or, equivalently, T −1 = −T ). The operator P − P ⊥ , where P is a projection operator on X and P ⊥ = IX − P , is an involution. The involution P − P ⊥ with band projections P is referred to as trivial . 4.13.2. Let X be a Dedekind complete vector lattice. Then there is no order bounded band preserving complex structure in X and there is no nontrivial order bounded band preserving involution in X. ⊳ An order bounded band preserving operator T on a universally complete vector lattice X with weak unit 1 is a multiplication operator: T x = ax (x ∈ X) for some a ∈ X. It follows that T is an involution if and only if a2 = 1 and so there is a band projection P on E with a = P 1 − P ⊥ 1 or T = P − P ⊥ . If T is a complex structure on E then the corresponding equation a2 = −1 has no solution. ⊲ 4.13.3. Theorem. Let F be a proper subfield of R and let B ⊂ R be a nonempty finite set. Then there exists a discontinuous F-linear function f : R → R such that f ◦ f = IR and f (x) = x for all x ∈ B. ∧ ⊳ Let E ⊂ R be a Hamel basis of R over R . Every x ∈ B can be written in the form x = e∈E λe (x)e, where λe (x) ∈ F for all e ∈ E . Put E (x) := {e ∈ E : λe (x) = 0} and E0 = x∈B E (x). Since B is ﬁnite, so is also E0 . Hence E \E0 has inﬁnite cardinality. There exists a decomposition E1 ∪ E2 = E \E0 , where E1 and E2 disjoint sets both having the same cardinality. Hence there exists a one-to-one mapping g0 from E1 onto E2 with the inverse g0−1 : E2 → E1 . Deﬁne the function g : E → E as follows: ⎧ ⎪ for e ∈ E1 , ⎨g0 (e), −1 g(e) = g0 (e), (4.1) for e ∈ E2 , ⎪ ⎩ e, for e ∈ E0 . Let f : R → R stand for the F-linear extension of g. For h ∈ E0 we have g(h)/h = 1, and for h1 ∈ E1 we have g(h1 ) = g0 (h1 ) ∈ E2 , so that g(h) = h and g(h)/h = 1. By 4.2.5 f is discontinuous. For arbitrary h ∈ E1 we have g(h) = g0 (h) ∈ E2 , whence g(g(h)) = g0−1 (g0 (h)) = h. Similarly, for h ∈ E2 we have g(h) = g0−1 (h) ∈ E1 and g(g(h)) = g0 (g0−1 (h)) = h. Obviously we have g(g(h)) = h for h ∈ E0 . Thus g(g(h)) = h for all h ∈ E . Now take an arbitrary x ∈ R and write 252 Chapter 4. Band Preserving Operators down the representation x = e∈E xe e with xe ∈ P. Using F-linearity of f and the relation f |E = g we deduce f (f (x)) = xe f (g(e)) = xe g(g(e)) = xe g(e) = x. e∈E e∈E e∈E Observe further that if x ∈ B then e ∈ E0 whenever xe = 0. Therefore, we have f (x) = xe e = x. xe g(e) = xe f (e) = e∈E e∈E0 e∈E0 Thus f (f (x)) = x for all x ∈ R and f (x) = x for x ∈ B. ⊲ 4.13.4. Theorem. Let F be a proper subfield of R. Then there exists a discontinuous F-linear function f : R → R such that f ◦ f = −IR . ⊳ The proof is similar to that of Theorem 4.13.3 with minor modiﬁcations: put E0 = ∅ and deﬁne 4 for e ∈ E1 , −g0 (e), g(e) = −1 g0 (e), for e ∈ E2 . If f : R → R is the F-linear extension of a function g and x = e∈E xe e then, taking it into account that f (g0 (e)) = e and f (g0−1 (c)) = −c for e ∈ E1 and c ∈ E2 , we get f (f (x)) = e∈E1 xe f (−g0 (e)) + xc f (g0−1 (c)) c∈E2 =− e∈E1 xe e − xc e = −x. c∈E2 Thus, f is the sought complex structure. ⊲ Interpreting Theorems 2.4.3 and 2.4.4 in a Boolean valued model yields the following result. 4.13.5. Theorem. Let X be a universally complete real vector lattice that is not locally one-dimensional. Then (1) For every nonempty finite set B ⊂ X there exists a band preserving involution T on X with T (x) = x for all x ∈ B. (2) There exists a band preserving complex structure on X. 4.13. Involutions and Complex Structures 253 ⊳ Assume that X = R↓. Take a one-to-one function ν : N → X with B = im(ν) and N := {1, . . . , N − 1}. The function σ := ν↑ : N ∧ → X may fail to be one-to-one within V(B) but B↑ is again ﬁnite, as B↑ = im(ν↑) by 1.2.7. By transfer, Theorem 4.13.3 is valid within V(B) , so there exists an R∧ -linear function τ : R → R such that τ ◦τ = IR and τ (x) = x for all x ∈ B↑ or, what is the same, τ ◦ σ = σ. From 1.2.3, 1.6.9, and 1.5.6 we now deduce 1 = [[(∀ x ∈ B↑)τ (x) = x]] = [[(∀ n ∈ N ∧ )τ (σ(n)) = σ(n)]] [[τ (ν(n)) = ν(n)]] [[τ (ν↑(n∧ )) = ν↑(n∧ )]] = = n∈N = n∈N [[τ ↓(ν(n)) = ν(n)]]. n∈N It follows that if T := τ ↓ then T ◦ T = IX by 1.2.4 and T (ν(n)) = ν(n) for all n ∈ {1, . . . , N − 1} as required in 4.13.5 (1). The second claim is proved in a similar way using Theorem 4.13.4. ⊲ 4.13.6. Corollary. Let X be a universally complete vector lattice. Then the following are equivalent: (1) X is locally one-dimensional. (2) There is no nontrivial band preserving involution on X. (3) There is no band preserving complex structure on X. 4.13.7. Corollary. Let X be a universally complete real vector lattice. Then X admits a structure of complex vector space with a band preserving complex multiplication. ⊳ A complex structure T on X allows us to deﬁne on X a structure of a vector space over the complexes C, by setting (α + iβ)x = αx + βT (x) for all z = α + iβ ∈ C and x ∈ X. If T is band preserving then the mapping x → zx (x ∈ X) is evidently band preserving for every ﬁxed z ∈ C. ⊲ 4.13.8. Corollary. If (Ω, Σ, μ) is an atomless Maharam measure space then L0 (Ω, Σ, μ) admits a structure of a complex vector space with band preserving complex multiplication. ⊳ This is immediate from 4.7.11 and Corollary 4.13.7. ⊲ 254 Chapter 4. Band Preserving Operators 4.14. Variations on the Theme In this section we brieﬂy consider the band preserving phenomenon in some natural environments (the endomorphisms of lattice ordered modules, bilinear operators on vector lattices, and derivations in AW ∗ algebras) and state some problems that may be viewed as versions of the Wickstead problem which are referred to as module, bilinear, and noncommutative Wickstead problem. 4.14.A. Lattice Ordered Modules This subsection deals with the module Wickstead problem stated as follows: 4.14.A.1. WP(A): When are all band preserving K-linear endomorphisms of a lattice ordered K-module X order bounded? Here K is a lattice ordered ring, and X is a lattice ordered module over K. Little is known about this problem. Boolean valued analysis provides the transfer principle which might translate WP(A) to WP. Below we describe the class of lattice ordered modules for which this transfer works perfectly. 4.14.A.2. An annihilator ideal of K is a subset of the form S ⊥ := {k ∈ K : (∀ s ∈ S)ks = 0} with a nonempty subset S ⊂ K. A subset S of K is called dense provided that S ⊥ = {0}; i.e., the equality k · S := {k · s : s ∈ S} = {0} implies k = 0 for all k ∈ K. A ring K is said to be rationally complete whenever, to each dense ideal J ⊂ K and each group homomorphism h : J → K such that h(kx) = kh(x) for all k ∈ K and x ∈ J, there is an element r in K satisfying h(x) = rx for all x ∈ J. A ring K is rationally complete if and only if K is selﬁnjective (cp. [249, Theorem 8.2.7 (3)]). 4.14.A.3. If K is an ordered ﬁeld within V(B) then K ↓ is a rationally complete semiprime f -ring, and there is an isomorphism χ of B onto the Boolean algebra B(K ↓) of the annihilator ideals (coinciding in the case under consideration with the Boolean algebra of all bands) of K ↓ such that b [[x = 0]] ⇐⇒ x ∈ χ(b∗ ) (x ∈ K, b ∈ B) (cp. [249, Theorem 8.3.1]). Conversely, assume that K is a rationally complete semiprime f -ring and B stands for the Boolean algebra B(K) of all annihilator ideals (bands) of K. Then there is K ∈ V(B) , called the Boolean valued representation of K, such that [[ K is an ordered 4.14. Variations on the Theme 255 ﬁeld ]] = 1 and the lattice ordered rings K and K ↓ are isomorphic (cp. [249, Theorem 8.3.2]). 4.14.A.4. A K-module X is separated provided that for every dense ideal J ⊂ K the identity Jx = {0} implies x = 0. Recall that a Kmodule X is injective whenever, given a K-module Y , a K-submodule Y0 ⊂ Y , and a K-homomorphism h0 : Y0 → X, there exists a Khomomorphism h : Y → X extending h0 . The Baer criterion says that a K-module X is injective if and only if for each ideal J ⊂ K and each K-homomorphism h : J → X there exists x ∈ X with h(a) = xa for all a ∈ J; see Lambek [276]. 4.14.A.5. Let X be a vector lattice over an ordered ﬁeld K within V(B) , and let χ : B → B(K ↓) be a Boolean isomorphism from 4.14.A.3. Then X ↓ is a separated unital injective lattice ordered module over K ↓ satisfying b [[x = 0]] ⇐⇒ χ(b)x = {0} (x ∈ X ↓, b ∈ B). Conversely, let K be a rationally complete semiprime f -ring, B := B(K), and let K be the Boolean valued representation of K. Assume that X is a unital separated injective lattice ordered K-module. Then there exists some X ∈ V(B) such that [[ X is a vector lattice over the ordered ﬁeld K ]] = 1 and there are algebraic and order isomorphisms j : K → K ↓ and ı : X → X ↓ such that ı(ax) = j(a)ı(x) (a ∈ K, x ∈ X) (cp. [249, Theorems 8.3.12 and 8.3.13]). Thus, the Boolean transfer principle is applicable to unital separated injective lattice ordered modules over rationally complete semiprime f -rings. 4.14.A.6. Consider an example. Let B be a complete Boolean algebra and let B be a complete subalgebra of B. We say that B is B-σdistributive if for every sequence (bn )n∈N in B we have ε(n)bn = 1, ε∈BN n∈N where ε(n)bn := ε(n) ∧ bn ∨ ε(n)∗ ∧ b∗n and b∗ is the complement of b ∈ B. Clearly, the {0,1}-σ-distributivity of B means that B is σ-distributive cp. 1.9.12 (3) . 256 Chapter 4. Band Preserving Operators 4.14.A.7. Theorem. Let X be a universally complete vector lattice with a fixed order unit 1 and let K be an order closed sublattice containing 1K := 1. Put B := C(1) and B := C(1K ). Then K is a rationally complete f -algebra, X is an injective lattice ordered K-module, and the following are equivalent: (1) B is B-σ-distributive. (2) Every element x ∈ X+ is locally K-constant; i.e., x = supξ∈Ξ aξ πξ 1 for some family (aξ )ξ∈Ξ of elements of K and a disjoint family (πξ )ξ∈Ξ of band projections in X. (3) Every band preserving K-linear endomorphism of X is order bounded. ⊳ We only sketch the proof. Let X and K be the same as in 4.14.A.5. There exist B ∈ V(B) such that [[ B is a complete Boolean algebra isomorphic to P(X ) ]] = 1 and B↓ is a complete Boolean algebra isomorphic to B (see 1.10.4). Moreover, B is B-σ-distributive if and only if B is σ-distributive within V(B) . We are done with interpreting 4.4.9 and 4.6.4 within V(B) . ⊲ 4.14.B. The Bilinear Wickstead Problem Let us characterize those universally complete vector lattice in which all band preserving bilinear operators are symmetric or order bounded. No new ideas are required here and all run along the lines of Section 4.6. The needed information about bilinear operators on vector lattices is in Bu, Buskes, and Kusraev [72]; also see Buskes and Kusraev [78]. 4.14.B.1. Let X be a vector lattice. A bilinear operator B : X ×X → X is separately band preserving provided that the mappings B(·, e) : x → B(x, e) and B(e, ·) : x → B(e, x) (x ∈ X) are band preserving for all e ∈ X or, which is the same, provided that B(L × X) ⊂ L and B(X × L) ⊂ L for every band L in X. 4.14.B.2. Assume that X is a vector lattice and B : X × X → X is a bilinear operator. Then the following are equivalent: (1) B is separately band preserving. (2) B(x, y) ∈ {x}⊥⊥ ∩ {y}⊥⊥ for all x, y ∈ X. (3) B(x, y) ⊥ z for all z ∈ X provided that x ⊥ z or y ⊥ z. If X has the principal projection property, then (1)–(3) are equivalent also to each of the two assertions: (4) πB(x, y) = B(πx, πy) for every π ∈ P(X) and all x, y ∈ X. 4.14. Variations on the Theme 257 (5) πB(x, y) = B(πx, y) = B(x, πy) for all π ∈ P(X) and x, y ∈ X. ⊳ We omit the routine arguments which are similar to 4.1.1 and 4.1.6. ⊲ 4.14.B.3. Let X and Y be vector lattices. Recall that a bilinear operator B from X × X to Y is orthosymmetric provided that |x| ∧ |y| = 0 implies B(x, y) = 0 for arbitrary x, y ∈ X (cp. Buskes and van Roij [81]). The diﬀerence of two positive orthosymmetric bilinear operators is orthoregular (cp. Buskes and Kusraev [78], and Kusraev [233]). Recall also that a bilinear operator b is symmetric or antisymmetric provided that B(x, y) = B(y, x) or B(x, y) = −B(y, x) for all x, y ∈ X. 4.14.B.4. The following important property of orthosymmetric bilinear operators was established in Buskes and van Rooj [81, Corollary 2]: If X and Y are vector lattices then every orthosymmetric positive bilinear operator from X × X into Y is symmetric. It is evident from 4.14.B.2 that a separately band preserving bilinear operator is orthosymmetric. Hence, all orthoregular separately band preserving operators are symmetric by the above result. At the same time an order bounded separately band preserving bilinear operator B is regular with B + (x, y) = B(x, y)+ and B − (x, y) = B(x, y)− for all x, y ∈ X (see Kusraev and Tabuev [257, Theorem 3.4]). This brings up the following question: 4.14.B.5. WP(B): Under what conditions are all separately band preserving bilinear operators in a vector lattice symmetric? Order bounded? In the case of a universally complete vector lattice the answer is similar to the linear case and is presented below in 4.14.B.7. The general case was not examined yet. 4.14.B.6. Let BLbp (X) stand for the set of all separately band preserving bilinear operators from X × X to X, where X := R↓. Clearly, BLbp (X) becomes a faithful unitary module over X provided that we deﬁne gB as gB : (x, y) → g · B(x, y) for all x, y ∈ X. Denote by BL(RR ) the element of V(B) that depicts the space of all R∧ -bilinear mappings from R × R into R. Then BL(RR ) is a vector space over R∧ within V(B) , and BL(RR )↓ is a faithful unitary module over X. Just as in 4.3.5 it can be proved that the modules BLbp (X) and BL(RR )↓ are isomorphic by sending each band preserving bilinear operator to its ascent. 4.14.B.7. Theorem. For a universally complete vector lattice X the following are equivalent: 258 Chapter 4. Band Preserving Operators (1) B(X) is σ-distributive. (2) There is no antisymmetric operator in BLbp (X). (3) All operators in BLbp (X) are symmetric. (4) All operators in BLbp (X) are order bounded. ⊳ The implication (1) =⇒ (4) can be proved as in 4.6.5, (4) =⇒ (3) is immediate from 4.14.B.4, while (3) =⇒ (2) is trivial. To prove the remaining implication (2) =⇒ (1), we can assume that X = R↓. Suppose that B is not σ-distributive. Then R∧ = R by 4.4.9 and a separately band preserving antisymmetric bilinear operator can be constructed on using the bilinear version of 4.2.8. Indeed, within V(B) , a Hamel basis E for R over R∧ contains at least two distinct elements e1 = e2 . Deﬁne the function β0 : E × E → R so that 1 = β0 (e1 , e2 ) = −β0 (e2 , e1 ), and β(e′1 , e′2 ) = 0 for all other pairs (e′1 , e′2 ) ∈ E × E in particular, 0 = β0 (e1 , e1 ) = β0 (e2 , e2 ) . Then β0 can be extended to an R∧ -bilinear function β : R × R → R. The descent B of β is a separately band preserving bilinear operator in X by 4.14.B.6, the bilinear version of 4.3.5. Moreover, B is nonzero and antisymmetric, since β is nonzero and antisymmetric by construction. This contradiction proves that R∧ = R and B is σ-distributive. ⊲ 4.14.B.8. (1) There exists a nonatomic universally complete vector lattice in which all separately band preserving bilinear operators are symmetric and order bounded. (2) If (Ω, Σ, μ) is an atomless Maharam measure space then there exists an essentially nontrivial antisymmetric separately band preserving bilinear operator in L0R (Ω, Σ, μ). ⊳ It follows from Theorems 4.14.B.7, 4.7.7 and 4.7.11. ⊲ 4.14.C. The Noncommutative Wickstead Problem The relevant information on the theory of Baer ∗-algebras and AW ∗ algebras can be found in Berberian [50], Chilin [89], and Kusraev [228]. 4.14.C.1. A Baer ∗-algebra is a complex involutive algebra A such that, for each nonempty M ⊂ A, there is a projection, i.e., a hermitian idempotent, p satisfying M ⊥ = pA where M ⊥ := {y ∈ A : (∀ x ∈ M ) xy = 0} is the right annihilator of M . Clearly, this amounts to saying that each left annihilator has the form ⊥ M = Aq for an appropriate projection q. To each left annihilator L in a Baer ∗-algebra there is a unique projection qL ∈ A such that x = xqL for all x ∈ L and qL y = 0 4.14. Variations on the Theme 259 whenever y ∈ L⊥ . The mapping L → qL is an isomorphism between the poset of left annihilators and the poset of all projections. Thus, the poset P(A) of all projections in a Baer ∗-algebra is an order complete lattice. Clearly, the formula q p ⇐⇒ q = qp = pq, sometimes pronounced as “p contains q,” speciﬁes some order on the set of projections P(A). An element z in A is central provided that z commutes with every member of A; i.e., (∀ x ∈ A) xz = zx. The center of a Baer ∗algebra A is the set Z (A) comprising central elements. Clearly, Z (A) is a commutative Baer ∗-subalgebra of A, with λ1 ∈ Z (A) for all λ ∈ C. A central projection of A is a projection belonging to Z (A). Put Pc (A) := P(A) ∩ Z (A). 4.14.C.2. A derivation on a Baer ∗-algebra A is a linear operator d : A → A satisfying d(xy) = d(x)y + xd(y) for all x, y ∈ A. A derivation d is inner provided that d(x) = ax − xa (x ∈ A) for some a ∈ A. Clearly, an inner derivation vanishes on Z (A) and is Z (A)-linear; i.e., d(ex) = ed(x) for all x ∈ A and e ∈ Z (A). Consider a derivation d : A → A on a Baer ∗-algebra A. If p ∈ A is a central projection then d(p) = d(p2 ) = 2pd(p). Multiplying this identity by p we have pd(p) = 2pd(p) so that d(p) = pd(p) = 0. Consequently, every derivation vanishes on the linear span of Pc (A), the set of all central projections. In particular, d(ex) = ed(x) whenever x ∈ A and e is a linear combination of central projections. Even if the linear span of central projections is dense in a sense in Z (A), the derivation d may fail to be Z (A)-linear. This brings up the natural question: Under what conditions is every derivation Z-linear on a Baer ∗-algebra A provided that Z is a Baer ∗-subalgebra of Z (A)? 4.14.C.3. An AW ∗ -algebra is a C ∗ -algebra with unity 1 which is also a Baer ∗-algebra. More explicitly, an AW ∗ -algebra is a C ∗ -algebra whose every right annihilator has the form pA, with p a projection. Clearly, Z (A) is a commutative AW ∗ -subalgebra of A. If Z (A) = {λ1 : λ ∈ C} then the AW ∗ -algebra A is an AW ∗ -factor. 4.14.C.4. A C ∗ -algebra A is an AW ∗ -algebra if and only if the following hold: (1) Each orthogonal family in P(A) has a supremum; (2) Each maximal commutative ∗-subalgebra of A0 ⊂ A is a Dedekind complete f -algebra (or, equivalently, coincides with the least norm closed ∗-subalgebra containing all projections of A0 ). 260 Chapter 4. Band Preserving Operators 4.14.C.5. Given an AW ∗ -algebra A, deﬁne the two sets C(A) and S(A) of measurable and locally measurable operators, respectively. Both are Baer ∗-algebras; cp. Chilin [89]. Suppose that Λ is an AW ∗ -subalgebra in Z (A), and Φ is a Λ-valued trace on A+ . Then we can deﬁne another Baer ∗-algebra, L(A, Φ), of Φ-measurable operators. The center Z (A) is a vector lattice with a strong unit, while the centers of C(A), S(A), and L(A, Φ) coincide with the universal completion of Z (A). If d is a derivation on C(A), S(A), or L(A, Φ) then d(px) = pd(x) p ∈ Pc (A) so that d can be considered as band preserving in a sense (cp. 4.1.1 and 4.10.4). The natural question arises concerning these algebras: 4.14.C.6. WP(C): When are all derivations on C(A), S(A), or L(A, Φ) inner? This question may be regarded as the noncommutative Wickstead problem. 4.14.C.7. The classiﬁcation of AW ∗ -algebras into types is determined from the structure of their lattices of projections P(A); see Kusraev [228] and Sakai [353]. We only recall the deﬁnition of type I AW ∗ algebra. A projection π ∈ A is abelian if πAπ is a commutative algebra. An algebra A has type I provided that each nonzero projection in A contains a nonzero abelian projection. A C ∗ -algebra A is B-embeddable provided that there is a type I AW ∗ algebra N and a ∗-monomorphism ı : A → N such that B = Pc (N ) and ı(A) = ı(A)′′ , where ı(A)′′ is the bicommutant of ı(A) in N . Note that in this event A is an AW ∗ -algebra and B is a complete subalgebra of Pc (A). 4.14.C.8. Theorem. Let A be a type I AW ∗ -algebra, let Λ be an AW ∗ -subalgebra of Z (A), and let Φ be a Λ-valued faithful normal semifinite trace on A. If the complete Boolean algebra B := P(Λ) is σ-distributive and A is B-embeddable, then every derivation on L(A, Φ) is inner. ⊳ We brieﬂy sketch the proof. Let A ∈ V(B) be the Boolean valued representation of A. Then A is a von Neumann algebra within V(B) . Since the Boolean valued interpretation preserves classiﬁcation into types, A is of type I. Let ϕ stand for the Boolean valued representation of Φ. Then ϕ is a C -valued faithful normal semiﬁnite trace on A and the descent of L(A , ϕ) is ∗-Λ-isomorphic to L(A, Φ); cp. Korol′ and Chilin [205]. Suppose that d is a derivation on L(A, Φ) and δ is the Boolean valued representation of d. Then δ is a C -valued C∧ -linear derivation on L(A , ϕ). Since B is σ-distributive, C = C∧ within V(B) 4.15. Comments 261 and δ is C -linear. But it is well known that every derivation on a type I von Neumann algebra is inner; cp. Albeverio, Ajupov, and Kudaybergenov [23]. Therefore, d is also inner. ⊲ 4.15. Comments 4.15.1. The theory of orthomorphisms stems from Nakano [320]. Orthomorphisms have been studied by many authors under various names (cp. Aliprantis and Burkinshaw [28]): dilatators (Nakano [320]), essentially positive operators (Birkhoﬀ [58]), polar preserving endomorphisms (Conrad and Diem [93]), multiplication operators (Buck [74] and Wickstead [408]), and stabilisateurs (Meyer [310]). The main stages of this development as well as the various aspects of the theory of orthomorphisms are reﬂected in the books: Abramovich and Kitover [8]. Bigard, Keimel, and Wolfenstein [57], Aliprantis and Burkinshaw [28], Zaanen [427, Chapter 20], de Pagter [327], etc.; also see the survey papers by Bukhvalov [75, Section 2.2] and Gutman [160, Chapter 6]. 4.15.2. (1) Functional equations occur practically in all branches of mathematics and have a wide variety of applications not only in mathematics but also in other disciplines. The ﬁrst functional equations for determining linear and quadratic functions appeared in the medieval centuries for using in applications. The ﬁrst systematic treatment of the theory of functional equations appeared in Cauchy [88]. For more historical details we refer to Aczél and Dhombres [14]. The state-of-the art of the theory can be grasped from the books: Aczél and Dhombres [14], Castillo and Ruiz-Cobo [87], Czerwik [100], Kuczma [211], Hyers, Isac, and Rassias [176], Kannappan [190], and Székelyhidi [373]. (2) Hamel [164] ﬁrst succeeded in proving the existence of discontinuous additive functions on R. Using the Zermelo Well-Ordering Theorem, Hamel showed that R, viewed as a vector space over the rationals Q, possesses a basis, a Hamel basis. Actually Hamel proved Theorem 4.2.2 for P = Q, whence the existence of a discontinuous additive function follows easily. Recall also that the Zermelo Well-Ordering Theorem, the Kuratowski–Zorn Lemma, and the axiom of choice are equivalent; see [180]. Blass [61] showed that the axiom of choice follows if we assume that each linear space over an arbitrary ﬁeld has a basis. (3) Theorem 4.2.4 is in Aczél and Dhombres [14, Theorem 2.3]. It is also true that the image of every open interval by a noncontinuous 262 Chapter 4. Band Preserving Operators solution of (L) is dense in R. These results show that solutions to (L) are either very regular or extremely pathological. 4.15.3. (1) The main result of Section 4.3 (Theorem 4.3.4) was established by Kusraev [229]. The problem whether or not the inverse of an injective band preserving operator on a vector lattice is also band preserving was posed by Abramovich in 1992. Huijsmans and Wickstead [175, Theorems 2 and 3] handled the problem under the additional assumption that the domain vector lattice either is uniformly complete or have the principal projection property. Later in Abramovich and Kitover [8, Theorem 7.4] the result was generalized to vector lattices with a coﬁnal family of band projections. Its ﬁnal form, stated in Theorem 4.3.6, was obtained by the same authors [9, Theorem 3.3]. Theorem 4.3.10 and Corollary 4.10.11 amount essentially to Theorem 14.8 in Abramovich and Kitover [8]. (2) It follows from 4.3.8 that every orthomorphism is order continuous. Order continuity of an extended orthomorphism was established independently by Bigard and Keimel in [56] and by Conrad and Diem in [93] using functional representation. A direct proof was found by Luxemburg and Schep [295]. Commutativity of every Archimedean f -algebra was proved by Birkhoﬀ and Pierce [60]; this paper also introduced the concept of f -algebra. The lattice ordered algebras were surveyed by Boulabiar, Buskes, and Triki [68, 69]. The fact that Orth(D, X) is a vector lattice under the pointwise algebraic and lattice operations was also obtained in Bigard and Keimel [56] and Conrad and Diem [93]. Extensive is the bibliography on the theory of orthomorphisms; and so we indicate only a portion of it: Abramovich, Veksler, and Koldunov [11], Abramovich and Wickstead [13], Bernau [51], Bigard and Keimel [56], Duhoux and Meyer [111], Gutman [161, 162], Huijsmans and de Pagter [173], Huijsmansand Wickstead [175], Luxemburg [291], Luxemburg and Schep [295], Mittelmeyer and Wolﬀ [312], de Pagter [329, 330], Wickstead [408, 410], and Zaanen [426]. 4.15.4. (1) In Section 4.4 we follow Kusraev [229]. The property of λ in 4.4.8 is usually referred to as absolute definability. Gordon [138] called a continuous function absolutely definable if it possesses an analogous property. For instance, the functions ex , log x, sin x, and cos x are absolutely deﬁnable. In particular, these functions reside in every Boolean valued universe, presenting the mappings from R to R that are continuations of the corresponding functions exp∧ (·), log∧ (·), sin∧ (·), and 4.15. Comments 263 cos∧ (·) from R∧ into R∧ . Practically all functions admitting a constructive deﬁnition are absolutely deﬁnable. (2) Instead of using continued fraction expansions in Section 4.4 we can involve binary expansions. In this event we have to construct a bijection of P(ω) onto some set of reals and apply 1.9.13 (3) in place of 1.9.13 (2). 4.15.5. (1) The terms “local linear independence” and “local Hamel basis” were coined in McPolin and Wickstead [309]. They appeared in Abramovich, Veksler and Koldunov [11] under the names d-independence and d-basis. Originally the concept was introduced by Cooper [94]. For this concept we choose the terms d-independence and d-basis, since it is somewhat weaker than that introduced in Kusraev [229] and presented in Section 4.5: A local Hamel basis in the sense of Deﬁnition 4.5.1 is what one gets interpreting a classical Hamel basis in a Boolean valued model, while a d-basis appears by interpreting a Hamel basis together with the zero element. (2) More precisely, consider a universally complete vector lattice X represented as the reals R in the Boolean valued universe V(B) with B = P(X); if E is an internal Hamel basis for R over R∧ , then E ↓ is a local Hamel basis in the sense of 4.5.1 (Theorem 4.5.7), while (E ∪ {0})↓ is a d-basis of X. Theorem 4.5.7, the main result of Section 4.5 was obtained in Kusraev [229, Proposition 4.6 (1)]. The representation in 4.5.3 and 4.5.4 is referred to as a d-expansion with respect to the local Hamel basis (d-basis). More details about d-bases and d-expansions are given in Abramovich and Kitover [8]. (3) The notions of d-independence and d-basis can be introduced in an arbitrary vector lattice (see Abramovich and Kitover [10]). A collection (xγ )γ∈Γ of elements in a vector lattice X is d-independent provided that for each band B in X, each ﬁnite subset {γ1 , . . . , γn } of Γ, n and each family of nonzero scalars c1 , . . . , cn the condition ı=1 cı xγı ⊥ B implies that xγı ⊥ B for ı = 1, . . . , n. A d-independent system (xγ )γ∈Γ is a d-basis provided that for each x ∈ X there is a full system (Bα )α∈A of pairwise disjoint bands in X and a system of elements (yα )α∈A in X such that each yα is a linear combination of elements in (xγ )γ∈Γ and (x − yα ) ⊥ Bα for all α ∈ A. (4) The dimension δ(R) of the vector space R over R∧ is a cardinal within V(B) . The object δ(R) carries important information on the interconnection of the Boolean algebra B and the reals R. Using 264 Chapter 4. Band Preserving Operators the properties of Boolean valued ordinals, we obtain the representation δ(R) = mixξ bξ α∧ξ , where (bξ ) is a partition of unity in B and (αξ ) is a family of standard cardinals. This representation is an instance of a “decomposition series” of B such that the principal ideals [0, bξ ] are “αξ -homogeneous” in a sense. 4.15.6. (1) For locally one-dimensional vector lattices the term essentially one-dimensional is also in use; see Abramovich and Kitover [8]. Proposition 4.6.2 establishes the Boolean valued status of locally one dimensionality: A laterally complete vector lattice is locally onedimensional if and only if its Boolean valued representation is a onedimensional vector space over the ﬁeld R∧ . Theorem 4.6.7 gives a negative answer to the following problem (Problem B in [9]): Is there a bijective disjointness preserving linear operator between vector lattices with a disjointness preserving inverse which is not order isomorphism? The existence of such an operator was demonstrated in Abramovich and Kitover [8, Theorem 13.4] with the help of d-basis. Theorems 4.3.4 and 4.5.7 enables us to reduce the problem to the easy exercise with a classical Hamel basis (see 4.6.6). Theorem 4.6.9 is due to Kusraev. (2) An orthomorphism is a band preserving operator that is orderbounded. In [408] Wickstead raised the question whether every band preserving operator must be order bounded automatically. Existence of an unbounded band preserving operator was announced for the ﬁrst time in [11, Theorem 1]. Later, it was clariﬁed that the situation described in the paper is typical in a sense. Namely, it was established by Abramovich, Veksler, and Koldunov in [12, Theorem 2.1] and by McPolin and Wickstead in [309, Theorem 3.2] that all band preserving operators in a universally complete vector lattice are automatically bounded if and only if this vector lattice is locally one-dimensional (Theorem 5.1.2). (The deﬁnitions of locally one-dimensional K-space and local Hamel basis, as well as the equivalence conditions (1)–(4) from 5.1.1, are presented in McPolin and Wickstead [309].) (3) It is seen from Theorem 4.3.4 and Corollary 4.3.5 that, at least in the case of a universally complete vector lattice, the claim of the Wickstead problem reduces to simple properties of reals and cardinals within V(B) . But even the reader who mastered the technique (of ascending and descending) of Boolean valued analysis might ﬁnd the above demonstration bulky as compared with the standard proof in Abramovich, Veksler, and Koldunov [12]; McPolin and Wickstead [309], and Gutman [161]. But the aim of the exposition is to demonstrate that the Boolean ap- 4.15. Comments 265 proach to the problem reveals new insights and new interconnections. (4) Wickstead’s problem admits diﬀerent answers depending on the spaces in which the operators in question are considered. There are many results that guarantee automatic boundedness for a band preserving operator in the particular classes of vector lattices. According to Abramovich, Veksler, and Koldunov [12, Theorem 2.1] (see also [12], [11]) every band preserving operator from a Banach lattice to a normed vector lattice is bounded. This claim remains valid if the Banach lattice of departure is replaced by a relatively uniformly complete vector lattice [12]. In McPolin and Wickstead [309] a similar result is obtained for the band preserving operators in a relatively uniformly complete vector lattice endowed with a locally convex locally solid topology. (5) Consider a band preserving operator S : R↓ → R↓ satisfying the Cauchy exponential equation: S(x + y) = S(x)S(y) for all x, y ∈ R↓. If, moreover, S enjoys the condition S(λx) = S(x)λ for all 0 < λ ∈ R and x ∈ R↓; then we call S an exponential operator. Say that S is order bounded if S sends order bounded sets to order bounded sets. If σ is the ascent of S then σ is exponential within V(B) . Therefore, in the class of functions bounded above on some nondegenerate interval we see that σ = 0 or σ(x) = ecx for all x ∈ R and some c ∈ R. This implies the following (see Gutman, Kusraev, and Kutateladze [163]): Each band preserving exponential operator S on R↓ is order bounded (and so S may be presented as S(x) = ecx for all x ∈ R↓ and some c ∈ R↓ or S is identically zero). (6) An analogous situation takes place if S satisﬁes the Cauchy logarithmic equation S(xy) = S(x) + S(y) for all 0 ≪ x, y ∈ R↓ and enjoys the condition S(xλ ) = λS(x) for all λ ∈ R and x ≫ 0. (The record 0 ≪ x means that 0 x and x⊥⊥ = R↓.) We call an S of this sort a logarithmic operator. We may now formulate another equivalent claim as follows: Every band preserving logarithmic operator S on {x ∈ R↓ : x ≫ 0} is order bounded (and, consequently, S may presented as S(x) = c log x for all 0 ≪ x ∈ R↓ with some c ∈ R↓). 4.15.7. (1) In Section 4.7 we follow Gutman [161]. The claim of 4.6.4 can be considered as a solution to the Wickstead problem about the order boundedness of all band preserving operators. But the new notion of locally one-dimensional vector lattice crept into the answer. The novelty of this notion led to the conjecture that it coincides with that 266 Chapter 4. Band Preserving Operators of a discrete (= atomic) vector lattice. In 1981 Abramovich, Veksler and Koldunov [12, Theorem 2.1] gave a proof for existence of an order unbounded band preserving operator in every nondiscrete universally complete vector lattice, thus seemingly corroborating the conjecture that a locally one-dimensional vector lattice is discrete (also cp. [4, Section 5]). But the proof was erroneous. Later in 1985, McPolin and Wickstead [309, Section 3] gave an example of a nondiscrete locally one-dimensional vector lattice, confuting the conjecture. But there was an error in the example. Finally, Wickstead [13] stated the conjecture as an open problem in 1993. (2) This problem was solved by Gutman [161]: He constructed an atomless Dedekind complete vector lattice with a singleton d-basis. Moreover, Gutman gave a purely algebraic description of locally one dimensional universally complete vector lattices (see Theorem 4.7.6). Proposition 4.7.11 is also due to Gutman (see Gutman, Kusraev, and Kutateladze [163]). 4.15.8. (1) It follows from Theorems 4.8.5 and 4.1.7 that a vector lattice with the projection property is locally one-dimensional if and only if each band preserving projection operator on it is a band projection. Thus, a vector lattice with the projection property is locally one-dimensional if and only if all band preserving projection operators are is order bounded. (2) Theorem 4.8.9, the main result of Section 4.8, is due to Abramovich and Kitover [7, Theorem 3.4]. But the description of the unbounded part of a band preserving projection operator P in [7, Theorem 3.4] relies upon Theorem 3.2 in [7] which is incorrect. Indeed, as can be seen from the proof of 4.8.7, a componentwise closed and laterally complete sublattice X0 ⊂ X admits inﬁnitely many band preserving projections P with X0 = im(P ) or X0 = ker(P ) each of which is deﬁned by the particular choice of X1 . (3) Since the space of R∧ -linear functions in R admits a complete description that uses a Hamel basis cp. 2.1.7 (2) ; therefore, EndN (R↓) may be described completely by means of a (strict) local Hamel basis. But this approach will evoke some problems of unicity. 4.15.9. (1) Theorem 4.9.8, the main result of Section 4.9, was proved by Kusraeva [259] using the d-basis machinery from Abramovich and Kitover [10, Theorem 3.4]. Our proof utilizes a Hamel basis within a Boolean valued model. Propositions 4.9.2, 4.9.3, and 4.9.7 are taken 4.15. Comments 267 from Boulabiar, Buskes, and Sirotkin (Lemma 3.1, Proposition 3.2, and Theorem 3.3 of [67], respectively). (2) Boulabiar, Buskes, and Sirotkin in [67, Theorem 4.6] proved among other things that an Archimedean vector lattice X is Kaplansky complete if and only if each locally algebraic orthomorphism on X is a strongly diagonal operator. Recall that a vector lattice X is Kaplansky complete if for every countable inﬁnite disjoint set E in X+ there exist u ∈ X+ and an inﬁnite set F ⊂ E such that u ∧ f = 0 for all f ∈ F , and a linear operator T on X is locally algebraic if for every u ∈ X, there exists a nonzero polynomial ϕ ∈ R(x) (depending on u) such that ϕ(T )(u) = 0. Thus Kaplansky completeness amounts to saying that every locally algebraic orthomorphism is algebraic. 4.15.10. The deﬁnition of complex vector lattice in Section 4.10 is due to Lotz [287]. Complex vector and Banach lattices are treated in the books by Abramovich and Aliprantis [5, Section 3.2], Meyer-Nieberg [311, Section 2.2], Schaefer [356, Chap. II, Section 11], and Zaanen [427, Sections 91 and 92] and [428, Chapter 6]. An axiomatic approach to complex vector lattice was used in Mittelmeyer and Wolﬀ [312]. 4.15.11. (1) Detailed presentation of a portion of ﬁeld theory in 4.11.1–4.11.9 is in Bourbaki [70, Chapter V], Van der Warden [405], and Zariski and Samuel [429, Chapter II]. Theorem 4.11.9 in the present form is from Gutman, Kusraev, and Kutateladze [163, Theorem 3.2.7]. (2) Two arbitrary transcendence bases for a ﬁeld over a subﬁeld have the same cardinality called the transcendence degree (cp. [429, Chapter II, Theorem 25]). Let τ (C ) be the transcendence degree of C over C∧ within V(B) . The Boolean valued cardinal τ (C ) carries some information on the connection between the Boolean algebra B and the complexes C . Each Boolean valued cardinal is a mixture of relatively standard cardinals; i.e., the representation τ (C ) = mixξ bξ α∧ξ holds, where (bξ ) is a partition of unity of B and (αξ ) is some family of cardinals (cp. 1.9.7 and 1.9.11). Moreover, for Bξ := [0, bξ ] we have V(Bξ ) |= τ (C ) = α∧ξ . In this connection, it would be interesting to characterize the complete Boolean algebras B such that τ (C ) = α∧ within V(B) for some cardinal α. (3) Given E ⊂ X, denote by X the set of elements of the form en1 1 · · · enk k , where e1 , . . . , ek ∈ E and k, n1 , . . . , nk ∈ N. A set E ⊂ X is locally algebraically independent provided that E is locally linearly independent in the sense of 4.5.1. This notion, presenting the external interpretation of the internal notion of algebraic independence (or tran- 268 Chapter 4. Band Preserving Operators scendence), seems to turn out useful in studying the descents of ﬁelds (cp. Kusraev and Kutateladze [249, Section 8.3]) or the general regular rings (cp. Goodearl [131]). 4.15.12. (1) Theorems 4.12.1 and 4.12.6 as well as Corollaries 4.12.7 and 4.12.11 were obtained by Kusraev [231] (see also [163, 232]). In particular, if μ is an atomless Maharam measure then the algebra L0C (Ω, Σ, μ) admits nontrivial derivations which are evidently not inner and also not continuous with respect to the topology of convergence in measure. Ber, Chilin, and Sukochev [48] proved independently that the algebra L0C ([0, 1]) of all (classes of equivalence of) measurable complex functions on the interval [0, 1] admit nontrivial derivations. Some extensions of this result and interesting related questions are discussed in Ber, de Pagter, and Sukochev [49]. (2) Using the same arguments as in Section 4.12, we can infer from 4.12.8 that if R∧ = R then there are nontrivial derivations on the real f -algebra R↓. Thus, in the class of universally complete real vector lattices with a ﬁxed structure of an f -algebra the absence of nontrivial derivations is equivalent to the σ-distributivity of the base of the algebra under consideration. At the same time there are no nontrivial band preserving automorphisms of the f -algebra R↓, regardless of the properties of its base (see 4.12.8). (3) It is well known that if Q is a compact space then there are no nontrivial derivations on the algebra C(Q, C) of continuous complex functions on Q; for example see Aczél and Dhombres [14, Chapter 19, Theorem 21]. At the same time, we see from 4.12.6 (1, 4) that if Q is an extremally disconnected compact space and the Boolean algebra of the clopen sets of Q is not σ-distributive then there is a nontrivial derivation on C∞ (Q, C). (4) Let L0C (Ω, Σ, μ) be the space of all (cosets of) measurable complex functions, and let L∞ C (Ω, Σ, μ) be the space of essentially bounded measurable complex functions. Then L∞ C (Ω, Σ, μ) is isomorphic to some C(Q, C); consequently, there are no nontrivial derivations on it. If the Boolean algebra B(Ω, Σ, μ) of measurable sets modulo negligible sets is not atomic (and therefore it is not σ-distributive; cp. 4.7.11); then, by 4.12.6 (4), there exist nontrivial derivations on L0C (Ω, Σ, μ) (cp. [48, 230, 219]). The same is true about the spaces L∞ (Ω, Σ, μ) and L0 (Ω, Σ, μ) of measurable real functions. (5) Consider a band preserving operator S : X → X with X := C ↓ satisfying the Cauchy functional equation S(u + v) = S(u)S(v) for all 4.15. Comments 269 270 Chapter 4. Band Preserving Operators u, v ∈ X. If, in addition, S satisﬁes the condition S(λu) = S(u)λ for arbitrary λ ∈ C and u ∈ X then we say that S is exponential. Say that S is order bounded if S sends order bounded sets to order bounded sets. If σ is the ascent of S then σ is exponential within V(B) ; therefore, in the class of functions bounded from above on a nonzero interval, we have either σ = 0 or σ(x) = ecx (x ∈ C ) for some c ∈ X; see Aczél and Dhombres [14, Chapter 5, Theorem 5]. Hence, we conclude that X is locally one-dimensional if and only if every band preserving exponential operator in X := C ↓ is order bounded (and consequently has the form S = 0 or S(x) = ecx , x ∈ C, for some c ∈ X). (6) Kurepa [214] proved that an additive function f : R → R is a derivation if and only if f (x) = −x2 f (1/x) for all 0 = x ∈ R. Interpreting this fact in a Boolean valued model and using the method of Section 4.12 we arrive at the following result: A band preserving linear operator T in a universally complete real vector lattice X is a derivation if and only if T x = −x2 T (x−1 ) for all invertible x ∈ X. In a similar fashion, we can prove some vector lattice counterparts of various characterizations and properties of derivations in R and C. 4.15.13. (1) See Theorem 4.13.3 in Kuczma [211, Theorem 12.5.2]. Theorem 4.13.5, the main result of Section 4.13, was proved by Kusraeva [260] using the technique of d-bases. In view of Corollary 2.4.7 a few points to note about spaces without complex structure should be made. (2) Recall that a (real) Banach space X is said to admit complex structure if there exists a bounded linear operator T on X with T 2 = −IX . This enables us to deﬁne on X a structure of vector space over C as in 4.13.7. Moreover, we can deﬁne a complex norm on X which is equivalent to the original. A ﬁnite-dimensional vector space admits complex structure if and only if the dimension of the space is even. In the inﬁnite-dimensional setting, there are real Banach spaces admitting no complex structure. This is the case of the James space, as it was shown by Dieudonné [104]. More examples of this kind have been constructed over the years, including uniformly convex examples (Szarek [371]), the hereditary indecomposable space of Gowers and Maurey [143], etc. (see also [142, 144]). (3) It is worth observing at this point that some complex Banach spaces cannot be obtained as the complexiﬁcation of any real Banach space. Bourgain [71] proved the existence of such space using probabilistic arguments; the ﬁrst explicit example was given by Kalton [189]. 4.15. Comments 271 A ﬁnite-dimensional version of this result was independently developed by Szarek [372]. 4.15.14. (1) Concerning the Boolean valued representation of lattice ordered rings and modules see Kusraev and Kutateladze [249], Ozawa [325], and Takeuti [384]. Deﬁnition 4.14.A.6 and Theorem 4.14.A.7 were introduced in Gutman, Kusraev, and Kutateladze [163, Theorem 4.3.6]. (2) Recently, Chilin and Karimov [90] obtained a classiﬁcation result for regular laterally complete modules over a universally complete f algebra Λ := C∞ (Q). They introduced the passport Γ(X) = (bγ )γ∈Γ for a such module X which is the uniquely deﬁned partition of unity in B := P(Λ) indexed by a set of pairwise diﬀerent cardinals with bγ X being strictly γ-homogeneous for all γ ∈ Γ (cp. 4.6.8). Then they proved that Λ-modules are isomorphic if and only if their passports coincide [90, Theorem 4.3]. It can easily be seen that a regular laterally complete (in the sense of [90]) Λ-module X is represented as X = X ↓ where X is a real vector space in V(B) . Moreover, dim(X ) ∈ V(B) , the algebraic dimension of X , is an internal cardinal and, according to 1.9.11, we have dim(X ) = mixγ∈Γ bγ γ ∧ where Γ is a set of cardinals and (bγ )γ∈Γ is the passport of X. Thus, the passport Γ(X) is the interpretation of the algebraic dimension dim(X ), whence the result ensues. (3) The characterization in Theorem 4.14.B.7 of a universally complete vector lattice in which all band preserving bilinear operators are symmetric is due to Kusraev [236]. The following corollary is immediate from this fact and 4.7.7: Let (Ω, Σ, μ) be a nonatomic Maharam measure space and let L0 (Ω, Σ, μ) be the vector space of all cosets of (almost everywhere equal) real measurable functions. Then there exists an essentially nontrivial separately band preserving antisymmetric bilinear operator in L0 (Ω, Σ, μ). (4) Theorem 4.14.C.8 is taken from Gutman, Kusraev, and Kutateladze [163, Theorem 4.3.6]. This fact lies in an interesting area of the theory of noncommutative integration stemming from Segal [362]. Considerable attention is given to derivations in various algebras of measurable operators associated with an AW ∗ -algebra and a central valued trace. We mention only the article [23] by Albeverio, Ajupov, and Kudaybergenov and the article [49] by Ber, de Pagter, and Sukochev. CHAPTER 5 Order Continuous Operators The approach of Chapter 3 is not applicable directly to order continuous operators since we lose order continuity in ascending an operator (see 3.3.2). The technique of ascending in Chapter 4 preserves order continuity, but this approach treats a narrow class of band preserving operators. In this chapter we pursue another approach that rests on Maharam’s ideas. This chapter focuses on Maharam operators in Dedekind complete vector lattices. The Maharam operators possess the most important properties of conditional expectation and enjoy the Radon–Nikodým type theorem and the Hahn type decomposition theorem. Surprisingly, each injective Banach lattice admits some Maharam operator that completely determines the structural particularities of the lattice. We will also consider some classes of operators whose deﬁnitions depend implicitly or explicitly on Maharam operators. 5.1. Order Bounded Module Homomorphisms Here we will address the Boolean valued representation of f -module homomorphisms and suggest some construction of f -modules that corresponds to the natural embedding into the order bidual in the case of order bounded functionals. 5.1.1. Assume that A is an f -algebra, while X and Y are f -modules over A. A linear mapping T : X → Y is called an A-module homomorphism or A-linear if T (ax) = aT x for all x ∈ X and a ∈ A. Denote by LA (X, Y ) the A-module of all A-linear operators from X to Y and deﬁne L∼ A (X, Y ) as the submodule of LA (X, Y ) consisting of order bounded ∼ ∼ operators; i.e., L∼ A (X, Y ) := LA (X, Y ) ∩ L (X, Y ). Let Ln,A (X, Y ) be ∼ the part of LA (X, Y ) comprising order continuous operators. If Y is ∼ Dedekind complete then L∼ A (X, Y ) and Ln,A (X, Y ) are f -modules over A as indicated in 2.11.1. 5.1. Order Bounded Module Homomorphisms 273 5.1.2. If X and Y are f -modules over A, with Y Dedekind complete; ∼ ∼ then L∼ A (X, Y ) and Ln,A (X, Y ) are bands in L (X, Y ). ⊳ Given a ∈ A, deﬁne â ∈ Orth(X) and ā ∈ Orth(Y ) by putting âx := ax (x ∈ X) and āy := ay (y ∈ Y ). Now deﬁne the endomorphisms Ra and La of L∼ (X, Y ) as Ra (T ) := T ◦ â, La (T ) := ā ◦ T (T ∈ L∼ (X, Y )). Observe that Ra and La are orthomorphisms in L∼ (X, Y ) (see 5.3.2 (1)) and an order bounded operator T ∈ L∼ (X, Y ) belongs to L∼ A (X, Y ) if and only if Ra (T ) = La (T ) for all a ∈ A, so that L∼ ker(Ra − La ). A (X, Y ) = a∈A Thus, L∼ A (X, Y ) is a band, since the kernel ker(Ra − La ) of the orthomorphism Ra − La is a band. It remains to note that L∼ n,A (X, Y ) is the intersection of the two bands in L∼ (X, Y ), namely, L∼ A (X, Y ) and L∼ (X, Y ). ⊲ n 5.1.3. Assume that X and Y are unital f -modules over a Dedekind complete f -algebra A with unit which we identify with an f -subalgebra and an order ideal in Au := R↓. Put B := P(A) and let X , Y ∈ V(B) stand for the Boolean valued representations of X and Y , respectively. Recall that [[X and Y are real vector lattices]] = 1. So, X ↓ and Y ↓ are f -modules over Au , and there are f -module isomorphisms ı from X to X ′ := X ↓ and j from Y to Y ′ := Y ↓ such that X ↓ = mix(ı(X)) and Y ↓ = mix(ı(Y )) (cp. 2.11.4 and 2.11.9). Let L(X , Y ) stand for the element in V(B) uniquely deﬁned by the relation: T ∈ L(X , Y )↓ if and only if [[T is a linear operator from X to Y ]] = 1. Then L(X , Y )↓ is an Au -module and, given T ∈ L(X , Y )↓, the descent T ↓ is an Au -linear operator from X ′ to Y ′ . The spaces L(X , Y ), L(X, Y ), and L(X ′ , Y ′ ) are considered as ordered vector spaces with the cones of positive operators. 5.1.4. Theorem. For each A-linear operator T : X → Y there exists a unique T ∈ V(B) such that [[T is a linear operator from X to Y ]] = 1, T ′ := T ↓ is an Au -linear operator from X ′ to Y ′ , and j ◦ T = T ′ ◦ ı. The mapping T → T is an order preserving injection from LA (X, Y ) into L(X , Y )↓ and the mapping T → T ↓ is an order preserving bijection from L(X , Y )↓ onto LA (X ′ , Y ′ ). 274 Chapter 5. Order Continuous Operators ⊳ By Theorem 2.11.9 we can assume without loss of generality that X ⊂ X ′ and Y ⊂ Y ′ with X ′ = mix(X) and Y ′ = mix(Y ), while ı and j are the embeddings. If b = [[x1 = x2 ]] and π := j(b) with j deﬁned as in 2.11.4, then πx1 = πx2 and so πT x1 = T (πx1 ) = T (π2 x2 ) = π2 T x2 . It follows that b [[T x1 = T x2 ]]. Thus T is extensional and we can deﬁne T := T ↑. In view of 1.6.5 T is a mapping from X to Y within V(B) , since X↑ = X and Y ↑ = Y . By 1.6.6 we have T ′ |X = T . If T¯ is one more linear operator from X to Y within V(B) and (T¯ ↓) ◦ ı = j ◦ T then T ′ and T¯ ↓ : X ′ → Y ′ coincide on ı(X) and so T ′ = T¯ ↓ by 1.6.5, since both T ′ and T¯ ↓ are extensional. It follows that T = T¯ . Let ⊙ and ⊕ stand respectively for scalar multiplication and addition on X , Y , and R. From the properties of ascents and descents it follows that the identities T x = T x, a ⊙ x = ax, a ⊙ y = ay, x1 ⊕ x1 = x1 + x2 , and y1 ⊕ y2 = y1 + y2 hold within V(B) for all x, x1 , x2 ∈ X, y1 , y2 ∈ Y , and a ∈ A. Combining these and the formula R↓ = mix(A), we deduce from 1.5.2 and 1.6.2 that T is linear within V(B) if and only if T is A-linear: [[(∀ a1 , a2 ∈ R)(∀ x1 , x2 ∈ X ) T (a1 ⊙ x1 ⊕ a2 ⊙ x2 ) = a1 ⊙ T (x1 ) ⊕ a2 ⊙ T (x2 )]] = [[T (a1 x1 + a2 x2 ) = a1 T (x1 ) + a2 T (x2 )]] = 1. a1 ,a2 ∈A x1 ,x2 ∈X The same argument implies that T is linear within V(B) if and only if T ′ is Au -linear. It follows from 1.6.7 that T → T ↓ is a bijection between L(X , Y )↓ and LA (X ′ , Y ′ ), while T → T ↑ is an injection from LA (X, Y ) into L(X , Y )↓. The linearity of these mappings may be proved by the argument similar to that in the proof of Theorem 3.3.3. Finally, by 1.5.1 we have [[T (X+ ) ⊂ Y+ ]] = 1 ⇐⇒ (T (X+ ))↓ ⊂ (Y+ )↓ ′ ) ⊂ Y+′ , ⇐⇒ T ↓((X ↓)+ ) ⊂ (Y ↓)+ ⇐⇒ T ′ (X+ whence T is positive within V(B) if and only if T ′ is positive. ⊲ 5.1.5. We make some additional remarks using the same notation as in Theorem 5.1.4. Note ﬁrst that, since LA (X ′ , Y ′ ) and L(X , Y )↓ are linearly and order isomorphic, T or T ′ is regular if and only if T is regular within V(B) . 5.1. Order Bounded Module Homomorphisms 275 (1) The order boundedness of T may be written as (∀ a ∈ X+ )(∃ b ∈ Y+ )T ([−a, a]) ⊂ [−b, b]. Interpreting this formula within V(B) and considering that T ([−a, a])↓ = T ′ ([−a, a]), it is easy to show that T ′ is order bounded if and only if so is T within V(B) . In particular, the order boundedness of T implies that T is order bounded within V(B) . But the converse is not true: If T is order bounded then T have the following “slicewise” order boundedness property: for each a ∈ X+ there exist a partition of unity (πξ ) in P(A) and a disjoint family (bξ ) in Y+ such that πξ T ([−a, a]) ⊂ [−bξ , bξ ] for all ξ. (2) A positive operator T ∈ LrA (X, Y ) is order continuous if and only if so is T ∈ Lr (X , Y ) within V(B) . ⊳ According to 3.1.2 (1) there is no loss of generality in assuming that T and T ′ are positive. Observe that T and T ′ are or are not order continuous simultaneously, since X and Y are order dense sublattices in X ′ and Y ′ , respectively. If [[A is a downward directed set in X and inf(A ) = 0]] = 1, then A := A ↓ is also downward directed and inf(A) = 0 in X ′ . The order continuity of T ′ implies that inf(T (A )) = inf(T ′ (A)) = 0 within V(B) . Conversely, if T is order continuous within V(B) and A ⊂ X ′ is downward directed with inf(A) = 0, then [[A := A↑ is upward directed and inf(A ) = 0]] = 1; consequently, for y := inf(T ′ (A)) we have [[a = inf(T (A )) = 0]] = 1, whence a = 0. ⊲ Given a real vector lattice X within V(B) , denote by X ∼ and ∼ Xn the internal vector lattices of order bounded and order continuous functionals on X , respectively. More precisely, [[σ ∈ X ∼ ]] = 1 and [[σ ∈ Xn∼ ]] = 1 mean that [[σ : X → R is an order bounded functional]] = 1 and [[σ : X → R is an order continuous functional]] = 1, respectively. 5.1.6. Corollary. Let Y be a Dedekind complete vector lattice and let X be a unital f -module over A := Z (Y ). For every T ∈ L∼ A (X, Y ) there exists a unique τ ∈ V(B) such that [[τ : X → R is R-linear and order bounded]] = 1, T ′ := τ ↓ is an order bounded Au -linear operator from X ′ to R↓, and j ◦ T = T ′ ◦ ı. The mapping T → τ is an injection ∼ ∼ ∼ from L∼ A (X, Y ) into (X )↓ and from Ln,A (X, Y ) into (Xn )↓. The ′ ′ ∼ mapping τ → τ ↓ is a lattice isomorphism of (X )↓ onto L∼ A (X , Y ) ′ ′ ∼ ∼ and (Xn )↓ onto Ln,A (X , Y ). 5.1.7. Theorem 5.1.4 and Corollary 5.1.6 enable us to treat some classes of module homomorphisms on f -modules over A as R-linear operators and, whenever A is the center of the range vector lattice, as 276 Chapter 5. Order Continuous Operators R-linear functionals. As an illustration consider the interpretation of Nakano’s results concerning the embedding a vector lattice into the order continuous bidual. To do this we need to recall some more deﬁnitions and facts. Consider an order ideal J in L∼ n,A (X, Y ). Note that then J is a f submodule of L∼ n,A (X, Y ). For an arbitrary x ∈ X deﬁne the operator x̂ : J → Y by x̂(T ) := T x for all T ∈ J . Clearly, x̂ ∈ L∼ A (J , Y ), since x̂ = (x+ )ˆ− (x− )ˆ, x̂ is a positive operator whenever x ∈ X+ , and x̂(aT ) = T (ax) = ax̂(T ) for all a ∈ A and T ∈ J . Moreover, x̂ ∈ L∼ n,A (J , Y ). Indeed, if a net Tα in J is decreasing and inf α Tα = 0, then o-limα x̂(Tα ) = 0, since |x̂(Tα )| = |Tα (x)| Tα (|x|) and o-lim Tα (|x|) = 0 in Y for all x ∈ X. The mapping x → x̂ is called the natural embedding of X to L∼ (J , Y ). A An f -module X over A := Z (Y ) is said to be Y -perfect with respect to J (or Y -perfect whenever J = L∼ n,A (X, Y )) if the natural embedding is a bijection of X onto L∼ n,A (J , Y ). In case Y = R we say that X is perfect with respect to J . 5.1.8. Theorem. Let X be a vector lattice, and let J be an ideal of Xn∼ . Then the embedding x → x̂ is an order continuous lattice homomorphism from X to Jn∼ whose range is an order dense vector sublattice of Jn∼ . The natural embedding is an injection if and only if J separates the points of X. ⊳ See Aliprantis and Burkinshaw [28, Theorem 1.70] and Zaanen [427, Theorem 109.3]. ⊲ 5.1.9. Theorem. A vector lattice X is perfect with respect to an order ideal J in Xn∼ if and only if Jn∼ separates the points of X and, given an increasing net (xα ) in X+ such that supα f (xα ) < ∞ for each 0 f ∈ J , there exists some x ∈ X satisfying x = supα xα in X. ⊳ See Aliprantis and Burkinshaw [28, Theorem 1.71] and Zaanen [427, Theorem 110.1]. ⊲ 5.1.10. Theorem. Let X be an f -module over A := Z (Y ) with Y a Dedekind complete vector lattice and J an order ideal in L∼ n,A (X, Y ) separating the points of X. Then (1) The natural embedding is an order continuous A-linear lattice isomorphism from X to L∼ n,A (J , Y ) whose range is an order dense sublattice of L∼ (J , Y ). n,A 5.1. Order Bounded Module Homomorphisms 277 (2) X is Y -perfect with respect to J if and only if, given an increasing net (xα ) in X+ with supα T xα existing in Y for all T ∈ J , there exists x ∈ X such that x = supα xα . ⊳ We use the same notation as in Theorem 5.1.4 and Corollary 5.1.6 and identify B with P(A). Assume that X = X ′ and Y = Y ′ . Put J := {T ↑ : T ∈ J }↑ and observe that J is an order ideal in Xn∼ . ′ ′ ∼ Indeed, according to Corollary 5.1.6, L∼ n,A (X , Y ) and (Xn )↓ are f module isomorphic under the ascent and the isomorphism sends J onto an order ideal in (Xn∼ )↓, say J¯; therefore, [[J = J¯↑ is an order ideal in (Xn∼ )]] = 1. By Corollary 5.1.6 the mapping τ → τ ↓ is a lattice ∼ ′ ′ ′ isomorphism of (J∼ n )↓ onto Ln,A (J , Y ) where J := J↓. Show that [[ J is point separating]] = 1. Take x ∈ X and put bx := b = [[x = 0]]. From the Kutatowski–Zorn Lemma it is easy to derive that there exist a partition (bξ )ξ∈Ξ∪{ξ0 } of unity in P(A) and a family (Tξ )ξ∈Ξ∪{ξ0 } in J such that Tξ0 = 0, bξ0 = b⊥ , and bξ [[Tξ x = 0]] for all ξ ∈ Ξ. Deﬁne Tx := T ∈ Xn∼ as T := mixξ∈Ξ∪{ξ0 } (bξ (Tξ ↑)). Clearly, bξ [[T = Tξ ↑]] ∧ [[Tξ ↑ ∈ J]] [[T ∈ J]] for all ξ ∈ Ξ ∪ {ξ0 } and bξ [[T x = Tξ ↑x]] ∧ [[Tξ ↑x = 0]] [[T x = 0]] for all ξΞ, so that [[Tx ∈ J]] = 1 and bx [[T x = 0]]. The result follows from simple calculation through 1.2.3, 1.5.2, and 1.6.2: [[ J is point separating ]] = [[(∀ x ∈ X ) x = 0 → (∃ τ ∈ J)τ (x) = 0 ]] = x∈X ′ [[x = 0]] ⇒ [[τ (x) = 0]] τ ∈J↓ bx ⇒ [[Tx (x) = 0]] = 1. x∈X ′ Let Φ and φ stand respectively for the natural embeddings of X to L∼ (J , Y ) and of X to J∼ n . Then [[φ(x)τ = τ x = T x = Φ(x)T ]] = 1 A for all x ∈ X, T ∈ J , and τ := T ↑. It follows that φ := Φ̂↑ with Φ̂ : X → J∼ n ↓ deﬁned as [[Φ̂(x)τ := Φ(x)(T ) = T x]] = 1 for τ = T ↑ and T ∈ J . From this we can easily derive that φ(X ) = {Φ(x)↑ : x ∈ X}↑ or, equivalently, φ(X )↓ = mix{Φ(x)↑ : x ∈ X}. By transfer, Theorems 5.1.8 and 5.1.9 are true within V(B) . Thus, by Theorem 5.1.8 [[ φ : X → J∼ n is an order continuous lattice isomorphism and φ(X ) is an order dense sublattice of J∼ n ]] = 1. Clearly Φ̂ = φ↓ is a lattice isomorphism and in view of 5.1.5 (2) it is also order continuous. Moreover, Φ̂(X) is an order dense sublattice in J∼ n ↓, since Φ̂(X) = φ(X )↓ by 1.5.3. It follows that Φ(X) is order dense sublattice ′ ′ in L∼ n,A (J , Y ). 278 Chapter 5. Order Continuous Operators The necessity in (2) is straightforward. To prove the suﬃciency, assume that for each increasing net (xα ) in X+ with supα T xα existing in Y for all T ∈ J there exists x ∈ X such that x = supα xα . Then it can easily be seen that, given an increasing net (xα ) in X+ such that supα τ (xα ) < ∞ for each 0 τ ∈ J, there exists some x ∈ X satisfying x = supα xα in X . By Theorem 5.1.9 we have [[ X is perfect]] = [[φ(X ) = J∼ Thus, Φ̂(X) = J∼ n ]] = 1. n ↓ and ∼ ′ ′ Φ(X) = Ln,A (J , Y ) by passing to descents. ⊲ 5.2. Maharam Operators Under discussion is some class of the order continuous positive operators that behave like functionals in many aspects. We establish a Radon– Nikodým-type Theorem for these operators. 5.2.1. Throughout this section X and Y are vector lattices with Y Dedekind complete. A positive operator T : X → Y is said to have the Maharam property (or T is said to be order interval preserving) whenever T [0, x] = [0, T x] for every 0 x ∈ X; i.e., if for every 0 x ∈ X and 0 y T x there is some 0 u ∈ X such that T u = y and 0 u x. A Maharam operator is an order continuous positive operator whose modulus enjoys the Maharam property. Say that a linear operator S : X → Y is absolutely continuous with respect to T and write S T if |S|x ∈ {|T |x}⊥⊥ for all x ∈ X+ . It can easily be seen that if S ∈ {T }⊥⊥ then S T , but the converse may be false. 5.2.2. The null ideal NT of an order bounded operator T : X → Y is deﬁned by NT := {x ∈ X : |T |(|x|) = 0. Observe that NT is indeed an ideal in X. The disjoint complement of NT is referred to as the carrier of T and is denoted by CT , so that CT := NT⊥ . An operator T is called strictly positive whenever X = CT ; i.e., 0 < x ∈ X implies 0 < |T |(x). Clearly, |T | is strictly positive on CT . Sometimes we ﬁnd it convenient to put XT := CT and YT := (im T )⊥⊥ . A positive operator T : X → Y is said to have the Levi property if sup xα exists in X for every increasing net (xα ) ⊂ X+ , provided that the net (T xα ) is order bounded in Y . For an order bounded order continuous operator T from X to Y denote by Dm (T ) the largest ideal of the universal completion X u onto which we can extend the operator 5.2. Maharam Operators 279 T by order continuity. For a positive order continuous operator T we have X = Dm (T ) if and only if T has the Levi property. The following theorem describes an important property of Maharam operators, enabling us to embed them into an appropriate Boolean valued universe as order continuous functionals. 5.2.3. Theorem. Let X and Y be some vector lattices with Y having the projection property and let T be a Maharam operator from X to Y . Then there exist an order closed subalgebra B of B(XT ) consisting of projection bands and a Boolean isomorphism h from B(YT ) onto B such that T (h(L)) ⊂ L for all L ∈ B(YT ). ⊳ Without loss of generality, we can assume that T is strictly positive and Y = YT . For each band L in Y , we put h(L) := {x ∈ X : T (|x|) ∈ L}. Clearly, h(L) is a vector subspace of X with h({0} ) = 0 and h(Y ) = X. Moreover, T (h(L)) ⊂ L for all L ∈ B(YT ) by the very deﬁnition of h. Prove that h(L) is a band in X for every L ∈ B(X). Indeed, if x ∈ h(L), u ∈ X, and |u| x then T (|u| ) T (x) ∈ L; i.e., u ∈ L, which proves that h(L) is an order ideal. Suppose that a set A ⊂ h(L) ∩ X+ is directed upwards and bounded from above by x0 ∈ X+ . Then the set T (A) ⊂ L+ is bounded above by T (x0 ). Consequently, taking ocontinuity of T into account, we obtain T (sup(A)) = sup{T (x) : x ∈ A} ∈ L. Thus, sup (A) ∈ L. Hence, h(L) is a band in X. It is easily seen that the mapping h : B(Y ) → B(X) is increasing: L1 ⊂ L2 implies h(L1 ) ⊂ h(L2 ). We now demonstrate that h is injective. To this end, we suppose that h(L1 ) = h(L2 ) for some L1 , L2 ∈ B(Y ) and, nevertheless, L1 = L2 , say L1 ∩ L⊥ 2 = ∅. Take an element 0 < y ∈ L1 such that y ⊥ L2 . Since y ∈ L1 ⊂ Y = T (X)⊥⊥ , there exist 0 < y1 ∈ Y and 0 < x ∈ X such that y1 y ∧ T (x). If y2 := T (x) − y1 then, by the Maharam property, x = x1 + x2 and T (xl ) = yl (l := 1, 2) for some 0 < xl ∈ X (l := 1, 2). But then x1 ∈ h(L1 ) and x1 ∈ / h(L2 ), contradicting the assumption h(L1 ) = h(L2 ). This proves the injectivity of h. Consider the inclusion ordered set B := im(h); i.e., B := {K ∈ B(X) : K = h(L), L ∈ B(Y )}. The above established fact means that h is an isomorphism of the ordered sets B(Y ) and B. Clarify what operations in B correspond to the Boolean operations in B(Y ) under the 280 Chapter 5. Order Continuous Operators order isomorphism h. First of all, observe that h(inf(U )) = h U = {h(L) : L ∈ U } (U ⊂ B(Y ) ). Further, let L1 ⊕ L2 be a disjoint decomposition of the vector lattice Y . Then h(L1 ) ∩ h(L2 ) = {0}. Given x ∈ X, we have the representation T x = y1 + y2 with yl := [Ll ](y) (l := 1, 2). Hence, by the Maharam property for T , there exist u1 and u2 ∈ X+ such that |x| = u1 + u2 and T (ul ) = yl (l := 1, 2). Furthermore, for some x1 , x2 ∈ X, we have x = x1 + x2 and |xl | = ul (l := 1, 2). This yields x1 ∈ h(L1 ) and x2 ∈ h(L2 ). Consequently, X is the algebraic direct sum of h(L1 ) and h(L2 ). Moreover, if xl ∈ h(Ll ) (l := 1, 2) then T (|x1 | ∧ |x2 | ) T (|x1 |)∧T (|x2 |) ∈ L1 ∩L2 = {0}. Hence T (|x1 |∧|x2 | ) = 0 and, since T is strictly positive, we obtain x1 ⊥ x2 . So, the bands h(L1 ) and h(L2 ) form a disjoint decomposition of the vector lattice X. Thus, h(L⊥ ) = h(L)⊥ for all L ∈ B(Y ). Since the mapping h : B(Y ) → B preserves inﬁma and complements, it is an order continuous monomorphism of B(Y ) onto an o-closed subalgebra B of the Boolean algebra B(X). The proof is complete. ⊲ 5.2.4. It is worth pointing out some corollaries to Theorem 5.2.3. Assume that X, Y , and T are as in 5.2.3. (1) If S : X → Y is a positive operator absolutely continuous with respect to T then S(h(L)) ⊂ L for all L ∈ B(YT ). ⊳ Given L ∈ B(YT ) and x ∈ h(L), we evidently have |S(x)| S(|x|) ∈ {T (|x|)}⊥⊥ ⊂ L and thus S(h(L)) ⊂ L. ⊲ (2) There exists a Boolean isomorphism h from P(Y ) onto an order closed subalgebra of P(X) such that πS = Sh(π) for all π ∈ P(Y ) whenever S : X → Y is a positive operator absolutely continuous with respect to T . ⊳ Let B be a Boolean algebra of projections onto the bands in B. Denote by the same symbol h the respective isomorphism from P(Y ) onto B ⊂ P(X). It follows from (1) that π ⊥ ◦ S ◦ h(π) = 0 or S ◦ h(π) = π ◦ S ◦ h(π). Replacing π by π ⊥ we obtain π ◦ S = π ◦ S ◦ h(π). We thus arrive at the sought relation π ◦ S = S ◦ h(π). ⊲ 5.2.5. Let X and Y be Dedekind complete vector lattices and T a Maharam operator from X to Y . Then there exists an f -module 5.2. Maharam Operators 281 structure over Z (Y ) on X such that an order bounded operator S from X to Y is absolutely continuous with respect to T if and only if S is Z (M )-linear. ⊳ We can assume that T is strictly positive, because otherwise the f -module multiplication on XT can be extended to the whole X by putting ax := aπx for all x ∈ X and a ∈ Z (Y ) where π is the band projection onto XT . The Boolean isomorphism h from 5.2.4 (2) can uniquely be extended to an f -algebra isomorphism from Z (Y ) onto an f -subalgebra in Z (X). Denote this isomorphism by the same symbol h. An f -module structure over Z (Y ) on X is induced by putting αx := h(α) x for all α ∈ Z (Y ) and x ∈ X. Take an operator S absolutely continuous with respect to T . If α := nl=1 λl πl , where λ1 , . . . , λn ∈ R+ and {π1 , . . . , πn } is a partition of unity in P(Y ), then, in view of 5.2.4 (2), πl ◦ α ◦ S = πl ◦ S (λl h(πl ) ) = πl ◦ S ◦ h(α) for all l. Summing over l yields α ◦ S = S ◦ h(α). By the Freudenthal Spectral Theorem the set of orthomorphisms α of the above form is uniformly dense in Z (Y ). Since S is uniformly continuous, we conclude that α ◦ S = S ◦ h(α) for all α ∈ Z (Y ). It follows that S is Z (Y )-linear. Conversely, assume that S is Z (Y )-linear, x ∈ X+ , and π is the band projection onto {T x}⊥. Then 0 = πT x = T h(π)x by 5.2.4 (2), so that h(π)x = 0 due to the strict positivity of T . Thus, πSx = Sπx = Sh(π)x = 0 and Sx ∈ {T x}⊥⊥ . ⊲ 5.2.6. Let X, Y , and T be as in 5.2.5. For a band K ∈ B(XT ) the following are equivalent: (1) T u = T v and u ∈ K imply that v ∈ K for all u, v ∈ X+ . ′ (2) T (K+ ) ⊂ T (K+ )⊥⊥ yields K ′ ⊂ K for all K ′ ∈ B(XT ). (3) K = h(L) for some L ∈ B(Y ). ′ ⊳ (1) =⇒ (2) Arguing for a contradiction, assume that T (K+ ) ⊂ ⊥⊥ ′ T (K+ ) for some K ∈ B(XT ) not contained in K. Then there exists ′ 0 < v ∈ K ′ with v ⊥ K. It follows that T v ∈ T (K+ )T (K+ )⊥⊥ = ⊥⊥ T (K) and, since T (K) is an order ideal in Y , we can choose 0 < u ∈ K such that 0 < T u T v. By the Maharam property there is 0 < u0 v with T u0 = T u. By (1) we have u0 ∈ K and this is a contradiction, since u0 ∈ K and u0 v ∈ K ′ imply u0 = 0. (2) =⇒ (3) Put L := T (K+ )⊥⊥ and observe that K ⊂ h(L) by deﬁnition of h (cp. 5.2.3). If K ′ := K ⊥ ∩ h(L) then K ′ ⊂ h(K) and ′ T (K+ ) ⊂ L = T (K+ )⊥⊥ , so that K ′ ⊂ K in view of (2). It follows that ′ K = {0} and K = h(L). 282 Chapter 5. Order Continuous Operators (3) =⇒ (1) If u ∈ K+ then T u ∈ L by the deﬁnition of h. Given v ∈ X+ with T u = T v, we have T v ∈ L and v ∈ K. ⊲ 5.2.7. A band K ∈ B(XT ) (as well as the corresponding band projection [K] ∈ P(XT )) is said to be T -saturated if (hence each) of the conditions 5.2.6 (1–3) is fulﬁlled. The set of all T -saturated bands (band projections) is denoted by BT (X) (respectively PT (X)). A band projection π ′ ∈ P(XT ) is T -saturated if and only if T π ′ T π implies π ′ π for all π ′ ∈ P(XT ). It follows that the isomorphism h : P(Y ) → PT (X) can be deﬁned as h(ρ) = {π ∈ P(X) : T π ρT } (ρ ∈ P(Y )). We now present the main result of this section stating that Maharam operators can be embedded into V(B) , turning thereby into order continuous functionals. This Boolean valued representation of Maharam operators enables one to obtain various facts about the Maharam operators from the corresponding ZFC theorems on functionals. 5.2.8. Theorem. Let X be a Dedekind complete vector lattice, Y := R↓, and let T : X → Y be a positive Maharam operator with Y = YT . Then there are X , τ ∈ V(B) satisfying the following: (1) [[ X is a Dedekind complete vector lattice and τ : X → R is an order continuous strictly positive functional with the Levi property ]] = 1. (2) X ↓ is a Dedekind complete vector lattice and a unital f -module over the f -algebra R↓. (3) τ ↓ : X ↓ → R↓ is a strictly positive Maharam operator with the Levi property and an R↓-module homomorphism. (4) There exists an order continuous lattice homomorphism ϕ : X → X ↓ such that ϕ(X) is an order dense ideal of X ↓ and T = τ ↓ ◦ ϕ. ⊳ Assume without loss of generality that T is strictly positive. By Corollary 5.1.6 and 5.2.5 there exist a Dedekind complete vector lattice X¯ and an order continuous R-linear functional τ̄ on X¯ within V(B) and there is a lattice isomorphism ϕ from X into X ′ := X¯ ↓ such that T = τ̄ ↓ ◦ ϕ. By transfer there exist an order ideal X in X¯ u , including X¯ , and a strictly positive order continuous functional τ : X → R with the Levi property such that τ |X¯ = τ̄ . Clearly, T = τ̄ ↓ ◦ ϕ = (τ |X¯ )↓ ◦ ϕ = τ ↓|X ′ ◦ ϕ = τ ↓ ◦ ϕ. 5.2. Maharam Operators 283 Moreover, ϕ(X) is an order dense ideal in X ′ and so in X ↓. Using Corollary 5.1.6 and 5.2.5 again, we conclude that τ ↓ is a Maharam operator. The Levi property for strict positiveness of τ ↓ are easily deduced from that of τ within V(B) . ⊲ 284 Chapter 5. Order Continuous Operators 5.3. Representation of Order Continuous Operators Theorem 5.2.8 together with the Boolean valued transfer principle enables us to assert that each fact about order continuous positive linear functionals on a Dedekind complete vector lattice has its counterpart for Maharam operators that can be demonstrated on using the descending– ascending machinery. The aim of this section is to prove an operator version of the next result. 5.3.1. Theorem. Let X be a vector lattice and let Xn∼ separate the points of X. Then there exist order dense ideals L and X ′ in X u and a linear functional τ : L → R such that the following hold: (1) X ′ = {x′ ∈ X ′ : xx′ ∈ L for all x ∈ X}. (2) τ is strictly positive, order continuous, and has the Levi property. (3) For every σ ∈ Xn∼ there exists a unique x′ ∈ X ′ such that σ(x) = τ (x · x′ ) (x ∈ X). (4) σ → x′ is a lattice isomorphism of Xn∼ onto X ′ . ⊳ The proof may be found in Vulikh and Lozanovskiı̆ [404, Theorem 2.1]. It can also be extracted from Vulikh [403, Theorem IX.3.1] or Kusraev [228, Theorem 3.4.8]. ⊲ 5.3.2. To translate Theorem 5.3.1 into a result for operators we need some preparations. Let X and Y be f -modules over an f -algebra A. Recall that for a ∈ A the orthomorphisms â ∈ Orth(X) and ā ∈ Orth(Y ) are deﬁned as â : x → ax (x ∈ X) and ā : y → ay (y ∈ Y ), while the mappings Ra and La on L∼ (X, Y ) are deﬁned by Ra (T ) := T â and La (T ) := āT ; see 3.1.2. (1) The maps a → Ra and a → La are f -algebra homomorphisms from A to Orth L∼ (X, Y ) . ⊳ It is easy to note that Ra ∈ Orth L∼ (X, Y ) whenever â ∈ Z (X)+ . For an arbitrary a ∈ A+ the sequence (πn ) in Z (X) with πn := â ∧ (nIX ) converges â2 -uniformly to â and, for all S, T ∈ L∼ (X, Y ) with S ∧(T πn ) = 0, we have S ∧(T πn ) = 0. Therefore, S ∧T = 0 implies S ∧ Ra (T ) = S ∧ (T â) = 0, since (T πn ) converges T â2 -uniformly to T â. It follows that Ra ∈ Orth L∼ (X, Y ) . Moreover, the mapping a → Ra is evidently a positive algebra homomorphism. It remains to observe that a positive algebra homomorphism is a lattice homomorphism. The case of the mapping a → La is treated similarly. ⊲ 5.3. Representation of Order Continuous Operators 285 (2) Let Y be a Dedekind complete vector lattice, A := Z (Y ), Ā := Orth(Y ), and A0 := St0 (P(Y )). If X is an f -module over Ā then ∼ ∼ L∼ A0 (X, Y ) = LA (X, Y ) = LĀ (X, Y ). ∼ ⊳ It suﬃces to ensure that L∼ A0 (X, Y ) ⊂ LĀ (X, Y ), because the ∼ converse inclusion is evident. Observe that LA0 (X, Y ) ⊂ L∼ A (X, Y ), since A0 is uniformly dense in A by the Freudenthal Spectral Theorem and every order bounded linear operator is uniformly continuous. An arbitrary a ∈ Ā+ is the a2 -uniform limit of the sequence (an ) in Z (Y ) 2 with an := a ∧ (nIY ). If T ∈ L∼ A (X, Y ) then T (ax) is the |T |(a x)uniform limit of (T (an x)) for all x ∈ X+ , so that aT x = T (ax). It follows that T ∈ L∼ (X, Y ). ⊲ Ā 5.3.3. Say that a set T ⊂ L∼ (X, Y ) separates the points of X or is point separating on X whenever, given nonzero x ∈ X, there exists T ∈ T such that T x = 0. In the case of Y Dedekind complete and T ⊂ L∼ (X, Y ) a sublattice, this is equivalent to saying that for every nonzero x ∈ X+ there is a positive operator T ∈ T with T x = 0. Assume that A := Z (Y ) and X is an f -module over A. If L∼ n,A (X, Y ) separates the points of X, then X is unital and {π̂ : π ∈ P(Y )} is an order closed subalgebra in P(X). ⊳ We have only to ensure that the Boolean homomorphism π → π̂ from P(Y ) to P(X) is order continuous. Take a decreasing family (πα ) in P(Y ) with inf α πα = 0 and suppose that 0 u π̂α x for all α with some ﬁxed x, u ∈ X+ . Then 0 = inf α πα T x = inf α T (π̂α x) T u 0 for all 0 T ∈ L∼ n,A (X, Y ). Hence u = 0 and inf α π̂α x = 0 for all x ∈ X+ . ⊲ 5.3.4. Denote X := X ↓ and A := R↓. The mapping assigning to each member σ ∈ X ∼ ↓ its descent S := σ↓ is a lattice isomorphism of ∼ X ∼ ↓ and Xn∼ ↓ onto L∼ A (X, R↓) and Ln,A (X, R↓), respectively. More∼ ∼ over, [[X ( resp. Xn ) separates the points of X ]] = 1 if and only if ∼ L∼ A (X, R↓) (resp. Ln,A (X, R↓)) separates the points of X. ⊳ The ﬁrst statement follows from Corollary 5.1.6, so that we need only verify the second one. Observe ﬁrst that L∼ A (X, R↓) coincides with the space LExt (X, R↓) of all extensional order bounded linear operators from X to R↓. Take x, y ∈ X and put b = [[x = y]]. Then χ(b)x = χ(b)y and, given S ∈ L∼ A (X, R↓), we have χ(b)Sx = S(χ(b)x) = S(χ(b)y) = χ(b)T y, so that b [[Sx = Sy]] and S ∈ LExt (X, R↓). Thus, L∼ A (X, R↓) ⊂ LExt (X, R↓) and the converse inclusion follows from 5.3.2 (2). 286 Chapter 5. Order Continuous Operators Now, formalize the claim that X ∼ is point separating: ϕ(X ) ≡ (∀ x ∈ X ) ((∀ σ ∈ X ∼ )σ(x) = 0) → x = 0 . In view of 1.5.2 and 1.5.6 [[ϕ(X )]] = bx ⇒ [[x = 0]] x∈X where bx := {[[Sx = 0]] : S ∈ LExt (X, R↓)}. Thus, X ∼ is point separating within V(B) if and only if bx [[x = 0]] for all x ∈ X. The latter is equivalent to saying that, given x ∈ X, we have χ(b)x = 0 when∼ ever χ(b)Sx = 0 for all S ∈ L∼ A (X, R↓). This implies that LA (X, R↓) separates the points of X. ⊲ 5.3.5. Theorem. Let X be an f -module over A := Z (Y ) with Y a Dedekind complete vector lattice and let L∼ n,A (X, Y ) separate the points of X. Then there exist an order dense ideal L in X u and a strictly positive Maharam operator T : L → Y such that the order ideal X ′ = {x′ ∈ X ′ : (∀ x ∈ X) xx′ ∈ L} ⊂ X u is lattice isomorphic to L∼ n,A (X, Y ). The isomorphism is implemented by assigning the operator ′ ′ Sx ′ ∈ L ∼ n,A (X, Y ) to an element x ∈ X by the formula Sx′ (x) = T (xx′ ) (x ∈ X). If there exists a strictly positive T0 ∈ L∼ n,A (X, Y ) then we can choose L and T such that X ⊂ L and T |X = T0 . ⊳ According to Theorem 3.1.10 L∼ n,A (X, Y ) is isomorphic to ), so that there is no loss of generality in assuming that X is Dedekind complete. By Gordon’s Theorem 2.4.2 we can assume also that Y u = R↓. Of course, in this event we can identify Au with Y u . In view of Theorem 2.11.9 there exists a Dedekind complete real vector lattice X within V(B) with B = P(Y ) such that X ↓ is an f module over Au , and there is an f -module isomorphism h from X to X ↓ satisfying X ↓ = mix(h(X)). By 5.3.4 Xn∼ separates the points of X . The transfer principle tells us that Theorem 5.3.1 is true within V(B) , so that there exist an order dense ideal L in X u and a strictly positive linear functional τ : L → R with the Levi property such that the order ideal X ′ = {x′ ∈ X u : x′ X ⊂ L } is lattice isomorphic to Xn∼ ; moreover, the isomorphism is implemented by assigning the functional σx′ ∈ Xn∼ to x′ ∈ X ′ using the rule σx′ (x) := τ (xx′ ) (x ∈ X ). Put X̂ := X ↓, L̂ := L ↓, T̂ := τ ↓, and X̂ ′ := X ′ ↓. By Theorem 2.11.9 we can identify the universally complete vector lattices X u , X̂ u , and δ L∼ n,A (X , Y 5.3. Representation of Order Continuous Operators 287 X u ↓ as well as X with a laterally dense sublattice in X̂. Then L̂ is an order dense ideal in X̂ u and an f -module over Au , while T̂ : L̂ → Y u is a strictly positive Maharam operator with the Levi property. Since the multiplication on X u is the descent of the internal multiplication in X u , we have the representation X̂ ′ = {x′ ∈ X u : x′ X̂ ⊂ L̂}. Moreover, X̂ ′ is u ′ f -module isomorphic to L∼ n,A (X̂, Y ) by assigning to x ∈ X̂ the operator ′ u Ŝx′ ∈ L∼ n,A (X̂, Y ) deﬁned as Ŝx′ (x) = T̂ (xx ) (x ∈ X̂). Putting L := {x ∈ L̂ : T̂ x ∈ Y }, ′ ′ ′ T := T̂ |L , ′ X := {x ∈ X̂ : x X ⊂ L}, we see that if x′ ∈ X ′ then Sx′ := Ŝx′ |X belongs to L∼ n,A (X, Y ). Con∼ versely, an arbitrary S ∈ Ln,A (X, Y ) has the representation Sx = T̂ (xx′ ) (x ∈ X) with some x′ ∈ X̂ ′ , so that T̂ (xx′ ) ∈ Y for all x ∈ X and so x′ ∈ X ′ , xx′ ∈ L for all x ∈ X, and Sx = T (xx′ ) (x ∈ X) by the above deﬁnitions. ⊲ 5.3.6. Corollary. Given π ∈ P(X), define π̂ ∈ P(L∼ n,A (X, Y )) as π̂ : S → S ◦ π. Under the hypotheses of 5.3.3 the mapping π → π̂ is a Boolean isomorphism of P(X) onto P(L∼ n,A (X, Y )). ′ ⊳ Let γ stand for the lattice isomorphism from L∼ n,A (X, Y )) onto X ′ in Theorem 5.3.5. Denote by π̃ the unique band projection on X which agrees with π on X ∩ X ′ . Then by Theorem 5.3.5 we have T (γ(π̂S)x) = π̂(S)x = S(πx) = T (γ(S)πx) = T (xπ̃γ(S)) for all x ∈ X, so that π̂(S) = γ −1 (π̃γ(S)) for all S ∈ L∼ n,A (X, Y ). It remains to observe that j(π) : S → γ −1 (π̃γ(S)) is a band projection in L∼ n,A (X, Y ) and the mapping π → j(π) is a Boolean isomorphism of P(X) onto P(L∼ n,A (X, Y )). ⊲ The following two results are immediate from Corollary 5.3.6. 5.3.7. Hahn Decomposition Theorem. Let T : X → Y be a Maharam operator. Then there is a band projection π ∈ P(X) such that T + = T ◦ π = |T | ◦ π and T − = −T ◦ π ⊥ = |T | ◦ π ⊥ . In particular, |T | = T ◦ (π − π ⊥ ) and T = |T | ◦ (π − π ⊥ ). ⊳ There is no loss of generality in assuming that |T | is strictly positive. Let [T + ] stands for the band projection onto the band in + L∼ In view of Corollary 5.3.6 there is n,A (X, Y ) generated by T . π ∈ P(X) such that π̂ = [T + ]. Now, by deﬁnitions we have T + = [T + ](T ) = π̂(T ) = T ◦ π and T − = −[T + ]⊥ (T ) = −π̂ ⊥ (T ) = −T ◦ π ⊥ . ⊲ 288 Chapter 5. Order Continuous Operators 5.3.8. Nakano Theorem. Let T1 , T2 : X → Y be order bounded operators such that T := |T1 | + |T2 | is a Maharam operator. Then T1 and T2 are disjoint if and only if so are their carriers; symbolically, T1 ⊥ T2 ⇐⇒ CT1 ⊥ CT2 . ⊳ Again, assume without loss of generality that T is strictly positive. By Corollary 5.3.6 there is πi ∈ P(X) such that π̂i = [Ti ]. Clearly, T1 ⊥ T2 if and only if π̂1 ⊥ π̂2 or, equivalently, π1 (X) ⊥ π2 (X). It remains to observe that Ti = π̂i (T ) and the carrier of π̂i (T ) coincides with the band πi (X) (i := 1, 2). ⊲ 5.3.9. Radon–Nikodým Theorem. Let X and Y be Dedekind complete vector lattices and let T be a positive Maharam operator. For an order bounded order continuous operator S from X to Y the following are equivalent: (1) S ∈ {T }⊥⊥. (2) S T . (3) There exists an orthomorphism ρ ∈ Orth∞ (X) such that Sx = T (ρx) for all x ∈ D(ρ). ⊳ The implication (1) =⇒ (2) is trivial. For the proof that (2) =⇒ (3), we can assume without loss of generality that T is strictly positive. In view of 5.2.5 there exists an f -module structure over A := Z (Y ) on X such that an order bounded order continuous operator S : X → Y is absolutely continuous with respect to T if and only if S is in Ln,A (X, Y ). Moreover, Ln,A (X, Y ) separates the points of X. Thus, by Theorem 5.3.5, there exist an order dense ideal L̂ in X u including X and a strictly positive Maharam operator T̂ : L → Y such that T̂ |X = T and S(x) = T̂ (xx′ ) (x ∈ X) for some x ∈ X u . It remains to put D(ρ) := {u ∈ X : ux′ ∈ X} and ρx = xx′ (x ∈ D(ρ)). To see (3) =⇒ (1), suppose that S0 ∈ {T }⊥ and 0 S0 |S| for some S0 ∈ L∼ (X, T )+ . Then S0 is absolutely continuous with respect to T and, by what has just been proved, there exists ρ0 ∈ Orth∞ (X) such that S0 x = T̂ (ρ0 x) for all x ∈ D(ρ0 ) ∩ X. From 3.1.5 we deduce 0 = (T ∧ S0 )x = inf T ((I − ρ0 )u + ρ0 x) T ρ0 x = S0 x 0ux for all 0 x ∈ X ∩ D(ρ0 ), so that S0 = 0 by order continuity of S. ⊲ 5.4. Conditional Expectation Type Operators 289 5.4. Conditional Expectation Type Operators Conditional expectation operators have many remarkable properties related to the order structure of the underlying function space. Boolean valued analysis enables us to demonstrate that these property are shared by a much more general class of operators. 5.4.1. Let Z be a universally complete vector lattice with (weak order) unit 1. Recall that a universally complete vector lattice Z is a semiprime f -algebra with a multiplicative unit 1. Assume that Φ : L1 (Φ) → Y is a strictly positive Maharam operator with the Levi property. We will write L0 (Φ) := Z whenever L1 (Φ) is an order dense ideal in Z. Also, denote by L∞ (Φ) the order ideal in Z generated by 1. Consider an order ideal X ⊂ Z and we will always assume that L∞ (Φ) ⊂ X ⊂ L1 (Φ). The associate space X ′ is deﬁned as the set of all x′ ∈ L0 (Φ) for which xx′ ∈ L1 (Φ) for all x ∈ X. Clearly, X ′ is also an order ideal in Z. Throughout this section (Ω, Σ, μ) is a ﬁnite measure space and L0 (Ω, Σ, μ) is the Dedekind complete vector lattice of Σ-measurable real functions on Ω with the usual identiﬁcation of μ-equivalent functions. The corresponding Lp -spaces Lp (Ω, Σ, μ) with 1 p ∞ are order dense ideals of L0 (Ω, Σ, μ). An ideal space (of measurable functions) is an order ideal X of the vector lattice L0 (Ω, Σ, μ), so that X is a Dedekind complete vector lattice. We will assume that L∞ (Ω, Σ,. μ) ⊂ X ⊂ L1 (Ω, Σ, μ). If ϕ : L1 (Ω, Σ, μ) → R is deﬁned as ϕ(x) := Ω x dμ then ϕ is an order continuous functional with the Levi property and, according to the above notation, Lp (ϕ) = Lp (Ω, Σ, μ) for 1 p ∞. 5.4.2. Let Φ : L1 (Φ) → Y be a Maharam operator, and let X0 be an order closed sublattice of L1 (Φ) with X ⊥⊥ = X0⊥⊥ . Put A := Orth(YΦ ) and Φ0 := Φ|X0 . Then the following are equivalent: (1) Φ0 has the Maharam property. (2) A band projection in L1 (Φ) is Φ-saturated if and only if its restriction onto X0 is a Φ0 -saturated band projection in X0 . (3) X0 is invariant under each Φ-saturated projection in L1 (Φ). (4) X0 is A-submodule and Φ0 is A-linear with respect to the f -module structure over A on X induced by Φ. 290 Chapter 5. Order Continuous Operators ⊳ There is no loss of generality in assuming Y = YΦ and X = CΦ . Note that the relation X ⊥⊥ = X0⊥⊥ implies Φ(X)⊥⊥ = Φ(X0 )⊥⊥ . Indeed, if L := Φ(X0 )⊥ = {0} then, by Theorem 5.2.3, h(L) ⊥ X0 and h(L) = {0} contradicting X0⊥ = {0}. Denote by h and h0 respectively the Boolean isomorphisms from P(Y ) onto the Boolean algebras of Φsaturated projections in L1 (Φ) and Φ0 -saturated projections in X0 existing by Theorem 5.2.3 and 5.2.7. It is easily seen that h0 (π) h(π)|X0 for all π ∈ P(Y ). At the same time Φ0 h0 (π) = πΦ0 = (πΦ)|X0 = (Φh(π))|X0 and so Φ(h(π)x − h0 (π)x) = 0 for every 0 x ∈ X0 . Since Φ is strictly positive, we conclude that h(π)x = h0 (π)x. It follows that h0 (π) = h(π)|X0 , so that the restriction of each Φ-saturated projection onto X0 is a Φ0 -saturated projection in X0 . The converse follows from the fact that a band projection in X0 has the unique extension to a band projection in L1 (Φ). Note that (2) =⇒ (3) is trivial, while (3) =⇒ (4) and (4) =⇒ (1) can easily be deduced by the argument similar to that in 5.2.5. ⊲ 5.4.3. Theorem. Let (Ω, Σ, μ) be a probability space and let X0 be a norm closed vector sublattice in L1 (Ω, Σ, μ) containing 1Ω . Then there exists a unique σ-subalgebra Σ0 of Σ such that X0 = L1 (Ω, Σ0 , μ0 ) with μ0 = μ|X0 . ⊳ See, for example, Douglas [110, Lemma 1]. ⊲ 5.4.4. Theorem. Let Φ : L1 (Φ) → Y be a strictly positive Maharam operator with Y = YΦ and let Z0 be an order closed sublattice in L0 (Φ). If 1 ∈ X0 := L1 (Φ) ∩ Z0 and the restriction Φ0 := Φ|X0 has the Maharam property then X0 = L1 (Φ0 ) and there exists an operator E(·|Z0 ) from L1 (Φ) onto L1 (Φ0 ) such that the following hold: (1) E(·|Z0 ) is an order continuous positive linear projection. (2) E(h(π)x|Z0 ) = h(π)E(x|Z0 ) for all π ∈ PΦ (X) and x ∈ L1 (Φ); i.e., E(·|Z0 ) commutes with all Φ-saturated projections. (3) Φ(xy) = Φ(yE(x|Z0 )) for all x ∈ L1 (Φ) and y ∈ L∞ (Φ0 ). (4) Φ0 (|E(x|Z0 )|) Φ(|x|) for all x ∈ L1 (Φ). (5) E(vE(x|Z0 )|Z0 ) = E(v|Z0 )E(x|Z0 ) for all x ∈ L1 (Φ) and v ∈ L (Φ); i.e., E(·|Z0 ) satisfies the averaging identity. ∞ ⊳ Put B := P(Y ). In view of Theorem 5.2.8 we can assume that Y = R↓, Φ = ϕ↓, L0 (Φ) = L0 (ϕ)↓, and L1 (Φ) = L1 (ϕ)↓ for some strictly positive order continuous functional ϕ : L1 (φ) → R in V(B) . Moreover, there exists a Boolean isomorphism χ from B onto PΦ (L1 (Φ)) 5.4. Conditional Expectation Type Operators 291 such that the relations b [[x y]] and χ(b)x χ(b)y are equivalent for all b ∈ B and x, y ∈ L1 (Φ). Say that a band projection π ∈ P(L0 (Φ)) is Φ-saturated whenever so is its restriction to L1 (Φ). By hypothesis, Z0 is a universally complete vector lattice. Moreover, Z0 is invariant under each Φ-saturated projection in L0 (Φ) because so is X0 . This two properties of Z0 amount to saying that χ(b)(Z0 ) ⊂ Z0 for all b ∈ B and o- ξ χ(bξ )zξ ∈ Z0 for every family (zξ ) in Z0 and every partition of unity (bξ ) in B. It follows that Z0 := Z0 ↑ is an internal order closed sublattice of L0 (ϕ) with 1 ∈ X0 := L1 (ϕ)∩Z0 and Z0 = Z0 ↓. Of course, X0 is an order closed sublattice of L1 (ϕ) and X0 ↓ = (L1 (ϕ) ∩ Z0 )↓ = L1 (ϕ)↓ ∩ Z0 ↓ = L1 (Φ) ∩ Z0 = X0 . In view of the Kakutani Representation Theorem we can assume further that L1 (ϕ) = L1 (Ω, Σ, μ) for some probability space (Ω, Σ, μ). By Theorem 5.4.3 X0 = L1 (Ω, Σ0 , μ0 ) = L1 .(ϕ0 ) for some σ-subalgebra Σ0 of Σ, where μ0 := μ|Σ0 and ϕ0 (x) = Ω x(ω) dμ0 (ω) for all x ∈ L1 (Ω, Σ, μ). In particular, Z0 = L0 (Ω, Σ, μ), ϕ0 = ϕ|X0 , and Φ0 = ϕ0 ↓. According to the classical Radon–Nikodým Theorem there exists the conditional expectation operator E (·|Σ0 ) with respect to Σ0 acting from L1 (ϕ) onto L1 (ϕ0 ). Denote by E(·|Z0 ) the descent of the internal conditional expectation operator E (·|Σ0 ). The required conditions 5.4.4 (1–5) can be obtained by interpretation of the elementary properties of the conditional expectation operator within V(B) . ⊲ 5.4.5. We will call the operator E(·|Z0 ) which is deﬁned by Theorem 5.4.4 the conditional expectation operator with respect to Z0 . Say that the sublattice Z0 in L0 (Φ) is Φ-ample whenever the conditions in Theorem 5.4.4 which ensures the existence of the conditional expectation operator E(·|Z0 ) with respect to Z0 are fulﬁlled. It follows from 5.4.2 that Z0 is Φ-ample if and only if Z0 is order closed in L0 (Φ), 1 ∈ L1 (Φ) ∩ Z0 , and Z0 ∩ L1 (Φ) is a submodule of L1 (Φ) with respect to the f -module structure over Z (Y ) on L1 (Φ) induced by Φ. Take w ∈ X ′ and observe that E(wx|Z0 ) ∈ L1 (Φ0 ) is well deﬁned for all x ∈ X. If additionally E(wx|Z0 ) ∈ X for every x ∈ X then we can deﬁne the linear operator T : X → X by putting T x = E(wx|Z0 ) (x ∈ X) called a weighted conditional expectation operator. Clearly, T is order bounded and order continuous. Moreover, for all x ∈ X+ we have T + x = E (w+ x|Z0 ), T − x = E (w− x|Z0 ), |T |x = E (|w|x|Z0 ). 292 Chapter 5. Order Continuous Operators In particular, T is positive if and only if so is w. Putting x := wx and y := 1 in 5.4.4 (3), we get Φ(wx) = Φ(wx1) = Φ(E (wx|Z0 )) = Φ(T x) for all x ∈ X. Now, x can be chosen to be a component of 1 with wx = w+ or wx = w− , so that T = 0 implies Φ(w+ ) = 0 and Φ(w− ) = 0, since Φ is strictly positive. Thus w ∈ X ′ is uniquely determined by T . Say that T satisﬁes the averaging identity, if T (y · T x) = T y · T x for all x ∈ X and y ∈ L∞ (Φ). Let us give a characterization of weighted conditional expectation operators on x. We start with the case of an ideal function space. 5.4.6. Theorem. Let (Ω, Σ, μ) be a finite measure space and let X be an order ideal in L1 (Ω, Σ, μ) including L∞ (Ω, Σ, μ). For a linear operator T on X the following are equivalent: (1) T is order continuous, satisfies the averaging identity, and keeps invariant L∞ (Ω, Σ, μ). (2) There exist w ∈ X ′ and a σ-subalgebra Σ0 of Σ such that T x = E (wx|Σ0 ) for all x ∈ X . ⊳ See Dodds, Huijsmans, and de Pagter [105, Proposition 3.1]. ⊲ 5.4.7. Theorem. Let Φ : L1 (Φ) → Y be a strictly positive Maharam operator and let X be an order dense ideal in L1 (Φ) including L∞ (Φ). For a linear operator T on X the following are equivalent: (1) T is order continuous, satisfies the averaging identity, leaves invariant L∞ (Φ), and commutes with all Φ-saturated projections. (2) There exist w ∈ X ′ and a Φ-ample sublattice Z0 in L0 (Φ) such that T x = E(wx|Z0 ) for all x ∈ X. ⊳ (1) =⇒ (2): Just as in the proof of Theorem 5.4.4, we can assume that X = X ↓, where X is an order ideal in L1 (Ω, Σ, μ) including L∞ (Ω, Σ, μ) for some probability space (Ω, Σ, μ). By hypotheses T commutes with all projections χ(b) (b ∈ B) and so it is extensional. Therefore, T := T ↑ is an internal mapping in X and T = T ↓. Moreover, T is linear, order continuous, satisﬁes the averaging identity, and keeps invariant L∞ (Ω, Σ, μ). By Theorem 5.4.6 there exist w ∈ X ′ and a σsubalgebra Σ0 of Σ such that T x = E (wx|Σ0 ) for all x ∈ X . It remains to note that E(·|Z0 ) is the descent of E (·|Σ0 ). (2) =⇒ (1): If the claims of (2) are true, then the operator T is well deﬁned on X by T x = E(wx|Z0 ) (x ∈ X). The required properties of T follow easily from Theorem 5.4.4. ⊲ 5.4. Conditional Expectation Type Operators 293 5.4.8. A linear operator T : X → X is a strictly positive order continuous projection if and only if T can be written uniquely in the form T = T1 + T2 with T1 and T2 satisfying the conditions: (1) There exist a σ-subalgebra Σ0 of Σ and a unique pair of functions 0 w ∈ X ′ and 0 k ∈ L1 (Ω, Σ, μ) such that E (wk|Σ0 ) = E (k|Σ0 ) = [k](1), [w] = [k], T1 x = kE (wx|Σ0 ) (x ∈ X ). (2) T1 is a positive order continuous operator with T1 T2 = T2 , T2 T1 = 0, and CT2 = (IX − [k])(X ). ⊳ See Dodds, Huijsmans, and de Pagter [105, Proposition 3.8 and Corollary 3.9]. ⊲ 5.4.9. A a linear operator T : X → X is a strictly positive order continuous projection commuting with all Φ-saturated band projections if and only if T can be written uniquely in the form T = T1 + T2 with T1 and T2 satisfying the conditions: (1) There exist an order closed sublattice Z0 of Z and a unique pair of elements 0 w ∈ X ′ and 0 k ∈ L1 (Φ) such that E(wk|Z0 ) = E(k|Z0 ) = [k](1), [w] = [k], T x = kE(wx|Z0 ) (x ∈ X). (2) T2 is a positive order continuous operator on X commuting with all Φ-saturated band projections such that T1 T2 = T2 , T2 T1 = 0, and CT2 = (IX − [k])(X). In particular, T1 = T RT , T2 = T (IX − RT ), and RT2 ⊂ RT1 . ⊳ The proof runs along the lines of the proof of Theorems 5.4.4 and 5.4.7 with obvious modiﬁcations. Apply Theorems 5.4.8 to T := T ↑ within V(B) and ﬁnd T1 , T2 ∈ V(B) such that [[T1 , T2 : X → X ]] = [[T1 + T2 = T ]] = 1, [[5.4.8 (1)]] = 1, and [[5.4.8 (2)]] = 1. Now observe that the two last identities are equivalent to 5.4.9 (1) 5.4.9 (2), respectively. ⊲ 5.4.10. Theorem. Let T : X → X be an order continuous positive projection commuting with Φ-saturated band projection. Put π := [CT ], π1 := πRT , π2 := π(IX − RT ), π3 := IX − π, and Xj := πj (X), and let Tıj stands for the restriction of πı T to Xj (ı, j := 1, 2, 3). Then the following hold: (1) π1 , π2 , and π3 are pairwise disjoint Φ-saturated band projections on X with π1 + π2 + π3 = IX . 294 Chapter 5. Order Continuous Operators (2) Tıj is a positive order continuous operator from Xj to Xı , T11 and T12 are strictly positive, T2j = Tı3 = 0 (ı, j := 1, 2, 3), and 2 = T11 , T11 T12 = T12 , T31 T11 = T31 , T31 T12 = T32 . T11 (3) There exist an order closed sublattice Z0 of Z and a unique pair of elements 0 w ∈ X ′ and 0 k ∈ L1 (Φ) such that E(wk|Z0 ) = E(k|Z0 ) = π1 1, π1 = [w] = [k], T x = kE(wx|Z0 ) (x ∈ X). Conversely, given operators πı and Tıj (ı, j := 1, 2, 3) satisfying (1)–(3), the operator T : X → X defined as πı T |Xj = Tıj (ı, j := 1, 2, 3) is a positive order continuous projection on X commuting with all Φ-saturated band projections. ⊳ Clearly, X1 , X2 , and X3 are invariant with respect to Φ-saturated band projections because π1 , π2 , and π3 are Φ-saturated. By deﬁnition π2 T = 0 and T π3 = 0, so T2j = Tı3 = 0 for all ı, j := 1, 2, 3. Note that T π = T and π1 T = πT , and so T31 T11 = π3 T π1 T |X1 = π3 T πT |X1 = T31 . Similarly, T31 T12 = T32 . The operator πT is a positive order continuous projection on X, as (πT )2 = π(T π)T = πT . Denote by T̃ the restriction of πT onto X0 := X3⊥ = π(X). If T̃ x = 0 for some 0 x ∈ X0 then 0 = T (T̃ x) = (T π)(T x) = T 2 x = T x and so x = 0, since T is the strictly positive on X0 . It follows that T̃ is strictly positive order continuous projection on X0 = X1 ⊕ X2 and by 5.4.9 T̃ is uniquely representable in the form T̃ = T̃1 + T̃2 with T̃1 and T̃2 satisfying 5.4.9 (1, 2). Observe now that π1 + π2 = π, RT̃ = πRT |X0 = π1 |X0 , and (IX0 − RT̃ ) = π(IX − RT )|X0 = π2 |X0 . It follows from this that T11 = π1 T˜1 |X1 and T12 = π1 T˜2 |X2 and so πı and Tıj obey (1)–(3). The converse is straightforward. ⊲ 5.5. Maharam Extension The main problem discussed in this section is the extension of an arbitrary positive operator to an order interval preserving order continuous operator; i.e., the Maharam extension. 5.5.1. Suppose that X is a vector lattice over a dense subfield F ⊂ R and ϕ : X → R is a strictly positive F-linear functional. There exist a Dedekind complete vector lattice X ϕ including X and a strictly 5.5. Maharam Extension 295 positive order continuous linear functional ϕ̄ : X ϕ → R having the Levi property and extending ϕ such that for every x ∈ X ϕ there is a sequence (xn ) in X with limn→∞ ϕ̄(|x − xn |) = 0. ⊳ Put d(x, y) := ϕ(|x − y|) and note that (X, d) is a metric space. Let X ϕ the completion of the metric space (X, d) and let ϕ̄ be the extension of ϕ to X ϕ by continuity. It is not diﬃcult to ensure that X ϕ is a Banach lattice having the additive norm &·&ϕ := ϕ̄(|·|) and including X as a norm dense F-linear sublattice. Thus, ϕ̄ is a strictly positive order continuous linear functional on X ϕ with the Levi property. ⊲ 5.5.2. Put L1 (ϕ) := X ϕ and let X̄ stand for the order ideal in L1 (ϕ) generated by X. Then (L1 (ϕ), &·&ϕ) is an AL-space and X̄ is a Dedekind complete vector lattice. Moreover, X is norm dense in L1 (ϕ) and so in X̄. For a nonempty subset U of a lattice L, we denote by U ↑ (respectively ↓ U ) the set of elements x ∈ L representable in the form x = sup(A) (x = inf(A)), where A is a nonempty upward (respectively downward) directed subset of U . Moreover, we put U ↑↓ := (U ↑ )↓ etc. If in the above deﬁnition A is countable, then we write U ↿ , U ⇃ , and U ↿⇃ instead of U ↑ , U ↓ , and U ↑↓ . Recall that for the Dedekind completion X δ we have X δ = X ↑ = X ↓ . 5.5.3. An element x̄ ∈ X̄ belongs to X ⇃↿ if and only if for all |x̄| y ∈ X and n ∈ N there exists un ∈ X ⇃ such that un x̄ and ϕ̄(x̄− un ) (1/n)ϕ(y). ⊳ Take x̄ ∈ X ⇃↿ and y ∈ X with |x̄| y and observe that the set A(x̄) := {u ∈ X ⇃ : u x̄, |u| y} is upward directed, since X ⇃ is a sublattice of X̄. Considering the identity x̄ = sup(A(x̄)) and order continuity of ϕ̄ we have ϕ̄(x̄) = sup ϕ(A(x̄)), so that for every n ∈ N there is un ∈ A(x̄) such that un x̄ and ϕ̄(x̄ − un ) (1/n)ϕ(y). Conversely, assume that for some x̄ ∈ X̄ we can choose a sequence (un ) in X ⇃ meeting the above conditions. Since X ⇃ is a sublattice of X̄, the sequence (un ) may be chosen increasing by replacing if need be un by u1 ∨ · · · ∨ un . Put u := supn∈N un and note that u x̄, ϕ̄(x̄ − u) = 0, and ϕ̄(x̄) = sup ϕ̄(un ) = ϕ̄(u). Since ϕ̄ is strictly positive, ϕ̄(x̄ − u) = 0 implies x̄ = u ∈ X ⇃↿ . ⊲ 5.5.4. X ⇃↿ is an order closed vector sublattice in X̄ and X ⇃↿ = X ↿⇃ . ⊳ We show ﬁrst that X ⇃↿ is closed under countable suprema and inﬁma. To this end note that (X ⇃↿ )↿ = X ⇃↿ holds trivially and we need only prove that (X ⇃↿ )⇃ = X ⇃↿ . Take z ∈ (X ⇃↿ )⇃ and pick y ∈ X with |z| y. For all 0 < ε ∈ R and n ∈ N we can choose vn ∈ X ⇃↿ with z vn 296 Chapter 5. Order Continuous Operators and ϕ̄(vn − z) (ε/2n )ϕ̄(y). By Proposition 5.5.3 for every n ∈ N there exists un ∈ X ⇃ such that un vn , |un | y, and ϕ̄(vn −un ) (ε/2n )ϕ̄(y). Put u := inf n∈N un and u′n := inf kn uk and observe that ∈ X ⇃ , |u| y, u n ′ ′ u z, u un un . Using the inequality |z − un | k=1 |z − un |, we deduce ϕ̄(|z − u′n |) n ϕ̄(|z − uk |) k=1 n ϕ̄(|z − vk |) + ϕ̄(|vk − uk |) 2εϕ̄(y). k=1 Considering that u = o-limn u′n we get ϕ̄(z − u) = lim ϕ̄(|z − u′n |) 2εϕ̄(y). n It follows from 5.5.3 that z ∈ X ⇃↿ and so (X ⇃↿ )⇃ ⊂ X ⇃↿ . Observe next that by the easy identities (A + B)⇃ = A⇃ + B ⇃ and (A + B)↿ = A↿ + B ↿ we have X ⇃↿ + X ⇃↿ = X ⇃↿ , X ⇃↿ − X ⇃↿ = X ⇃↿ + X ↿⇃ = X ⇃↿⇃ = X ⇃↿ . This shows that X ⇃↿ is a vector sublattice in X̄. ⊲ 5.5.5. The identities X̄ = X ⇃↿ = X ↿⇃ and L1 (ϕ) = X ⇃↿ = X ↿⇃ hold with both (·)⇃↿ and (·)↿⇃ taken in X̄ and L1 (ϕ), respectively. {x}⊥⊥ : x ∈ X+ . Therefore, given ⊳ Note that X̄ = X ⊥⊥ = 0 x̄ ∈ X̄, we can pick a disjoint family (uξ )ξ∈Ξ in X̄ and a family (xξ )ξ∈Ξ in X+ such that x̄ = supξ∈Ξ uξ and uξ ∈ {xξ }⊥⊥ for all ξ ∈ Ξ. Moreover uξ = 0 holds for at most countably many ξ, since ϕ̄(x̄) = ⊥⊥ ⊂ X ⇃↿ (with disjoint complements taken in X̄) ξ∈Ξ ϕ̄(uξ ). If {xξ } ⇃↿ then xξ ∈ X and so x̄ ∈ X ⇃↿↿ = X ⇃↿ . Consequently, by the Freudenthal Spectral Theorem and 5.5.4 it suﬃces to show that C(X̄, u0 ) ⊂ X ⇃↿ for all 0 u0 ∈ X, where C(X̄, u0 ) stands for the Boolean algebra of components of u0 in X̄. Assume now that u0 = u1 + u2 for some disjoint u1 , u2 ∈ C(X̄, u) and put ϕ̄ı := ϕ̄ ◦ [uı ] (ı = 0, 1, 2). Then ϕ̄1 and ϕ̄2 are disjoint components of ϕ̄0 . If ϕı stands for the restriction of ϕ̄ı onto X ⇃↿ , then ϕ0 is an order continuous functional by 5.5.4. Moreover, ϕ1 and ϕ2 are disjoint components of ϕ0 , so that by the Nakano Theorem u0 = v1 + v2 for some 5.5. Maharam Extension 297 disjoint v1 , v2 ∈ C(X ⇃↿ , u) with ϕ1 (v2 ) = ϕ2 (v1 ) = 0 and ϕı strictly positive on {vı }⊥⊥ . From this we deduce that ϕ̄([u1 ]v2 ) = ϕ̄1 (u2 ∨ v2 ) = ϕ̄1 (u2 ) + ϕ1 (v2 ) − ϕ̄1 (u2 ∧ v2 ) = 0 and so [u1 ]v2 = 0 or u1 ⊥ v2 . Similarly, u2 ⊥ v2 and we obtain u1 = v1 and u2 = v2 . It follows that C(X̄, u0 ) = C(X ⇃↿ , u0 ) ⊂ X ⇃↿ and the proof is complete. ⊲ 5.5.6. Let X be a vector lattice within V(B) and ∅ = U ⊂ X . Then (U )↓ = (U ↓)⇃ and (U ↿ )↓ = (U ↓)↿ . ⇃ ⊳ The two required relations are handled similarly, so that we restrict demonstration to the second. For an arbitrary x ∈ X ↓ we have the equivalence within V(B) : x ∈ U ↿ ↔ (∃ σ : N∧ → U )(σ is increasing and x = sup(im(σ))) According to the maximum principle, [[x ∈ U ↿ ]] = 1 if and only if there exists σ ∈ V(B) with the properties [[σ : N∧ → U ]] = 1, [[σ is increasing]] = 1, and [[x = sup(im(σ))]] = 1. Putting s := σ↓ and using 1.5.9 and 1.6.8 we arrive at the assertion: x ∈ U ↿ ↓ if and only if there exists an increasing function s : N → U ↓ such that x = sup(im(s)). This gives the required result. ⊲ 5.5.7. Theorem. Let X and Y be vector lattices with Y Dedekind complete and T a positive linear operator from X to Y . There exist a Dedekind complete vector lattice X̄ and a strictly positive Maharam operator T̄ : X̄ → Y satisfying the conditions: (1) There exist a lattice homomorphism ι : X → X̄ and an f -algebra homomorphism θ : Z (Y ) → Z (X̄) such that αT x = T̄ (θ(α)ι(x)) (x ∈ X, α ∈ Z (Y )). (2) ι(X) is a majorizing sublattice in X̄ and θ(Z (Y )) is an o-closed sublattice and subring of Z (X̄). (3) The representation X̄ = (X ⊙ Z (Y ))↓↑ holds, n where X ⊙ Z (Y ) is a subspace of X̄ consisting of all finite sums k=1 θ(αk )ι(xk ) with x1 , . . . , xn ∈ X and α1 , . . . , αn ∈ Z (Y ). ⊳ Assume without loss of generality that T is strictly positive, since otherwise we can replace T by its restriction to the carrier CT . By Theorem 3.3.3, there exists a positive R∧ -linear functional τ : X ∧ → R 298 Chapter 5. Order Continuous Operators such that [[ T (x) = τ (x∧ ) ]] = 1 for all x ∈ X. It is easy to see that τ is strictly positive within V(B) : ∧ )τ (x) = 0 → x = 0]] [[(∀ x ∈ X+ [[τ (x∧ ) = 0 → x = 0]] = [[T (x) = 0 → x = 0]] = 1. = x∈X+ x∈X+ By 5.5.1 there exists a Dedekind complete vector lattice X := (X ∧ )τ including X ∧ and a strictly positive order continuous linear functional τ̂ : X → R with the Levi property extending τ . Moreover, X ∧ is dense in X with respect to norm &·&τ := τ̂ (|·|)). By Theorem 5.2.8 τ̂ ↓ : X ↓ → R↓ is a strictly positive Maharam operator with the Levi property and an R↓-module homomorphism. Moreover there exists a lattice isomorphism ι from X into X ↓ such that T = τ̂ ↓ ◦ ι = τ ↓. Denote by X̄ and T̄ the order ideal in X ↓ generated by ι(X) and the restriction of τ̂ ↓ onto X̄, respectively. Then im(T̄ ) ⊂ Y and T̄ : X̄ → Y is a strictly positive Maharam operator. The R↓-module structure on X ↓ induces an f -algebra homomorphism θ from R↓ into Orth(X ↓) such that θ(a)(x) = ax for all a ∈ R↓ and x ∈ X ↓. Identify Z (Y ) with the sublattice of corresponding multipliers in R↓ and denote the restriction of θ to Z (Y ) by the same symbol we get (1), since αT x = α(T̄ ◦ι)x = T̄ (θ(α)ι(x)) for all α ∈ Z (Y ) and x ∈ X. The assertion (2) follows from 2.11.9 and it remains to prove (3). Let X¯ stand for the order ideal in X generated by X ∧ . In view of 5.5.5 X¯ is a Dedekind complete vector lattice, X¯ = (X ∧ )⇃↿ = (X ∧ )↿⇃ , and from 5.5.6 we get X¯ ↓ = (mix(X))⇃↿ = (mix(X))↿⇃ = (mix(X))↓↑ = (mix(X))↑↓ . Denote by M the subset in X̄ consisting of the elements of the form of ξ∈Ξ θ(πξ )xξ , where (πξ )ξ∈Ξ is a partition of unity in P(Y ) and (xξ )ξ∈Ξ is an order bounded family in X. It is a routine exercise to check that X̄ = (M )⇃↿ = (M )↿⇃ = (M )↓↑ = (M )↑↓ . Let n M0 stand for the part of M consisting of the ﬁnite sums↓ πn ∈ P(Y ). Then M ⊂ M0 k=1 θ(πk )xk with pairwise disjoint π1 , . . . , (and, of course, M ⊂ M0↑ ). Indeed, if x̄ = ξ∈Ξ θ(πξ )xξ and |xξ | y 5.5. Maharam Extension 299 (ξ ∈ Ξ) for some y ∈ X+ , then x̄ = inf ξ∈Ξ {πξ xξ + π ⊥ y}. It follows therefore that x̄ = inf α∈A uα , where A is the collection of all ﬁnite subsets of Ξ and uα := ξ∈α θ(πξ )xξ ∈ M0 . Hence X̄ ⊃ M0↓↑ = (M0↓ )↓↑ ⊃ M ↓↑ = X̄. Consequently, X̄ = (X ⊙ Z (Y ))↓↑ , since M0 ⊂ X ⊙ Z (Y ). ⊲ 5.5.8. Theorem. If a strictly Maharam operator S̄ from a Dedekind complete vector lattice Z to Y , while lattice homomorphisms κ : X → Z and η : Z (Y ) → Z (Z) satisfy the conditions 5.5.7 (1–3) in place of T̄ , ι, and θ respectively, then there exists a lattice isomorphism h from X̄ onto an order closed sublattice in the order ideal of Z generated by κ(X) such that κ = h ◦ ι and T̄ = S̄ ◦ h. ⊳ Consider the bilinear operators B and D from X × Z (Y ) to X̄ and Z respectively deﬁned as B(x, α) := θ(α)ι(x) and D(x, α) := η(α)κ(x). Let B̄ and D̄ stand for the lattice homomorphisms from Z ⊗ Z (Y ) to Z̄ and Z, respectively, uniquely determined by B̄⊗ = B and D̄⊗ = D (cp. 3.2.6 (1)). Observe that B(x, α) > 0 and D(x, α) > 0 whenever 0 < x ∈ X and 0 < α ∈ Z (Y ), so that B̄ and D̄ are lattice isomorphisms of Z ⊗ Z (Y ) onto the vector sublattices in X̄ and Z generated by B(X × Z (Y )) and D(X × Z (Y )), respectively (cp. 3.2.7 (1)). Under the hypotheses in (4) we have T̄ (θ(α)ι(x)) = S̄(η(α)κ(x)) for all x ∈ X and α ∈ Z (Y ) and so T̄ ◦ B̄⊗ = S̄ ◦ D̄⊗. It follows from 3.2.6 (3) that T̄ ◦ B̄ = S̄ ◦ D̄. Deﬁne h : im(B̄) → im(D̄) as h := D̄ ◦ B̄ −1 and note that h a lattice isomorphism with T̄ (u) = S̄(h(u)) for all u ∈ im(B̄). Moreover, h is order continuous. Indeed, given a downward directed set A ⊂ im(B̄) with inf(A) = 0 in X̄, we have S̄(inf(h(A))) = inf(S̄(h(A))) = inf T̄ (A) = 0 and so inf h(A) = 0, since S̄ is strictly positive. Note also that h ◦ ι = κ, since denoting by I the unit element of the f -algebra Z (Y ) we have h(ι(x)) = h(B(x, I)) = (D̄ ◦ B̄ −1 ◦ B̄)(x ⊗ I) = D(x, I) = κ(x). Extend h from im(B̄) to X̄ = B(X ⊗ Z (Y ))↓↑ . If (uα ) is a downward directed set in B(X ⊗ Z (Y )), u = inf α uα , and there is x ∈ X with |uα | ιx for all α, then |h(uα )| κ(x) for all α and there exists inﬁmum of a downward directed net (h(uα )) in Z. Put h(u) := inf α h(uα ). Similarly, if (vα ) upward directed net in B(X ⊗ Z (Y ))↓ with v = supα vα , then we can deﬁne h(v) := supα h(uα ). The deﬁnition is sound and the relation S̄ ◦ h = T̄ holds because of the order continuity and strict positivity of T̄ and S̄. For the same reason, h is a lattice isomorphism of X̄ onto a sublattice in Z. 300 Chapter 5. Order Continuous Operators ¯ the order ideal of Z generated by κ(X) and ensure Denote by X̄ ¯ . Since κ = h ◦ ι implies that im(h) is an order closed sublattice in X̄ ¯ im(h) ⊂ X̄ , all we need to do is to check that im(h) contains suprema of ¯ . For such V all upward directed sets V ⊂ im(h) with v0 = sup(V ) ∈ X̄ we can choose x ∈ X+ with |v| κ(x) for all v ∈ V ∪ {v0 }. Note that for arbitrary u ∈ U := h−1 (V ) we have h(|u|) κ(x) = h(ι(x)) and so |u| ι(x). It follows that u0 := sup(U ) ∈ X̄ and h(u0 ) = sup(h(U )) = ¯. ⊲ sup(V ) = v0 ∈ X̄ 5.5.9. The pair (X̄, T̄ ) (or T̄ for short) is called a Maharam extension of T if it satisﬁes 5.5.7 (1–3). The pair (X̄, ι) is also called a Maharam extension space for T . Two Maharam extensions T1 and T2 of T with the respective Maharam extension spaces (X1 , ι1 ) and (X2 , ι2 ) are said to be isomorphic if there exists a lattice isomorphism h of X1 onto X2 such that T1 = T2 ◦ h and ι2 = h ◦ ι1 . Theorem 5.5.8 tells us that a Maharam extension is unique up to isomorphism. 5.5.10. Two simple additional remarks follow. (1) As was shown in the proof of Theorem 5.5.7, X̄ = (M )⇃↿ . It is evident from this that X̄ = (X ⊙Z (Y ))⇃↿ whenever Y is order separable. (Recall that a vector lattice is said to be order separable whenever every set in it having a supremum contains a ﬁnite or countable subset with the same supremum.) (2) Put W := {w1 − w2 : w1 , w2 ∈ (X ⊙ Z (Y ))↓ }. Clearly, W is a sublattice and a vector subspace of X̄. Moreover, W is a majorizing vector sublattice, since ι(X) ⊂ W and (w1 − w2 )+ = w1 ∨ w2 − w2 ∈ W for all w1 , w2 ∈ (X ⊙ Z (Y ))↓ . Observe also that W is an order dense in X̄. Indeed, if 0 < x̄ ∈ X̄ then there exists an upward directed set A ⊂ (X ⊙ Z (Y ))↓ such that x̄ = sup(A). Because of x̄ = sup{a+ : a ∈ A}, we can pick a ∈ A with 0 < a+ x̄. Thus, X̄ is the Dedekind δ completion of W ; i.e., X̄ = (X ⊙ Z (Y ))↓ − (X ⊙ Z (Y ))↓ . 5.6. Properties of Maharam Extension Now we discuss some additional structural properties of Maharam extension. In particular, description of the Boolean algebra of band projections in the Maharam extension space is presented. As an application, approximation of the Boolean algebra of components of a positive operator by elementary fragments is also given. 5.6. Properties of Maharam Extension 301 5.6.1. Let X and Y be vector lattices with Y Dedekind complete, T : X → Y a positive operator and (X̄, T̄ ) a Maharam extension of T . Consider a universal completion X̄ u of X̄ with a ﬁxed f -algebra structure. Let L1 (T ) be the greatest order dense ideal in X̄ u onto which T̄ can be extended by order continuity. In more detail, L1 (T ) := {x ∈ X̄ u : T̄ ([0, |x|] ∩ X̄) is order bounded in Y }, T̂ x := sup{T̄ u : u ∈ X̄, 0 u x} T̂ x = T̂ x+ − T̂ x− (x ∈ L1 (T )+ ), (x ∈ L1 (T )). Deﬁne the Y -valued norm · on L1 (T ) by u := T̂ (|u|). In terms of lattice normed spaces (L1 (T ), · ) is a Banach–Kantorovich lattice; see 5.8.4 below (cp. Kusraev [228, Chapter 2]). In particular, au = |a| u (a ∈ Z (Y ), u ∈ L1 (T ). 5.6.2. Let X and Y be vector lattices with Y Dedekind complete and T a positive linear operator from X to Y . Then there exist an AL-space L within V(B) and a lattice isomorphism h from L1 (T ) onto an order dense ideal in L ↓ such that the following hold: (1) [[The functional τ̂ : L → R defined as τ̂ (x) := &x+ & − &x− & (x ∈ L ) is order continuous and has the Levi property]] = 1. (2) [[(h(ι(X)))↑ is a norm dense R∧ -linear sublattice in L ]] = 1. (3) T̂ = τ̂ ↓ ◦ h, T = T̂ ◦ h ◦ ι, and · = & · &↓ ◦ h. ⊳ This fact can be extracted from the proof of Theorem 5.5.7. ⊲ 5.6.3. Theorem. For a positive T : X → Y the following hold: (1) L1 (T ) is an f -module over Z (Y ) and X̄ is its f -submodule. (2) T̂ : L1 (T ) → Y is a Maharam operator extending T̄ . (3) The sublattice ι(X) is dense in L1 (T ) in the sense that for each u ∈ L1 (T ) and 0 < ε ∈ R for each there exist a partition (πξ )ξ∈Ξ of [ u ] in P(Y ) and a family (xξ )ξ∈Ξ in X such that u− πξ ι(xξ ) ε u . ξ∈Ξ ⊳ Clearly, (1) and (2) follow from Theorem 5.2.8 and so all we have to show is (3). According to 5.6.2 we can assume that L1 (T ) ⊂ L ↓ and 302 Chapter 5. Order Continuous Operators h is the embedding. For an arbitrary u ∈ L1 (T ) we have [[u ∈ L ]] = 1. If u = 0 there is nothing more to prove, if not b = [[|u| > 0]] = 0. Moreover, passing from V(B) to a relative Boolean valued model V([0,b]) if necessary, we can assume b = 1. Interpreting the fact that (ι(X))↑ is norm dense in L within V(B) , we deduce 1 = [[(∀ 0 < ε ∈ R∧ )(∃ x ∈ ι(X)↑X ∧ )(&u − x& ε|u|)]] [[&u − ι(x)& ε∧ |u|]]. = 0<ε∈R x∈X It follows that for every 0 < ε ∈ R there exists a partition on unity (bξ )ξ∈Ξ in B and a family (xξ )ξ∈Ξ in X such that bξ [[&u − ι(xξ )& ε∧ |u|]] for all ξ ∈ Ξ. If πξ := χ(bξ ) then πξ u − ι(xξ ) ε πξ u = πξ u by 2.2.4 (G). Summing up over ξ ∈ Ξ yields the desired result. ⊲ 5.6.4. Theorem. For every operator S ∈ {T }⊥⊥ there exists a unique operator S̄ ∈ {T̄ }⊥⊥ such that S = S̄ ◦ ι. The mapping S → S̄ implements an isomorphism of the vector lattices {T }⊥⊥ and {T̄ }⊥⊥ . ⊳ Observe that the mapping R : S̄ → S̄ ◦ ι from {T̄ }⊥⊥ to L∼ (X, Y ) is linear and positive and sends the order ideal generated by T̄ into the order ideal generated by T . Moreover, im(R) ⊂ {T }⊥⊥, since S̄α ↑ S̄ implies S̄α ◦ ι ↑ S̄ ◦ ι for every increasing family (S̄α ) of positive operators in {T̄ }⊥⊥ . So, all we have to show is that every S ∈ {T }⊥⊥ admits the unique extension to S̄ ∈ {T̄ }⊥⊥ such that S = S̄ ◦ ι. There is no loss of generality in assuming that S is positive. Let S lie in the order ideal generated by T ; i.e., 0 S λT for some λ ∈ R. Then 0 S ◦ ι−1 λT̄ |ι(X) , so that by Theorem 3.1.8 there exists a positive extension S̄ of S ◦ ι−1 to X̄ such that 0 S̄ λT̄ . Clearly, S̄ ◦ ι = S and S̄ ∈ {T̄ }⊥⊥ . Take an increasing net (Sα ) of positive operators in the order ideal generated by T such that S := sup Sα ∈ {T }⊥⊥. On account of what was just proved there is a family (S̄α ) in {T̄ }⊥⊥ such that S̄α ◦ ι = Sα for all α. If z ∈ X̄ then |z| ι(x) for some x ∈ X and we may, therefore, estimate |S̄α z| S̄α (|z|) Sx. Thus it is possible to deﬁne some positive operator by putting S̄z := sup S̄α z (z ∈ X̄+ ). Obviously, S̄ = sup S̄α ∈ {T̄ }⊥⊥ and S̄ ◦ ι = S. It remains to show that for an operator S ∈ {T }⊥⊥ there exists at most one S̄ ∈ {T̄ }⊥⊥ with 5.6. Properties of Maharam Extension 303 S̄ ◦ι = S. Assume that S̄1 ◦ι = S = S̄2 ◦ι for some S̄1 , S̄2 ∈ {T̄ }⊥⊥ . Then S̄1 and S̄2 coincide on ι(X). By Theorem 5.2.5 S̄1 and S̄2 are Maharam operators and so they coincides on X ⊙ Z (Y ) due to Z (Y )-linearity and coincide on X̄ = (X ⊙ Z (Y ))↓↑ due to order continuity. ⊲ The following result is a variant of the Radon–Nikodým Theorem for positive operators. 5.6.5. Theorem. Let X and Y be vector lattices with Y Dedekind complete and let T be a positive linear operator from X to Y . For every operator S ∈ {T }⊥⊥ there is a unique element z = zT ∈ X̄ u satisfying Sx = T̂ (z · ı(x)) (x ∈ X). The mapping T → zT establishes a lattice isomorphism between the band {T }⊥⊥ and the order dense ideal in X̄ u defined by {z ∈ X̄ u : z · ı(X) ⊂ L1 (T )}. ⊳ The proof is immediate from 5.3.5 and 5.6.4. ⊲ 5.6.6. According to 5.6.4 and 5.3.6 the vector lattices X̄, L1 (T ), {T }⊥⊥, and {T̄ }⊥⊥ have isomorphic Boolean algebras of projections. Below we will give a detailed description for bases for X̄ and {T }⊥⊥. As usual, we denote by [ιx] the band projection in X̄ onto {ι(x)}⊥⊥ . Given an order ideal G in X and a positive operator T ∈ L∼ (X, Y ), denote by πG (T ) the least extension of T |G (cp. 3.1.9). Clearly, πG (T )x = sup{T (x ∧ g) : g ∈ G} for all x ∈ X+ . Put πe := πG whenever G is an order ideal generated by e ∈ X+ . The following representation for πe is straightforward: πe T x = supn T (ne ∧ x) (x ∈ E + , T ∈ L+ (E, F )), πe T x = πe T x+ − πe T x− πe T = πe T + − πe T − (x ∈ E, T ∈ L+ (E, F )), (T ∈ L∼ (E, F )). Denote by S (X̄) and S (T ) the sets of all projections in X̄ and the set of all components of T , respectively, representable as n k=1 ρk [ιxk ] and n k=1 ρk πxk (T ), 304 Chapter 5. Order Continuous Operators where x1 , . . . , xn ∈ X+ , ρ1 , . . . , ρn ∈ P(Y ), n ∈ N. Given a band K in X, denote by K the band projection in X̄ onto (ιK)⊥⊥ ; i.e., K := [ιK]. Put x := [ι({x}⊥⊥ )] and πx := π{x}⊥⊥ (x ∈ X). Let C (X̄) and C (T ) denote the sets of band projections in X̄ and components of T representable respectively as n ρk · xk and k=1 n k=1 ρk · πxk , where n ∈ N, ρ1 , . . . , ρn ∈ P(Y ), and x1 , . . . , xn ∈ X. In the case of a vector lattice X with the principal projection property we may consider one more set A (T ) consisting of the components of T representable as n ρk ◦ T ◦ [xk ] (ρ1 , . . . , ρn ∈ P(Y ), x1 , . . . , xn ∈ X), k=1 where n ∈ N and [xk ] is the band projection in X onto {xk }⊥⊥ . 5.6.7. For all x ∈ X+ and K ∈ B(X) the representations hold: (1) πx (T ) = T̄ ◦ [ιx] ◦ ι. (2) πK (T ) = T̄ ◦ K ◦ ι. (3) πx (T ) = T̄ ◦ x ◦ ι. ⊳ Indeed, using the order continuity of Φ̄, we deduce πx (T )y = sup{T (y ∧ nx) : n ∈ N} = sup{T̄ (ι(y) ∧ nι(x)) : n ∈ N} = T̄ (sup{ι(y) ∧ nι(x)) = T̄ ◦ [ιx](ι(y))). The proof of (2) is similar and (3) is a particular case of (2). ⊲ 5.6.8. Let W be a vector lattice with a weak order unit u and the principal projection property. If w ∈ W+ and w = inf(V ) for some V ⊂ W then + 1 [w] = . v− u n n∈N v∈V 5.6. Properties of Maharam Extension 305 ⊳ We may assume that W = R↓ with R ∈ V(B) and B := P(W ). Then V ↑ is a numerical set and w = inf(V ↑) within V(B) ; therefore, w = 0 ↔ 0 < w ↔ (∃ n ∈ N∧ )(∀ v ∈ V ↑)(v − (1/n) u)+ = 0. Calculating the Boolean truth values and considering 2.4.9 we deduce for traces (see 2.4.8) ew = [[w = 0]] = e(v−(1/n)u)+ . n∈N v∈V The claim follows from this formula, since the band projection [w] is represented in R↓ as multiplication by the trace ew , while multiplication is an order continuous lattice homomorphism. ⊲ 5.6.9. Theorem. The following are valid: (1) P(X̄) = S (X̄)↓↑ ; (2) P(X̄) = C (X̄)↑↓↑ . n ⊳ (1): Recall that M0 stands for the set of ﬁnite sums k=1 θ(πk )xk with pairwise disjoint π1 , . . . , πn ∈ P(Y ). By deﬁnition [ιy] ∈ S (X̄) for each y ∈ M0 . If 0 y ∈ M0↓ , then we can choose x ∈ X+ and V ⊂ M0 so that ιx v y for all v ∈ V and y = inf(V ). Applying 5.6.8 with w := y and u := ιx, we have + 1 [y] = v − ιx . n n∈N v∈V + Since yn,v = v − (1/n)ιx belongs to M0 , it follows that [yn,v ] ∈ S and [y] ∈ S ↓↑ . An arbitrary projection π ∈ P(X̄) has the representation π = sup{[y] : y ∈ X̄+ , πy = y}. Thus, taking 5.5.7 (3) into consideration ↑ we arrive at the desired containment π ∈ (S ↓↑ )↑ = S ↓↑ . (2): It suﬃces to show that [ιx] ∈ C ↑↓ for every x ∈ X+ . Then S (X̄) ⊂ C (X̄)↑↓ , so that ↓↑ = C (X̄)↑↓↑ ⊂ P(X̄). P(X̄) = S (X̄)↓↑ ⊂ C (X̄)↑↓ Thus, what we need is only to justify the representation: (nx − t)+ . [ιx] = t∈X+ n∈N 306 Chapter 5. Order Continuous Operators Put σt := n (nx − t)+ and σ = t σt . It is not diﬃcult to observe that σt [ιx] for all t ∈ X+ For an arbitrary projection ρ ∈ P(X̄) with ρ ∧ [ιx] = 0 put ρt := ρ ∧ [ιt] (t ∈ X+ ). Then ρt [ι(t − nx)+ ] (t − nx)+ for every n ∈ N. Since (t− nx)+ ∧ (nx − t)+ = 0 it follows ρt ∧ (nx − t)+ = 0 and ρt ∧ σt = n (ρt ∧ (nx − t)+ = 0. From this we obtain ρt ∧ σ = 0, ρ ∧ σ = sup ρt ∧ σ = 0. Putting ρ = [ιx]⊥ , we arrive at the desired inequalities [ιx] σ [ιx]. ⊲ 5.6.10. The following are valid: (1) C(T ) = S (T )↓↑ ; (2) C(T ) = C (T )↑↓↑ . If X has the principal projection property then (3) C(T ) = A (T )↑↓↑ . ⊳ This is immediate from 5.6.7 and 5.6.9. ⊲ 5.7. Banach Lattices and Banach f -Modules In this section we consider some interplay between the lattice norm and the f -module structure on a vector lattice. 5.7.1. A norm &·& on a vector lattice X is called monotone or a lattice norm if |x| |y| implies &x& &y& for all x, y ∈ X. A normed lattice is a vector lattice equipped with a monotone norm. A normed lattice complete with respect to the norm is called a Banach lattice. In a normed lattice X the lattice operations are continuous and the positive cone X+ is closed. Every two norms making a vector lattice a Banach lattice are equivalent. The norm dual X ′ of a normed lattice X is a Dedekind complete Banach lattice. Moreover, X ′ is an order ideal of X ∼ and X ′ = X ∼ ′ whenever X is a Banach lattice. For arbitrary x0 ∈ X+ and x′0 ∈ X+ we have &x′0 & = sup{x, x′0 : x ∈ X+ , &x& 1}, ′ , &x′ & 1}. &x0 & = sup{x, x′0 : x′ ∈ X+ 5.7.2. One of the important features of Banach lattice theory is the interplay between the norm and order. A Banach lattice X is said to have 5.7. Banach Lattices and Banach f -Modules 307 (1) an order continuous norm if limα &xα & = 0 for every decreasing net (xα ) with inf α xα = 0; (2) the Levi property or a Levi norm if supα xα exists in X for every increasing net (xα ) in X+ with &xα & 1 for all α; (3) the Fatou property or a Fatou norm if limα &xα & = &x& for for every increasing net (xα ) in X+ with supα xα = x; (4) property (P ) if there exists a contractive positive projection in X ′′ onto X. We will use also the expressions “X is an order continuous (Levi, Fatou) Banach lattice.” A Banach lattice with order continuous, Levi, or Fatou norm is also called order continuous, order semicontinuous, or monotonically complete, respectively. A Dedekind complete Banach lattice X with a separating order continuous dual has property (P ) if and only if X has the Levi and Fatou properties. A Banach lattice X is said to be a Kantorovich–Banach space (or brieﬂy a KB-space) whenever every increasing norm bounded sequence of X+ is norm convergent. This is equivalent to saying that X has an order continuous Levi norm. Let us list some properties of order continuous norms and KB-spaces. 5.7.3. Theorem. For an arbitrary Banach lattice X the following are equivalent: (1) The norm on X is order continuous. (2) X is Dedekind σ-complete and sequentially order continuous. (3) Every monotone order bounded sequence in X is convergent. (4) Every disjoint order bounded sequence in X+ is norm convergent to zero. (5) Each closed order ideal of X is a projection band. (6) The null ideal N (x′ ) is a band for every x′ ∈ X ′ . (7) The natural embedding X → X ′′ sends X onto an ideal of X ′′ . (8) All norm continuous linear functionals on X are order continuous. ⊳ The proof can be found in Aliprantis and Burkinshaw [28, Theorems 4.9, 4.14] and Meyer-Niberg [311, Theorem 2.4.2, Corollary 2.4.4]. ⊲ 5.7.4. Theorem. For a Banach lattice X the following hold: (1) X is a KB-space if and only if the natural embedding X → X ′′ sends X onto a band of X ′′ . 308 Chapter 5. Order Continuous Operators (2) X is reflexive if and only if X and X ′ are both KB-spaces. ⊳ See Aliprantis and Burkinshaw [28, Theorems 4.60 and 4.70] and Meyer-Niberg [311, Theorems 2.4.12 and 2.4.15]. ⊲ 5.7.5. Two classes of Banach lattices play a signiﬁcant role in Banach lattice theory. A Banach lattice X is said to be (1) an AL-space if &x + y& = &x& + &y& for all x, y ∈ X+ with x ∧ y = 0; (2) an AM -space if &x ∨ y& = max{&x&, &y&}) for all x, y ∈ X+ with x ∧ y = 0. An AM -space has a (strong order) unit u 0 if the order interval [−u, u] is the unit ball of X. Each AL-space is a KB-space and an AM -space has an order semicontinuous norm. A Banach lattice X is an AL-space (respectively AM space) if and only if X ′ is an AM -space (AL-space). A lattice isometry is a lattice isomorphism that is also an isometry. Banach lattices are lattice isometric if there exists a one-to-one lattice isometry between them. 5.7.6. Kakutani–Kreı̆ns Representation Theorem. An AM space is lattice isometric to a sublattice of C(Q) for some Hausdorﬀ compact topological space Q. Moreover, if an AM -space X has a strong order unit then X is lattice isometric to the whole of C(Q). ⊳ See Aliprantis and Burkinshaw [28, Thorem 4.29], Meyer-Niberg [311, Theorem 2.1.3], and Semadeni [363, Theorem 13.2.3]. ⊲ 5.7.7. Nakano–Stone Completeness Theorem. Let K be a Hausdorﬀ compact topological space. The vector lattice C(Q) is Dedekind complete if and only if Q is extremally disconnected (≡ the closure of every open set in K is open).3 ⊳ See Meyer-Niberg [311, Propositions 2.1.4 and 2.1.5] and Semadeni [363, Theorem 24.7.1]. ⊲ 5.7.8. Assume that a measure space (Ω, Σ, μ) is semi-ﬁnite, that is, if A ∈ Σ and μ(A) = ∞ then there exists B ∈ Σ with B ⊂ A and 0 < μ(A) < ∞. The vector lattice L0 (Ω, Σ, μ) (of μ-cosets) of μmeasurable functions on Ω is Dedekind complete if and only if (Ω, Σ, μ) is 3 An extremally disconnected Hausdorﬀ compact space is often referred to as Stonean; cp. 2.8.6. 5.7. Banach Lattices and Banach f -Modules 309 localizable. In this event Lp (Ω, Σ, μ) is also Dedekind complete. (A measure space (Ω, Σ, μ) is localizable or Maharam if it is semi-ﬁnite and, for every A ⊂ Σ, there is a B ∈ Σ such that (i) A \ B is negligible for every A ∈ A ; (ii) if C ∈ Σ and A\C is negligible for every A ∈ A , then B\C is negligible (cp. Fremlin [126]).) Observe that P(L0 (Ω, Σ, μ)) ≃ Σ/μ−1 (0). 5.7.9. Kakutani Representation Theorem. A Banach lattice is an AL-space if and only if it is lattice isometric to L1 (Ω, Σ, μ) for some localizable measure space (Ω, Σ, μ). ⊳ See Aliprantis and Burkinshaw [28, Theorem 4.27], Meyer-Niberg [311, Theorem 2.7.1], and Semadeni [363, §2.3]. ⊲ 5.7.10. Theorem. If X is a Banach lattice, then Orth(X) under the order unit norm is an AM -space with unit IX , the identity operator on X. In particular, Orth(X) = Z (X) and &T & = &T &∞ := inf{0 < λ ∈ R : |T | λIX } (T ∈ Z (X)). ⊳ See Aliprantis and Burkinshaw [28, Thorem 4.77] and Meyer-Niberg [311, Theorem 3.1.12]. ⊲ 5.7.11. A Banach f -module over an f -algebra A is a Banach lattice that is simultaneously an f -module over A. By Deﬁnition 2.11.1 and Theorem 5.7.10, X is a Banach f -module over an f -algebra A if and only if there exists an f -algebra homomorphism h : A → Z (X) such that ax = h(a)x for all a ∈ A and x ∈ X. Thus, A is considered as an f -subalgebra of Z (X) with the induced order unit norm &a& := &h(a)&∞ (a ∈ A). In particular, &ax& &a&&x& for all a ∈ A and x ∈ X. Given Banach f -modules X and Y over A, denote by L (X, Y ) and LA (X, Y ) respectively the spaces of all continuous linear and A-linear ∼ operators from X to Y and put Ln,A (X, Y ) := L (X, Y ) ∩ L∼ n,A (X, Y ). If Y is Dedekind complete then LA (X, Y ) and Ln,A (X, Y ) are bands in L∼ (X, Y ) and Banach f -modules over A. 5.7.12. We can produce Banach f -modules by distinguishing a complete Boolean algebra of M -projections in a Banach lattice. A band projection π in a Banach lattice X is called an M -projection if &x& = max{&πx&, &π ⊥ x&} for all x ∈ X, where π ⊥ := IX − π. The collection of all M -projections forms the subalgebra M(X) of the Boolean algebra of all band projections P(X) in X. It is easily seen by induction 310 that Chapter 5. Order Continuous Operators 3 n 3 3 3 3 3 πk x3 = max &πk x& 3 3 3 k:=1,...,n k=1 for x ∈ X and every ﬁnite partition of unity π1 , . . . , πn in M(X). An M -module over A is a Banach f -module over A satisfying &ax ∨ by& = max{&ax&, &by&} for all x, y ∈ X and a, b ∈ A with a ⊥ b. If A has the projection property then the f -algebra homomorphism in 5.7.11 maps P(A) into M(X); i.e., the multiplication by each π ∈ P(A) is an M -projection in X. Assume that X is a Banach lattice and B is a complete subalgebra of the complete Boolean algebra B(X) consisting of projection bands and denote by B the corresponding Boolean algebra of band projections. Let Λ := Λ(B) stand for a Dedekind complete AM -space with unit such that P(Λ) is isomorphic to B. A Boolean isomorphism h from P(Λ) onto B can be extended to a unital f -algebra isomorphism from Λ into Z (X). Thus h induces an f -module structure over Λ on X. 5.7.13. We will identify P(Λ) and B and write B ⊂ L(X). If (bξ )ξ∈Ξ is a partition of unity in B and (xξ )ξ∈Ξ is a family in X, then there is at most one element x ∈ X with bξ xξ = bξ x for all ξ ∈ Ξ. This element is called the mixture of (xξ ) by (bξ ) and is denoted by x = mixξ∈Ξ (bξ xξ ). Clearly, x = o- ξ∈Ξ bξ xξ . A Banach lattice X is said to be B-cyclic or B-complete if the mixture of every family in the unit ball U (X) of X by each partition of unity in B (with the same index set) exists in U (X). 5.8. Lattice Normed Spaces In this section we consider the structural properties of a vector space equipped with some norm taking values in a vector lattice. The most important peculiarities of such space are connected with the norm decomposability property. 5.8.1. Consider a vector space X and a real vector lattice Λ. A mapping · : X → Λ+ is a vector (Λ-valued ) norm if the following hold: (1) x = 0 ⇐⇒ x = 0 (x ∈ X); (2) λx = |λ| x (λ ∈ R, x ∈ X); (3) x + y x + y (x, y ∈ X). 5.8. Lattice Normed Spaces 311 A vector norm is called a decomposable norm or a Kantorovich norm if (4) given λ1 , λ2 ∈ Λ+ and x ∈ X with x = λ1 + λ2 , there exist x1 , x2 ∈ X such that x = x1 + x2 and xk = λk (k := 1, 2). If (4) is valid only for disjoint λ1 , λ2 ∈ Λ+ , then the norm is said to be disjointly decomposable or, in short, d-decomposable. In the case that X is a vector lattice, the vector norm is said to be monotone or a lattice norm whenever (5) |x| |y| =⇒ x y (x, y ∈ X). or in brief X is called a lattice normed space over A pair X, · Λ if · is a Λ-valued norm on a vector space X. If the norm · is decomposable then the space X is called decomposable as well. Put X := { x : x ∈ X}. 5.8.2. Say that the elements x, y ∈ X are norm disjoint and write x⊥ ⊥ y whenever x ∧ y = 0. A metric band in X is a subset of the form ⊥ M⊥ := {x ∈ X : (∀ y ∈ M ) x ⊥ ⊥ y} with ∅ = M ⊂ X. (1) If x, y ∈ X are norm disjoint, then x + y = x + y . ⊳ Indeed, the relations x ∧ y = 0 and x x + y + y imply x x+y + y ∧ x x+y ∧ x x+y . Similarly, y x + y ; therefore, x + y = x ∨ y x + y . ⊲ (2) A Boolean algebra of projections in a vector space X is a set P of commuting idempotent linear operators on X in which the Boolean operations have the following form: π ∧ ρ := π ◦ ρ = ρ ◦ π, π ∨ ρ = π + ρ − π ◦ ρ, π ∗ = IX − π (π, ρ ∈ P), and the zero and identity operators in X serve as the top and the bottom elements of the Boolean algebra P. If X is a normed space then we will assume additionally that P consists of contractive projections and speak of a Boolean algebra of projections in a normed space X. Suppose that P is isomorphic to a Boolean algebra B. In this event we will identify the Boolean algebras P and B, writing B ⊂ L(X). (3) Let B stand for the set of all metric bands ordered by inclusion. It is not diﬃcult to check that if every band of the vector lattice Λ contains the norm of some nonzero element, then B is a complete Boolean algebra 312 Chapter 5. Order Continuous Operators ⊥ with the mapping K → K ⊥ (K ∈ P) as Boolean complementation; ⊥ see [228, 2.1.2]. Decomposability of X implies that X = K ⊕ K ⊥ for all K ∈ P, so that B deﬁnes an isomorphic Boolean algebra of projections on X. 5.8.3. Suppose that X is a d-decomposable lattice normed space, ⊥⊥ Λ is a vector lattice with the projection property, and Λ := X . Then there exists a complete Boolean algebra P of projections in X and an isomorphism h from P(Λ) onto P such that b x = h(b)x b ∈ P(Λ), x ∈ X . Moreover, if X is a vector lattice and · is monotone and decomposable, then P is a complete subalgebra of the Boolean algebra P(X). ⊳ Given L ∈ B(Λ), we let by deﬁnition h(L) := x ∈ X : x ∈ L . Clearly, the mapping h : L → h(L) from B(Λ) to B preserves the intersection of every nonempty family of bands. Therefore, h preserves inﬁma, since in the algebras consideration under they coincide with intersections. Moreover, h {0} = {0} and h Λ) = X. Observe that ⊥ ⊥ h(L⊥ ) = h(L)⊥ for all L ∈ B(Λ). The inclusion h(L⊥ ) ⊂ h(L)⊥ is ⊥ ⊥ obvious. If 0 = x ∈ h(L) then x is disjoint from all the elements of the form y in L. At the same time, x ∈ / h(L⊥ ) implies that 0 < e x for some e ∈ L+ . Therefore, in the band {e}⊥⊥ there are no elements of ⊥⊥ the form y , which contradicts our assumption Λ := X . It follows from the d-decomposability assumption that X is the direct sum of K ⊥ and K ⊥ for every metric band K ∈ B. Thus, to each K ∈ B there corre⊥ sponds the projection πK in X along K ⊥ . Assign P := {πK : K ∈ B}. It is clear that P is a complete Boolean algebra of projections isomorphic to B. Denote by the same letter h the mapping sending a band projection ρ ∈ P(Λ) to πK ∈ P with K := h(ρΛ). Then h is an isomorphism of the Boolean algebras P(Λ) and P. By the deﬁnition of h, we have h(π)x ∈ K := h(πΛ); i.e., h(π)x ∈ πΛ. Thus, π ⊥ h(π)x = 0, or π h(π)x = h(π)x . Since h(π)x and h(π ⊥ )x are norm disjoint, by 5.8.2 (1) we have π x = π h(π)x + h(π ⊥ )x = π h(π)x . Consequently, π x = π h(π)x = h(π)x . Assume now that X is a vector lattice. From the monotonicity of the vector norm it is easily seen that x ⊥ ⊥ y implies x ⊥ y for all x, y ∈ Y , so that h(L) ⊥ h(L⊥ ) for every L ∈ B(Λ). Thus we have h(L⊥ ) ⊂ h(L)⊥ . To prove the converse inclusion, assume that x ⊥ h(L) and x ∈ / h(L⊥ ). 5.8. Lattice Normed Spaces 313 Then x ∈ / L⊥ and we can choose 0 < e ∈ L with e x . According to the decomposability of X there exist u, v ∈ X such that x = u + v, u = e, and v = x −e. Since u ∈ h(L) by deﬁnition of h, we have x ⊥ u and so |x| |v|. It follows that x v = x − e and we get a contradiction 0 < e 0. Thus, we have proved that h(L⊥ ) = h(L)⊥ . Replacing in this identity L by L⊥ yields h(L) = h(L⊥ )⊥ . Therefore, h(L) ∈ B(X) ⊥ and B ⊂ B(X). By the above we get h(L)⊥ = h(L⊥ ) = h(L)⊥ , so that Boolean complement in B is induced from B(X). Since in both algebras B(X) and B inﬁma coincide with set-theoretic intersections, we conclude that B is a complete subalgebra of B. ⊲ 5.8.4. Take some λ ∈ Λ+ . A sequence (xn ) in X is said to be λuniformly convergent to x ∈ X (respectively, λ-uniformly Cauchy) if for each 0 < ε ∈ R there exists n(ε) ∈ N such that x − xn ελ for all n(ε) n ∈ N (respectively, xn − xm ελ for all n(ε) n, m ∈ N). A sequence (xn ) in X is said to be Λ-uniformly convergent to x ∈ X (respectively, Λ-uniformly Cauchy) if there exists λ ∈ Λ+ such that (xn ) converges λ-uniformly to x ∈ X (respectively, is λ-uniformly Cauchy). Say that X is Λ-uniformly complete whenever every Λ-uniformly Cauchy sequence is uniformly convergent. A subset A ⊂ X is called norm order bounded if the set { x : x ∈ A} is order bounded in Λ. A lattice normed space X over Λ is called laterally complete whenever, given a partition of unity (bξ ) in P(Λ) and a norm order bounded family (xξ ) in X there exists x ∈ X such that bξ x = bξ xξ for all ξ ∈ Ξ. A lattice normed space X over a Dedekind complete vector lattice Λ is said to be a Banach–Kantorovich space if X is decomposable, Λ-uniformly complete, and laterally complete. 5.8.5. Let X be a decomposable uniformly Λ-complete lattice normed ⊥⊥ space over a vector lattice Λ with Λ = X and P is as in 5.8.3. Then X admits the structure of a faithful unital module over Z (Λ) such that the following hold: (1) The natural representation of Z (Λ) in X defines an isomorphism of P(Λ) and P from 5.8.3. (2) ax = |a| x for all a ∈ Z (Λ) and x ∈ X. If, in addition, X is a vector lattice with monotone norm, then (3) P is a complete subalgebra of the Boolean algebra P(X). (4) X is an f -module over Z (Λ). n ⊳ Let a ∈ A := Z (Λ) be a simple element; i.e., a = k=1 λk πk where λ1 , . . . , λn ∈ R and π1 , . . . , πn is a ﬁnite partition of unity in P(Λ). Then 314 Chapter 5. Order Continuous Operators we put ax := n k=1 λk h(πk )x. Considering 5.8.2 (1) and 5.8.3, we have ax = n λk h(πk )x = k=1 n |λk |πk x = a x . k=1 By the Freudenthal Spectral Theorem an arbitrary a ∈ A is the uniform limit of an increasing sequence of simple elements (an ) ⊂ A. The sequence (an x) ⊂ X is uniformly Λ-fundamental, since an x − am x = |an − am | x . Therefore, we can put ax := Λ- lim an x. Moreover, ax = Λ- lim an x = r-lim |an | x = a x . The remaining part of the proof is straightforward. ⊲ 5.8.6. Let Λ be a vector lattice and let X and Y be lattice normed spaces over Λ. A linear operator T is said to be order norm bounded if there exists an orthomorphism S ∈ Orth(Λ) such that T (x) S( x ) for all x ∈ X. Put A := Z (Λ). If X and Y are decomposable and uniformly Λ-complete, then an order norm bounded linear operator T : X → Y is A-linear with respect to the module structures on X and Y defined as in 5.8.5. In particular, T π = πT for all π ∈ P(Λ). ⊳ If T is order norm bounded then, in view of 5.8.3, for all x ∈ X and π ∈ P(Λ) we have πT (π ⊥ x) = π T (π ⊥ x) πS (π ⊥ x) = ππ ⊥ S( x ) = 0. This implies πT π ⊥ = 0 or πT = πT π ⊥ . Replacing π ⊥ by π in the latter identity yields T π = πT π ⊥ , so that T π = πT . Further, we argue as in 5.8.5 using the Freudenthal Spectral Theorem. ⊲ 5.8.7. Assume now that Λ is a Banach lattice and X is a lattice normed space over Λ. Then, the Λ-valued norm · enables us to deﬁne some mixed norm on X by putting |||x||| := & x & (x ∈ X). In this situation, the normed space (X, |||·|||) is called a space with mixed norm. In view of the inequality | x − y | x − y and monotonicity of the norm on Λ we have & x − y & |||x − y||| (x, y ∈ X), so that · is a norm continuous mapping from (X, |||·|||) into Λ. 5.8. Lattice Normed Spaces 315 A Banach space with mixed norm over Λ is a pair (X, · ) such that · is a vector norm on X with values in a Banach lattice Λ and X is a Λuniformly complete lattice normed. The following proposition justiﬁes this deﬁnition (see Kusraev [228, Proposition 7.1.2]). (1) Let Λ be a Banach lattice and let · be a Λ-valued norm on X. Then (X, |||·|||) is a Banach space if and only if the lattice normed space (X, · ) is Λ-uniformly complete. Combining (1) and 5.8.5 we obtain the following. (2) Let Λ be a Dedekind complete vector lattice and let X be a decomposable Banach space with mixed norm over Λ. Then X admits a structure of a faithful unital module over A := Z (Λ) such that |||ax||| &a&∞ |||x||| for all a ∈ A and x ∈ X. In particular, X is a Banach space with the Boolean algebra of projections P(Λ). 5.8.8. Let X be a Banach space and let B be a complete Boolean algebra of projections on X. Given a partition of unity (bξ )ξ∈Ξ in B and a family (xξ )ξ∈Ξ , we refer to x ∈ X satisfying the condition bξ x = bξ xξ for all ξ ∈ Ξ as a mixture of (xξ ) by (bξ ) and use the notation x := mixξ∈Ξ (bξ xξ ). The mixture is unique if (∀ ξ ∈ Ξ)bξ x = 0 implies x = 0. A Banach space X is said to be B-cyclic or mix-complete whenever, given a partition of unity (bξ ) in B and a norm bounded family (xξ ) in X, we can ﬁnd the unique element x ∈ X such that x = mixξ∈Ξ bξ xξ and &x& = sup{&bξ xξ & : ξ ∈ Ξ}. Clearly, this deﬁnition agrees with that in 5.7.13. 5.8.9. Let X and Y be Banach spaces with B ⊂ L (X) and B ⊂ L (Y ). An operator T : X → Y is called B-linear, if it is linear and commutes with all projections from B. Denote the set of all bounded B-linear operators from X into Y by LB (X, Y ). Clearly LB (X, Y ) is a B-cyclic Banach space whenever so is Y . A one-to-one B-linear operator is called a B-isomorphism and an isometric B-isomorphism is called a B-isometry. In the case of Banach lattices, a B-isometric lattice homomorphism is referred to as lattice B-isometry. The space X # := LB (X, Λ) is called the B-dual of X whenever Λ = Λ(B); see 5.7.12. 5.8.10. Let Λ = R⇓ be the bounded part of the universally complete vector lattice R↓; i.e., Λ is the order-dense ideal in R↓ generated by the unity 1 := 1∧ ∈ R ↓. Take a Banach space X within V(B) and put X ⇓ := {x ∈ X ↓ : x ∈ Λ}. Then X ⇓ is a Banach–Kantorovich space called the bounded descent of X . Since Λ is an order complete AM - 316 Chapter 5. Order Continuous Operators space with unity, X ⇓ is a Banach space with mixed norm over Λ, hence, a B-cyclic Banach space (cp. Kusraev [228, 7.3.3]). 5.8.11. Theorem. For a Banach space X and a complete Boolean algebra B the following are equivalent: (1) X is B-cyclic with respect to a complete Boolean algebra B of projections on X. (2) X is linearly isometric to a Banach space with mixed norm defined by a Banach–Kantorovich space over the unital Dedekind complete AM -space Λ = Λ(B). (3) X is linearly isometric to the restricted descent X ⇓ of a Banach space X within V(B) . ⊳ The proof can be extracted from Kusraev [228, Theorems 7.3.2, 7.3.3 (1), 8.3.1, and 8.3.2]. ⊲ 5.8.12. Theorem. The bounded descent of the Banach space L (X , Y ) and the B-cyclic Banach space LB (X, Y ) are isometrically B-isomorphic. Some isomorphism is given by sending a bounded Blinear operator T : X → Y to the T := T ↑ defined by the relations [[T : X → Y ]] = 1 and [[T (x) = T (x)]] = 1 (x ∈ X). In particular, X ∗ ⇓ and X # are isometrically B-isomorphic. ⊳ See Kusraev [228, Theorem 8.3.6]. ⊲ 5.9. Boolean Valued Banach Lattices In this section we discuss brieﬂy the question: What is the Boolean valued interpretation of Banach lattice theory? We restrict discussion only to some basic facts needed in the sequel. Some of the proofs can be extracted from Gordon [133, 134] but we will give independent proofs for the sake of completeness. 5.9.1. Theorem. The bounded descent of a Banach lattice within V(B) is a B-cyclic Banach lattice. Conversely, if X is a B-cyclic Banach lattice, then in V(B) there exists a Banach lattice X that is unique up to the isometric isomorphism and whose bounded descent is lattice Bisometric to X. Moreover, the mapping π → π⇓ is an isomorphism of the Boolean algebras M(X )↓ and M(X); in symbols, M(X )↓ ≃ M(X ⇓). ⊳ The Banach part of the claim follows from Theorem 5.8.11. Assume that X is a B-cyclic Banach lattice and put X+ := X↑. Given an 5.9. Boolean Valued Banach Lattices 317 extensional mapping f , we have f (A)↑ = f ↑(A↑) where A ⊂ dom(f ) (cp. 1.6.3 and 1.6.5). Applying this successively to the addition f : (x, y) → x + y (x, y ∈ X) with A := X+ × X+ and to the Λmultiplication f : (λ, x) → λx (λ ∈ Λ, x ∈ X) with A := Λ+ × X+ we ﬁnd [[X+ + X+ = X ]] = 1 and [[R+ · X+ = X+ ]] = 1; i.e., [[X+ is a convex cone]] = 1. Moreover, [[X+ is pointed]] = 1, since [[±x ∈ X+ and &x& 1]] = 1 imply ±x ∈ X+ ↓ ∩ X ⊂ X+ . Deﬁne the order on X as [[(∀ x, y ∈ X )(x y ↔ y − x ∈ X+ )]] = 1. By transfer (X , X+ ) is an ordered Banach space within V(B) . Moreover, for all x, y ∈ X the relations x y and [[x y]] = 1 are equivalent. Consider the sentence σ ≡ (∀ a ∈ {0, 1})(∀ x, y ∈ X ) (ax ay ↔ (a = 1∨x y)) which is a very simple ZF-theorem. By transfer [[σ]] = 1. Calculating the Boolean truth values for quantiﬁers we ﬁnd that this is equivalent to saying that [[ax ay]] = [[a = 1]]∗ ∨ [[x y]] for all a ∈ {0, 1}↓ and x, y ∈ X ↓. Using the Boolean isomorphism χ : B → {0, 1}↓, we can replace a ∈ {0, 1}↓ by χ(b) for b ∈ B and write b∗ ∨ [[x y]] = [[χ(b)x χ(b)y]]. Now it is easy to see that b [[x y]] ⇐⇒ χ(b)x χ(b)y (b ∈ B, x, y ∈ X ↓). The last relation allows us to treat the interplay between X and X . As an example we prove that X is a vector lattice; i.e., the sentence (∀ x ∈ X )(∃ y ∈ X )y = sup{x, −x} is true within V(B) . Using the rules for calculating Boolean truth values (see 1.6.2) and the maximum principle we have to prove that for every x ∈ X there exists y ∈ X for which [[y = sup{x, −x}]] = 1. Put y = |x| and note that [[±x y]] = 1. Thus, it remains to check that [[(∀ u ∈ X )(±x u → y u)]] = 1. Again by and 1.2.3 and 1.6.2 it is equivalent to the relation [[±x u]] [[y u]] (u ∈ X). If b = [[±x u]] then ±χ(b)x χ(b)u and χ(b)y χ(b)u. It follows that b [[y u]]. The Λ-valued · of X is the descent of the norm & · &X of X 3 3 norm and &x&X = 3 x 3∞ (x ∈ X). Therefore, & · &X is a lattice norm if and only if |x| |y| implies x y for all x, y ∈ X. Let & · &X be a lattice norm. If [[|x| |y|]] = 1 for some x, y ∈ X then |x| |y|. Now, if x y were false, there would be 3 π ∈3 B and3 0 < 3ε ∈ R with π x > π( y + ε1). Therefore, &πx&X = 3π x 3∞ 3π y 3∞ + ε > &πy&X , which contradicts the hypothesis. Thus, (X , X+ ) is a Banach lattice. Assume that π is an M -projection in X and Π is the restriction of π↓ to X. Then [[π ◦ π = π]] = 1, [[0 πx x (x ∈ X+ )]] = 1, and &x& = max{&πx&, &π ⊥ x&} (x ∈ X ). By 1.5.5 (1) and 1.5.6 π↓ = (π ◦ π)↓ = π↓ ◦ π↓ and hence Π = Π ◦ Π. Since [[πx = Πx]] = 1 (x ∈ X), we 318 Chapter 5. Order Continuous Operators have 0 Πx x for all x ∈ X. Finally, the relations [[&x& = ⊥ max &πx&, &π ⊥ x& (x ∈ X )]] = 1 and x = max 3 Πx , Π⊥ x 3 (x ∈ X) are equivalent, whence we deduce &x& = 3 Πx ∨ Π x 3∞ = max{&Πx&, &Π⊥ x&}. Thus, Π is an M -projection in X; i.e., Π ∈ M(X). Conclusions in the reverse direction are similar. The remaining details are obvious. ⊲ 5.9.2. The element X ∈ V(B) from Theorem 5.9.1 is said to be the Boolean valued representation of X. Let X and Y be the Boolean valued representations of B-cyclic Banach lattices X and Y , respectively. Let L (X , Y ) and L r (X , Y ) denote the elements in V(B) which represent the spaces of all bounded linear operators and regular operators from X into Y . 5.9.3. Corollary. Let X be the Boolean valued representation of a B-cyclic Banach lattice X. Then B = M(X) if and only if [[M(X ) = {0, IX }]] = 1. ⊳ This is immediate from Theorem 5.9.1, since B is the descent of the two-element Boolean algebra {0, IX } (see 1.8.1). ⊲ 5.9.4. Corollary. For a Banach lattice X and a complete Boolean algebra B the following are equivalent: (1) X is lattice isometric to the bounded descent of some Banach lattice X within V(B) . (2) X is lattice isometric to a Banach lattice with mixed norm defined by a Banach–Kantorovich lattice over a unital Dedekind complete AM space Λ = Λ(B). (3) X is B-cyclic relative to the complete Boolean algebra of M projections B. ⊳ See Theorems 5.8.11 and 5.9.1. ⊲ 5.9.5. Let X be a Banach space and B ⊂ L (X). A net (xα )α∈A in X is said to be B-convergent to x ∈ X if for every 0 < ε ∈ R there exists a partition of unity (πα )α∈A in B with &πα (x − xβ )& ε for all α, β ∈ A, β α. In this event x is called the B-limit of (xα ). Let BX0 stand for all x ∈ X representable as x := mixξ∈Ξ (bξ xξ ) with an arbitrary family (xξ ) in X0 and a partition of unity (bξ ) in B. A subset X0 ⊂ X is B-dense in X if every x ∈ X is the B-limit of some family in X0 . Equivalently, X0 is B-dense in X if BX0 is norm dense in X. Now take a B-cyclic Banach lattice X. A decreasing net (xα )α∈A in X is B-convergent to zero if for every 0 < ε ∈ R there exists a partition 5.9. Boolean Valued Banach Lattices 319 of unity (πα )α∈A in B such that &πα xα & ε for all α ∈ A. The norm on X is said to be B-continuous if every decreasing net (xα )α∈A in X with inf α xα = 0 is B-convergent to zero. If A = N in this deﬁnition, we say that the norm on X is σ-B-continuous. Write Xn# for the space of all norm bounded order continuous B-linear operators from X to Λ. 5.9.6. Theorem. Suppose that X is a B-cyclic Banach lattice and X ∈ V(B) is its Boolean valued representation. Then the following hold: (1) X is Dedekind complete ⇐⇒ [[X is Dedekind complete ]] = 1. (2) X is Fatou (Levi) ⇐⇒ [[X is Fatou (Levi) ]] = 1. (3) X is order B-continuous ⇐⇒ [[X is order continuous ]] = 1. (4) X is order B-continuous and Levi ⇐⇒ [[ X is a KB-space ]] = 1. (5) S ∈ Xn# ⇐⇒ [[ σ := S↑ ∈ Xn′ ]] = 1. ⊳ (1): Just as in the proof of Theorem 2.2.4 we can show that for A ⊂ X+ there exists a = sup(A) if and only if [[there exists sup(A↑)]] = 1 and in this case [[a = sup(A↑)]] = 1. Thus, the Dedekind completeness of X within V(B) implies that X is Dedekind complete. Conversely, suppose that X is Dedekind complete and take a set A ⊂ X+ bounded above by u ∈ X . There is no loss of generality in assuming that [[&u& 1]] = 1. Then A := A ↓ lies in X and, taking the cancelation rule A ↓↑ = A (see 1.6.6) into account, we get the following: there exists a = sup(A ↓) if and only if [[there exists sup(A )]] = 1 and in this case [[a = sup(A )]] = 1. (2): We may assume without loss of generality that the upward directed sets in the deﬁnitions of Fatou norm and Levi norm are taken from the unit balls B(X) and B(X ). Moreover, if A ⊂ X is upward directed then [[A↑ is upward directed]] = 1 and [[A ⊂ X is upward directed]] = 1 implies that A ↓ is upward directed. Finally, observe that B(X )↓ = x ∈ X ↓ : x 1 = B(X). Let X have a Levi norm and take an upward directed set A ⊂ B(X). It follows that a : a ∈ A ⊂ [−1, 1] and thus [[{&a& : a ∈ A↑} ⊂ [−1, 1] ]] = 1; i.e., [[A↑ ⊂ B(X )]] = 1. By hypothesis a = sup(A↑) exists in X , whence a = sup(A). The argument for the converse is similar. To ensure the claim concerning the Fatou norm it suﬃces 3 3to observe3that3 b = sup a : a ∈ A in Λ if and only if &b& = 3 b 3∞ = sup 3 a 3∞ : a ∈ A , since the AM -space Λ has a Levi norm. (3): Using the above remarks in (2) it is easy to see that [[ X has an order continuous norm ]] = 1 if and only if for every downward directed set 320 Chapter 5. Order Continuous Operators A ⊂ X+ with inf(A) = 0 we have inf a : a ∈ A = 0 in Λ. By Theorem 2.6.1 the latter property amounts to the following: for every ε > 0 there exists a partition of unity (πa )a∈A in B such that πa a = πa a < ε1 for all a ∈ A. Thus, we arrive at the desired result, since the relations πa a < ε1 and &πa a& < ε are equivalent. (4): This is immediate from (2) and (3). (5): By Theorem 5.8.12 S ∈ X # if and only if [[ σ := S↑ ∈ X ∗ ]] = 1. Moreover, S and σ are positive or not simultaneously. Thus, we can conﬁne demonstration to the case of S positive. Observe also that if [[A ⊂ X+ ]] = 1 and A = A ↓ then S(A) = σ(A )↓ by 1.5.5 (1) and 1.5.6 and if A ⊂ X+ and A = A↑ then [[σ(A) = S(A)↑]] = 1 by 1.6.3 and 1.6.5. Use the same argument as in (1), but with inﬁmum instead of supremum. We see that if inf(A) = 0 and S is order continuous then [[inf σ(A ) = 0]] = 1 and if [[inf(A ) = 0 and σ is order continuous]] = 1 then inf S(A) = 0. ⊲ 5.9.7. Corollary. For every B-cyclic Banach lattice X the following are equivalent: (1) The norm on X is B-continuous. (2) X is order σ-complete and the norm on X is σ-B-continuous. (3) Every monotone order bounded sequence in X is B-convergent. (4) Every disjoint order bounded sequence in X+ is B-convergent to zero. (5) Every norm closed B-complete order ideal of X is a band. (6) The null ideal N (x# ) is a band for every x# ∈ X # . (7) Every norm continuous B-linear operator from X to Λ is order continuous; i.e., X # = Xn# . (8) The natural embedding of X into X ## sends X onto an order ideal of X ##. ⊳ This is proved by interpreting Theorem 5.7.3 within V(B) and making use of Theorem 5.9.6. For example, the equivalence (1) ⇐⇒ (8) of Theorem 5.7.3 together with Theorem 5.9.6 (3) implies that X is Bcontinuous if and only if [[X ′ = Xn′ ]] = 1. To ensure that the latter is equivalent to 5.9.7 (8), it is suﬃcient to observe that the B-cyclic Banach lattices Xn# and Xn′ ⇓ are lattice B-isometric. The natural embedding x → x̂ of X into X ## := (X # )# is deﬁned by putting x̂(T ) = T x for all T ∈ X # . The equivalence (1) ⇐⇒ (7) of Theorem 5.7.3 together with Theorem 5.9.6 (3) shows 5.10. Injective Banach Lattices 321 that X is B-continuous if and only if [[the natural embedding sends X onto an order ideal of X ′′ ]] = 1. This is equivalent to saying that the natural embedding sends X onto an order ideal of X ## , since X ## and X ′′ ⇓ are lattice B-isometric. ⊲ 5.9.8. Theorem. For a B-cyclic Banach lattice X the following hold: (1) The natural embedding of X into X ## sends X onto a band of ## X if and only if X has a B-continuous Levi norm. (2) The natural embedding of X into X ## sends X onto X ## if and only if X and X # have both B-continuous Levi norms. ⊳ Interpret Theorem 5.7.4 in V(B) making use of Theorem 5.9.6. ⊲ A B-cyclic Banach lattice X is said to be B-reflexive if X = X ## (or, more precisely, the natural embedding sends X onto X ## ). 5.9.9. Corollary. A B-cyclic Banach lattice X is B-reflexive if and only if X and X # have order B-continuous Levi norms. 5.10. Injective Banach Lattices In this section we present several analytical and geometric characterizations of injective Banach lattices. 5.10.1. A real Banach lattice X is said to be injective if, for every Banach lattice Z, every closed vector sublattice Y ⊂ Z, and every positive linear operator T : Y → X there exists a positive linear extension T̂ : Z → X with &T & = &T̂ &. This deﬁnition is illustrated by the commutative (T = T̂ ◦ ι) diagram: ? X _❄ ❄ ⑧⑧ ⑧ ❄ T̂ T ⑧⑧ ❄ ⑧ ⑧ ❄ ⑧⑧ ⑧ ❄ ⑧ /Z Y ι 5.10.2. We now state two elementary properties of injective Banach lattices which are immediate from the deﬁnition. (1) If X is an injective Banach lattice and a closed vector sublattice X0 ⊂ X is the range of a contractive positive projection P then X0 is an injective Banach lattice. ⊳ We need only take P T̂ in Deﬁnition 5.10.1 in case im(T ) ⊂ X0 . ⊲ 322 Chapter 5. Order Continuous Operators (2) If (Xα ) is a family of injective Banach lattices then their l∞ product (X, & · &∞ ) is also an injective Banach lattice (where X consists of all families x = (xα ) with xα ∈ Xα and &x&∞ := supα &xα & < ∞). ⊳ Let Pα : X → Xα stand for the natural projection x = (xα ) → xα . Then Pα is a contractive positive projection as X is equipped with the product order and Pα T : Y → Xα admits a positive extension T̂α : Z → Xα with &Pα T & = &T̂α &. Deﬁne T̂ : Z → X as T̂ x := (T̂α z) and note that T̂ is a positive extension of T and &T̂ & = sup sup &T̂α z& = sup sup &Pα T y& = &T &. ⊲ α z1 y1 α Next, we consider two important examples. 5.10.3. Theorem. A Dedekind complete AM -space with unit is an injective Banach lattice. ⊳ Let X be a Dedekind complete AM -space with unity 1, let Y0 ⊂ Y be a closed vector sublattice of a Banach lattice Y , and let T0 : Y0 → X be a positive linear operator. Deﬁne p : Y → X by putting p(y) := &T0 &&y + &1 (y ∈ Y ). Observe that p is a sublinear operator and T0 (y) = T0 (y + ) − T0 (y − ) T0 (y + ) &T0 &&y + &1 = p(y) (y ∈ Y0 ). By the Hahn–Banach–Kantorovich Theorem there exists a linear extension T : Y → X of T0 such that T y p(y) for all y ∈ Y . Evidently T is positive, since −T (y) = T (−y) p(−y) = &T0 &&(−y)+ & = 0 whenever y 0 and &T & &T0 & because of |T y| T (|y|) p(|y|) = &T0 &&y&1 for all y ∈ Y . ⊲ 5.10.4. Theorem. Each AL-space is an injective Banach lattice. ⊳ Slightly diﬀerent proofs can be found in Lotz [288, Proposition 3.2], Haydon [169, Proposition 2A], Meyer-Niberg [311, Theorem 3.2.5], and Schaefer [357, Theorem 4.2]. ⊲ 5.10.5. Each Banach lattice L is lattice isometric to a closed vector sublattice of an injective Banach lattice. ⊳ Given α ∈ L′ , put Iα := {x ∈ L : |x|, α = 0}, and equip the quotient vector lattice L/Iα with the norm &x̃α &α := |x|, α where x̃α := x + Iα is a coset of x ∈ L. This norm is additive on the positive cone, the completion Xα of (L/Iα , &·&α ) is an AL-space. The l∞ -product X of the family {Xα : 0 α ∈ L′ , &α& 1} is an injective Banach 5.10. Injective Banach Lattices 323 lattice by 5.10.2 (2). It remains to observe that the mapping x → (x̃α ) is a lattice isometry from L into X. ⊲ 5.10.6. Theorem. For a Banach lattice X the following are equivalent: (1) X is injective. (2) If X is lattice isometrically embedded into a Banach lattice Y and T0 is a positive linear operator from X to a Banach lattice Z then there exists a positive linear extension T : Y → Z with &T0 & = &T &. (3) If X is lattice isometrically embedded into a Banach lattice Y then there exists a contractive positive projection from Y onto X. ⊳ (1) =⇒ (3) and (2) =⇒ (3): To ensure that (3) is a special case of both (1) and (2), we need only to take Y := X and Z := Y in (1), Z := X in (2), and T0 the identity operator in both cases. (3) =⇒ (1): By 5.10.5 we can assume that X is a closed vector sublattice of an injective Banach lattice, say L. It follows that a positive linear operator T0 from a closed vector sublattice Y of a Banach lattice Z to X ⊂ L admits a positive linear extension T̄ : Z → L with &T̄ & = &T0 &. By (3) there exists a contractive positive projection P from L onto X. The operator T := P ◦ T̄ : Z → X has the desired properties. (3) =⇒ (2): If Y , Z, and T0 are given as in (2) then by (3) there exists a contractive positive projection P from Y onto X and the operator T := T0 ◦ P is the desired extension. ⊲ 5.10.7. Corollary. An injective Banach lattice is Dedekind complete and has the Fatou and Levi properties. ⊳ For every Banach lattice X the natural embedding κ : X → X ′′ is a lattice isometry and κ(X) is a closed sublattice in X ′′ . If X is injective, then there exists a positive contractive projection from X ′′ onto κ(X); see Theorem 5.10.6 (3). Given an order or norm bounded set U in X, there exists y := sup κ(U ) in X ′′ , since X ′′ is Dedekind complete and has the Levi property. Moreover, the identities x := κ −1 (P y) = sup(U ) and &x& = supu∈U &u& evidently are true in X because X ′′ has the Fatou property too. ⊲ 5.10.8. Corollary. The Banach lattice of continuous function C(K) on a Hausdorﬀ compact topological space K is injective if and only if K is extremally disconnected. ⊳ This is immediate from 5.10.3 and 5.10.7 on using the Kakutani– 324 Chapter 5. Order Continuous Operators Kreı̆ns Representation Theorem and the Nakano–Stone Completeness Theorem. ⊲ 5.10.9. (1) A Banach lattice X has the Cartwright property if, given x1 , x2 , y ∈ X+ with &x1 & 1, &x2 & 1, and &x1 + x2 + y& 2, there exist y1 , y2 ∈ X+ such that y1 +y2 = y, &x1 +y1 & 1, and &x2 +y2 & 1. (2) A Banach lattice X has the splitting property if, given x1 , x2 , y ∈ X+ and 0 < r1 , r2 ∈ R with &x1 & r1 , &x2 & r2 , and &x1 + x2 + y& r1 + r2 , there exist y1 , y2 ∈ X+ such that y1 + y2 = y, &x1 + y1 & r1 , and &x2 + y2 & r2 . (3) A Banach lattice X has the finite order intersection property if, given z ∈ X+ and ﬁnite collections x1 , . . . , xn ∈ X+ , y1 , . . . , ym ∈ X+ and strictly positive reals r1 , . . . , rn ∈ R+ , s1 , . . . , sm ∈ R+ such that &xı & rı , &yj & sj , and &xı + yj + z& rı + sj for all ı := 1, . . . , n and j := 1, . . . , m, there exist u, v ∈ X+ with z = u + v, &xı + u& rı , and &yj + v& sj for all ı := 1, . . . , n and j := 1, . . . , m. 5.10.10. Theorem. A Banach lattice has the Cartwright property if and only if it has the splitting property if and only if it has the finite order intersection property. ⊳ See Cartwright [85, Theorem 2.9]. ⊲ 5.10.11. Theorem. A Banach lattice has the Cartwright property if and only if its bidual is injective. A Banach lattice is injective if and only if it has the Cartwright property and property (P ). ⊳ See Cartwright [85, Theorem 3.6 and Corollary 3.8]. ⊲ 5.10.12. Theorem. A Banach lattice is injective if and only if it has the Cartwright, Fatou and Levi properties. ⊳ See Haydon [169, Theorem 3.6 and Corollary 5.D]. ⊲ 5.11. Injectives: M -Structure In this section we will demonstrate that injective Banach lattices carry M -structure in addition to their structure as Banach lattices, which determines important peculiar properties. We start with some elementary facts concerning M -projections. 5.11.1. For a projection π in a Banach space X the following are equivalent (with π ⊥ := IX − π) : (1) &x& = max{&πx&, &π ⊥ x&} (x ∈ X). (2) &πu + π ⊥ v& = max{&πu&, &π ⊥v&} (u, v ∈ X). 325 5.11. Injectives: M -Structure (3) &πu + π ⊥ v& max{&u&, &v&} (u, v ∈ X). ⊳ The equivalence (1) ⇐⇒ (2) is immediate: Putting x := πu + π ⊥ v in (1) yields (2) and, conversely, (1) is the particular case of (2) with x = u = v. It is easily seen from (1) that π and π ⊥ are contractive, which shows that (2) =⇒ (3). For the implication (3) =⇒ (1) observe that taking u := πx and v := π ⊥ y in (3) yields &x& max{&πx&, &π ⊥ x&} and the reverse inequality is also true, since π and π ⊥ are evidently contractive under the assumption (3). ⊲ 5.11.2. Theorem. Assume that a Banach lattice X has the Fatou and Levi properties. Then M(X) is an order closed subalgebra of the complete Boolean algebra P(X). In particular, a Banach lattice having the Fatou and Levi properties is B-cyclic with B := M(X). ⊳ It is immediate from 5.7.12 that π and π ⊥ are M -projections or not simultaneously. If π and ρ are M -projections then, from 5.11.1 (1, 2) we deduce &x& = max{&πx&, &π ⊥ x&} = max{max{&ρπx&, &ρ⊥ πx&}, &π ⊥ x&} = max{&ρπx&, max{&π(x − ρx)&, &π ⊥ (x)&}} = max{&ρπx&, &(I − ρπ)x&}, so that πρ is an M -projection. It follows easily by induction that &x& = sup{&πα x& : α ∈ A} for every ﬁnite partition of unity (πα )α∈A in M(X) and for all x ∈ X. Observe that the last identity is true for an arbitrary partition of unity (πα )α∈A provided X has the Fatou property. Indeed, if Θ stands for the collection of all ﬁnite subsets of A and ρθ := supα∈θ πα then the family (ρθ |x|)θ∈Θ is upward directed with |x| = supθ∈Θ ρθ |x| and taking the Fatou property into account we deduce &x& = sup &ρθ |x| & = sup sup &πα |x| & = sup &πα x&. θ∈Θ θ∈Θ α∈θ α∈A Assume now that π ∈ P(X) lies in the order closure of M(X) in P(X). Then there exist a partition of unity (πα )α∈A in M(X) and a subset A0 ⊂ A such that π = supα∈A0 πα and π ⊥ = supα∈A′0 πα with A′0 = A \ A0 . 326 Chapter 5. Order Continuous Operators From the above we get &x& = sup &πα x& = max α∈A sup &πα x&, sup &πα x& α∈A0 α∈A′0 = max{&πx&, &π ⊥ x&}. Thus π ∈ M(X) and M(X) is an order closed subalgebra of P(X). ⊲ 5.11.3. Let B be a band in a Banach lattice X. An element x ∈ X is called maximal in B if x is a maximal element of the set {y ∈ X : &y& = &x&} ∩ B. We say that x is maximal if B = X and relatively maximal if B = {x}⊥⊥ . Given u ∈ X+ , put Mu := {x ∈ X : &u + y& > &u& for all 0 < y |x|}. It is immediate from the deﬁnition that Mu is solid; i.e., x ∈ Mu and |y| |x| imply y ∈ Mu for all x, y ∈ X. In particular, 0 ∈ Mu and M0 = X. Also, it can easily be seen that u is maximal in B if and only if B = Mu . 5.11.4. Let a Banach lattice X have the Levi and Fatou properties. Given a band X0 , 0 < ε ∈ R ∪ {∞}, and x ∈ X0 , there exists a maximal element of the set Vε (x) := {y ∈ X0 : y x, &y& = &x&, &x − y& ε}. ⊳ It is an easy consequence of the Kuratowski–Zorn Lemma. We need only observe that a linearly ordered subset A ⊂ Vε (x) is norm bounded and has supremum ā = sup(Vε (x)) y in X by the Levi property, while the Fatou property implies &ā& = supa∈A &a& = &x& and &ā − y& = supa∈A &a − y& ε, so that ā ∈ Vε (x). ⊲ 5.11.5. If a Banach lattice X has the Cartwright property then &u + v1 + v2 & = &u& + &v1 + v2 & for all 0 v1 , v2 ∈ Mu . ⊳ Put s := &v1 + v2 &, t := &u&, and r := &v1 + v2 + u& − t. Note that r s and apply the Cartwright property with x1 := v1 + v2 , x2 := u, and y := (1 − (r/s))(v1 + v2 ). By 5.10.9 (2) there exist y1 , y2 ∈ X+ such that y1 + y2 = y, &u + y2 & = t, and &v1 + v2 + y2 & = s. If y2 = 0 then either x := y2 ∧v1 or x := y2 ∧v2 is nonzero and t = &u& &u+x& &u+y2 & = t. At the same time 0 < x v1 or 0 < x v2 and so x ∈ Mu , implying 5.11. Injectives: M -Structure 327 that &u + x& > &u& by deﬁnition. This contradiction shows that y2 = 0 and y1 = y. Hence, we arrive at the equation &v1 + v2 & = s = &v1 + v2 + y1 & = (1 + (1 − r/s))&v1 + v2 & which implies that r = s. ⊲ 5.11.6. If a Banach lattice X has the Cartwright property then Mu is a band and &u + |x| & = &u& + &x& for all x ∈ X. ⊳ It is an immediate consequence of 5.11.5 and the deﬁnition of Mu that Mu is an order ideal in X. Assume that x0 = sup(A) ∈ X for some upward directed set A ⊂ (Mu )+ . For an arbitrary 0 < y0 x0 choose x ∈ A such that 0 < x ∧ y0 =: y. Then 0 < y x ∈ Mu and by deﬁnition &y0 + u& &y + u& > &u&, so that x0 ∈ Mu and Mu is a band. ⊲ 5.11.7. Let X be a Banach lattice with the Cartwright, Levi, and Fatou properties and 0 < u ∈ X. Then the representation holds (with |x|/&x& = 0 for x := 0): 3 3 3 &u& 3 3 = &u& . Mu⊥ := x ∈ X : 3 u + |x| 3 &x& 3 ⊳ Without loss of generality we can assume that &u& = 1, since Mλu = Mu for all 0 < λ ∈ R. Now it suﬃces to prove that the element x ∈ X+ with &x& = 1 lies in Mu⊥ if and only if &u+x& = 1. If &u+x& = 1 and 0 x0 x for some x0 ∈ Mu then by 5.11.6 we have 1 &u + x0 & = &u& + &x0 & = 1 + &x0 &; it follows that x0 = 0 and x ⊥ Mu . Conversely, assume that 0 x ∈ Mu⊥ and &x& = 1. Then the set U (x) := {y ∈ X : 0 < y x, &u + y& = 1} is nonempty, since otherwise x ∈ Mu , contradicting the choice of x. Note that for an upward directed set A ⊂ U (x) we have y0 := sup(A) ∈ U (x), since by the Fatou property &u + y0 & = supy∈A &u + y& = 1. By the Kuratowski–Zorn Lemma there exists a maximal element ȳ ∈ U (x). Put z := x − ȳ and observe that &ȳ + z& = &x& = 1 and &u + ȳ + ȳ + z& = &u + ȳ + x& 2. Applying the Cartwright property 5.10.9 (1) with x1 := u + ȳ, x2 := ȳ, y := z, we can split z as z = z1 + z2 , where &u + ȳ + z1 & 1 and &ȳ + z2 & 1. The maximality of ȳ in U (x) implies z2 = 0 and &u+x& = &u+ ȳ +z1 & = 1. ⊲ 5.11.8. Let X be a Banach lattice with the Cartwright, Levi, and Fatou properties. Then Mu is an M -band for every u ∈ X+ . ⊳ Given u ∈ X+ with &u& = 1, take y ∈ Mu and z ∈ Mu⊥ with max{&y&, &z&} 1. By 5.11.7 &u + z& = 1, and so &u + z + y& 2. By the Cartwright property there exist y1 , y2 ∈ X+ with y1 + y2 = y, 328 Chapter 5. Order Continuous Operators &u + y1 & 1, and &z + y2 & 1. Since y ∈ Mu , we have y1 = 0, and so y = y2 and &y + z& 1. ⊲ 5.11.9. Theorem. Let X be a Banach lattice with the Cartwright, Levi, and Fatou properties. If M(X) = {0, IX } then X is an AL-space. ⊳ Assume that X is not an AL-space. Then there exist x, y ∈ X+ with &x + y& < &x& + &y&. Note that x = 0 and y = 0. Take 0 < ε < &x& + &y& − &x + y&, form Vε (x) as in 5.11.4 with X0 = X, and denote by u a maximal member of Vε (x). Prove that Mu is a nontrivial proper M -band. If y ∈ Mu then &u + y& = &u& + &y& by 5.11.6 and this yields a contradiction: &u + y& &x + y& + &u − x& &x + y& + ε < &x& + &y& = &u& + &y&. Thus y ∈ / Mu and Mu = X. Observe now that x is not maximal, since otherwise x = u = 0 and we again arrive at a contradiction Mu = X. If u − x were maximal, then we would have Mu−x = X and 5.11.6 would imply &u& = &(u − x) + x& = &u − x& + &x& > &x& = &u&. ⊥ ⊥ Thus, Mu−x = {0} and by 5.11.6 we can choose 0 < v ∈ Mu−x with &v& = &u − x& and &u − x + v& = &u − x& ε. If v ∈ / Mu then there would exist 0 < z y such that &u + z& = &u& = &x&. This contradicts maximality of u, because &u + z − x& &u + v − x& ε. It follows that v ∈ Mu = {0}. It remains to apply 5.11.8 to conclude that [Mu ] is a nontrivial proper M -projection; i.e., M(X) = {0, IX }. ⊲ 5.11.10. Corollary. For an injective Banach lattice X the following are equivalent: (1) X is an AL-space. (2) M(X) = {0, IX }. (3) Zm (X) is one-dimensional. ⊳ Evidently, (2) and (3) are equivalent for every Banach lattice, while (2) =⇒ (1) is just Theorem 5.11.9. The remaining implication (1) =⇒ (2) is easy and can be extracted from Harmand, Werner, and Wener [166, Example 1.6 (a) or Theorem 1.8]. ⊲ 5.12. Representation of Injective Banach Lattices The results above allow us to get the representation results for injective Banach lattices. 5.12.1. Theorem. Let X be a B-cyclic Banach lattice and let X be its Boolean valued representation in V(B) . Then the following hold: 5.12. Representation of Injective Banach Lattices 329 (1) V(B) “X is injective” if and only if X is injective. (2) V(B) “X is an AM -space” if and only if X is an AM -space. (3) V(B) “X is an AL-space” if and only if X is injective and B ≃ M(X). ⊳ (1): Theorem 5.10.2 is valid within V(B) by transfer. In view of Theorem 5.9.6 (2) we only have to show that [[X has the splitting property]] = 1 if and only if X has the splitting property. It is easy to see that [[X has the splitting property]] = 1 is equivalent to the following property: For all x, y, z ∈ X+ with x 1, y 1, and x+y +z 21, there exist u, v ∈ X+ such that z = u + v, x + u 1 and y + v 1. But the latter amounts to the splitting property in X, since the relations x C1 and &x&X = & x &∞ C are equivalent. (2): Since the Λ-valued norm · in X is the restricted descent of the norm & · &X and the join (x, y) → x ∨ y in X is the descent of the similar operation on X , it follows that [[ & · &X is an M -norm]] = 1 if and only if x ∨ y = x ∨ y for all x, y ∈ X + . Since (Λ, & · &∞ ) is an AM -space, we deduce &x ∨ y&X = & x ∨ y &∞ = & x &∞ ∨ & y &∞ = &x&X ∨ &y&X . (3): By transfer and Theorem 5.11.9 we can claim that [[X is an ALspace if and only if X is injective and M(X ) = {0, IX }]] = 1. Therefore, the result is immediate from (1), Theorem 5.9.1, and 1.8.1. ⊲ 5.12.2. Corollary. Let X be a Banach lattice with the Fatou and Levi properties and B an isomorphic copy of the complete Boolean algebra M(X). Then X is injective if and only if X is lattice B-isometric to the bounded descent of some AL-space X from V(B) . ⊳ It is immediate from Theorems 5.11.2 and 5.12.1 (3). ⊲ 5.12.3. A positive operator T : X → Y is said to have the Levi property if sup xα exists in X for every increasing net (xα ) ⊂ X+ , provided that the net (T xα ) is order bounded in Y . A Maharam operator T is an order continuous order interval preserving (≡ T ([0, x] = [0, T x]) for all x ∈ X+ ) operator (cp. 5.2.1). 5.12.4. Consider vector lattices X and Y , with Y order complete, and an operator Φ ∈ L+ (X, Y ). Suppose that Φ is strictly positive (≡ x > 0 implies Φ(x) > 0) and put x := Φ(|x|) (x ∈ X). Then (X, · ) is a lattice normed space. The bo-completion of X denoted by L1 (Φ) is a Banach–Kantorovich lattice (cp. [228, Theorems 2.2.8 and 2.2.11]). It is easy to 330 Chapter 5. Order Continuous Operators see that L1 (Φ) = X if and only if Φ is a strictly positive Maharam operator with the Levi property. 5.12.5. Theorem. Let X be a Banach lattice with the complete Boolean algebra B = M(X) of M -projections, and let Λ be a Dedekind complete unital AM -space such that P(Λ) is isomorphic to B. Then the following are equivalent: (1) X is injective. (2) X is lattice B-isometric to the bounded descent of some AL-space from V(B) . (3) There exists a strictly positive Maharam operator Φ : X → Λ with the Levi property such that X = L1 (Φ) and &x& = &Φ(|x|)&∞ for all x ∈ X. (4) There is a Λ-valued additive norm 3 on 3 X such that (X, · ) is a Banach–Kantorovich lattice and &x& = 3 x 3∞ for all x ∈ X. ⊳ (1) ⇐⇒ (2): This follows from Corollary 5.12.2 and Theorem 5.12.1 (3). (2) =⇒ (3): Assume that the Boolean valued representation X of X is an AL-space within V(B) . Working within V(B) and using the transfer principle, we can ﬁnd a strictly positive order continuous functional φ : X → R with the Levi property such that &x&X = φ(|x|) for all x ∈ X . The descent Φ′ := φ↓ as well as its restriction Φ := Φ′ |X : X → Λ is a strictly positive Maharam operator with the Levi property (cp. 5.2.8). Since · = (&·&X )↓ we have x 3 =3Φ(|x|) for all x ∈ X. By the deﬁnition of restricted descent &x&X = 3 x 3∞ = &Φ(|x|)&∞ . (3) =⇒ (4): If (3) is true then some Λ-valued additive norm on X is deﬁned by x := Φ(|x|) (x ∈ X). The fact that (X, · ) is a Banach– Kantorovich space follows from Theorem 5.5.7. (4) =⇒ (2): This is immediate from Theorems 5.8.11, 5.9.1, and 5.12.1 (3). ⊲ 5.12.6. Corollary. If Φ is a strictly positive Maharam operator with the Levi property taking values in a Dedekind complete AM -space Λ with unit and |||x||| = &Φ(|x|)&∞ (x ∈ L1 (Φ)), then (L1 (Φ), |||·|||) is an injective Banach lattice and there is a Boolean isomorphism ϕ from P(Λ) onto M(L1 (Φ)) with π ◦ Φ = Φ ◦ ϕ(π) for all π ∈ B. Conversely, each injective Banach lattice X is lattice B-isometric to (L1 (Φ), |||·|||) for some strictly positive Maharam operator Φ with the Levi property taking values in a Dedekind complete AM -space Λ with unity, where B = P(Λ) ≃ M(L1 (Φ)). 5.12. Representation of Injective Banach Lattices 331 5.12.7. Corollary. An injective Banach lattice has an order Bcontinuous norm with B the complete Boolean algebra of its M -projections. ⊳ It is immediate from 5.9.6 (3) and 5.12.5 (2). ⊲ 5.12.8. Corollary. An injective Banach lattice X has an order continuous norm if and only if X is a finite l∞ -product of AL-spaces. ⊳ It is clear from the representation &x&X = &Φ(|x|)&∞ (x ∈ X) that X has an order continuous norm if and only if Λ has an order continuous norm. But the latter occurs only if Λ is ﬁnite dimensional. ⊲ A Maharam operator Φ in Theorem 5.12.5 (3) and Corollary 5.11.6 is not unique. If σ is an automorphism of B then there exists a unique lattice isomorphism σ of Λ onto itself such that σ (π1) = σ(π)1. The operator σ is called the shift by σ. If σ is the shift in Λ by an automorphism of B and β is an invertible positive orthomorphism in L1 (Φ) then Ψ = σ ◦ Φ ◦ β is a strictly positive Maharam operator with the Levi property and the Banach lattices L1 (Φ) and L1 (Ψ) coincide. The following result tells us that this example is exhaustive. 5.12.9. Theorem. Let X be an injective Banach lattice and let Φ, Ψ : X → Λ be strictly positive Maharam operators with the Levi property such that &Φ(|x|)&∞ = &x&X = &Ψ(|x|)&∞ for all x ∈ X. Then there exist an automorphism σ of B and an invertible positive orthomorphism β in X such that Ψ = σ ◦ Φ ◦ β. ⊳ If the conditions above are satisﬁed then Φ(X) = Λ = Ψ(X) and by Corollary 5.2.4 (2) there are order continuous Boolean homomorphisms ̺, τ : B → M(X) such that π ◦ Φ = Φ ◦ ̺(π) and π ◦ Ψ = Ψ ◦ τ (π) for all π ∈ B. Observe that σ := τ −1 ◦ ̺ is an automorphism of a Boolean algebra B. Let σ stand for the corresponding shift operator on Λ. It can easily be seen from the deﬁnitions (see Kusraev [228, 5.3.2, 5.3.3]) that σ (πλ) = σ(π) σ (λ) for all π ∈ B and λ ∈ Λ. Put Φ1 = σ ◦ Φ and note that Φ1 is also a strictly positive Maharam operator with the Levi property. Moreover, Ψ is absolutely continuous with respect to Φ1 . Indeed, for π ∈ B and x ∈ X we have σ(π)Φ1 (x) = σ(π) σ (Φ(x)) = σ (πΦ(x)) =σ (Φ(̺(π)x)) = Φ1 (̺(π)x) = Φ1 (τ σ(π)x), and so π ◦ Φ1 = Φ1 ◦ τ (π) (π ∈ B). If πΦ1 (x) = 0 for some π ∈ B and x ∈ X+ then Φ1 (τ (π)x) = 0 and τ (π)x = 0, as Φ1 is strictly positive. 332 Chapter 5. Order Continuous Operators It follows that πΨ(x) = Ψ(τ (π)x) = 0 and so Ψ(x) ∈ Φ1 (x)⊥⊥ for all x ∈ X+ . By the Radon–Nikodým Theorem 5.3.9 there exists a sequence (βn ) of positive orthomorphisms in X such that Ψ(x) = supn Φ1 (βn x) for all x ∈ X+ . The positive orthomorphism β = supn βn is well deﬁned in X, since Φ1 has the Levi property. Moreover, by order continuity we have Ψ(x) = Φ1 (βx) (x ∈ X) or Ψ = σ ◦ Φ ◦ β. ⊲ 5.12.10. The construction of the Maharam extension of positive operators (see Section 5.5) together with Corollary 5.12.6 supplies plenty of injective Banach lattices. Recall some notation. Given a subset M of a vector lattice X, denote by M ↓ the collection of all elements x ∈ X that can be written as x = inf(A), where A is a downward directed subset of M . The set M ↑ is deﬁned similarly on using upward directed sets. We also put M ↓↑ := (M ↓ )↑ . Write Zm (X) for the order closed f -subalgebra of the center Z (X) generated by M(X). We recall also that a subspace X0 of a B-cyclic Banach space X is said to be B-dense if for all x ∈ X and 0 < ε ∈ R there are xε ∈ X, a partition of unity (πξ )ξ∈Ξ in M(X), and a family (xξ )ξ∈Ξ in X0 such that &x − xε & ε and πξ xε = πξ xξ (ξ ∈ Ξ). 5.12.11. Theorem. Let L be a vector lattice, let Λ be a Dedekind complete AM -space with unit, and let Φ : L → Λ be a positive operator. Then there exists a unique (up to lattice isometry) injective Banach lattice X such that the following hold: (1) M(X) ≃ P(Λ). (2) There are a lattice homomorphism ι from L to X and an f -algebra isomorphism h from Z (Λ) onto Zm (X) such that &σΦ(x)&∞ = &h(σ)ι(x)& (x ∈ L+ , σ ∈ Z (Λ)+ ). (3) ι(L) is B-dense in X. (4) X = X0↓↑ , where X0 comprises all finite sums πk ∈ M(X) and xk ∈ L (k = 1, . . . , n ∈ N). n k=1 πk ι(xk ) with ⊳ The Maharam extension Φ̄ of Φ is a strictly positive Maharam operator by Theorem 5.5.7. If X = L1 (Φ̄) is the domain of Φ̂ (see 5.6.1) and &x& = &Φ̃(|x|)&∞ (x ∈ X) then X is an injective Banach lattice by Theorem 5.12.5. The properties (1)–(4) are immediate from Theorems 5.5.7 and 5.6.3. ⊲ 5.12.12. Theorem. Let X be an order B-continuous B-cyclic Banach lattice and let X u be its universal completion. There exists an order dense ideal L in X u which is an injective Banach lattice with 5.13. Operators Factorable Through Injective Banach Lattices 333 M(L) ≃ B. Moreover, if L = L1 (Φ) for a strictly positive Maharam operator Φ : L → Λ = Λ(B) with the Levi property, then the mapping assigning the operator Sx′ : X → Λ to an element x′ ∈ X ′ by Sx′ (x) = Φ (x · x′ ) (x ∈ X) is a lattice B-isometry from X ′ onto X # . ⊳ The Boolean valued representation X of X is an order continuous Banach lattice within V(B) (cp. Theorem 5.9.6). Working within V(B) we can ﬁnd an order continuous strictly positive functional φ : L1 (φ) → R having the Levi property, with L1 (φ) an order dense ideal in the universal completion X u of X . Put X ′ := {x′ ∈ X u : x′ · X ⊂ L1 (φ)}. Then assigning to every element x′ ∈ X ′ the functional σx′ : x → φ(xx′ ) (x ∈ X ) yields a lattice isometry from X ′ onto the dual X ∗ . It is easy to see that X u ↓ = X u . Deﬁne Φ as the restriction of φ↓ to L := L1 (Φ) := {x ∈ X u : φ↓(x) ∈ Λ}. Clearly, Φ is a strictly positive Maharam operator with the Levi property and so L1 (φ)⇓ = L1 (Φ). It remains to observe that identifying X with an order dense ideal in X ↓ we have X ∗ ⇓ = X # , X ′ ⇓ = X ′ , and Sx′ = σx ↓. ⊲ 5.12.13. Corollary. An injective Banach lattice X is lattice Bisometric to Ln,B (Z (X), Λ) := LB (Z (X), Λ) ∩ Ln (Z (X), Λ). If X is represented as L1 (Φ) for a strictly positive Maharam operator Φ with the Levi property, then the lattice B-isometry is carried out by assigning to each x ∈ X the operator Sx : π → Φ(πx) (π ∈ Z (X)). ⊳ Put X := L1 (Φ) in Theorem 5.12.12. Then X = L1 (φ) and so X ′ = L∞ (φ). Consequently we can deduce X ′ = X ′ ⇓ = L∞ (φ)⇓ = L∞ (Φ). It remains to note that L∞ (Φ) and Z (X) are lattice B-isometric and X # = Ln,B (X, Λ) by Corollary 5.9.7 (7). ⊲ 5.12.14. Corollary. Let X be an injective Banach lattice, while Y and Λ be Dedekind complete AM -spaces with unit such that P(Y ) ≃ P(X) and B := P(Λ) ≃ M(X). Then X is lattice B-isometric to Ln,B (Y, Λ). 5.13. Operators Factorable Through Injective Banach Lattices This section treats the operators that admit factorization through injective Banach lattices. In fact we implement the Boolean valued interpretation of a portion of the theory of cone absolutely summing operators. 334 Chapter 5. Order Continuous Operators 5.13.1. Let X be a Banach lattice and let Y be a B-cyclic Banach space. Denote by Prtσ := Prtσ (B) and Pfin (X) the set of all countable partitions of unity in B and the collection of all ﬁnite subsets of X, respectively. Given T ∈ L (X, Y ), put σ(T ) := σB (T ) := sup inf sup n (πk )∈Prtσ k∈N ı=1 &πk T xı & : 3 3 3 3 n 3 3 (xı ) ∈ Pfin (X), 3 |xı |3 1 . ı=1 An operator T ∈ L (X, Y ) is said to be B-summing if σ(T ) < ∞. Thus, T is B-summing if and only if there exists a positive constant C such that for every ﬁnite collection x1 , . . . , xn ∈ X there is a countable partition of unity (πk )k∈N in B with sup n k∈N ı=1 3 3 n 3 3 3 |xı |3 &πk T xı & C 3 3. ı=1 Denote by SB (X, Y ) the set of all B-summing operators. 5.13.2. Observe that if B = {0, 1} ≃ {0, IY } then S (X, Y ) := S{0,1} (X, Y ) is the space of cone absolutely summing operators (cp. [356, Ch. 4, §3, Proposition 3.3 (d)]) or (which is the same) (1, 1)-concave operators. In this case σ(T ) takes the form sup n ı=1 ς(T ) := σ{0,1} (T ) := 3 3 n 3 3 3 3 |xı |3 1 . &T xı & : (xı ) ∈ Pfin (X), 3 ı=1 A linear operator T is cone absolutely summing if and only if for every norm summable sequence (xn ) in X+ , the sequence (T xn ) is absolutely summable in Y (cp. [356, Ch. 4, §3, Proposition 3.3]). Let X and Y stand for the Boolean valued representation of X and Y , respectively. Write S (X , Y ) for the space of all cone absolutely summing operators from X to Y within V(B) . If T ∈ S (X , Y )⇓ and T = T ⇓ then [[σ(T ) = ς(T )]] = 1. 5.13.3. Suppose that Q is a Stonean space and X is a Banach space. Let C∞ (Q, X) be the set of cosets of continuous vector-functions u that 5.13. Operators Factorable Through Injective Banach Lattices 335 act from comeager subsets dom(u) ⊂ Q into X. (Recall that a set is called comeager if its complement is of ﬁrst category.) Vector-functions u and v are equivalent if u(q) = v(q) whenever q ∈ dom(u) ∩ dom(v). The set C∞ (Q, X) is endowed, in a natural way, with the structure of a module over the f -algebra C∞ (Q). Moreover, the continuous extension of the pointwise norm q → &u(q)& deﬁnes a decomposable norm u → u ∈ C∞ (Q) on C∞ (Q, X). Moreover, C∞ (Q, X) is a Banach– Kantorovich space (cp. 5.8.4). Denote by C# (Q, X) the part of C∞ (Q, X) that consists of vectorfunctions u satisfying u ∈ C(Q) endowed with the norm &u& := & u &∞ (cp. Kusraev [228, 2.3.3]). Suppose that Q is a Stonean space and X is a Banach lattice. Then the space C# (Q, X) is a B-cyclic Banach lattice with B isomorphic to the Boolean algebra Clop(Q) of clopen subsets of Q. For U ∈ Clop(Q) the corresponding M -projection in C# (Q, X) is given by u → 1U u. 5.13.4. Theorem. Suppose that X is a Banach lattice and X is the completion of the metric space X ∧ within V(B) . Then [[ X is a Banach lattice ]] = 1 and X ⇓ is B-isomorphic to C# (Q, X). Moreover, if Y is a B-cyclic Banach lattice, then T → T ◦ h is a lattice B-isometry from LB (C# (Q, X), Y ) onto L (X, Y ), where h is the lattice isometry from X into C# (Q, X) defined as h(x) := 1Q x. ⊳ The proof is a due modiﬁcation of Kusraev [228, 8.3.4]. ⊲ 5.13.5. Corollary. A Banach lattice X is an AL-space if and only if C# (Q, X) is an injective Banach lattice with M(C# (Q, X)) isomorphic to the Boolean algebra Clop(Q). The Boolean isomorphism sends a clopen set G ⊂ Q to the M -projection u → u1G (u ∈ C# (Q, X)). ⊳ Assume that X is an AL-space. By restricted transfer (X ∧ , & · &∧ ) is a normed vector lattice over R∧ within V(B) . Moreover, the norm & · &∧ is additive on the positive cone (X ∧ )+ and so X is an AL-space within V(B) . Thus, C# (Q, X) is an injective Banach lattice with M(C# (Q, X)) isomorphic to Clop(Q) by Theorems 5.12.1 (3) and 5.13.4. Conversely, the mapping x → 1Q x is a lattice isometry from X onto a closed vector sublattice in C# (Q, X). Therefore, X is injective whenever C# (Q, X) is injective. If π ∈ M(X) then u → π ◦ u1Q (u ∈ C# (Q, X)) is an M projection in C# (Q, X) and M(C# (Q, X)) ≃ Clop(Q) implies π = 0 or π = IX . Thus, X is an AL-space by Theorem 5.11.10. ⊲ 5.13.6. Theorem. Let X, X , and Y be as in Theorem 5.13.4, and let Y be the Boolean valued representation of Y . For every 336 Chapter 5. Order Continuous Operators T ∈ SB (X, Y ) there exists a unique T := T↑∈ V(B) determined from the formulas [[ T ∈ S (X , Y ) ]] = 1, [[ T x∧ = T x ]] = 1 (x ∈ X). The mapping T → T is a B-isometry from SB (X, Y ) onto the restricted descent S (X , Y )⇓. ⊳ Suppose that T ∈ LB (X, Y ) and its Boolean valued representation T (see Theorem 5.8.12) is cone absolutely summing; i.e., T ∈ S (X , Y )⇓. Then ς(T ) ∈ Λ and we can assume ς(T ) C1 for some 0 < C ∈ R. Moreover, the relation 3 3 3 3 ∧3 ∀ θ ∈ Pfin (X ) &T x& C 3 |x|3 (∗) 3 x∈θ x∈θ holds in V(B) and so its Boolean truth value is 1. Recall that X ∧ is a dense sublattice in X . Replacing X and θ ∈ Pfin (X ) by X ∧ and θ ∈ Pfin (X ∧ ) respectively and using the formula Pfin (X ∧ ) = Pfin (X)∧ , we can replace the universal quantiﬁer in (∗) over ﬁnite subsets of X ∧ within V(B) by the external inﬁmum over θ ∈ Pfin (X) and get 3 3 3 3 (B) ∧3 &T x& C 3 |x|3 (∗∗) V |= 3. x∈θ ∧ x∈θ ∧ Recall that if Q is the rationals then Q may be considered as the internal rationals. Denote by B(Y ) the unit ball of Y . Given 0 < ε ∈ R and θ ∈ Pfin (X) we have the sentence that is a formal presentation of the fact that &T x& rx (1 + ε)&T x& for a suitable rational rx : ∧ (∀ x ∈ θ∧ )(∃ rx ∈ Q∧ )(rx (1 + ε∧ )&T x&) ∧ (T x ∈ rx B(Y )). Replacing quantiﬁers by inﬁmum over θ and supremum over Q we deduce that for every x ∈ θ there are a countable partition of unity (πx,k ) and a sequence of rationals (rx,k ) such that ∧ ∧ πx,k [[(rx,k (1 + ε∧ )&T x∧ &) ∧ (T x∧ ∈ rx,k B(Y ))]] (k ∈ N). Let (πk ) be a common reﬁnement of the ﬁnite collection of partitions of unity (rx,k ) (x ∈ θ). Then for every x ∈ θ there is k(x) ∈ N such that ∧ πk [[rx,k(x) (1 + ε∧ )&T x∧ &]], ∧ πk [[T x∧ ∈ rx,k(x) B(Y )]] (k ∈ N). (∗∗∗) 337 ∧ Since [[T x∧ = T x]] = 1 and rx,k(x) B(Y ) = rx,k(x) B(Y ) ⇓, the second relation in (∗∗∗) implies πk T x ∈ rx,k(x) B(Y ) or &πk T x& rx,k(x) . The last inequality together with (∗∗) and the ﬁrst relation in (∗∗∗) yields ∧ ∧ rx,k(x) &πk T x& 5.13. Operators Factorable Through Injective Banach Lattices x∈θ x∈θ = ∧ rx,k(x) x∈θ 3 3 3 3 ∧ 3 3 (1 + ε) &T x & ((1 + ε)C) 3 |x |3 x∈θ x∈θ 3 3 3 3 ∧ = (1 + ε)C 3 |x|3 3 3 . ∧ ∧ ∧ x∈θ It follows that for every ﬁnite subset θ ⊂ X we have 3 3 3 3 |x|3 sup &πk T x& (1 + ε)C 3 inf 3. 3 (πk )∈Prtσ k∈N x∈θ x∈θ Thus, T is B-summing and σ(T ) C, since ε > 0 is arbitrary. Conversely, assume that T ∈ SB (X, Y ) and C is a positive constant in Deﬁnition 5.13.1. Them for a ﬁnite subset θ ⊂ X there is a countable partition of unity (πk ) in B such that 3 3 3 3 πk T x &πk T x&πk 1 C 3 |x|3 3 31. k∈N x∈θ k∈N x∈θ x∈θ Using the deﬁnition of T , we deduce from the last inequality 00 // T x ∧ [[(∀ x ∈ θ)T x = πk T x]] &T x∧ & = πk x∈θ // x∈θ ∧ &T x & = x∈θ x∈θ 3 3 00 3 3 πk T x C ∧ 3 |x∧ |3 . x∈θ Finally, for every θ ∈ Pfin (X) we have 3 3 3 ∧ 3 3 |x | πk &T x∧ & C ∧ 3 1= 3 3 k∈N x∈θ x∈θ and so we arrive at (∗), which implies that T ∈ S (X , Y ) and [[ς(T ) C ∧ ]] = 1. ⊲ 5.13.7. Corollary. Let X be a Banach lattice and let Y be a Bcyclic Banach lattice. An operator T ∈ LB (X, Y ) is B-summing with 338 Chapter 5. Order Continuous Operators σ(T ) C if and only if there exists λ ∈ Λ such that &λ&∞ C and for every finite collection x1 , . . . , xn ∈ X we have 3 n 3 n 3 3 3 T xı λ3 |xı |3 3. ı=1 ı=1 5.13.8. Theorem. Let X be a Banach lattice and let Y be a B-cyclic Banach lattice. For T ∈ LB (X, Y ) the following are equivalent: (1) T is B-summing and σ(T ) C. (2) There exists a linear operator S ∈ L (X, Λ) such that &S& C and &πT x& &πS(|x|)&∞ for all x ∈ X and π ∈ P(Λ). (3) There exist an injective Banach lattice L, a lattice homomorphism T1 ∈ L (X, L) with B-dense range, and T2 ∈ L (L, Y ) such that &T1 & C, &T2 & 1, and T = T2 ◦ T1 . ⊳ (1) =⇒ (2): Assume that T ∈ LB (X, Y ) with σ(T ) C and T is deﬁned as in Theorem 5.13.6. Then T ∈ SB (X , Y ) and by Schaefer [356, Ch. 4, § 3, Proposition 3.3 (b)] there is σ ∈ V(B) such that [[ σ ∈ X ′ , &σ& C ∧ and &T x& |x|, σ for all x ∈ X ]] = 1. If S is the bounded descent of σ then &S& C and T x S(|x|) for all x ∈ X. The last inequality is equivalent to (∀ π ∈ P(Λ)) &πT x& &πS(|x|)&∞ . (2) =⇒ (3): Using Theorem 5.12.11 with Φ := S, we only have to put L := X, T1 := ι : X → L, and deﬁne T2 : L → Y by T2 x := limε→0 T2 xε and πξ T2 xε = πξ T ι(xξ ) (ξ ∈ Ξ). Evidently, by 5.12.11 (2, 3) we have &T1 & C, T2 ∈ LB (L, Y ) and &T2 & 1. Moreover, T = T2 ◦ ι = T2 ◦ T1 by deﬁnition and T1 (X) is B-dense in L by 5.12.11 (3). (3) =⇒ (1): Let T = T2 ◦T1 be a factorization claimed in (3). Observe that the relation T x S(|x|) (x ∈ X) implies T2 u u (u ∈ L). For every ﬁnite collection x1 , . . . , xn ∈ X+ we have n ı=1 T2 ◦ T1 xı n ı=1 T1 xı = n Φ ◦ T1 xı ı=1 = Φ ◦ T1 n ı=1 xı 3 3 n 3 3 3 C3 x ı 31 3 ı=1 and (1) follows from Corollary 5.13.7. ⊲ 5.13.9. Corollary. Let X0 be a Banach sublattice of a Banach lattice X and let Y be a B-cyclic Banach space. If T0 ∈ SB (X0 , Y ) then T0 admits an extension T ∈ SB (X, Y ) with σ(T0 ) = σ(T ). 5.14. Variations on the Theme 339 5.13.10. Theorem. Let X be a Banach lattice and let Y be a Bcyclic Banach lattice. The following are equivalent: (1) SB (X, Y ) is an injective Banach lattice with a Boolean algebra of M -projections isomorphic to B. (2) X is an AM -space and Y is an injective Banach lattice with B = M(Y ). ⊳ In order to ensure the claim, we interpret in V(B) the corresponding result for cone absolutely summing operators (due to Schlotterbeck; see Schaefer [356, Ch. 4, Proposition 4.5]) saying that S (X , Y ) is an ALspace if and only if X is an AM -space and Y is an AL-space. By Theorems 5.12.1 (3) and 5.13.6 SB (X, Y ) is an injective Banach lattice with M(SB (X, Y )) isomorphic to B if and only if [[S (X , Y ) is an ALspace]] = 1. Thus, the latter is equivalent to the conjunction of the two assertions: [[X is an AM -space]] = 1 and [[Y is an AL-space]] = 1. The claim is immediate from Theorem 5.12.1 (1, 2). ⊲ 5.14. Variations on the Theme In this section we sketch some further applications of the Boolean valued approach to nonassociative Radon–Nikodým type theorems, integration with respect to a measure taking values in a Dedekind complete vector lattice, and transfer in harmonic analysis. 5.14.A. The Radon–Nikodým Theorem for JB-Algebras 5.14.A.1. Let A be a vector space over some ﬁeld F. Say that A is a Jordan algebra, if there is given a (generally) nonassociative binary operation A × A ∋ (x, y) → xy ∈ A on A, called multiplication and satisfying the following for all x, y, z ∈ A and α ∈ F: (1) xy = yx; (2) (x + y)z = xz + yz; (3) α(xy) = (αx)y; (4) (x2 y)x = x2 (yx). An element e of a Jordan algebra A is a unit element or a unit of A, if e = 0 and ea = a for all a ∈ A. 5.14.A.2. Recall that a JB-algebra A is simultaneously a real Banach space and a Jordan algebra with unit 1 such that 340 Chapter 5. Order Continuous Operators (1) &xy& &x& · &y& (x, y ∈ A), (2) &x2 & = &x&2 2 (x ∈ A), 2 (3) &x & &x + y 2 & (x, y ∈ A). The set A+ := {x2 : x ∈ A}, presenting a convex cone, determines the structure of an ordered vector space on A such that the unit 1 of the algebra A serves as a strong order unit, and the order interval [−1, 1] := {x ∈ A : −1 x 1} serves as the unit ball. Moreover, the inequalities −1 x 1 and 0 x2 1 are equivalent. The intersection of all maximal associative subalgebras of A is called the center of A and denoted by Z (A). The element a belongs to Z (A) if and only if (ax)y = a(xy) for arbitrary x, y ∈ A. If Z (A) = R · 1, then A is said to be a JB-factor. The center Z (A) is an associative JB-algebra, and such an algebra is isometrically isomorphic to the real Banach algebra C(Q) of continuous functions on some compact space Q. 5.14.A.3. The idempotents of a JB-algebra are also called projections. The set of all projections P(A) forms a complete lattice with the order deﬁned as π ρ ⇐⇒ π ◦ ρ = π. The sublattice of central projections Pc (A) := P(A) ∩ Z (A) is a Boolean algebra. Given a complete Boolean algebra B denote by Λ(B) a unital Dedekind complete AM -space with B ≃ P(Λ(B)) (which is unique up to lattice isometry). Assume that B is a subalgebra of the Boolean algebra Pc (A) or, equivalently, Λ (B) is a subalgebra of the center Z (A). Then we say that A is a B-JB-algebra if, for every partition of unity (eξ )ξ∈Ξ in B and every family (xξ )ξ∈Ξ in A, there exists a unique B-mixture x := mixξ∈Ξ (eξ xξ ); i.e., the only element x ∈ A such that eξ xξ = eξ x for all ξ ∈ Ξ. If Λ (B) = Z (A), then a B-JB-algebra is also referred to as centrally extended JB-algebra. The unit ball of a B-JB-algebra is closed under B-mixing. Consequently, each B-JB-algebra is a B-cyclic Banach space. 5.14.A.4. Theorem. The restricted descent of a JB-algebra within V(B) is a B-JB-algebra. Conversely, for every B-JB-algebra A there exists a unique (up to isomorphism) JB-algebra A within V(B) whose restricted descent is isometrically B-isomorphic to A. Moreover, [[A is a JB-factor ]] = 1 if and only if Λ (B) = Z (A). ⊳ See Kusraev and Kutateladze [249, Theorem 12.7.6] and Kusraev [226, Theorem 3.1]. ⊲ 5.14.A.5. Now we give two applications of the above Boolean valued representation result to B-JB-algebras. Theorems 5.14.A.7 and 5.14.A.11 5.14. Variations on the Theme 341 below appear by transferring the corresponding facts of the theory of JBalgebras. Let A be a B-JB-algebra and put Λ := Λ (B). An operator Φ ∈ A# is called a Λ-valued state if Φ 0 and Φ(1) = 1. A state Φ is said to be normal if, for every increasing net (xα ) in A satisfying x := sup xα , we have Φ(x) = o-lim Φ(xα ). If A is the Boolean valued representation of A, then the ascent ϕ := Φ ↑ is a bounded linear functional on A by Theorem 5.8.12. Moreover, ϕ is positive and order continuous; i.e., ϕ is a normal state on A . The converse is also true: if [[ϕ is a normal state on A ]] = 1, then the restriction of the operator ϕ ↓ to A is a Λ-valued normal state. Now we will characterize B-JB-algebras that are B-dual spaces. Toward this end, it suﬃces to give Boolean valued interpretation for the following result. 5.14.A.6. Theorem. A JB-algebra is a dual Banach space if and only if it is monotone complete and has a separating set of normal states. ⊳ See Shultz [364, Theorem 2.3]. ⊲ 5.14.A.7. Theorem. Let B be a complete Boolean algebra and let Λ be a Dedekind complete unital AM -space with B ≃ P(Λ). A B-JBalgebra A is a B-dual space if and only if A is monotone complete and admits a separating set of Λ-valued normal states. If one of these equivalent conditions holds, then the part of A# consisting of order continuous operators serves as a B-predual space of A. ⊳ See Kusraev and Kutatelaze [249, Theorem 12.8.5] and Kusraev [226, Theorem 4.2]. ⊲ 5.14.A.8. An algebra A satisfying one of the equivalent conditions 5.14.A.7 is called a B-JBW -algebra. If, moreover, B coincides with the set of all central projections, then A is said to be a B-JBW -factor. It follows from Theorems 5.14.A.4 and 5.14.A.7 that A is a B-JBW algebra (B-JBW -factor) if and only if its Boolean valued representation A ∈ V(B) is a JBW -algebra (JBW -factor). A mapping Φ : A+ → Λ∪{+∞} is a (Λ-valued) weight if the following are satisﬁed (under the assumptions that λ + (+∞) := +∞ + λ := +∞, λ·(+∞) =: λ for all λ ∈ Λ, while 0·(+∞) := 0 and +∞+(+∞) := +∞): (1) Φ(x + y) = Φ(x) + Φ(y) for all x, y ∈ A+ ; (2) Φ(λx) = λΦ(x) for all x ∈ A+ and λ ∈ Λ+ . A weight Φ is said to be a trace provided that (3) Φ(x) = Φ(Us x) for all x ∈ A+ and s ∈ A with s2 = 1. 342 Chapter 5. Order Continuous Operators Here, Ua is the operator from A to A deﬁned for a given a ∈ A as Ua : x → 2a(ax) − a2 (x ∈ A). This operator is positive; i.e., Ua (A+ ) ⊂ A+ . If a ∈ Z (A), then Ua x = a2 x (x ∈ A). A weight (trace) Φ is called normal if Φ(x) = supα Φ(xα ) for every increasing net (xα ) in A+ with x = supα xα ; semifinite if there exists an increasing net (aα ) in A+ with supα aα = 1 and Φ(aα ) ∈ Λ for all α; and bounded if Φ(1) ∈ Λ. Given two Λ-valued weights Φ and Ψ on A, say that Φ is dominated by Ψ if there exists λ ∈ Λ+ such that Φ(x) λΨ(x) for all x ∈ A+ . 5.14.A.9. We need a few additional remarks on descents and ascents. Fix +∞ ∈ V(B) . If Λ = R⇓ and Λu = R↓ then (Λu ∪ {+∞})↑ = (Λ ∪ {+∞})↑ = Λ↑ ∪ {+∞}↑ = R ∪ {+∞}. At the same time, Λ⋆ := (R ∪ {+∞})↓ = mix(R↓ ∪ {+∞}) consists of all elements of the form λπ := mix(πλ, π ⊥ (+∞)) with λ ∈ Λu and π ∈ P(Λ). Thus, Λu ∪ {+∞} is a proper subset of Λ⋆ , since xπ ∈ Λ ∪ {+∞} if and only if π = 0 or π = IΛ . Assume now that A = A ↓ with A a JB-algebra within V(B) and B isomorphic to P(A). Every bounded weight Φ : A → Λ is evidently extensional: b := [[x = y]] implies bx = by, which in turn yields bΦ(x) = Φ(bx) = Φ(by) = bΦ(y) or, equivalently, b [[Φ(x) = Φ(y)]]. But an unbounded weight may fail to be extensional. Indeed, if Φ(x0 ) = +∞ and Φ(x) ∈ Λ for some x0 , x ∈ A and b ∈ P(A) then Φ(mix(bx, b⊥ x0 )) = mix(bΦ(x), b⊥ (+∞)) ∈ / Λ ∪ {+∞}. Given a semiﬁnite weight Φ on A, we deﬁne its extensional modiﬁ : A → Λ⋆ as follows. If Φ(x) ∈ Λ we put Φ(x) cation Φ := Φ(x). If Φ(x) = +∞ then x = sup(D) with D := {a ∈ A : 0 a x, Φ(a) ∈ Λ} by semiﬁniteness. Let b stand for the greatest element of P(Λ) such that Φ(bD) is order bounded in Λu and put λ := sup(Φ(bD)). We deﬁne Φ(x) ⊥ ⊥ ⊥ as λb = mix(bλ, b (+∞)); i.e., bΦ(x) = λ and b Φ(x) = b (+∞). It is is extensional. Thus, for ϕ := Φ↑ we have not diﬃcult to check that Φ [[ϕ : A → R ∪ {+∞}]] = 1 and, according to 1.6.6, Φ = ϕ↓ = Φ. But if we deﬁne ϕ⇓ as ϕ⇓(x) = ϕ↓(x) whenever ϕ↓(x) ∈ Λ and ϕ⇓(x) = +∞ otherwise, then Φ = (Φ↑)⇓. 5.14.A.10. Theorem. Let A be a JBW -algebra and let τ be a normal semifinite real-valued trace on A . For each real-valued weight ϕ 343 5.14. Variations on the Theme on A dominated by τ there exists a unique positive element h ∈ A such that ϕ(a) = τ (Uh1/2 a) for all a ∈ A+ . Moreover, ϕ is bounded if and only if τ (h) is finite and ϕ is a trace if and only if h is a central element of A . ⊳ This fact was proved in King [199]. ⊲ 5.14.A.11. Theorem. Let A be a B-JBW -algebra and let T be a normal semifinite Λ-valued trace on A. For each weight Φ on A dominated by T there exists a unique positive h ∈ A such that Φ(x) = T(Uh1/2 x) for all x ∈ A+ . Moreover, Φ is bounded if and only if T(h) ∈ Λ and Φ is a trace if and only if h is a central element of A. ⊳ We present a sketch of the proof. Taking into consideration the and τ = T↑. Then within V(B) the remarks in 5.14.A.9, we put ϕ = Φ↑ following hold: τ is a semiﬁnite normal real-valued trace on A and ϕ is real-valued weight on A dominated by τ . By transfer we can apply Theorem 5.14.A.10 and ﬁnd h ∈ A such that ϕ(x) = τ (Uh1/2 x) for all x ∈ A+ . Actually, h ∈ A and ϕ⇓(x) = τ ⇓(Uh1/2 x) for all x ∈ A+ . It remains to note that Φ = ϕ⇓ and T = τ ⇓. The details of the proof are left to the reader. ⊲ 5.14.B. Vector Measures and Integrals 5.14.B.1. Let Z be a universally σ-complete vector lattice with unit 1 and let Y be an arbitrary vector lattice. Fix a subalgebra A of the σ-complete Boolean algebra C(1) of all components of 1 in Z. A Y valued measure on A is a mapping μ : A → Y ∪ {+∞} such that μ(A ) ⊂ Y+ ∪ {+∞}, μ(0) = 0 and +∞ μ an n=1 = o- +∞ μ(an ) := n=1 +∞ n μ(ak ) n=1 k=1 for an arbitrary disjoint sequence (an ) in A . Here, M stands for the supremum in Y whenever it exists and +∞ otherwise. A measure μ is called semifinite if μ(a) = sup{μ(b) : b ∈ A , b a, μ(b) ∈ Y } for all a∈A. Denote by S(A ) the vector sublattice of Z comprising allA -simple elements; i.e., x ∈ S(A ) means that some representation x = nk=1 αk ak holds with α1 , . . . , αn ∈ R and pairwise disjoint a1 , . . . , an ∈ A . Put ! n αk μ(ak ) (x ∈ S A ) . Iμ (x) := x dμ := k=1 344 Chapter 5. Order Continuous Operators It is clear that this formula correctly deﬁnes the positive linear operator . . Iμ : S(A ) → Y and | x dμ| |x| dμ for all x ∈ S(A ). 5.14.B.2. Let us deﬁne the integral at the elements that can be approximated by A -simple elements. We say that a positive element x ∈ Z is integrable with respect to μ or μ-integrable if there is an increasing sequence (xn )n∈N of positive . elements in S(A ) o-converging in Z to x and the supremum supn∈N xn dμ exists in Y . For such a sequence (xn ) the sequence of the integrals (Iμ (xn ))n∈N is o-fundamental, since 6! 6 ! ! 6 6 6 xn dμ − xm dμ6 |xn − xm | dμ 6 6 ! ∞ ! xk dμ − xp dμ −→ 0, k=1 p→∞ where p = min{m, n}. Now we can deﬁne the integral of x by putting ! ! Iμ (x) := x dμ := o-lim xn dμ. n→∞ The deﬁnition is sound. An element x ∈ E is integrable (= μ-integrable) if its positive part x+ and negative part x− are both integrable and in this event we put ! ! ! Iμ (x) := x dμ := x+ dμ − x− dμ. 5.14.B.3. Denote by L 1 (μ) and L ∞ (μ) the set of all integrable elements in Z and the order ideal in S(A ) generated by the order unit, respectively. It can easily be checked that L 1 (μ) is an order dense ideal in S(A ) and Iμ : L 1 (μ) → Y is a positive linear operator. Deﬁne in L 1 (μ) the Y -valued seminorm ! x ∈ L 1 (μ) . x 1 := |x| d μ We say that two elements x, y ∈ G are μ-equivalent if there is a unit element e ∈ C(1) with μ(1 − e) = 0 and [e]x = [e]y. The set N (μ) of all elements that are μ-equivalent to zero is a sequentially o-closed order ideal in L 1 (μ). It follows from the deﬁnition of integral that N (μ) = {x ∈ L 1 (μ) : x 1 = 0}. Deﬁne the Dedekind σ-complete 5.14. Variations on the Theme 345 vector lattice L1 (μ) as the quotient space of L 1 (μ) by the σ-ideal N (μ). The coset of x ∈ L 1 (μ) will be denoted by x̃. The Y -valued norm 1 1 on L x ∈ L x̃ x (μ) is introduced by setting := (μ) . Thus, 1 1 L (μ), · is a lattice normed space over Y . 5.14.B.4. Put A◦ := {a ∈ A : μ(A) = +∞} and N (μ) := {a ∈ A : μ(a) = 0}. Let A˜ and φ denote the quotient algebra A /N (μ) and the natural quotient mapping A /N (μ) → A˜, respectively. There is a unique measure μ̃ : A˜ → Y such that μ̃ ◦ φ = μ. Given a Boolean homomorphism h : B := P(Y ) → A˜, we say that μ is modular with respect to h, or h-modular if bμ̃(φa) = μ̃(h(b) ∧ φ(a)) for all a ∈ A◦ and b ∈ B. Clearly, the modularity of μ means that bμ(a) = μ(b′ ∧ a′ ) for all a ∈ A◦ , a′ ∈ φ(a) and b′ ∈ h(b). Moreover, the modularity of μ amounts to the modularity of μ̃; i.e., bμ̃(ã) = μ̃(h(b) ∧ ã) for all b ∈ B and ã ∈ A˜◦ . Let e := {b ∈ B : (∀ a ∈ A ) bμ(a) = 0}. Then eμ(A ) = {0} and μ(A ) ⊂ (1 − e)Y . Moreover, bμ(A ) = {0} if and only if h(b) ∈ N (μ). Thus, h is injective on [0, 1−e]. In the sequel we agree that μ(A◦ )⊥⊥ = Y and in this event h is an isomorphic embedding of B into A˜. An h-modular measure μ is said to be ample (with respect to h) if for every partition of unity (bξ )ξ∈Ξ in B and an arbitrary family (aξ )ξ∈Ξ in A there exists a unique (up to equivalence) element a ∈ A such that bξ μ(a△aξ ) = 0 for all ξ ∈ Ξ. Because of the h-modularity of μ this amounts to saying that h(bξ ) ∧ φ(a) = h(bξ ) ∧ φ(aξ ) for all ξ ∈ Ξ. In particular, if μ is ample with respect to h, then h is a complete isomorphism of B into A˜. Say that μ is Maharam if μ is semiﬁnite and A˜ is Dedekind complete. It can easily be checked that a bounded modular measure is Maharam if and only if it is ample. For an unbounded measure only the necessity is true: a modular Maharam measure is ample. A vector measure algebra is a triple (A , μ, Y ), where A is a Dedekind σ-complete Boolean algebra and μ : A → Y ∪{+∞} is a strictly positive countably additive measure. If Y = R we speak of a (scalar) measure algebra (A , μ). A measure algebra (A , μ, Y ) is also called Maharam or h-modular whenever so is the measure μ. 5.14.B.5. For the lattice normed space L1 (μ), · the following hold: (1) L1 (μ) is uniformly Y -complete. (2) L1 (μ) is disjointly decomposable if and only if μ is modular. 346 Chapter 5. Order Continuous Operators (3) L1 (μ) is a Banach–Kantorovich lattice if and only if μ is Maharam. ⊳ See Kusraev[228, 6.1.8 and 6.1.9 (3, 4)]. ⊲ 5.14.B.6. Theorem. Let (A , μ, Y ) be an h-modular Maharam measure algebra with Y := R↓ and h a Boolean isomorphism from B := P(Y ) into A . Then there exist A, m ∈ V(B) such that the following hold: (1) [[(A, m) is a Maharam scalar measure algebra]] = 1. (2) If μ′ := m⇓ and A ′ := A↓ then (A ′ , μ′ , Y ) is a h′ -modular Maharam vector measure algebra with h′ a Boolean isomorphism from B into A ′ . (3) There exists a Boolean isomorphism j from A onto A ′ such that μ = μ′ ◦ j and h′ = j ◦ h. ⊳ Apply Theorem 1.10.4 with D := A and put A := D, A ′ := D′ , j := H, h′ := ı′ , and μ′ := μ ◦ j−1 . The h-modularity assumption implies that μ′ is h′ -modular, which in turn implies the estimate [[a1 = a2 ]] [[μ′ (a1 ) = μ′ (a2 )]] provided that μ′ (a1 ), μ′ (a2 ) ∈ Y . Since μ′ is semiﬁnite, just as in 5.14.A.9 we deﬁne the extensional modiﬁcation μ ′ : A ′ → Λ⋆ ′ ′ ′ of μ . Thus, we can deﬁne m := μ ↓ and ensure that μ = m⇓. Since (A ′ , μ′ , Y ) is evidently an h′ -modular Maharam vector measure algebra, it can be deduced using the ascending–descending machinery that (A, m) is a Maharam scalar measure algebra within V(B) . ⊲ To state the next theorem we use the notations from 5.14.B.6. 5.14.B.7. Theorem. Suppose L1 (m), Im ∈ V(B) have the properties that [[L1 (m) is a Banach lattice of m-integrable spectral systems from S(A )]] = 1 and [[Im. is an order continuous linear functional on L1 (m) defined as Im : x → x dm (x ∈ L1 (m))]] = 1. Then the following hold: (1) L1 (m)↓ is a Dedekind complete Banach–Kantorovich lattice over Y and Im ↓ is a strictly positive Maharam operator from L1 (m)↓ to Y . (2) There exists an isometric lattice isomorphism g from L1 (m)↓ onto L1 (μ) such that (Im ↓) ◦ g = Iμ . ⊳ Since L1 (m) and L1 (μ) are Dedekind complete, we can reduce demonstration to the case of ﬁnite measures by decomposition into a direct sum of bands with order unit. So we will assume below that m and μ take values in R and Y = R↓ respectively. Since L1 (m) is a Banach lattice within V(B) , we see that L1 (m)↓ is a Banach–Kantorovich lattice 5.14. Variations on the Theme 347 (cp. Kusraev [228, Chapter 2]). Let x : R → A be a summable spectral system in A and y := x↑. Then [[y : R∧ → A is an increasing mapping ∧ satisfying y(R ) = 1 and y(R∧ ) = 0]] = 1. Deﬁne x̄ ∈ V(B) by the formula y(s) : s ∈ R∧ , s < t x̄(t) = (t ∈ R). Clearly, x̄ is a spectral system in A and [[x̄(t∧ ) = x(t)]] = 1 (t ∈ R∧ ). Show that [[Iμ (x) = Im (x̄)]] = 1 for all x ∈ L1 (μ). To this end, take ε > 0 and let νn and tn (±n ∈ ω) enjoy the conditions −∞ ←− . . . ν−k < . . . < ν0 < . . . < νk . . . → +∞, sup (νn+1 − νn ) < ε. tn ∈ [νn , νn+1 ) (±n ∈ ω), ±n∈ω Deﬁne σ, σn ∈ Y = R↓ by σ := ∞ tn μ(x(νn+1 ) − x(νn )), −∞ n σn := tk μ(x(νk+1 ) − x(νk )). k=−n Without loss of generality, we can assume that e := μ(1) is an order unit of Y . It is easy that σn is an integral sum for Im (x̄) within V(B) ; i.e., ∧ σn = n k=−n∧ tk m(x̄(νk+1 ) − x̄(νk )) = 1. ∧ ∧ Since σ = o-lim σn ; we have by 2.4.5 that [[σ = limn→∞ σn ]] = 1, imn→∞ plying that [[ |σ − Im (x̄)| < ε∧ ]] = 1. Moreover, |Iμ (x) − σ| εe, and so [[ |Im (x̄) − Iμ (x)| 2 ε∧ e ]] = 1. Since ε > 0 is arbitrary, we see that [[Im (x̄) = Iμ (x)]] = [[x̄ ∈ L1 (m)]] = 1. If g : L1 (μ) → L1 (m)↓ is deﬁned as g(x) = x̄, then (Im ↓) ◦ g = Iμ . Given x ∈ L1 (μ), we see within V(B) that &x̄& = Im (|x̄|) = Im ( |x| ) = Iμ (|x|) = x . Hence, [[ &g(x)& = x ]] = 1 and we conclude that g is an isometry. The linearity of g and the preservation of the meets and joins of nonempty 348 Chapter 5. Order Continuous Operators ﬁnite sets under g are proved by similar arguments. Thus, we will show only that g is additive. To this end, take another spectral system y ∈ L1 (μ). Recall that x(s) ∧ y(t) : s, t ∈ R, s + t = r . (x + y)(r) = Observe also that x + y(r∧ ) = (x + y)(r), x̄(r∧ ) = x(r) and ȳ(r∧ ) = y(r) (r ∈ R). So we have the following within V(B) : = x + y(r∧ ) = (x + y)(r) x(s) ∧ y(t) : s, t ∈ R, s + t = r = x̄(s∧ ) ∧ ȳ(t∧ ) : s, t ∈ R, s + t = r = x̄(s) ∧ ȳ(t) : s, t ∈ R∧ , s + t = r∧ = (x̄ + ȳ)(r∧ ). Consequently, x + y and x̄ + ȳ coincide on the dense subset R∧ ⊂ R. Since each spectral system is left continuous, x + y = x̄ + ȳ. Take an arbitrary z ∈ L1 (m)↓ and put y := z|R∧ ; i.e., y ∈ V(B) is the restriction of the spectral system z : R → A to R∧ . If x := y↓, then x : R → A is a spectral system in A . It is easy that x̄ = z. Moreover, the m-summability of z implies the μ-summability of x. Hence, [[Im (z) = Iμ (x)]] = 1. ⊲ 5.14.B.8. Theorem. A Banach lattice X is injective if and only if there exists a modular Maharam vector measure algebra (A , μ, Λ), where Λ is a Dedekind complete unital AM -space with P(Λ) isomorphic to M(X), such that X is lattice P(Λ)-isometric to L1 (μ). Moreover, X admits a Banach f -module structure over Z (Λ) and the lattice P(Λ)isometry between X and L1 (m) is an f -module isomorphism too. ⊳ This is immediate from Theorems 5.7.9, 5.12.1 and 3.14.B.7. ⊲ 5.14.C. Transfer in Harmonic Analysis In what follows, G is a locally compact abelian group, τ its topology, τ (0) a basic neighborhood system of 0 in G, and G′ stands for the dual group. Then G is also the dual group of G′ and we write g, γ := γ(g) (g ∈ G, γ ∈ G′ ). We consider G as an additive group. 5.14.C.1. By restricted transfer G∧ is a group within V(B) . At the same time τ (0)∧ may fail to be a topology of G∧ . But G∧ becomes a topological group by deﬁning the basic neighborhood system of 0 := 0∧ to be τ (0)∧ . This topological group is again denoted by G∧ . Clearly, 349 5.14. Variations on the Theme G∧ may fail to be locally compact. Let U be a neighborhood of 0 such that U is compact. Then U is totally bounded. It follows by restricted transfer that U ∧ is also totally bounded, since total boundedness can be expressed by a restricted formula. Therefore the completion of G∧ is again locally compact. The completion of G∧ is denoted by G , and by the above observation G is a locally compact abelian group within V(B) . 5.14.C.2. Let Y be a Dedekind complete vector lattice and let YC be its complexiﬁcation. A vector-valued function ϕ : G → Y is said to be uniformly order continuous on a set K if inf sup{|ϕ(g1 ) − ϕ(g2 )| : g1 , g2 ∈ K, g1 − g2 ∈ U } = 0. U ∈τ (0) This amounts to saying that ϕ is order bounded on K and, if e ∈ Y is an arbitrary upper bound of ϕ(K), then for arbitrary 0 < ε ∈ R there exists a partition of unity (πα )α∈τ (0) in P(Y ) such that πα |ϕ(g1 ) − ϕ(g2 )| εe for all α ∈ τ (0) and g1 , g2 ∈ K, g1 − g2 ∈ α. If, in this deﬁnition we put g2 = 0, then we arrive at the deﬁnition of a mapping ϕ order continuous at zero. Let us introduce the class of dominated mappings with values in a vector lattice Y . A mapping ψ : G → YC is called positive definite if n ψ(gj − gk ) cj ck 0 j,k=1 for all ﬁnite collections g1 , . . . , gn ∈ G and c1 , . . . , cn ∈ C (n ∈ N). For n = 1, the deﬁnition readily implies that ψ(0) ∈ Y+ . For n = 2, we have the inequality |ψ(g)| ψ(0) (g ∈ G). If we introduce the structure of an f -algebra with unit ψ(0) in the order ideal of Y generated by ψ(0) then, for n = 3, from the above deﬁnition we can deduce one more inequality |ψ(g1 ) − ψ(g2 )|2 2ψ(0)(ψ(0) − Re ψ(g1 − g2 )) (g1 , g2 ∈ G). It follows that every positive deﬁnite mapping ψ : G → YC o-continuous at zero is order bounded (by the element ψ(0)) and uniformly ocontinuous. A mapping ϕ : G → Y is called dominated if there exists a positive deﬁnite mapping ψ : G → YC such that 6 n 6 n 6 6 6 6 ψ(gj − gk )cj ck ϕ(gj − gk )cj ck 6 6 6 6 j,k=1 j,k=1 350 Chapter 5. Order Continuous Operators for all g1 , . . . , gn ∈ G, c1 , . . . , cn ∈ C, and n ∈ N. In this case we also say that ψ is a dominant of ϕ. It is easy to show that if ϕ : G → YC has a dominant order continuous at zero then ϕ is order bounded and uniformly order continuous. We denote by D(G, YC ) the vector space of all dominated mappings from G into YC whose dominants are order continuous at zero. We also consider the set D(G, YC )+ of all positive deﬁnite mappings from G into YC . This set is a cone in D(G, YC ) and deﬁnes the order compatible with the structure of a vector space on D(G, YC ). Actually, D(G, YC ) is a Dedekind complete complex vector lattice (see 5.14.C.13 below). We also deﬁne D(G , C ) ∈ V(B) to be the set of functions ϕ : G → C with the property that [[ϕ has a dominant continuous at zero]] = 1. 5.14.C.3. Let Y = R↓. For every ϕ ∈ D(G, YC ) there exists a unique ϕ̃ ∈ V(B) such that [[ϕ̃ ∈ D(G , C )]] = 1 and [[ϕ̃(x∧ ) = ϕ(x)]] = 1 for all x ∈ G. The mapping ϕ → ϕ̃ is a linear and order isomorphism from D(G, Y ) onto D(G , C )↓. 5.14.C.4. Deﬁne C0 (G) as the space of all continuous complex functions f on G vanishing at inﬁnity. The latter means that for every 0 < ε ∈ R there exists a compact set K ⊂ G such that |f (x)| < ε for all x ∈ G \ K. Denote by Cc (G) the space of all continuous complex functions on G having compact support. Evidently, Cc (G) is dense in C0 (G) with respect to the norm & · &∞ . 5.14.C.5. Let us introduce one simple class of majorized or dominated operators. Let X be a normed complex vector space and let Y be a complex Banach lattice. A linear operator T : X → Y is called majorized or dominated if T sends the unit ball of X into an order bounded subset of Y . This amounts to saying that there exists c ∈ Y+ such that |T x| c&x&∞ for all x ∈ X. The set of all dominated operators from X to Y is denoted by Lm (X, Y ). If Y is Dedekind complete then the element T := {|T x| : x ∈ X, &x& 1} exists and is called the abstract norm of T . Moreover, if X is a vector lattice and Y a Dedekind complete vector lattice then Lm (X, Y ) is a vector sublattice of L∼ (X, Y ). Given a positive operator T ∈ Lm (C0 (G′ ), Y ), we can deﬁne the mapping ϕ : G → Y by putting ϕ(g) = T (g, ·) for all g ∈ G, since the function γ → g, γ lies in C0 (G′ ) for every g ∈ G. It is not diﬃcult to 5.14. Variations on the Theme 351 ensure that the so-deﬁned mapping ϕ is order continuous at zero and positive deﬁnite. The converse is also true; see 5.14.C.8. 5.14.C.6. Consider a metric space (M, r). The deﬁnition of metric space can be written as a bounded formula, say ϕ(M, r, R), so that [[ϕ(M ∧ , r∧ , R∧ )]] = 1 by restricted transfer. In other words, (M ∧ , r∧ ) is a metric space within V(B) . Moreover we consider the internal function r∧ : M ∧ × M ∧ → R∧ ⊂ R as an R-valued metric on M ∧ . Denote by (M , ρ) the completion of the (M ∧ , r∧ ); i.e., [[(M , ρ) is a complete metric space and M ∧ is a dense subset of M ]] = 1 and [[r(x)∧ = ρ(x∧ )]] = 1 for all x ∈ M . If (X, & · &) is a real (or complex) normed vector space then [[X ∧ is a vector space over the ﬁeld R∧ (or C∧ ) and & · &∧ is a norm on X ∧ with values in R∧ ⊂ R]] = 1. So, we will consider X ∧ as an R∧ -vector space with an R-valued norm within V(B) . Let X ∈ V(B) stand for the (metric) completion of X ∧ within V(B) . It is not diﬃcult to see that [[X is a real (complex) Banach space including X ∧ as an R∧ (C∧ )-linear subspace]] = 1, since the metric (x, y) → &x − y& on X ∧ is translation invariant. Clearly, if X is a real (complex) Banach lattice then [[X is a real (complex) Banach lattice including X ∧ as an R∧ (C∧ )-linear sublattice]] = 1 (see 5.13.3–5.13.5). 5.14.C.7. Theorem. Let Y = C ↓ and let X ′ be the topological dual of X within V(B) . For every T ∈ Lm (X, Y ) there exists a unique τ ∈ X ′ ↓ such that [[τ (x∧ ) = T (x)]] = 1 for all x ∈ X. The correspondence T → φ(T ) := τ defines an isomorphism between the C ↓-modules Lm (X, Y ) and X ′ ↓. Moreover, T = φ(T ) for all T ∈ Lm (X, Y ). If X is a normed lattice then [[X ′ is a Banach lattice ]] = 1, X ′ ↓ is a vector lattice and φ is a lattice isomorphism. ⊳ It suﬃces to settle the case of the real scalars. Apply Kusraev [228, Theorem 8.3.2] to the lattice normed space X := (X, · ), where x = &x&1. By [228, Theorem 8.3.4 (1) and Proposition 8.3.4 (2)] the spaces X ′ ↓ := L (B) (X , R)↓ and Lm (X, Y ) are linearly isometric. To complete the proof, refer to [228, Proposition 5.5.1 (1)]. ⊲ 5.14.C.8. Theorem. A mapping ϕ : G → YC is order continuous at zero and positive definite if and only if there exists a unique positive operator T ∈ Lm (C0 (G′ ), YC ) such that ϕ(g) = T (g, ·) for all g ∈ G. ⊳ By transfer, 5.14.C.3, and Theorem 5.14.C.7, we can replace ϕ and T by their Boolean valued representations ϕ̃ and τ . The norm completion of C0 (G′ )∧ within V(B) coincides with C0 (G ′ ). (This can be 352 Chapter 5. Order Continuous Operators proved by the reasoning similar to that in Takeuti [380, Proposition 3.2].) Application of the classical Bochner Theorem (see Loomis [286, Section 36A]) to ϕ̃ and τ yields the desired result. ⊲ 5.14.C.9. We now specify the vector integral of 5.14.B for elements of some abstract Dedekind σ-complete vector lattice. Take as a universally σ-complete vector lattice Z the vector lattice RQ of all real functions on a nonempty set Q. Let A be a σ-algebra of subsets of Q; i.e., A ⊂ P(Q). We identify this algebra with the isomorphic algebra of the characteristic functions {1A := χA : A ∈ A } so that S(A ) is the space of all A -simple functions on Q; i.e., f ∈ S(A ) means that f = nk=1 αk χAk for some α1 , . . . , αn ∈ R and disjoint A1 , . . . , An ∈ A . Let a measure μ be deﬁned on A and take values in a Dedekind complete vector lattice Y . We suppose that μ is order bounded. If f ∈ S(A ) then we put by deﬁnition Iμ := ! f dμ = n αk μ(Ak ). k=1 As was described in 5.14.B, the integral Iμ can be extended to the spaces of μ-summable functions L 1 (μ) for which the more informative notations L 1 (Q, μ) and L 1 (Q, A , μ) are also used. On identifying equivalent functions, we obtain the Dedekind σ-complete vector lattice L1 (μ) := L1 (Q, μ) := L1 (Q, A , μ). 5.14.C.10. Assume now that Q is a topological space. Denote by F (Q), K (Q), and B(Q) the collections of all closed, compact, and Borel subsets of Q. A measure μ : B(Q) → Y is said to be quasi-Radon (quasiregular ) if μ is order bounded and |μ|(U ) = sup{|μ|(K) : K ∈ K (Q), K ⊂ U } (|μ|(U ) = sup{|μ|(K) : K ∈ F (Q), K ⊂ U }) for every open set U ⊂ Q. If these equalities are fulﬁlled for all Borel U ⊂ Q then we speak about Radon and regular measures. Say that μ = μ1 + iμ2 : B(Q) → YC has one of the above properties whenever the property is enjoyed by μ1 and μ2 . We denote by qca(Q, Y ) the vector lattice of all σ-additive quasi-Radon measures on B(Q) with values in YC . If Q is locally compact or (even completely regular) then qca(Q, Y ) is a vector lattice; see [228, Theorem 6.2.2]. The variation |μ| of a YC valued (in particular, C-valued) Borel measure μ is deﬁned as the least positive measure ν : B(Q) → Y with |μ(A)| ν(a) for all A ∈ B(Q). 5.15. Comments 353 5.14.C.11. Theorem. Let Y be a real Dedekind complete vector lattice and let Q be a locally compact topological space. Then for each T in Lm (C0 (Q), YC ) there exists a unique measure μ := μT ∈ qca(Q, YC ) such that ! T (f ) = f dμ (f ∈ C0 (Q)). Q Moreover, T → μT is a lattice isomorphism from Lm (C0 (Q), YC ) onto qca(Q, YC ). ⊳ See Kusraev and Malyugin [255, Theorem 2.5]. ⊲ 5.14.C.12. Theorem. Assume that G is a locally compact abelian group, G′ is the dual group of G, and Y is a Dedekind complete real vector lattice. For ϕ : G → YC the following are equivalent: (1) ϕ has a dominant order continuous at zero. (2) There exists a unique measure μ ∈ qca(G′ , YC ) such that ! ϕ(g) = χ(g) dμ(χ) (g ∈ G). G′ ⊳ This is immediate from 5.14.C.8 and 5.14.C.11. ⊲ 5.14.C.13. Corollary. The Fourier transform establishes a lattice isomorphism between the space of measures qca(G′ , Y ) and the space of dominated mappings D(G, YC ). In particular, D(G, YC ) is a Dedekind complete complex vector lattice. 5.15. Comments 5.15.1. (1) The fact is well known in the context of lattice normed spaces and operator algebras that the module homomorphisms become linear operators when ascended to a suitable Boolean valued universe; cp. [248, 249]. Here we ﬁrstly publish analogous results about Boolean valued representation of bounded homomorphisms of f -modules in Theorem 5.1.4 and Corollary 5.1.6. (2) Theorem 5.1.10 is the Boolean valued interpretation of a portion of Nakano duality theory (Theorems 5.1.8 and 5.1.9). This result was obtained for the ﬁrst time by Luxemburg and de Pagter using standard 354 Chapter 5. Order Continuous Operators tools [293, Theorem 4.9]. Proposition 5.1.2 is taken also from Luxemburg and de Pagter [293, Lemma 4.4]. 5.15.2. (1) In [299]–[302] Maharam created a powerful approach to studying positive operators (also see the survey by Maharam [303]). The concept of Maharam operator and the main ideas of Sections 5.1–5.8 stem from these papers. The concept of interval preserving operator was introduced by Maharam under the name full-valued F -integral (= fullvalued function-valued integral). The Maharam idea was that we need full-valuedness for transferring the results of the classical integration theory to operators in function lattices. (2) Luxemburg in the joint articles with Schep [295] and de Pagter [293] extended some portion of Maharam’s theory to the case of positive operators in Dedekind complete vector lattices. The terms Maharam property and Maharam operator were introduced by Luxemburg and Schep in [295] and by Kusraev in [217] (see more details in Kusraev [228]). The Maharam ideas were transplanted to the environment of convex operators by Kusraev [217, 219]. This theory is presented in Kusraev and Kutateladze [247]. (3) Theorem 5.2.8 states that each Maharam operator is an interpretation of some order continuous linear functional in an appropriate Boolean valued model. This Boolean valued status of the concept of Maharam operator was announced in [217] and proved in [220] by Kusraev (see also [228]). It is worth emphasizing that Maharam’s approach is notable within Boolean valued analysis for the clarity and simplicity of the idea, because a considerable part of the theory reduces to manipulating numerical measures and integrals in a suitable Boolean valued model. (4) Therefore, the Maharam operators must play the same role in the theory of Banach f -modules as the Lebesgue integral in the theory of Banach spaces. For instance, we can introduce an analog of the Lebesgue scale of function spaces. To this end, consider a Dedekind complete vector lattice Y and a universally complete vector lattice Z with a unit (= a universally complete unital f -algebra). Let Φ : L1 (Φ) → Y be a strictly positive Maharam operator with L1 (Φ) an order dense ideal of Z. Take IY p ∈ Λ := Z (Y ) and deﬁne the vector lattice Lp (Φ) ⊂ Z and the Y -valued norm · p on Lp (Φ) as Lp (Φ) := z ∈ Z : |z|p ∈ L1 (Φ) , 1 z := Φ(|x|p ) p z ∈ Lp (Φ) . 5.15. Comments 355 The expression |z|p makes sense on evoking the generalized functional calculus as deﬁned in Haydon, Levy, and Raynaud [170] and Tasoev [389]. It can be showed that Lp (Φ) is a Banach–Kantorovich lattice. The scale can be studied on using the Boolean valued representation or the straightforward sectionwise techniques of continuous Banach bundles (cp. Kusraev [228, Sections 2.4 and 2.5]). 5.15.3. Operator variants of the Hahn Decomposition Theorem (Theorem 5.3.7), the Nakano Theorem (Theorem 5.3.8), and the Radon– Nikodým Theorem (Theorem 5.3.9) were obtained by Luxemburg and Schep in [295]. Maharam established Theorem 5.3.9 for a full-valued integral acting between spaces of measurable functions [302]. Theorem 5.3.5 due to Kusraev [217, 219]. The proof given in 5.3.5 is just a Boolean valued interpretation of the corresponding scalar result, i.e. Theorem 5.3.1. This latter result for functionals in the order ideal generated by Φ was proved by Vulikh (see Kantorovich, Vulikh, and Pinsker [196] and Vulikh [403]); the general case was announced by Lozanovskiı̆ in [289] and proved in Vulikh and Lozanovskiı̆ [404]. Another proof in the scalar case was given by Rice [345]. 5.15.4. (1) A detailed discussion of the properties of conditional expectation can be found in Neveu [322] and Rao [343]. Conditional expectation operators on an Lp space were characterized as averaging operators by Moy [314] and Rota [351] and as contractive projections by Douglas [110] and Ando [30]. Positive projections on a rearrangement invariant KB-space were characterized in terms of conditional expectation by Kulakova [212]. Dodds, Huijsmans, and de Pagter [105] extended the Kulakova characterization to arbitrary ideals of measurable functions. This result [105, Theorem 3.10] gives a complete description of order continuous positive projections in terms of weighted conditional extension operators. (2) Theorem 3.10 in Dodds, Huijsmans, and de Pagter [105] is a particular case Y = R of Theorem 5.4.10, and so L∞ (Ω, Σ, μ) ⊂ X ⊂ L1 (Ω, Σ, μ). At the same time Theorem 5.4.10, the main result of Section 5.4 is itself is nothing else but the Boolean valued interpretation of [105, Theorem 3.10]. (3) Grobler and de Pagter [153] introduced the class of multiplication conditional expectation representable (M CE-representable) operators on ideals of measurable functions. Grobler and Rambane [155] characterized the class of order continuous order bounded operators on ideals 356 Chapter 5. Order Continuous Operators of measurable functions, showing that multiplication operators, Riesz homomorphisms, and conditional expectations constitute the building blocks of every order continuous operator. Of course, similar results with a conditional expectation type operator of Section 5.4 can be obtained by transfer. 5.15.5. (1) The construction of Section 5.5 stems from the Maharam theory of positive operators [301, 303]. In this section we follow the articles by Akilov, Kolesnikov, and Kusraev [20, 21] of 1988. Therein the three diﬀerent approaches to describing the Maharam extension were suggested: the ﬁrst uses the technique of completion of a lattice normed space (see Kusraev [228, Section 2.2]); the second treats the Maharam extension as a space of ﬁlters; and on the embedding x → x̂ the third bases of a vector lattice X to L∼ (L∼ (X, Y ), Y by the formula x̂(T ) := T x (T ∈ L∼ (X, Y )). The last approach is accomplished also by Luxemburg and de Pagter in the voluminous paper [293], where the problem of extending a positive operator to a Maharam operator was thoroughly studied. Regarding the functional representation of the Maharam extension space see Kolesnikov and Kusraev [202] and Kusraev [228, Section 6.3]. (2) The main result of Section 5.5 is the construction of a Maharam extension of a given positive operator. The article by Luxemburg and de Pagter [293] treats a more general situation, where J is a given ideal of operators in L∼ (X, Y ), and a Dedekind complete vector lattice X̄ is constructed such that each operator T ∈ J has the Maharam extension T̄ : X̄ → Y , and T → T̄ is a lattice homomorphism. It should be also mentioned that the main result on the Maharam extension in [293, Theorem 5.4] was presented by Luxemburg at the conference in honor of Dorothy Maharam and was announced in [292] without proof. 5.15.6. (1) The properties of the Maharam extension in Section 5.6 have their natural framework in Dedekind complete vector lattices. It would be worthy to look for topological and metric aspects of the Maharam extension. Theorems 5.6.3–5.6.5 are taken from [228, Section 3.5]. More details can be found in Akilov, Kolesnikov, and Kusraev [21] and Luxemburg and de Pagter [293]. (2) In 5.6.9 (1, 2) and 5.6.10 (1–3) every component of a positive operator is obtained from its simpler fragments by up and down procedures. Similar assertions are often referred to as up-down theorems. The ﬁrst up-down theorem (5.6.10 (3)) was established by de Pagter [328]; also see Aliprantis and Burkinshaw [27, 28]. But it involved the two essential 5.15. Comments 357 constraints: Y was assumed to admit a separating set of o-continuous functionals, and X was order complete (or at least with the principal projection property). The ﬁrst constraint was eliminated by Kusraev and Strizhevskiı̆ in [256] and the second, by Akilov, Kolesnikov, and Kusraev in [21]. Of course, in the latter case the set of simple fragments is essentially diﬀerent (see 5.6.7). (3) A set of projections P ⊂ P(L∼ (X, Y )) is said to be generating if for all T ∈ L+ (X, Y ) and x ∈ X we have T x+ = sup{pT x : p ∈ P}. A general up-down theorem was obtained by Kutateladze [269]. Namely, if P is a generating set of projections in L∼ (X, Y ) (where X and Y are vector lattices with Y order complete) then E(Φ) = P ∨ (Φ)↑↓↑ , where ∞ ∨ P (Φ) comprises the components representable as k=1 πk ◦(ρk Φ) with pairwise disjoint πk ∈ P(Y ) and arbitrary ρk ∈ P. All formulas from 5.6.10 can be deduced from Kutateladze’s Up-Down Theorem by specifying generating sets. 5.15.7. (1) The material in 5.7.1–5.7.10 is traditional for the theory of normed lattices and can be found in Aliprantis and Burkinshaw [28], Kantorovich and Akilov [195], Meyer-Nieberg [311], Schaefer [356], and Schwarz [361]. As examples of Banach spaces with some Boolean algebra of M -projections we mention the Banach spaces with mixed norm: Lp,∞ (μ ⊗ ν) and L∞ (μ, X), where 1 p ∞, X is a Banach lattice, and μ and ν are ﬁnite or σ-ﬁnite measures. (2) The following suﬃcient condition on a measure space (Ω, Σ, μ) under which L0 (Ω, Σ, μ) is Dedekind complete (and hence universally complete) is used rather often. A measure space (Ω, Σ, μ) is said to have the direct sum property if Σ includes a family (Aξ )ξ∈Ξ of pairwise disjoint measurable sets of ﬁnite measure such that for every measurable set A ∈ Σ of ﬁnite measure there exist a countable set of indices Θ⊂ Ξ and a measurable set A0 ∈ Σ with μ(A0 ) = 0 satisfying A = A0 ∪ ξ∈Θ (A ∩ Aξ ) . If a measure space (Ω, Σ, μ) has the direct sum property then the associate vector lattice L0 (Ω, Σ, μ) (as well as the Boolean algebra Σ/μ−1 (0)) is Dedekind complete; see Kantorovich and Akilov [195] and Kusraev [228]. 5.15.8. (1) The concept of lattice normed space was introduced for the ﬁrst time by Kantorovich in 1936 [192]. These are vector spaces normed by elements of a vector lattice. Somewhat earlier, Kurepa [212] considered espaces pseudodistanciés, i.e. spaces with a metric that takes values in an ordered vector space. The ﬁrst applications of vector norms 358 Chapter 5. Order Continuous Operators and metrics were related to successive approximations in numerical analysis; see Kantorovich [192, 194, 196], Kollatz [203], and Schröder [360]. The theory of lattice normed spaces and dominated operators on them is presented in Kusraev [228]. (2) It is worth stressing that Kantorovich [193] is the very paper in which the unusual decomposability axiom (see 5.8.1 (4)) for an abstract norm appeared for the ﬁrst time. Paradoxically, this axiom was often omitted as inessential in the further research by various authors. The profound importance of 5.8.1 (4) was rediscovered in connection with Boolean valued analysis (see Kusraev [221] and [222]). Namely, the decomposability axiom implies the existence of a Boolean algebra of linear projections in a lattice normed space and so it leads to a Boolean valued representation as a normed lattice. The spaces with a ﬁxed Boolean algebra of linear projections and a coordinated order (the so-called coordinated spaces) were studied by Cooper [94, 95]. 5.15.9. (1) The tools of Section 5.9 are some combinations of those stemming from Gordon [133]–[137] and Takeuti [379, 381, 383, 384]. In particular, Theorem 5.9.1 is a combination of Theorems 2.2.4 and ˙ 5.8.11((1) ⇐⇒ (3)). The B-convergence in 5.9.5 is essentially the piecewise convergence or m-convergence by Takeuti (cp. [379, 383]). (2) Order continuity does not pique much interest in the context of B-cyclic Banach lattices. If a B-cyclic Banach lattice (X, & · &) is order continuous, then B is a ﬁnite Boolean algebra and so there are ﬁnitely many order continuous Banach lattices (Xk , & · &k ) (k := 1, . . . , n) such that X = X1 ⊕ · · · ⊕ Xn and &x& = max{&xk &k : k = 1, . . . , n} (x = x1 + · · · + xn , xk ∈ Xk ). Indeed, assuming that B is inﬁnite and denoting the Stone space of B by Q, we can choose an decreasing net (eα ) in C(Q) such that inf α eα = 0 and λ := inf α &eα &∞ > 0. By Corollary 5.9.4 X is a Banach–Kantorovich lattice over C(Q) and so X is a C(Q)module. The B-completeness of X implies the existence of x0 ∈ X+ with x0 = 1 := 1Q . The net (eα x0 ) is decreasing, inf α eα x0 = 0, and &eα x0 & = &eα x0 &∞ = &eα &∞ λ > 0, so X is not order continuous. 5.15.10. (1) The concept of injective Banach lattice was introduced by Lotz in [288]. In this article he also proved Theorems 5.10.3 and 5.10.4. Theorem 5.10.3 was earlier obtained by Abramovich [1]. A Banach lattice X is called λ-injective if &T & λ&T̂ & is required in 5.10.1. In this book injective means 1-injective; the λ-injective Banach lattices (λ > 1) are not considered. Concerning λ-injective Banach lattices 5.15. Comments 359 (λ > 1) we refer to Lindenstrauss and Tzafriri [282], Lindenstrauss and Wulbert [283], and Mangheni [307]. (2) By 5.10.1 the injective Banach lattices are the injective objects of the category of Banach lattices with positive contractions as morphisms. Arendt [35, Theorem 2.2] proved that the injective objects are the same if the regular operators with contractive modulus are taken as morphisms. (3) The Banach space C(Q) with Q extremally disconnected is the only (up to isometric isomorphism) injective object in the category of Banach spaces and linear contractions (see Goodner [132], Kelley [198], and Nachbin [315]). Hasumi [168] treated the complex case. Thus, Theorem 5.10.4 shows that there is an essential diﬀerence between injective Banach lattices and injective Banach spaces. (4) A separable Banach lattice X is said to be separably injective if for every pair of separable Banach lattices Y ⊂ Z and every positive (continuous) linear operator from Y to X, there exists a norm preserving positive linear extension from Z to Y . In [77, Theorem 3] Buskes made the observation that every separably injective Banach lattice is injective. (5) The injective objects in the category of Banach spaces can be also characterized geometrically in terms of the binary intersection property: a Banach space is injective if and only if each collection of pairwise intersecting closed balls {x ∈ X : &x − xi & ri } has nonempty intersection; see Nachbin [315]. An important contribution to the study of injective Banach lattices was made by Cartwright [85] who founded the order intersection property (Deﬁnition 5.10.9 (3)) and proved Theorems 5.10.10 and 5.10.11. 5.15.11. (1) Another signiﬁcant advance is due to Haydon [169]. He discovered that an injective Banach space has a mixed AM -AL-structure and, in particular, proved Theorem 5.11.9. It was also proved (Theorem 5.10.12) in this article that a Banach lattice with the Cartwright, Levi, and Fatou properties is necessarily injective; see Haydon [169, Corollary 5D]. Thus, the conjunction of the Levi and Fatou properties is equivalent to property (P) for a Banach lattice with the Cartwright property. It follows that Theorem 5.11.9 gives a completely intrinsic characterization of injective Banach lattices, while Theorem 5.10.11 contains an extrinsic property (P ). In Section 5.11 we follow Haydon [169]. (2) The notion of M -projection goes back to Alfsen and Eﬀros [24] and Ando [31] and plays a crucial role in the theory of injective Banach lattices. The dual concept of L-projection is deﬁned by the norm condi- 360 Chapter 5. Order Continuous Operators tion &x& = &πx& + &(I − π)x& (x ∈ X). In a wider context of the general Banach space theory the concepts are presented in Behrends [41] and Harmand, Werner, and Wener [166]. A closed subspace J of a Banach space X is called an M -ideal if J ⊥ := {x′ ∈ X ′ : J ⊂ ker(x′ )} is the range of an L-projection on X ′ . The main idea is to study the structure of a Banach space by means of the collections of its M -ideals. (3) A natural generalization of the concept of M -projection is the concept of Lp -projection, p 1, introduced by Beherends [40]. A linear projection π on a Banach space X is called an Lp -projection if &x&p = &πx&p + &(I − π)x&p for all x ∈ X. An L1 -projection is referred to as Lprojection. Every two Lp -projections commute and the collection of all Lp -projections forms a complete Boolean algebra. Moreover, there is a complete duality between Lp -projections in X and Lq -projections in X ′ with q = p/(p−1). Detailed presentation of this concept is in Beherends, Danckwerts, Evans, Göbel, Greim, Meyfarth, and Müller [42]. (4) A version of Theorem 5.11.2 for general Banach spaces is also true (see Cunningham [98]): The set of all M -projections forms a (generally not complete) Boolean algebra. The set of all L-projections forms a complete Boolean algebra. The closed linear span of the set of L-projections on X is called the Cunningham algebra of X and denoted by Cun(X). The centralizer Z (X) of X is a commutative unital C ∗ -algebra which is dual to the Cunningham algebra: T ∈ Cun(X) ⇐⇒ T ′ ∈ Z (X ′ ) and T ′ ∈ Cun(X ′ ) ⇐⇒ T ∈ Z (X). The L-structure of X provides the integral module representation of X such that the operators in the Cunningham algebra correspond to the multiplication operators; see Beherends, Danckwerts, Evans, Göbel, Greim, Meyfarth, and Müller [42]. Similar consideration on using the M -structure leads to the maximal function module representation so that the operators from Z (X) correspond to multiplication operators; see Beherends [41] and Cunningham [99]. 5.15.12 (1) In Section 5.12 we follow Kusraev [240, 242]. Theorem 5.12.1 states that each injective Banach lattice embeds into an appropriate Boolean valued model, becoming an AL-space. According to this fact and the principles of Boolean valued analysis, each theorem about the AL-space within Zermelo–Fraenkel set theory has an analog in the original injective Banach lattice interpreted as a Boolean valued AL-space. This transfer principle is a new powerful tool in studying injective Banach lattices; see Kusraev [240]–[243]. (2) Corollary 5.12.14 is essentially the Main Representation Theorem by Haydon in [169, Theorem 5C]. Another representation result by 5.15. Comments 361 Haydon [169, Theorem 6H] (see Theorem 5.14.B.8) tells us that an injective Banach lattice can be represented as L1 (m) for some Λ-valued modular Maharam measure m. This is immediate from 5.12.5, since the . mapping Φ : L1 (m) → Λ deﬁned by Φ : f → f dm is a Maharam operator with Levi property and L1 (m) = L1 (Φ); see [228, Theorem 6.1.10] and subsection 5.15.B. The Haydon Third Representation Theorem [169, Theorem 7B] can also be deduced from Theorem 5.12.5 on using the bundle representation of Banach–Kantorovich spaces; see Kusraev [228, Section 2.4] and [239]. 5.15.13. In Section 5.13 we follow Kusraev [240] and [242]. The cone absolutely summing operators were introduced by Levin [278] and independently but later by Schlotterbeck; see Schaefer [356, Ch. 4]. The meticulous exposition of the general theory of p-summing operators and their relatives with various interconnections and applications can be found in Diestel, Jarchow, and Tonge [103]. Observe that if B = {0, IY } then SB (X, Y ) is the space of cone absolutely summing operators; see [356, Ch. 4, §3, Proposition 3.3 (d)] or (which is the same) 1-concave operators; see [103, p. 330]. 5.15.14.A. (1) JB-algebras are nonassociative real analogs of C ∗ algebras and von Neumann operator algebras. The theory of these algebras stems from Jordan, von Neumann, and Wigner [186] and exists as a branch of functional analysis since the mid 1960s. The stages of its development are reﬂected in Alfsen, Shultz, and Størmer [25]. The theory of JB-algebras undergoes intensive study, and the scope of its applications widens. Among the main areas of research are the structure and classiﬁcation of JB-algebras, nonassociative integration and quantum probability theory, the geometry of states of JB-algebras, etc. (see Hanshe-Olsen and Störmer [165], Ajupov [15, 16], Ajupov, Usmanov, and Rakhimov [19] as well as the references therein). (2) The Boolean valued approach to JB-algebras was charted by Kusraev in the article [226] which contains Theorems 5.14.A.4 and 5.14.A.7 (also see [227]). These theorems are instances of the Boolean valued interpretation of the results by Shulz [364] and by Ajupov and Abdullaev [17]. In [226] Kusraev introduced the class of B-JBW -algebras which is broader than the class of JBW -algebras. The principal distinction is that a B-JBW -algebra has a faithful representation as an algebra of selfadjoint operators on some AW ∗ -module rather than on a Hilbert space as in 362 Chapter 5. Order Continuous Operators the case of JBW -algebras (cp. Kusraev and Kutateladze [249]). The class of AJW -algebras was ﬁrstly mentioned by Topping in [393]. Theorem 5.14.A.11 was never published before. 5.15.14.B. (1) In 5.14.B we brieﬂy present a Boolean valued approach to Wright’s theory of Stone-algebra-valued measures and integrals [419, 418, 420]. The material of this subsection (excluding Theorem 5.14.B.8) is taken from Kusraev and Malyugin [252]. We can easily reveal that Wright’s theory is a measure theoretic incarnation of Maharam’s ideas for positive operators. Theorem 5.14.B.8 is essentially Haydon’s Representation Theorem [169, Theorem 6H]. (2) According to Wright [418] a measure μ : A → C(Q) (with Q extremally disconnected and compact) is modular with respect to an algebra homomorphism π : C(Q) → L∞ (μ) if Iμ (π(a)f ) = aIμ (f ) for all a ∈ C(Q) and f ∈ L1 (μ)). Equivalence of this deﬁnition to that in 5.14.B.4 follows from 5.14.B.5 (2) and 5.8.3. It follows from 5.14.B.5 (3), deﬁnition of Banach–Kantorovich space in 5.8.4, and [228, Theorem 7.4.4] that μ is ample if and only if L1 (μ) is a Banach–Kantorovich space as well as if and only if L2 (μ) is a Kaplansky–Hilbert module. Thus, Wright’s ample measure as deﬁned in [418] is the same as that in 5.14.B.4. (3) Wright [418] showed in [418, Theorem 4.1] that the Radon– Nikodým Theorem is true for ample measures. This was done by applying the Kaplansky Theorem [197, Theorem 5] (with X := L2 (μ)) which is read as follows: If X is a Kaplansky–Hilbert module over Λ and f : X → Λ a continuous Λ-linear operator, then there exists a unique element y ∈ X such that f (x) = x | y for all x ∈ X. An improved version of the fact was obtained by Haydon [169, 6G]. This result is immediate from Theorems 4.14.B.7 and 5.3.10. (4) In [411] Wickstead constructed an integral with respect to a vector measure with range a universally complete vector lattice admitting the Radon–Nikodým Theorem. The resultant space of integrable functions is rather similar to the construction of a Maharam extension space of Section 5.5. The article [411] lucidly shows the obstacles to constructing an integral with values in a vector lattice. Namely, the deﬁnition of integral with range a Dedekind complete vector lattice needs as an adequate construction the completion that has some mixed structure of order and topology. This means that the completion appears in the two stages: ﬁrstly we supplement the space with all mixtures—the order stage, and secondly we adjoin the limits with respect to relative uniform 5.15. Comments 363 convergence—the topological stage; see [228, Theorems 2.2.2 and 3.2.8]. The problem arises then to ﬁnd an appropriate functional realization of such a completion; cp. [228, Section 6.3]. 5.15.14.C. (1) In [380] Takeuti introduced the Fourier transform for the mappings deﬁned on a locally compact abelian group and having as values pairwise commuting normal operators in a Hilbert space. By applying the transfer principle, he developed a general technique for translating classical results to operator-valued functions. In this way he in particular established a version of the Bochner Theorem describing the set of all inverse Fourier transforms of positive operator-valued Radon measures. Given a complete Boolean algebra B of projections in a Hilbert space H, denote by (B) the space of all selfadjoint operators on H whose spectral resolutions are in B; i.e., ! A ∈ (B) ⇐⇒ A = λ dEλ with (∀ λ ∈ R) Eλ ∈ B . If Y := (B) then Theorem 5.14.C.8 is essentially Takeuti’s result [380, Theorem 1.3]. (2) Kusraev and Malyugin in [255] developed Takeuti’s results in the following directions: First, they considered more general arrival spaces, namely, norm complete lattice normed spaces. So the important particular cases of Banach spaces and Dedekind complete vector lattices were covered. Second, the class of dominated mappings was identiﬁed with the set of all inverse Fourier transforms of order bounded quasi-Radon vector measures. Third, the construction of a Boolean valued universe was eliminated from the deﬁnitions and statements of results. In particular, Theorem 5.14.C.12 and Corollary 5.14.C.13 correspond to [255, Theorem 4.3] and [255, Theorem 4.4]; while their lattice normed valued versions, to [255, Theorem 4.1] and [255, Theorem 4.5]. (3) Theorem 5.15.C.7 is due to Gordon [134, Theorem 2]. Proposition 3.3 in Takeuti [380] is essentially the same result stated for the particular departure and arrival spaces; i.e., X = L1 (G) and Y = (B). (4) Theorem 5.14.C.11 is taken from Kusraev and Malyugin [255]. In the case of Q compact, it was proved by Wright in [424, Theorem 4.1]. In this result μ cannot be chosen regular rather than quasiregular. Wright in [421, Theorem T] obtained the characterization of order complete vector lattices for which this choice is always possible: A Dedekind complete vector lattice Y is weakly (σ, ∞)-distributive if and only if each Y -valued 364 Chapter 5. Order Continuous Operators Baire measure on an arbitrary compact space can be extended to a regular Y -valued Borel measure if and only if every Y -valued quasiregular Borel measure on an arbitrary compact space is regular. (5) Quasiregular measures were introduced by Wright in [419]. He also introduced quasi-Radon measures in [425]. Wright discovered the principal distinction between Radon and quasi-Radon measures and the role of the distinction in the problem of extending the measures and operators that arrive in Dedekind complete vector lattices; see [420]– [425]. The theory of Radon vector measures was then further developed by Kusraev and Malyugin; see [254, 255] and Malyugin [305, 306]. REFERENCES 1. Abramovich Yu. A., Injective envelopes of normed lattices, Dokl. Acad. Nauk SSSR, 12 (1971), 511–514. 2. Abramovich Y. A., Weakly compact sets in topological Dedekind complete vector lattices, Teor. Funkciı̆, Funkcional. Anal. i Priložen, 15 (1972), 27– 35. 3. Abramovich Y. A., On spaces of operators acting between Banach lattices, in: Investigations on Linear Operators and Theory of Functions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 73:8 (1977), 182–193. 4. Abramovich Yu. A., Multiplicative representation of disjointness preserving operators, Indag. Math. (N. S.), 45:3 (1983), 265–279. 5. Abramovich Y. A. and Aliprantis C. D., An Invitation to Operator Theory, Graduate Stud. in Math., 50, Amer. Math. Soc., Providence, R. I., 2002. 6. Abramovich Yu. A., Arenson E. L., and Kitover A. K., Banach C(K)-Modules and Operators Preserving Disjointness, Longman Scientific & Technical, Harlow, 1992.(Pitman Research Notes in Mathematics Series, 277.) 7. Abramovich Y. A. and Kitover A. K., d-Independence and d-bases in vector lattices, Rev. Romaine de Math. Pures et Appl., 44 (1999), 667–682. 8. Abramovich Y. A. and Kitover A. K., Inverses of Disjointness Preserving Operators, Mem. Amer. Math. Soc., 143:679, Providence, R. I., 2000. 9. Abramovich Y. A. and Kitover A. K., Inverses and regularity of disjointness preserving operators, Indag. Math. (N. S.), 13:2 (2002), 143–167. 10. Abramovich Y. A. and Kitover A. K., d-Independence and d-bases, Positivity, 7:1 (2003), 95–97. 11. Abramovich Yu. A., Veksler A. I., and Koldunov A. V., On disjointness preserving operators, Dokl. Akad. Nauk SSSR, 289:5 (1979), 1033–1036. 12. Abramovich Yu. A., Veksler A. I., and Koldunov A. V., Disjointness-preserving operators, their continuity, and multiplicative representation, in: Linear Operators and Their Appl., Leningrad Ped. Inst., Leningrad, 1981. 13. Abramovich Yu. A. and Wickstead A. W., The regularity of order bounded operators into C(K). II, Quart. J. Math. Oxford, 44:3 (1993), 257–270. 14. Aczél J. and Dhombres J., Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge etc., 1989. 15. Ajupov Sh. A., Jordan Operator Algebras, in: Itogi Nauki i Tekhniki, Ser. Modern Problems in Mathematics. Recent Advances [in Russian], 27, VINITI, M., 1985, 67–98. 16. Ajupov Sh. A., Classification and Representation of Ordered Jordan Algebras, Fan, Tashkent, 1986. 17. Ajupov Sh. A. and Abdullaev R. Z., Radon–Nikodým theorem for weights on semi-finite JBW-algebras, Math. Z., 188 (1985), 475–484. 18. Ajupov Sh. A. and Kudaybergenov K. K., Derivations and automorphisms of algebras of bounded operators on Banach–Kantorovich spaces, to appear. 366 References 19. Ajupov Sh. A, Usmanov Sh. M., and Rakhimov A. A., Jordan Real and Li Structures in Operator Algebras, Kluwer Academic Publishers, Dordrecht, 2001. 20. Akilov G. P., Kolesnikov E. V., and Kusraev A. G., The Lebesgue extension of a positive operator, Dokl. Akad. Nauk SSSR, 298:3 (1988), 521–524. 21. Akilov G. P., Kolesnikov E. V., and Kusraev A. G., On the order continuous extension of a positive operator, Siberian Math. J., 29:5 (1988), 24–35. 22. Akilov G. P. and Kutateladze S. S., Ordered Vector Spaces, Nauka, Novosibirsk, 1978. 23. Albeverio S., Ajupov Sh. A., and Kudaybergenov K. K., Derivations on the algebras of measurable operators aﬃliated with a type I von Neumann algebra, Siberian Adv. Math., 18:2 (2008), 86–94. 24. Alfsen A. and Eﬀros E., Structure in real Banach spaces, Ann. of Math., 96:1 (1972), 98–128. 25. Alfsen E. M., Shultz F. W., and Störmer E., A Gel’fand–Neumark theorem for Jordan algebras, Adv. Math., 28:1 (1978), 11–56. 26. Aliprantis Ch. D. and Burkinshaw O., Locally Solid Riesz Spaces, Academic Press, New York, 1978. 27. Aliprantis C. D. and Burkinshaw O., The components of a positive operator, Math. Z., 184:2 (1983), 245–257. 28. Aliprantis C. D. and Burkinshaw O., Positive Operators, Academic Press, New York, 1985. 29. Amemiya I., A general spectral theory in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Math., 12 (1953), 111–156. 30. Ando T., Contractive, projections in Lp -spaces, Pacific J. Math., 17 (1966), 391–405. 31. Ando T., Closed range theorems for convex sets and linear liftings, Pacific J. Math., 44 (1973), 393–410. 32. Antonovskiı̆ M. Ya., Boltjanskiı̆ V. G., and Sarymsakov T. A., A Survey of the theory of topological semifields, Uspekhi Mat. Nauk, 21:4 (1966), 185–218. 33. Araujo J., Beckenstein E., and Narici L., Biseparating maps and homeomorphic realcompactifications, J. Math. Anal. Appl., 12 (1995), 258–265. 34. Arendt W., Spectral theory of Lamperti operators, Indiana Univ. Marh. J., 32 (1983), 199–215. 35. Arendt W., Factorization by lattice homomorphisms, Math. Z., 185:4 (1984), 567–571. 36. Arendt W. and Hart D. R., The spectrum of quasi-invertible disjointness preserving operators, J. Funct. Anal., 33 (1986), 149–167. 37. Aron R., Downey L., and Maestre M., Zero sets and linear dependence of multilinear forms, Note di Matematica, 25:1 (2005–2006), 49–54. 38. Azouzi Y., Boulabiar K., and Buskes G., The de Schipper formula and squares of Riesz spaces, Indag. Math. (N. S.), 17:4 (2006), 265–279. 39. Badé W. G., On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc., 80 (1955), 343–359. 40. Behrends E., Lp -Struktur in Banachräumen, Studia Math., 55 (1976), 71–85. 41. Behrends E., M -Structure and the Banach–Stone Theorem, Springer-Verlag, Berlin etc., 1979. References 367 42. Behrends E., Danckwerts R., Evans R., Göbel S., Greim P., Meyfarth K., and Müller W., Lp -Structure in Real Banach Spaces, Lecture Notes in Mathematics, 613, Springer-Verlag, Berlin–Heidelberg–New York, 1977. 43. Bell J. L., Boolean Valued Models and Independence Proofs in Set Theory, Clarendon Press, New York, 1985. 44. Benamor F., Riesz spaces of order bounded disjointness preserving operators, Comment. Math. Univ. Carolinae, 48 (2007), 607–622. 45. Benamor F. and Boulabiar K., On the modulus of disjointness preserving operators on complex vector lattices, Algebra Univ., 54 (2005), 185–193. 46. Benamor F. and Boulabiar K., Maximal ideals of disjointness preserving operators, J. Math. Anal. Appl., 322 (2006), 599–609. 47. Ber A. F., Chilin V. I., and Sukochev F. A., Derivations in regular commutative algebras, Math. Notes, 75 (2004), 418–419. 48. Ber A. F., Chilin V. I., and Sukochev F. A., Non-trivial derivation on commutative regular algebras, Extracta Math., 21 (2006), 107–147. 49. Ber A. F., de Pagter B., and Sukochev F. A., Derivations in algebras of operator-valued functions, J. Operator Theory, 66:2 (2011), 261–300. 50. Berberian S. K., Baer ∗-Rings, Springer-Verlag, Berlin, 1972. 51. Bernau S. J., Orthomorphisms of Archimedean vector lattices, Math. Proc. Cambridge Phil. Soc., 89 (1981), 119–126. 52. Bernau S. J., Sums and extensions of vector lattice homomorphisms, Acta Appl. Math., 27 (1992), 33–45. 53. Bernau S. J., Huijsmans C. B., and de Pagter B., Sums of lattice homomorphisms, Proc. Amer. Math. Soc., 115:1 (1992), 151–156. 54. Bernau S. J. and Lacey E. H., The range of a contractive projection on an Lp -space, Pacific J. Math., 53:1 (1974), 21–41. 55. Beukers F., Huijsmans C. B., and de Pagter B., Unital embedding and complexification of f -algebras, Math. Z., 183 (1983), 131–144. 56. Bigard A. and Keimel K., Sur les endomorphismes conservant les polaires d’un groupe réticulé archimédien, Bull. Soc. Math. France, 97 (1969), 381–398. 57. Bigard A., Keimel K., and Wolfenstein S., Groupes et Anneaux Réticulés, Springer-Verlag, Berlin etc., 1977. 58. Birkhoﬀ G., Lattice Theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., 25, Providence, R. I., 1967. 59. Birkhoﬀ G. and von Neumann J., The logic of quantum mechanics, Ann. of Math., 37 (1936), 823–843. 60. Birkhoﬀ G. and Pierce R. S., Lattice-ordered rings, Ann. Acad. Brasil Ciên, 28 (1956), 41–68. 61. Blass A., Existence of basis implies the axiom of choice, Contemp. Math., 31 (1984), 31–33. 62. Bonnice W. and Silvermann R., The Hahn–Banach extension and the least upper bound properties are equivalent, Proc. Amer. Math. Soc., 18:5 (1967), 843–850. 63. Boulabiar K., Recent trends on order bounded disjointness preserving operators, Irish Math. Soc. Bulletin, 62 (2008), 43–69. 64. Boulabiar K. and Buskes G., Polar decomposition of order bounded disjointness preserving operators, Proc. Amer. Math. Soc., 132 (2004), 799–306. 65. Boulabiar K. and Buskes G., Vector lattice powers: f -algebras and functional calculus, Communications in Algebra, 34 (2006), 1435–1442. 368 References 66. Boulabiar K., Buskes G., and Sirotkin G., A power of algebraic Lamperti operators, Electron. Res. Announc. Amer. Math. Soc., 9 (2003), 94–98. 67. Boulabiar K., Buskes G., and Sirotkin G., Algebraic order bounded disjointness preserving operators and strongly diagonal operators, Integral Equations and Operator Theory, 54 (2006), 9–31. 68. Boulabiar K., Buskes G., and Triki A., Recent results in lattice ordered algebras, Contemp. Math., 32 (2003), 99–133. 69. Boulabiar K., Buskes G., and Triki A., Results in f -algebras, in: Positivity, Birkhaüser, Boston etc., 2007, 73–96. 70. Bourbaki N., Algebra (Polynomials and Fields, Ordered Groups), Hermann, Paris, 1967. 71. Bourgain J., Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc., 96 (1986), 221–226. 72. Bu Q., Buskes G., and Kusraev A. G., Bilinear maps on products of vector lattices. A survey, in: Positivity, Birkhaüser, Boston etc., 2007, 97–126. 73. Bu Q., Buskes G., Popov A., Tcacius A. and Troitsky V. G., 2-concavification of a Banach lattice equals the diagonal of Fremlin tensor square, Positivity, 17 (2013), 283–298. 74. Buck R. C., Multiplication operators, Pacific J. Math., 11 (1961), 95–103. 75. Bukhvalov A. V., Order bounded operators in vector lattices and spaces of measurable functions, in: Math. Anal., VINITI, M., 1988, 3–63. 76. Buskes G., Separably-injective Banach lattices are injective, Proc. Roy. Irish Acad. Sect. A, 85:2 (1985), 185–186. 77. Buskes G., The Hahn–Banach Theorem Surveyed, Inst. Mat. PAN, Warszawa, 1993 (Diss. Math., CCCXXVII). 78. Buskes G. and Kusraev A. G., Representation and extension of orthoregular bilinear operators, Vladikavkaz Math. J., 9:1 (2007), 16–29. 79. Buskes G., de Pagter B., and van Rooij A., Functional calculus in Riesz spaces, Indag. Math. (N.S.), 4:2 (1991), 423–436. 80. Buskes G. and Redfield R. H., Square means and geometric means in latticeordered groups and vector lattices, to appear. 81. Buskes G. and van Rooij A., Almost f -algebras: commutativity and the Cauchy–Schwarz inequality, Positivity, 4:3 (2000), 227–231. 82. Buskes G. and van Rooij A., Squares of Riesz spaces, Rocky Mountain J. Math., 31:1 (2001), 45–56. 83. Buskes G. and van Rooij A., Bounded variation and tensor products of Banach lattices, Positivity, 7:1–2 (2003), 47–59. 84. Cartwright D. I., The order completeness of some spaces of vector-valued functions, Bull. Austral. Math. Soc., 11 (1974), 57–61. 85. Cartwright D. I., Extension of positive operators between Banach lattices, Memoirs Amer. Math. Soc., 164 (1975), 1–48. 86. Cartwright D. I. and Lotz H. P., Some characterizations of AM - and ALspaces, Math. Z., 142 (1975), 97–103. 87. Castillo E. and Ruiz-Cobo M. R., Functional Equations and Modelling in Science and Engineering, Marcel Dekker, New York, 1992. 88. Cauchy A. L., Cuors d’analyse de l’Ecole Polytechnique Royale Ire Party. Analyse algébraique (first publ.), de Bure, Paris, 1821; Oeuvres, Sér. 2, 3, Gauthier-Villars, Paris, 1897. References 369 89. Chilin V. I., Partially ordered involutive Baer algebras, in: Contemporary Problems of Math. Recent Advances, 27, VINITI, M., 1985, 99–128. 90. Chilin V. I. and Karimov J. A., Laterally complete C∞ (Q)-modules, Vladikavkaz Math. J., 16:2 (2014), 69–78. 91. Cignoli A., A Hahn–Banach theorem for distributive lattices, Rev. Un. Mat. Argentina, 25 (1971), 335–342. 92. Cohen P. J., Set Theory and the Continuum Hypothesis, W. A. Benjamin Inc., New York etc., 1966. 93. Conrad P. F. and Diem J. E., The ring of polar preserving endomorphisms of an abelian lattice-ordered group, Illinois J. Math., 15 (1971), 222–240. 94. Cooper J. L. B., Coordinated spaces, Proc. London Math. Soc., 3:3 (1953), 305–327. 95. Cooper J. L. B., On generalization of the Köthe coordinated spaces, Math. Ann., 162:3 (1966), 351–363. 96. Coppel W. A., Foundations of Convex Geometry, Cambridge Univ. Press, Cambridge, 1988. 97. Cristescu R., Ordered vector spaces and linear operators, Tunbridge, Abacus Press, 1976 (Translated from the Romanian). 98. Cunningham F., L-structure in L-spaces, Trans. Amer. Math. Soc., 95 (1960), 274–299. 99. Cunningham F., M -structure in Banach spaces, Proc. Cambridge Phil. Soc., 63 (1967), 613–629. 100. Czerwic S., Functional Equations an Inequalities in Several Variables, World Scientific Publishing Co. Pte. Ltd., New Jersey etc., 2002. 101. Dales H. and Woodin W., An Introduction to Independence for Analysts, Cambridge Univ. Press, Cambridge, 1987. 102. Dalla Chiara M. L., Giuntini R., and Rédei M., The history of quantum logic, in: The Many Valued and Nonmonotonic Turn in Logic, 8 (Handbook of the History of Logic) (Gabbay, D. M., Woods, J. (eds.)), Elsevier, Amsterdam etc., 2007, 205–283. 103. Diestel J., Jarchow H., and Tonge A., Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995. 104. Dieudonné J., Complex Structures on Real Banach Spaces, Proc. Amer. Math. Soc., 3 (1952), 162–164. 105. Dodds P. G., Huijsmans C. B., and de Pagter B., Characterizations of conditional expectation–type operators, Pacific J. Math., 141:1 (1990), 55–77. 106. Dodds P. G. and de Pagter B., Orthomorphisms and Boolean algebras of projections, Math. Z., 187 (1984), 361–381. 107. Dodds P. G. and de Pagter B., Algebras of unbounded scalar-type spectral operators, Pacific J. Math., 130:1 (1987), 41–74. 108. Dodds P. G., de Pagter B., and Ricker W. J., Reflexivity and order properties of scalar-type spectral operators in locally convex spaces, Trans. Amer. Math. Soc., 293 (1986), 355–380. 109. Dodds P. G. and Ricker W. J., Spectral measures and the Bade reflexivity theorem, J. Funct. Anal., 61 (1985), 136–163. 110. Douglas R. G., Contractive projections on an Lp space, Pacific J. Math., 15 (1965), 443–462. 111. Duhoux M. and Meyer M., A new proof of the lattice structure of orthomorphisms, J. London Math. Soc., 25:2 (1982), 375–378. 370 References 112. Dundord H., Schwartz J. T., Linear Operators III. Spectral operators, WileyInterscience, New York etc., 1971. 113. Enflo P. and Starbird T., Subspaces of L1 containing L1 , Stud. Math., 65:2 (1979), 203–225. 114. Engesser K., Gabbay D. M., and Lehmann D., A New Approach to Quantum Logic, 8 (Ser. Stud. in Logic), College Publ., 2007. 115. Engesser K., Gabbay D. M., and Lehmann D. (eds.), Handbook of Quantum Logic and Quantum Structures, Elsevier, Amsterdam etc., 2009. 116. Fakhoury H., Préduaux de L-espace: notion de centre, J. Funct. Anal., 9:2 (1972), 189–207. 117. Foulis D. J., A note on orthomodular lattice, Portugal. Math., 21 (1962), 65–72. 118. Foulis D. J., Greechie R. J., Dalla Chiara M. L., and Giuntini, R., Quantum Logic, in: Encyclopedia of Applied Physics (Trigg G. (ed.)), VCH Publishers, 1996, 229–255. 119. Fourman M. P. and Scott D. S., Sheaves and logic, in: Appl. of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977), Springer-Verlag, Berlin etc., 1979, 302–401. 120. Fraenkel A. A., Bar-Hillel Y., and Lévy A., Foundations of Set Theory, Stud. Logic Foundations Math. 67, North-Holland Publishing Co., AmsterdamLondon, second revised edition, 1973. 121. Fremlin D. H., Tensor product of Archimedean vector lattices, Amer. J. Math., 94 (1972), 777–798. 122. Fremlin D. H., A characterization of L-spaces, Indag. Math., 36 (1974), 270–275. 123. Fremlin D. H., Tensor products of Banach lattices, Math. Ann., 211 (1974), 87–106. 124. Fremlin D. H., Topological Riesz Spaces and Measure Theory, Cambridge Univ. Press, London, 1974. 125. Fremlin D. H., Measure Theory. 2. Broad Foundations, Cambridge Univ. Press, Cambridge, 2001. 126. Fremlin D. H., Measure Theory, 3, Measure Algebras, Cambridge Univ. Press, Cambridge, 2002. 127. Freudenthal H., Teilwise geordnete moduln, Proc. Konink. Nederl. Akad. Wetensch., 39 (1936), 641–651. 128. Ganiev I. G. and Kudajbergenov K. K., Banach–Steinhaus uniform boundedness principle for operators in Banach–Kantorovich spaces over L0 , Math. Tr., 9:1 (2006), 21–33. 129. Gierz G., Darstellung von Banachverbänden durch Schnitte in Büdeln, Mitt. Math. Sem. Giessen, 125 (1977). 130. Givant S. and Halmos P., Introduction to Boolean algebras, Springer Science+Business Media, New York, 2009. 131. Goodearl K. R., Von Neumann Regular Rings, Pitman, London, 1979. 132. Goodner D. B., Projections in normed linear spaces, Trans. Amer. Math. Soc., 69 (1950), 89–108. 133. Gordon E. I., Real numbers in Boolean valued models of set theory and Kspaces, Dokl. Akad. Nauk SSSR, 237:4 (1977), 773–775. 134. Gordon E. I., K-spaces in Boolean valued models of set theory, Dokl. Akad. Nauk SSSR, 258:4 (1981), 777–780. References 371 135. Gordon E. I., To the theorems of identity preservation in K-spaces, Sibirsk. Mat. Zh., 23:5 (1982), 55–65. 136. Gordon E. I., Rationally Complete Semiprime Commutative Rings in Boolean Valued Models of Set Theory [in Russian], Gor′ kiı̆, VINITI, No. 3286-83 Dep, 1983. 137. Gordon, E. I., Elements of Boolean Valued Analysis [in Russian], Gor′ kiı̆ State University, Gor′ kiı̆, 1991. 138. Gordon E. I., Nonstandard Methods in Commutative Harmonic Analysis, Amer. Math. Soc., Providence, R. I., 1997. 139. Gordon E. I. and Lyubetskiı̆ V. A., Some applications of nonstandard analysis in the theory of Boolean valued measures, Dokl. Akad. Nauk SSSR, 256:5 (1981), 1037–1041. 140. Gordon E. I. and Lyubetskiı̆ V. A., Boolean extensions of uniform structures, in: Studies on Non-Classical Logics and Formal Systems, Nauka, Moscow, 1983, 82–153. 141. Gordon H., Decomposition of linear functionals on Riesz spaces, Duke Math J., 27 (1960), 597–606. 142. Gowers W. T., A solution to Banach’s hyperplane problem, Bull. London Math. Soc., 26 (1994), 523–530. 143. Gowers W. T. and Maurey B., The unconditional basic sequence problem, J. Amer. Math. Soc., 6 (1993), 851–874. 144. Gowers W. T. and Maurey B., Banach spaces with small spaces of operators, Math. Ann., 307 (1997), 543–568. 145. Gödel K., The Consistency of the Continuum Hypothesis, Ann. of Math. Studies, 3, Princeton University Press, Princeton, N. J., 1940. 146. Gönüllü U., Trace class and Lidskiı̆ trace formula on Kaplansky–Hilbert modules, Vladikavkaz Math. J., 16:2 (2014), 29–37. 147. Gönüllü U., Schatten type classes of operators on Kaplansky–Hilbert modules, in: Studies in Mathematical Analysis (Korobeı̆nik Yu. F. and Kusraev A. G. (eds.)), VSC RAS, Vladikavkaz, 2014, 227–238. 148. Gönüllü U., The Rayleigh–Ritz minimax forlmula in Kaplansky–Hilbert modules, Positivity, to appear (DOI 10.1007/s11117-014-0301-9). 149. Grayson R. J., A Sheaf Approach to Models of Set Theory, M. Sc. Thesis, Oxford Univ., 1975. 150. Grayson R. J., Heyting-valued models for intuitionistic set theory, in: Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977), Springer-Verlag, Berlin, 1979, p. 40. 151. Grobler J. J. and Huijsmans C. B., Disjointness preserving operators on complex Riesz spaces, Positivity, 1 (1997), 155–164. 152. Grobler J. J. and Labuschagne C., The tensor product of Archimedean ordered vector spaces, Math. Proc. Cambridge Philos. Soc., 104:2 (1988), 331–345. 153. Grobler J. J. and de Pagter B., Operators representable as multiplication conditional expectation operators, J. Operator Theory, 48 (2002), 15–40. 154. Grobler J. J., de Pagter B., and Rambane D. T., Lattice properties of operators defined by random measures, Quaestiones Math., 26 (2003), 307–319. 155. Grobler J. J. and Rambane D. T., Operators Represented by Conditional Expectations and Random Measures, Positivity, 9 (2005), 369–383. 156. Grothendieck A., Une caracterisation vectorielle-metrique des espaces L1 , 372 References Canad. J. Math., 4 (1955), 552–561. 157. Gutman A. E., Banach bundles in the theory of lattice-normed spaces. I. Continuous Banach bundles, Siberian Adv. Math., 3:3 (1993), 1–55. 158. Gutman A. E., Banach bundles in the theory of lattice-normed spaces. II. Measurable Banach bundles, Siberian Adv. Math., 3:4 (1993), 8–40. 159. Gutman A. E., Banach bundles in the theory of lattice-normed spaces. III. Approximating sets and bounded operators, Siberian Adv. Math., 4:2 (1994), 54–75. 160. Gutman A. E., Banach bundles in the theory of lattice-normed spaces, in: Linear Operators Compatible with Order [in Russian], Sobolev Institute Press, Novosibirsk, 1995, 63–211. 161. Gutman A. E., Locally one-dimensional K-spaces and σ-distributive Boolean algebras, Siberian Adv. Math., 5:2 (1995), 99–121. 162. Gutman A. E., Disjointness preserving operators, in: Vector Lattices and Integral Operators (S. S. Kutateladze (ed.)), Kluwer Academic Publishers, Dordrecht etc., 1996, 361–454. 163. Gutman A. E., Kusraev A. G., and Kutateladze S. S., The Wickstead Problem, Sib. Elektron. Mat. Izv., 5 (2008), 293–333. 164. Hamel G., Eine Basis alter Zahlen und die unstetige Lösungen der Funktionalgleichung f (x + y) = f (x) + f (y), Math. Ann., 60 (1905), 459–462. 165. Hanshe-Olsen H. and Störmer E., Jordan Operator Algebras, Pitman Publ. Inc., Boston, 1984. 166. Harmand P., Werner D., and Wener W., M -Ideals in Banach Spaces and Banach Algebras, Springer-Verlag, Berlin etc., 1993. 167. Hart D. R., Some properties of disjointness preserving operators, Indag. Math., 47 (1985), 183–197. 168. Hasumi M., The extension property for complex Banach spaces, Tôhoku Math. J., 10 (1958), 135–142. 169. Haydon R., Injective Banach lattices, Math. Z., 15 (1977), 19–47. 170. Haydon R., Levy M., and Raynaud Y., Randomly Normed Spaces, Hermann, Paris, 1991. 171. Henriksen M. and Smith F. A., A look at biseparating maps from an algebraic point of view, Contemp. Math. Amer. Math. Soc., 253 (2000), 125–144. 172. Hofstedter D. R., Gödel, Escher, Bach: an Eternal Golden Braid, Vintage Books, New York, 1980. 173. Huijsmans C. B. and de Pagter B., Ideal theory in f -algebras, Trans. Amer. Math. Soc., 269 (1982), 225–245. 174. Huijsmans C. B. and de Pagter B., Disjointness preserving and diﬀuse operators, Comp. Math., 79 (1991), 351–374. 175. Huijsmans C. B. and Wickstead A. W., The inverses of band preserving and disjointness preserving operators, Indag. Math., 3:2 (1992), 179–183. 176. Hyers D. H., Isac G., and Rassias Th. M., Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998. 177. Ioﬀe A. D., A new proof of the equivalence of the Hahn–Banach extension and the least upper bound properties, Proc. Amer. Math. Soc., 82:3 (1981), 385–389. 178. Jameson G. J. O., Ordered Linear Spaces, Springer-Verlag, Berlin etc., 1970. 179. Jech T. J., Lectures in Set Theory, with Particular Emphasis on the Method of Forcing, Lecture Notes in Mathematics, 217, Springer-Verlag, Berlin–New References 373 York, 1971. 180. Jech T. J., The Axiom of Choice, North Holland, Amsterdam etc., 1973. 181. Jech T. J., Abstract theory of abelian operator algebras: an application of forcing, Trans. Amer. Math. Soc., 289:1 (1985), 133–162. 182. Jech T. J., First order theory of complete Stonean algebras (Boolean-valued real and complex numbers), Canad. Math. Bull., 30:4 (1987), 385–392. 183. Jech T. J., Boolean-linear spaces, Adv. Math., 81:2 (1990), 117–197. 184. Jech T. J., Set Theory, Springer-Verlag, Berlin, 1997. 185. Jonge De and Van Rooij A. C. M., Introduction to Riesz Spaces, Mathematisch Centrum, Amsterdam, 1977. 186. Jordan P., von Neumann J., and Wigner E., On an algebraic generalization of the quantum mechanic formalism, Ann. Math., 35 (1944), 29–64. 187. Kalmbach G., Orthomodular logic, Z. Math. Logik Grundl. Math., 20 (1974), 395–406. 188. Kalmbach G., Orthomodular Lattices, Academic Press, London, 1983. 189. Kalton N. J., An elementary example of a Banach space not isomorphic to its complex conjugate, Canad. Math. J., 38 (1995), 218–222. 190. Kannappan Pl., Functional Equations an Inequalities with Applications, Springer-Verlag, Dordrecht etc., 2009. 191. Kantorovich L. V., On semiordered linear spaces and their applications to the theory of linear operations, Dokl. Akad. Nauk SSSR, 4:1–2 (1935), 11–14. 192. Kantorovich L. V., On a class of functional equations, Dokl. Akad. Nauk SSSR, 4:5 (1936), 211–216. 193. Kantorovich L. V., To the general theory of operations in semiordered spaces, Dokl. Akad. Nauk SSSR, 1:7 (1936), 271–274. 194. Kantorovich L. V., The method of successive approximation for functional equations, Acta Math., 71 (1939), 63–97. 195. Kantorovich L. V. and Akilov G. P., Functional Analysis, Nauka, Moscow, 1984. 196. Kantorovich L. V., Vulikh B. Z., and Pinsker A. G., Functional Analysis in Semiordered Spaces, Gostekhizdat, Moscow and Leningrad, 1950. 197. Kaplansky I., Modules over operator algebras, Amer. J. Math., 75:4 (1953), 839–858. 198. Kelley J. L., Banach spaces with the extension property, Trans. Amer. Math. Soc., 72 (1952), 323–326. 199. King W. P. C., Semi-finite traces on JBW -algebras, Math. Proc. Cambridge Philos. Soc., 72 (1983), 323–326. 200. Kolesnikov E. V., The shadow of a positive operator, Sibirsk. Mat. Zh., 37:3 (1996), 513–592. 201. Kolesnikov E. V., Diﬀuse and atomic components of a positive operator, Sibirsk. Mat. Zh., 39:2 (1998), 333–342. 202. Kolesnikov E. V. and Kusraev A. G., On functional representation of the Lebesgue extension, Sibirsk. Mat. Zh., 32:1 (1991), 190–194. 203. Kollatz L., Funktionalanalysis und Numerische Mathematik, Springer-Verlag, Berlin etc., 1964. 204. Koppelberg S., Free constructions, in: Handbook of Boolean algebras (Monk, J. D., Bonnet R. (eds.)), 1, Elsevier Sci. Publ. B. V., North-Holland, 1989, 129–172. 205. Korol′ A. M. and Chilin V. I., Measurable operators in a Boolean-valued model 374 References of set theory, Dokl. Akad. Nauk UzSSR, 3 (1989), 7–9. 206. Krasnosel′ skiı̆ M. A., Positive Solutions to Operator Equations. Chapters of Nonlinear Analysis, Fizmatgiz, Moscow, 1962. 207. Krasnosel′ skiı̆ M. A., Lifshits E. A., and Sobolev A. V., Positive Linear Systems. The Method of Positive Operators, Nauka, Moscow, 1985. 208. Krasnosel′ skiı̆ M. A., Zabreı̆ko P. P., Pustylnik E. I., and Sobolevskiı̆ P. E., Integral Operators in Spaces of Summable Functions, Nauka, Moscow, 1966. 209. Kriger H. J., Beiträge zur Theorie Positiver Operatoren, Acad. Verlag, Berlin, 1969. (Schriftenreihe der Institute für Math. Reihe A, Heft 6.) 210. Krivine J. L., Théorèmes de factorisation dans les espaces réticulés, Seminar Maurey–Schwartz, 1973/74, École Politech., Exposé 22–23. 211. Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, Birkhäuser, Basel etc., 2009. 212. Kulakova V. G., Positive projections on symmetric KB-spaces, Proc Steklov Inst. Math., 155 (1981) (Eng. transl. 1983), 93–100. 213. Kurepa G., Tableaux ramifiès d’ensembles. Espaces pseudodistanciès, C. R. Acad. Sci. Paris Sér. A, 198 (1934), 1563–1565. 214. Kurepa S., The Cauchy functional equation and scalar product on vector spaces, Glasnik Mat.-Fiz. Astronom. Ser. II Društvo Mat. Fiz. Hrvatske 19 (1934), 23–36. 215. Kusraev A. G., On a Property of the Base of the K-space of Regular Operators and Some of Its Applications, Sobolev Inst. of Math., Novosibirsk, 1977. 216. Kusraev A. G., Boolean valued analysis of duality between universally complete modules, Dokl. Akad. Nauk SSSR, 267:5 (1982), 1049–1052. 217. Kusraev A. G., General desintegration formulas, Dokl. Akad. Nauk SSSR, 265:6 (1982), 1312–1316. 218. Kusraev A. G., On categories and functors of Boolean valued analysis, Dokl. Akad. Nauk SSSR, 271:2 (1983), 283–286. 219. Kusraev A. G., Abstract disintegration in Kantorovich spaces, Sibirsk. Mat. Zh., 25:5 (1984), 79–89. 220. Kusraev A. G., Order continuous functionals in Boolean valued models of set theory, Siberian Math. J., 25:1 (1984), 57–65. 221. Kusraev A. G., On Banach–Kantorovich spaces, Sibirsk. Mat. Zh., 26:2 (1985), 119–126. 222. Kusraev A. G., Vector Duality and Its Applications, Nauka, Novosibirsk, 1985. 223. Kusraev A. G., Numeric systems in Boolean-valued models of set theory, in: Proceedings of the VIII All-Union Conference on Mathematical Logic (Moscow), Moscow, 1986, p. 99. 224. Kusraev A. G., Linear operators in lattice-normed spaces, in: Studies on Geometry in the Large and Mathematical Analysis, 9, Trudy Inst. Mat. (Novosibirsk) [in Russian], Novosibirsk, 1987, 84–123. 225. Kusraev A. G., Kantorovich spaces and the metrization problems, Sibirsk. Mat. Zh., 34:4 (1993), 107–116. 226. Kusraev A. G., Boolean-valued analysis and JB-algebras, Sibirsk. Mat. Zh., 35: 1 (1994), 124–134. 227. Kusraev A. G., On the structure of type I2 AJW -algebras, Sibirsk. Mat. Zh., 40:4 (1999), 124–134. 228. Kusraev A. G., Dominated Operators, Kluwer Academic Publishers, Dor- References 375 drecht, 2000. 229. Kusraev A. G., On band preserving operators, Vladikavkaz Math. J., 6:3 (2004), 47–58. 230. Kusraev A. G., Automorphisms and derivations in the algebra of complex measurable function, Vladikavkaz Math. J., 7:3 (2005), 45–49. 231. Kusraev A. G., Automorphisms and derivations in extended complex f algebras, Siberian Math. J., 47:1 (2006), 97–107. 232. Kusraev A. G., Analysis, algebra, and logic in operator theory, in: Complex Analysis, Operator Theory, and Mathematical Modeling (Korobeı̆nik Yu. F. and Kusraev A. G. (eds.)), VSC RAS, Vladikavkaz, 2006, 171–204. 233. Kusraev A. G., On the structure of orthosymmetric bilinear operators in vector lattices, Dokl. RAS, 408:1 (2006), 25–27. 234. Kusraev A. G., Orthosymmetric bilinear operators on vector lattices, in: Studies in Modern Analysis and Mathematical Modeling (Korobeı̆nik Yu. F. and Kusraev A. G. (eds.)), VSC RAS, Vladikavkaz, 2006, 171–204. 235. Kusraev A. G., Orthosymmetric Bilinear Operators, Preprint No. 1, VSC RAS, Vladikavkaz, 2007. 236. Kusraev A. G., When are all separately band preserving bilinear operators symmetric? Vladikavkaz Math. J., 9:2 (2007), 22–25. 237. Kusraev A. G., Functional calculus and Minkowski duality in vector lattices, Vladikavkaz Math. J., 11 (2009), 31–42. 238. Kusraev A. G., A Radon–Nikodým type theorem for orthosymmetric bilinear operators, Positivity, 14:2 (2010), 225–238. 239. Kusraev A. G., Measurable bundles of Banach lattices, Positivity, 14:4 (2010), 225–238. 240. Kusraev A. G., Boolean Valued Analysis Approach to Injective Banach lattices, Preprint No. 1, SMI VSC RAS, Vladikavkaz, 2011. 241. Kusraev A. G., Boolean Valued Analysis Approach to Injective Banach Lattices. II, Preprint No. 1, SMI VSC RAS, Vladikavkaz, 2012. 242. Kusraev A. G., Boolean-valued analysis and injective Banach lattices, Dokl. Ross. Akad. Nauk, 444:2 (2012), 143–145; Engl. transl.: Dokl. Math., 85:3 (2012), 341–343. 243. Kusraev A. G., The classification of injective Banach lattices, Dokl. Ross. Akad. Nauk, 453:1 (2013), 12–16; Engl. transl.: Dokl. Math., 88:3 (2013), 1–4. 244. Kusraev A. G. and Kutateladze S. S., Nonstandard Methods of Analysis, Nauka, Novosibirsk, 1990; Kluwer Academic Publishers, Dordrecht, 1994. 245. Kusraev A. G. and Kutateladze S. S., Nonstandard methods for Kantorovich spaces, Siberian Adv. Math., 2:2 (1992), 114–152. 246. Kusraev A. G. and Kutateladze S. S., Nonstandard methods in geometric functional analysis, Amer. Math. Soc. Transl. Ser. 2, 151 (1992), 91–105. 247. Kusraev A. G. and Kutateladze S. S., Subdiﬀerential Calculus: Theory and Applications, Nauka, Moscow, 2007 [in Russian]; Engl. transl. Subdiﬀerentials: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1995. 248. Kusraev A. G. and Kutateladze S. S., Boolean Valued Analysis, Nauka, Novosibirsk, 1999; Kluwer Academic Publishers, 1999. 249. Kusraev A. G. and Kutateladze S. S., Introduction to Boolean Valued Analysis, Nauka, Moscow, 2005. 376 References 250. Kusraev A. G. and Kutateladze S. S., Envelopes and inequalities in vector lattices, Positivity, 15:4 (2011), 661–676. 251. Kusraev A. G. and Kutateladze S. S., On order bounded disjointness preserving operators, Siberian Math. J., 55:5 (2014). 252. Kusraev A. G. and Malyugin S. A., Some Questions of the Theory of Vector Measures [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1988. 253. Kusraev A. G. and Malyugin S. A., On atomic decomposition of vector measures, Sibirsk. Mat. Zh., 30:5 (1989), 101–110. 254. Kusraev A. G. and Malyugin S. A., The product and projective limit of vector measures, in: Contemporary Problems of Geometry and Analysis [in Russian], Nauka, Novosibirsk, 1989, 132–152. 255. Kusraev A. G. and Malyugin S. A., Fourier transform of dominated mappings, Sibirsk. Mat. Zh., 35:6 (1994), 1287–1304. 256. Kusraev A. G. and Strizhevskiı̆ V. Z., Lattice-normed spaces and dominated operators, in: Studies on Geometry and Functional Analysis, 7 [in Russian], Trudy Inst. Mat. (Novosibirsk), Novosibirsk, 1987, 132–158. 257. Kusraev A. G. and Tabuev S. N., On disjointness preserving bilinear operators, Vladikavkaz Math. J., 6:1 (2004), 58–70. 258. Kusraev A. G. and Tabuev S. N., On multiplicative representation of disjointness preserving bilinear operators, Siberian Math. J., 49:2 (2008), 357–366. 259. Kusraeva Z. A., Algebraic band preserving operators, Vladikavkaz Math. J., 15:3 (2013), 54–57. 260. Kusraeva Z. A., Involutions and complex structures on real vector lattices, to appear. 261. Kutateladze S. S., Choquet boundaries in K-spaces, Uspekhi Mat. Nauk, 30:5 (1975), 107–146. 262. Kutateladze S. S., Supporting sets of sublinear operators, Dokl. Akad. Nauk SSSR, 230:5 (1976), 1029–1032. 263. Kutateladze S. S., Extreme points of subdiﬀerentials, Dokl. Akad. Nauk SSSR, 242:5 (1978), 1001–1003. 264. Kutateladze S. S., Convex operators, Uspekhi Mat. Nauk, 34:1 (1979), 167– 196. 265. Kutateladze S. S., The Kreı̆n-Mil′man theorem and its converse, Sibirsk. Mat. Zh., 21:1 (1980), 130–138. 266. Kutateladze S. S., On convex analysis in modules, Sibirsk. Mat. Zh., 22:4 (1981), 118–128. 267. Kutateladze S. S., Descents and ascents, Dokl. Akad. Nauk SSSR, 272:2 (1983), 521–524. 268. Kutateladze S. S., On the technique of descents and ascents, Optimizatsija, 33 (1983), 17–43. 269. Kutateladze S. S., Fragments of a positive operators, Sibirsk. Mat. Zh., 30:5 (1989), 111–119. 270. Kutateladze S. S., Fundamentals of Functional Analysis, Kluwer Academic Publishers, Dordrecht, 1995. 271. Kutateladze S. S., Nonstandard tools for convex analysis, Math. Japon, 43:2 (1996), 391–410. 272. Kutateladze S. S. (ed.), Vector Lattices and Integral Operators, Kluwer Academic Publishers, Dordrecht, 1996. 273. Kutateladze S. S., On diﬀerences of lattice homomorphisms, Siberian Math. References J., 46:2 (2005), 393–396. 377 378 References 274. Kutateladze S. S., On Grothendieck subspaces, Siberian Math. J., 46:3 (2005), 620–624. 275. Lacey H. E., The Isometric Theory of Classical Banach Spaces, SpringerVerlag, Berlin etc., 1974. 276. Lambek J., Lectures on Rings and Modules, Blaisdell, Toronto, 1966. 277. Lavrič B., On Freudenthal’s spectral theorem, Indag. Math. (N. S.), 48 (1986), 411–421. 278. Levin V. L., Tensor products and functors in categories of Banach spaces determined by KB-lineals, Dokl. Acad. Nauk SSSR, 163:5 (1965), 1058– 1060. 279. Lévy A., Basic Set Theory, Berlin etc., Springer-Verlag, 1979. 280. Lindenstrauss J. and Tzafriri L., Classical Banach Spaces. Vol. 1. Sequence Spaces, Springer-Verlag, Berlin etc., 1977. 281. Lindenstrauss J. and Tzafriri L., Classical Banach Spaces. Vol. 2. Function Spaces, Springer-Verlag, Berlin etc., 1979. 282. Lindenstrauss J. and Tzafriri L., On the isomorphic classification of injective Banach lattices, Advances Math., 7 (1981), 489–498. 283. Lindenstrauss J. and Wulbert D. E., On the classification of the Banach spaces whose duals are L1 -spaces, J. Funct. Anal., 4 (1969), 332–349. 284. Lipecki Z., Extensions of vector lattice homomorphisms, Proc. Amer. Math. Soc., 79 (1980), 247–248. 285. Locher J. L. (ed.), The World of M. C. Escher, Abradale Press, New York, 1988. 286. Loomis L. H., An Introduction to Abstract Harmonic Analysis, D. Van Nostrand Company Inc., Princeton, 1953. 287. Lotz, H. P., Über das Spektrum positiver Operatoren, Math. Z., 108 (1968), 15–32. 288. Lotz H. P., Extensions and liftings of positive linear mappings on Banach lattice, Trans. Amer. Math. Soc., 211 (1975), 85–100. 289. Lozanovskiı̆ G. Ya., On Banach lattices of Calderon, Dokl. Akad. Nauk SSSR, 172:5 (1967), 1018–1020. 290. Lozanovskiı̆ G. Ya., The functions of elements of vector lattices, Izv. Vyssh. Uchebn. Zaved. Mat., 4 (1973), 45–54. 291. Luxemburg W. A. J., Some Aspects of the Theory of Riesz Spaces, Univ. Arkansas Lect. Notes Math., 4, Fayetteville, Arkansas, 1979. 292. Luxemburg W. A. J., The work of Dorothy Maharam on kernel representation of linear operators, in: Measures and measurable dynamics, Rochester, New York, 1987, Amer. Math. Soc, Providence, 1989, 177–183. 293. Luxemburg W. A. J. and de Pagter B., Maharam extension of positive operators and f -algebras, Positivity, 6:2 (2002), 147–190. 294. Luxemburg W. A. J. and de Pagter B., Representation of positive projections I, Positivity, 9:2 (2005), 293–325. 295. Luxemburg W. A. J. and Schep A., A Radon-Nikodým type theorem for positive operators and a dual, Indag. Math., 40 (1978), 357–375. 296. Luxemburg W. A. J. and Schep A., An extension theorem for Riesz homomorphisms, Indag. Math., 41 (1979), 145–154. 297. Luxemburg W. A. J. and Zaanen A. C., Riesz Spaces, 1, North Holland, Amsterdam–London, 1971. 298. Luxemburg W. A. J., Zaanen A. C., The linear modulus of an order bounded References 379 linear transformation, Proc. Ned. Acad. Wetensch., A74 (1971), 422–447. 299. Maharam D., The representation of abstract measure functions, Trans. Amer. Math. Soc., 65:2 (1949), 279–330. 300. Maharam D., Decompositions of measure algebras and spaces, Trans. Amer. Math. Soc., 69:1 (1950), 142–160. 301. Maharam D., The representation of abstract integrals, Trans. Amer. Math. Soc., 75:1 (1953), 154–184. 302. Maharam D., On kernel representation of linear operators, Trans. Amer. Math. Soc., 79:1 (1955), 229–255. 303. Maharam D., On positive operators, Contemp. Math., 26 (1984), 263–277. 304. Maltsev A. I., Algebraic systems, Nauka, M., 1970; Engl. transl.: Berlin etc., Springer-Verlag, 1973. 305. Malyugin S. A., Quasi-Radon measures, Sibirsk. Mat. Zh., 32:5 (1991), 101– 111. 306. Malyugin S. A., On the lifting of quasi-Radon measures, Sibirsk. Mat. Zh., 42:2 (2001), 407–413. 307. Mangheni P. J., The classification of injective Banach lattices, Israel J. Math., 48 (1984), 341–347. 308. Maslyuchenko O. V., Mykhaylyuk V. V., and Popov M. M., A lattice approach to narrow operators, Positivity, 13 (2009), 459–495. 309. McPolin P. T. N. and Wickstead A. W., The order boundedness of band preserving operators on uniformly complete vector lattices, Math. Proc. Cambridge Philos. Soc., 97:3 (1985), 481–487. 310. Meyer M., Le stabilisateur d’un espace vectoriel réticulé, C. R. Acad. Sci. Ser. A., 283 (1976), 249–250. 311. Meyer-Nieberg P., Banach Lattices, Springer-Verlag, Berlin etc., 1991. 312. Mittelmeyer G. and Wolﬀ M., Über den Absolutbetrag auf komplexen Vektorverbänden, Math. Z., 137 (1974), 87–92. 313. Monteiro A., Généralization d’un théorème de R. Sikorski sur les algèbres Bool, Bull. Sci. Math., 89:2 (1965), 65–74. 314. Moy S.-T. C., Characterization of conditional expectation as transformations on function spaces, Pacific J. Math., 4 (1954), 47–54. 315. Nachbin L., A theorem of Hahn–Banach type for linear transformation, Trans. Amer. Math. Soc., 68 (1950), 28–46. 316. Nagel R. (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin–New York, 1986. 317. Nakano H., Über die Charakterizierung des allgemainen C-Raumen, Proc. Imp. Acac. Tokyo, 17 (1941), 301–307. 318. Nakano H., Modern Spectral Theory, Maruzen Co., Tokyo, 1950. 319. Nakano H., Modulated Semi-Ordered Linear Spaces, Maruzen Co., Tokyo, 1950. 320. Nakano H., Teilweise geordnete algebra, Japan J. Math., 17 (1950), 425–511. 321. Nakano H., Product spaces of semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser. I, 12 (1953), 163–210. 322. Neveu J., Calcul des Probabilités, Masson, Paris, 1964. 323. Ozawa M., Boolean valued interpretation of Hilbert space theory, J. Math. Soc. Japan, 35:4 (1983), 609–627. 324. Ozawa M., A classification of type I AW ∗ -algebras and Boolean valued analysis, J. Math. Soc. Japan, 36:4 (1984), 589–608. 380 References 325. Ozawa M., Boolean valued interpretation of Banach space theory and module structure of von Neumann algebras, Nagoya Math. J., 117 (1990), 1–36. 326. Ozawa M., Transfer principle in quantum set theory, J. Symbolic Logic, 72:2 (2007), 625–648. 327. de Pagter B., f -Algebras and Orthomorphisms, Ph. D. Dissertation, University of Leiden, 1981. 328. de Pagter, B., The components of a positive operator, Indag. Math. (N. S.), 48:2 (1983), 229–241. 329. de Pagter B., A note on disjointness preserving operators, Proc. Amer. Math. Soc., 90:4 (1984), 543–549. 330. de Pagter B., The space of extended orthomorphisms in Riesz space, Pacific J. Math., 112:1 (1984), 193–210. 331. de Pagter B. and Schep A. R., Band decomposition for disjointness preserving operators, Positivity, 4:3 (2000), 259–288. 332. Pavičić M. and Megill N. D., Binary orthologic with modus ponens is either orthomodular or distributive, Helv. Phys. Acta, 71 (1998), 610–628. 333. Pavičić M. and Megill N. D., Non-orthomodular models for both standard quantum logic and standard classical logic: Repercussions for quantum computers, Helv. Phys. Acta, 72 (1999), 189–210. 334. Pavičić M. and Megill N. D., Is quantum logic a logic? in: Handbook of Quantum Logic and Quantum Structures: Quantum logic (Engesser, K., Gabbay, D. M., and Lehmann, D. (eds.)), Elsevier, 2009, 23–47. 335. Peressini A. L., Ordered Topological Vector Spaces, Harper and Row, New York–London, 1967. 336. Piron C. P., Foundations of Quantum Physics, Benjamin Inc., New York, 1976. 337. Plichko A. M. and Popov M. M., Symmetric function spaces on atomless probability spaces, Dissertationes Math. (Rozprawy Mat.) 306 (1990), 1–85. 338. Popov M. M. and Randrianantoanina B., Narrow Operators on Function Spaces and Vector Lattices, De Gruyter, Berlin, 2012 (de Gruyter Studies in Mathematics, 45). 339. Ptak P. and Pulmannova S., Orthomodular Structures as Quantum Logics, Kluwer Academic Publishers, Dordrecht, 1991. 340. Radnaev V. A., On Metric n-Indecomposability in Ordered Lattice Normed Spaces and Its Applications, PhD Thesis, Sobolev Institute Press, Novosibirsk, 1997. 341. Radnaev V. A., On n-disjoint operators, Siberian Adv. Math., 7:4 (1997), 44–78. 342. Rao K. P. S. B. and Rao M. B., Theory of Charges. A Study of Finitely Additive Measures, Academic Press, London–New York, 1983. 343. Rao M. M., Conditional Measure and Applications, Marcel Dekker, New York, 1993. 344. Rasiowa H. and Sikorski R., The Mathematics of Metamathematics, Panstwowe Wydawnictwo Naukowe, Warszawa, 1963. 345. Rice N. M., Multiplication in vector lattices, Canad. J. Math., 20 (1968), 1136–1149. 346. Riesz F., Sur la décomposition des opérations fonctionnelles, Atti Congresso Internaz, Bologna 1928, 3 (1930), 143–148. 347. Riesz F., Sur quelques notions fundamentales dans la théorie générale des opérations linéaires, Ann. Math, 41 (1940), 147–178. References 381 348. van Rooij A. C. M., When do the regular operators between two Riesz spaces form a Riesz space? Dokl. Technical Report 8410, Katholike Universiteit, Nijmegen, 1984. 349. Rosenthal H. P., Embeddings of L1 in L1 , Contemp. Math., 26 (1984), 335– 349. 350. Rosser J. B., Simplified Independence Proofs. Boolean Valued Models of Set Theory, Academic Press, New York–London, 1969. 351. Rota G. C., On the representation of averaging operators, Rend. Math. Padova, 30 (1960), 52–64. 352. Rubinov A. M., Sublinear operators and their application, Uspekhi Mat. Nauk, 32:4 (1977), 113–174. 353. Sakai S., C ∗ -Algebras and W ∗ -Algebras, Springer-Verlag, Berlin etc., 1971. 354. Schaefer H. H., Spectral measures in locally convex algebras, Acta Math. 107 (1962), 125–173. 355. Schaefer H. H., Topological Vector Spaces, Springer-Verlag, New York–London, 1966. 356. Schaefer H. H., Banach Lattices and Positive Operators, Springer-Verlag, Berlin etc., 1974. 357. Schaefer H. H., Aspects of Banach lattices, in: MAA Stud. Math. (Bartle R. C., ed.), Math. Asoc. of America, Washington, 1980. 358. Schaefer H. H., Positive bilinear forms and the Radon–Nikod’ym theorem, in: Funct. Anal.: Survey and Recent Results, 3, Amsterdam e. a., 1984, 135–143. (Proc 3rd Conf. Paderborn, 24–29 May, 1983.) 359. Schep A., Factorization of positive multilinear maps, Illinois J. Math., 28 (1984), 579–591. 360. Schröder J., Das Iterationsferfahren bei allgemeinieren Abstandsbegriﬀ, Math. Z., 66:1 (1956), 111–116. 361. Schwarz H.-V., Banach Lattices and Operators, Teubner, Leipzig, 1984. 362. Segal I. E., A non-commutative extension of abstract integration, Ann. of Math., 57 (1953), 401–457. 363. Semadeni Zb., Banach Spaces of Continuous Functions, Polish Scientific Publishers, Warszawa, 1971. 364. Shultz F. W., On normed Jordan algebras which are Banach dual spaces, J. Funct. Anal., 31:3 (1979), 360–376. 365. Sikorski R., Boolean Algebras, Springer-Verlag, Berlin etc., 1964. 366. Sobczyk A. and Hammer P. C., A decomposition of additive set functions, Duke Math. J., 11 (1944), 839–846. 367. Sobolev V. I., On a poset valued measure of a set, measurable functions and some abstract integrals, Dokl. Akad. Nauk SSSR, 91:1 (1953), 23–26. 368. Solovay R. M., A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math., 92:2 (1970), 1–56. 369. Solovay R. M.,, Real-valued measurable cardinals, in: Axiomatic Set Theory (Proc. Sympos. Pure Math., 13:1, Univ. California, Los Angeles, 1967), Amer. Math. Soc., Providence, R. I., 1971, 397–428. 370. Solovay R. and Tennenbaum S., Iterated Cohen extensions and Souslin’s problem, Ann. Math., 94:2 (1972), 201–245. 371. Szarek S. J., A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc., 97 (1986), 437–444. 382 References 372. Szarek S. J., On the existence and uniqueness of complex structure and spaces with “few” operators, Trans. Amer. Math. Soc., 293 (1986), 339–353. 373. Székelyhidi L., Convolution Type Functional Equations on Topological Abelian Groups, World Scientific, Singapore, 1991. 374. Szulga J., (p, r)-convex functions on vector lattices, Proc. Edinburg Math. Soc., 37:2 (1994), 207–226. 375. Tabuev S. N., On a characterization of diﬀuse operators on vector lattices, Vladikavkaz Math. J., 2:1 (2000), 18–16. 376. Tabuev S. N., Decomposition of an atomic operator, Vladikavkaz Math. J., 5:1 (2003), 53–56. 377. Tabuev S. N., Bilinear Operators on Vector Lattices, PhD Thesis, Rostov-onDon, 2009. 378. Takano M., Strong completeness of lattice-valued logic, Archive for Math. Logic, 41 (2002), 497–505. 379. Takeuti G., Two Applications of Logic to Mathematics, Iwanami and Princeton Univ. Press, Tokyo and Princeton, 1978. 380. Takeuti G., A transfer principle in harmonic analysis, J. Symbolic Logic, 44:3 (1979), 417–440. 381. Takeuti G., Boolean valued analysis, in: Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977), Springer-Verlag, Berlin etc., 1979, 714–731. 382. Takeuti G., Quantum set theory, in: Current Issues in Quantum Logic (Beltrametti, E., van Frassen, B. C. (eds.)), Plenum, New York, 1981, 303–322. 383. Takeuti G., Boolean completion and m-convergence, in: Categorical Aspects of Topology and Analysis (Ottawa, Ont. 1980), Springer-Verlag, Berlin etc., 1982, 333–350. 384. Takeuti G., C ∗ -algebras and Boolean valued analysis, Japan. J. Math. (N. S.), 9:2 (1983), 207–246. 385. Takeuti G., Von Neumann algebras and Boolean valued analysis, J. Math. Soc. Japan, 35:1 (1983), 1–21. 386. Takeuti G. and Titani S., Heyting-valued universes of intuitionistic set theory, in: Logic Symposia Hakone (1979), Springer-Verlag, Berlin–New York, 1981, 189–306. 387. Takeuti G. and Titani S., Globalization of intuitionistic set theory, Annals of Pure and Applied Logic, 33 (1987), 195–211. 388. Takeuti G. and Zaring W. M., Axiomatic set Theory, Springer-Verlag, New York, 1973. 389. Tasoev B. B., Generalized functional calculus on lattice ordered algebras, Siberian Math. J., 44:8 (2014), 1185–1194. 390. Titani S., Lattice valued set theory, Archive for Mathematical Logic, 38:6 (1999), 395–421. 391. Titani S. and Kozawa H., Intern, J. Theor. Physics, 42:11 (2003), 2576–2602. 392. To T-O., The equivalence of the least upper bound property in ordered vector spaces, Proc. Amer. Math. Soc., 30:2 (1970), 287–296. 393. Topping D. M., Jordan algebras of self-adjoint operators, Mem. Amer. Math. Soc., 53 (1965). 394. Von Neumann J., Mathematische Grundlagen der Quantenmechanik, Dover Publ., New York, 1943; First edition: Springer-Verlag, Heidelberg, 1932. References 383 395. Veksler A. I., On representation partial multiplications in linear structures, Izv. Akad. Nauk SSSR, Ser. Mat., 31 (1960), 1203–1228. 396. Veksler A. I., On the homomorphisms between classes of regular operators in K-lineals and their completions, Izv. Vyssh. Uchebn. Zaved. Mat., 14:1 (1960), 48–57. 397. Veksler A. I., Projection properties of vector lattices and the Freudenthal theorem, Math. Nachr., 74 (1976), 7–25. 398. Veksler A. I. and Geyler V. G., Order and disjoint completeness of linear partially ordered spaces, Sib. Math. J., 13 (1972), 30–35. 399. Vladimirov D. A., Boolean Algebras, Nauka, Moscow, 1969. 400. Vopěnka P., General theory of ▽-models, Comment. Math. Univ. Carolin, 7:1 (1967), 147–170. 401. Vopěnka P, The limits of sheaves over extremally disconnected compact Hausdorﬀ spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 15:1 (1967), 1–4. 402. Vulikh B. Z., On linear multiplicative operations, Dokl. Akad. Nauk., 41 (1943), 148–151. 403. Vulikh B. Z., Introduction to the Theory of Partially Ordered Spaces [in Russian], Fizmatgiz, Moscow, 1961. 404. Vulikh B. Z. and Lozanovskiı̆G. Ya., On representation of complete linear and regular functionals in semiordered spaces, Mat. Sbornik (N. S.), 84:3 (1971), 331–354. 405. Van der Waerden B. L., Algebra, Springer-Verlag, Berlin etc., 1964; Engl. transl.: New York, 1949. 406. Weis L., Decompositions of positive operators and some of their applications, in: Functional Analysis: Survey and Recent Results, 3 (Proc. 3rd Conf., Paderborn, 24–29 May, 1983), Amsterdam, 1984. 407. Weis L., On the representation of order continuous operators by random measures, Trans. Amer. Math. Soc., 285:2 (1984), 535–563. 408. Wickstead A. W., Representation and duality of multiplication operators on Archimedean Riesz spaces, Comp. Math., 35:3 (1977), 225–238. 409. Wickstead A. W., Extensions of orthomorphisms, J. Austral Math. Soc., Ser A., 29 (1980), 87–98. 410. Wickstead A. W., Relatively central operators on archimedean vector lattices, Proc. Royal Irish Acad. Sect A, 80:2 (1980), 191–208. 411. Wickstead A. W., Stone algebra valued measures: integration of vector-valued functions and Radon–Nikodým type theorems, Proc. London Math. Soc., 45:2 (1982), 193–226. 412. Wickstead A. W., The injective hull of an archimedean f -algebra, Comp. Math., 62:4 (1987), 329–342. 413. Wickstead A. W., AL-spaces and AM -spaces of operators, Positivity, 4:3 (2000), 303–311. 414. Wickstead A. W., Regular operators between Banach lattices, in: Positivity (Boulabiar, K., Buskes, G., and Tiki, A. (eds.)), Birkhäuser, Basel etc., 2007, 255–279. 415. Wolﬀ M., Disjointness preserving operators on C*-algebras, Arch. Math., 62 (1994), 248-253. 416. Wittstock G., Eine Bemerkung über Tensorprodukte von Banachverbänden, Arch. Math., 25 (1974), 627–634. 384 References 417. Wittstock G., Ordered normed tensor products, Foundations of quantum mechanics and ordered linear spaces, Springer, Berlin, 1974, 67–84 (Lecture Notes in Phys; 29). 418. Wright J. D. M., A Radon–Nikodým theorem for Stone algebra valued measures, Trans. Amer. Math. Soc., 139 (1969), 75–94. 419. Wright J. D. M., Stone algebra valued measures and integrals, Proc. London Math. Soc., 19:3 (1969), 107–122. 420. Wright J. D. M., The measure extension problem for vector lattices, Ann. Inst. Fourier (Grenoble), 21 (1971), 65–68. 421. Wright J. D. M., The solution of a problem of Sikorski and Mattes, Bull. London Math. Soc., 3 (1971), 52–54. 422. Wright J. D. M., Vector lattice measure on locally compact spaces, Math. Z., 120 (1971), 1903–203. 423. Wright J. D. M., An algebraic characterization of vector lattices with the Borel regularity property, J. London Math. Soc, 7:2 (1973), 277–285. 424. Wright J. D. M., An extension theorem, J. London Math. Soc., 7:2 (1973), 531–539. 425. Wright J. D. M., Products of positive vector measures, Quart. J. Math., 24:94 (1973), 189–206. 426. Zaanen A. C., Examples of orthomorphisms, J. Approx. Theory, 13:2 (1975), 192–204. 427. Zaanen A. C., Riesz Spaces, 2, North Holland, Amsterdam etc., 1983. 428. Zaanen A. C., Introduction to Operator Theory in Riesz Spaces, SpringerVerlag, Berlin etc., 1997. 429. Zariski O. and Samuel P., Commutative Algebra, Springer-Verlag, Berlin, 1991. NAME INDEX Abdullaev R. Z., 356 Abramovich Yu. A., 110, 185, 188, 189, 258–264, 353 Aczél J., 258, 265, 266 Ajupov Sh. A., 112, 267, 356 Akilov G. P., 53, 110, 185, 351, 352 Albeverio S., 267 Alfsen E. M., 354, 356 Aliprantis C. D., 110, 118, 176, 185, 258, 264, 272, 302–304, 351, 352 Ando T., 350, 354 Antonovskiı̆ M. Ya., 116 Araujo J., 188 Arendt W., 188, 354 Arenson E. L., 185, 188, 189 Bade W. G., 114 Bar-Hillel Y., 45 Beckenstein E., 188 Behrends E., 355 Bell J. L., 1, 45–49, 111 Benamor F., 193 Ber A. F., 265, 267 Berberian S. K., 255 Bernau S. J., 185, 188, 190, 259 Bernays P. I., 45 Bigard A., 258, 259 Birkhoﬀ G., 50, 51, 109, 258, 259 Blass A., 258 Boltjanskiı̆ V. G., 116 Bonnice W., 185 Boulabiar K., 186, 188, 193, 259, 264 Bourbaki N., 264 Bourgain J., 266 Bu Q., 186, 187, 253 Buck R. C., 58 Bukhvalov A. V., 258 Burkinshaw O., 110, 118, 176, 185, 258, 272, 302–304, 351, 352 Buskes G., 115, 126, 185–188, 253, 254, 259, 264, 354 Cantor G., 44, 45 Cartwright D. I., 319, 354 Castillo E., 258 Cauchy A. L., 258 Chilin V. I., 255, 257, 265, 267 Church A., 48 Cignoli A., 50 Cohen P. J., 45, 46 Conrad P. F., 258, 259 Cooper J. L. B., 353, 260 Coppel W. A., 221 Cristescu R., 186 Cunningham F., 355 Czerwik S., 258 Dales H., 111 386 Name Index Dalla Chiara M. L., 50 Danckwerts R., 355 Dedekind R., 44 Dhombres J., 258, 265, 266 Diem J. E., 258, 259 Diestel J., 356 Dieudonné J., 266 Diskant H., 51 Dodds P. G., 114, 287, 288, 350 Douglas R. G., 285, 350 Duhoux M., 259 Dunford N., 114 Eﬀros E., 354 Enflo P., 193 Engesser K., 51 Escher M. C., 47 Evans R., 355 Foulis D. J., 50 Fourman M. P., 50 Fraenkel A. A., 45 Frege G., 44 Fremlin D. H., 125, 126, 186, 304 Freudenthal H., 109, 113 Gabbay D. M., 51 Ganiev I. G., 116 Geyler V. A., 73 Giuntini R., 50 Givant S., 6 Goodearl K. R., 265 Goodner D. B., 354 Gordon E. I., 110–113, 115, 117, 259, 311, 353, 358 Gordon H., 192 Gowers W. T., 266 Göbel S., 355 Gödel K., 45, 46 Gönüllü U., 116 Grayson R. J., 50 Greechie R. J., 50 Greim P., 355 Grobler J. J., 125, 126, 188, 350 Grothendieck A., 190 Gutman A. E., 116, 188, 191, 258, 259, 261–264, 267 Halmos P., 6, 48 Hamel G., 258 Hammer P. C., 193 Hanshe-Olsen H., 112, 356 Harmand P., 323, 355 Hart D. R., 188, 193 Hasumi M., 354 Haydon R., 115, 116, 317, 319, 354, 356, 357 Henriksen M., 188 Hofstedter D. R., 47 Huijsmans C. B., 188, 190, 192, 193, 259, 287, 288, 350, 259 Hyers D. H., 258 Ioﬀe A. D., 185 Isac G., 258 Jameson G. J. O., 185 Jarchow H., 356 Jech T., 1, 45, 49, 111–114 Jordan P., 356 Kalton N., 266 Kalmbach G., 51 Kannappan Pl., 258 Name Index Kantorovich L. V., 109, 110, 112, 113, 115, 116, 184, 185, 187, 188, 350, 352, 353 Kaplansky I., 357 Karimov J. A., 267 Keimel K., 258, 259 Kelley J. L., 354 King W., 338 Kitover A. K., 185, 188, 189, 258, 259, 260, 261, 263 Koldunov A. V., 188, 259–263 Kolesnikov E. V., 188, 192, 351, 352 Kollatz L., 116, 353 Koppelberg S., 49 Korol′ A. M., 257 Krasnosel′ skiı̆ M. A., 186 Kriger H. J., 186 Krivine J. L., 115 Kuczma M., 258, 266 Kudajbergenov K. K., 116, 267 Kulakova V. G., 350 Kurepa G., 115, 266, 352 Kusraev A. G., 1, 47, 48, 53, 110, 112–116, 185, 186, 188–191, 193, 253–255, 257, 259–265, 267, 279, 310, 311, 330, 335, 336, 341, 342, 346, 349–353, 355–359 Kusraeva Z. A., 263, 266 Kutateladze S. S., 1, 47, 48, 53, 110, 112–115, 171, 185, 188– 190, 193, 262, 264, 265, 267, 335, 336, 349, 352, 357 Labuschagne C., 125, 126 Lacey H. E., 110, 186 Lambek J., 252 Lavrič B., 113 387 Lehmann D., 51 Levi B., 45 Levin V. L., 356 Levy M., 115, 116 Lévy A., 45 Lifshits E. A., 186 Lindenstrauss J., 110, 115, 186, 187, 190, 354 Lipecki Z., 185 Lisovskaya S. A., 116 Locher J. L., 47 Loomis L. H, 347 Lotz H. P., 264, 317, 353 Lozanovskiı̆ G. Ya., 115, 279, 350 Luxemburg W. A. J., 53, 98, 110, 113, 115, 185, 186, 191, 259, 348–351 Lyubetskiı̆ V. A., 117 Maharam D., 340, 349–351, 357 Maltsev A. I., 48 Malyugin S. A., 114, 193, 348, 357, 358, 359 Mangheni P. J., 354 Maslyuchenko O. V., 192, 193 Maurey B., 266 McPolin P. T. N., 260–263 Megill N. D., 51, 52 Meyer M., 188, 258, 259 Meyer-Nieberg P., 54, 110, 118, 186, 188, 264, 302–304, 317, 352 Meyfarth K., 355 Mittelmeyer G., 259, 264 Monteiro A., 50 Mostowski A., 48 Moy S.-T. C., 350 Müller W., 355 388 Name Index Mykhaylyuk V. V., 192, 193 Nachbin L., 354 Nagel R., 186 Nakano H., 109, 113, 186, 258 Narici L., 188 von Neumann J., 45, 50, 51, 114, 356 Neveu J., 350 Ozawa M., 267 de Pagter B., 113–115, 187, 188, 190–192, 258, 259, 265, 267, 287, 288, 348–351 Pavičić M., 51, 52 Peano G., 45 Peressini A. L., 186 Pierce R. S., 258 Pinsker A. G., 110, 112, 115, 116, 186, 350 Piron C. P., 51 Plichko A. M., 192 Popov A., 187 Popov M. M., 192, 193 Ptak P., 51 Pulmannova S., 51 Pustylnik E. I., 186 Radnaev V. A., 190 Rakhimov A. A., 356 Rambane D. T., 350 Randrianantoanina B., 192 Rao K. P. S. B, 181 Rao M. B., 181 Rao M. M., 350 Rasiowa H., 46, 48 Rassias Th. M., 258 Raynaud Y., 115, 116 Rédei M., 50 Rice M. M., 350 Ricker W. J., 114 Riesz F., 109, 115, 184 van Rooij A., 115, 126, 186, 187, 254 Rosenthal H. P., 192 Rosser J. B., 111 Rota G. C., 350 Rubinov A. M., 190 Ruiz-Cobo M. R., 258 Russell B., 44 Sakai S., 257 Sasaki U., 51 Samuel P., 264 Sarymsakov T. A., 116 Schaefer H. H., 54, 110, 114, 186, 190, 236, 264, 317, 333, 334, 352, 356 Schep A. R., 185, 187, 190, 191, 259, 349, 350 Schlotterbeck U., 356 Schröder J., 353 Schwartz J. T., 114 Schwarz H.-V., 54, 110, 186, 352 Scott D. S., 46, 47, 50, 111 Segal I. E., 267 Semadeni Zb., 303, 304 Shultz F. W., 336, 356 Sikorski R., 6, 46, 48, 223, 225 Silvermann R., 185 Sirotkin G., 264 Skolem T., 45 Smith F. A., 188 Name Index Sobczyk A., 193 389 Usmanov Sh. M., 356 Sobolev V. I., 114, 115, 186 Sobolevskiı̆ P. E., 186 Solovay R., 46–50, 111 Starbird T., 193 Störmer E., 112, 356 Strizhevskiı̆ V. Z., 352 Veksler A. I., 73, 113, 188, 259, 260, 261, 262, 263 Vladimiriv D. A, 6 Vopěnka P., 46 Vulikh B. Z., 54, 110, 112, 113, 115, 116, 185, 186, 188, 279, 350 Sukochev F. A., 265, 267 Székelyhidi L., 258 Szarek S. J., 266, 267 Szulga J., 115, 187 Tabuev S. N., 188, 191–193, 254 Takano M., 51 Takeuti G., 1, 45, 50, 51, 111, 112, 117, 267, 347, 353, 358 Tarski A., 48 Tasoev B. B., 115 Tcacius A., 187 Tennenbaum S., 48–50 Titani S., 50, 51 To T.-O., 185 Tonge A., 356 Topping D. M., 357 Triki A., 186, 259 Troitsky V. G., 187 Tzafriri L., 110, 115, 186, 187, 354 van der Waerden B. L., 264 Weis L., 192 Wener W., 323, 355 Werner D., 323, 355 Wickstead A. W., 194, 259, 260– 263, 357 Wigner E., 356 Wittstock G., 186 Wolfenstein S., 258 Wolﬀ M., 189, 259, 264 Woodin W., 111 Wright J. D. M., 357–359 Wulbert D. E., 190, 354 Zaanen A. C., 53, 54, 98, 110, 113, 185, 186, 236, 258, 259, 264, 272 Zabreı̆ko P. P., 186 Zaring W. M., 1, 45, 111 Zariski O., 264 Zermelo E., 44, 45 SUBJECT INDEX Abelian projection, 257 Absolute definability, 259 Absolutely continuous, 274 — definable, 259 Abstract norm, 345 Algebraic operator, 232, 240 — B-system, 22 — closure, 240 Algebraically closed, 240 — independent, 240 Ample vector measure, 340 Antisymmetric operator, 254 Approximating sequence, 107 Archimedean vector lattice, 54 Ascent of a set, 18 — — a correspondence, 18 Assignment operator, 1 Associate space, 284 Atomic formula, 181 — measure, 2 — vector lattice, 57 Atom of a measure, 181 Averaging identity, 287 Axiom of Choice, 5, 42 — — extensionality, 5 — — foundation, 5, 42 — — infinity, 5, 42 — — pair, 42 — — pairing, 44 — — — — — — powerset, 5, 42 — replacement, 42 — separation, 44 — union, 5, 42 schema of replacement, 5 Baer ∗-algebra, 255 — criterion, 252 Banach f -module, 304 — lattice, 301 — space with mixed norm, 310 Banach–Kantorovich lattice, 296 — space, 308 Band of a vector lattice, 56, 236 — preserving operator, 195, 237 — projection, 56, 236 Bidisjoint collections, 190 Binary intersection property, 354 Boolean algebra of projections, 306 — antimorphism, 32 — domain, 43 — semimetric, 30 — set, 20 — truth value, 7 — valued representation of a B-set, 21 — — — of a vector lattice, 73 — — — of an f -ring, 251 — — — of a B-cyclic Banach lattice, 313 Subject Index — — — of a JB-algebra, 335 — — — of band preserving operators, 130 — — — of order bounded operators, 205 — — — of a Maharam operator, 278 — — universe, 7 Bound variable, 2 Bounded descent, 311 — formula, 10 — measure, 180 — trace, 337 — weight, 337 Brouwer lattice, 35 Calculus of o-bounded operators, 120 Calibrating function, 86 Canonical embedding, 13 — sublinear operator, 190 Cardinal, 27 — comparability principle, 27 Cardinality, 27 Carrier of an operator, 274 Carrier of a system, 22 Cartwright property, 319 Categorical, 52 Cauchy B-filter, 103 — functional equation, 198 — logarithmic equation, 262 Cayley numbers, 112 Center of a ∗-algebra, 256 — — a JB-algebra, 335 Central element, 256 — projection of a ∗-algebra, 256 — projections of a JB-algebra, 335 391 Centrally extended JB-algebra, 335 Characteristic function, 224 — of an element, 70 Church schema, 3 Classifier, 2 Cofinal family of band projections, 196 — — — projection bands, 196 Collapsed cardinal, 48 Compatible elements, 37 Completely additive function, 167 — disjoint elements, 113 — metrizable, 103 Completion, 225 Complex f -algebra, 65, 238 — structure, 248 — vector lattice, 64, 235 Complexication of an f -algebra, 65 — — a solution set, 201 — — a space of operators, 236 — — a vector lattice, 64 Component of a Boolean algebra, 56 — — an operator, 130 Composition of correspondences, 14 Cone, 137 — absolutely summing operators, 329 Containment, 2 Continued fraction expansion, 212 Continuous vector lattice, 57 Contractive mapping, 21 Cover of a Boolean algebra, 223 Cyclic hull, 13 — set, 13 — subsequence, 108 Subject Index Cyclically compact set, 108 Decomposable Λ-metric space, 101 — norm, 306 Decomposition theorem, 139 Decreasing net, 67 Dedekind complete vector lattice, 58 — completion, 99, 209 — cut, 208 Definor, 1 Dense subalgebra, 225 — subset of a ring, 251 Derivation extension theorem, 241 — of an algebra, 238 — — a field, 241 — — a ∗-algebra, 256 Descent of an algebraic system, 25 — — an element, 15 — — a correspondence, 16 — — a relation, 17 Diagonal operator, 233 Distinct elements, 213 Diﬀuse measure, 180 — operator, 130, 164 — vector lattice, 57 Dilatator, 258 Direct sum property, 352 Discrete element, 57 — vector lattice, 57 Disjoint complement, 56 — elements, 55 — sets, 55 Disjointly-decomposable norm, 306 Disjointness preserving bilinear operator, 173 — — measure, 180 392 — — operator, 130, 189 — — set, 176 Domain of a quantifier, 2 — — a correspondence, 14 Dominant of a linear operator, 119 — of a mapping, 345 Dominated mapping, 344 — operator, 119, 345 Eﬀective domain, 137 Element, 2 Enflo–Starbird function, 193 Equipollent sets, 27 Equipotent sets, 27 Escher rules, 19, 47 Espaces pseudodistanciés, 115, 352 Essentially nontrivial authomorphism, 247 — nontrivial derivation, 247 — positive operators, 258 Exhaustion principle, 9 Exponential operator, 262, 266 Extended orthomorphism, 195 Extension of a field, 240 Extensional mapping, 16 — correspondence, 16, 18 Extremal space, 86 Extremally disconnected space, 86 Fatou norm, 302 — property, 302 Field of reals, 58 Finite order intersection property, 319 — ordinal, 27 — set, 15 — signature, 21 Subject Index Finitely additive, 180 Formula valid within V(B) , 8 Fragment of an element, 56 — of an operator, 130 Free variable, 2 Fremlin tensor product, 125 Freudenthal spectral theorem, 82 Function integrable with respect to a spectral measure, 89 Generating set of projections, 352 Global intuitionistic set theory, 50 Gordon theorem, 59 Grothendieck subspace, 141 Hahn decomposition theorem, 282 Hahn–Banach theorem for Boolean homomorphisms, 33 Hahn–Banach–Kantorovich theorem, 121 Hamel basis, 258 Hammer–Sobczyk decomposition theorem, 181 Heuristic transfer principle, 110 Hexagon, 37 Heyting algebra, 35 Homomorphism of B-systems, 23 Ideal, 57 — center, 195 — space, 284 Image of a correspondence, 14 Increasing net, 67 — operator, 138 Induction principle, 7 Infinite cardinal, 27 — distributive laws, 55 393 Injective Banach lattice, 316 — module, 252 Inner derivation, 256 Integrable element of a vector lattice, 339 Interpretation mapping, 22 Intuitionistic predicate calculus IL, 38 Inverse correspondence, 14 Involution, 248 Involutory operator, 248 Isomorphic lattices, 123 — Maharam extensions, 295 Isomorphism extension theorem, 241 — of B-systems, 24 Join, 54 Jordan algebra, 334 Kakutani representation theorem, 304 Kakutani–Kreı̆ns representation theorem, 303 Kantorovich lemma, 119 — norm, 306 — principle, 185 — space, 58 Kantorovich–Banach space, 302 Kaplansky complete, 264 Lamperti operators, 188 Lateral completion, 73 Laterally complete algebraic B-system, 25 — — lattice normed space, 308 — — Λ-metric space, 101 — — module, 267 Subject Index — — vector lattice, 58 Lattice B-isometry, 310 — bimorphism, 125, 171 — homomorphism, 58, 237 — isometry, 303 — isomorphism, 123 — monomorphism, 123 — norm, 301, 306 — normed space, 306 — ordered algebra, 63 Least extension, 122 Levi norm, 302 — property, 274, 302, 324 Limit ordinal, 27 Linear multiplicative operation, 188 Local Hamel basis, 213 Localizable measure space, 304 Locally algebraic operator, 264 — algebraically independent, 264 — constant element, 217 — linearly independent, 213 — one-dimensional, 217 Logarithmic operator, 262 Loomis–Sikorski theorem, 92 Maharam extension, 295 — extension space, 295 — measure space, 304 — operator, 274, 324 — property, 274, 349 Majorant of an operator, 119 Majorizing operator, 345 — sublattice, 57 Massive sublattice, 57 Maximal element, 321 — n-disjoint set, 176 394 Maximum principle, 12 Meet, 54 Membership relation, 2 Metric, 100 — band, 306 Meyer theorem, 133 Minorizing sublattice, 57 Mixed norm, 309 Mix-compact set, 107 Mix-complete B-set, 21 — Banach lattice, 305 — Banach space, 310 — Λ-metric space, 101 Mixing, 12 — principle, 13 Mixture in a B-set, 21 — in a Banach lattice, 305 — — — Boolean valued universe, 13 — — — vector lattice, 96 — — — Λ-metric space, 102 — of a family, 310 Modified ascent, 20 — descent, 17 Modular measure, 340 Modulus of an element, 54 Modus ponens, 5, 38 Monotone norm, 301, 306 Monotonically complete Banach lattice, 302 Moreau–Rockafellar formula, 138 Multiplication operator, 258 Nakano Theorem, 283 Nakano–Stone completeness theorem, 303 Natural number, 27 Subject Index — embedding, 272 Naturals, 27 Negative part, 54 Nonatomic measure, 167 Noncommutative Wickstead problem, 255, 257 Nonexpanding mapping, 21 Norm disjoint elements, 306 — order bounded, 308 Normal weight, 337 — filter, 46 Normed lattice, 301 Octonions, 112 Operator of order bounded variation, 124 Ogasawara theorem, 121 Order bounded operator, 119 — — additive function, 201 — — bilinear operator, 124 — — set, 57, 236 — calculus, 120 — complete vector lattice, 58 — continuous operator, 120 — — mapping, 344 — — measure, 167 — — norm, 302 — dense ideal, 57 — — sublattice, 57 — ideal, 57 — intersection property, 354 — interval, 55 — — preserving, 274 — narrow operator, 192 — norm bounded operator, 309 — projection, 56 — semicontinuous, 302 395 — separable, 295 — summable family, 67 — unit, 57 — σ-complete vector lattice, 58 Ordered algebra, 63 — vector space, 54 Ordinal, 26 — number, 26 Orthocomplementation, 36 Orthocomplemented lattice, 36 Orthogonal elements, 36 Ortholattice, 36 Orthomodular lattice, 37 Orthomorphism, 123, 176, 195, 238 Orthoregular bilinear operator, 126, 254 Orthosymmetric bilinear operator, 126 Partition of unity, 12 Perfect f -module, 272 Point separating set, 280 Polar preserving endomorphism, 258 Polydisjoint operator, 146 Positive bilinear operator, 123 — cone, 54 — definite mapping, 344 — decomposition, 138 — element, 54 — integer, 27 — linear operator, 118 — measure, 180 — operator in a complex vector lattice, 236 — part, 54 Subject Index — semidefinite bilinear operator, 126, 171 Principal band, 56 — projection, 57 — — property, 57 Principle of cardinal comparability, 27 Projection band, 56 — property, 57 — in a JB-algebra, 335 Proper axiom, 5 Pseudocomplement, 35 Pseudoembedding operator, 130 Pseudo-Boolean algebra, 35 Pure extension of a field, 240, 245 Purely non-σ-distributive Boolean algebra, 247 Quantification laws, 38 Quantum logic, 37 — predicate calculus QL, 41 — set theory, 41 Quasi-Radon measure, 347 Quasiextremal space, 86 Quasiextremally disconnected space, 86 Quasiregular measure, 347 Quaternions, 111 Radon measure, 347 Radon–Nikodým theorem, 283 — — theorem for JB-algebras, 334 Range projection, 151 Rationally complete ring, 251 Reals, 59 Refined subset, 223 396 — function from a cover, 223 — function from a sequence of covers, 224 Regular bilinear operator, 124 — linear operator, 119, 237 — measure, 347 Regulator of convergence, 67 Relation preservation, 110 Relative pseudocomplement, 34 — uniform convergence, 67 Relatively maximal, 321 — uniformly complete vector lattice, 67 Resolution of the identity, 83 Restricted Cauchy functional equation, 202 — formula, 10 — transfer principle, 14 Riesz decomposition property, 55 Riesz–Kantorovich theorem, 120 Rules for canceling arrows, 19 Sandwich theorem, 33 Sasaki hook, 41 Semifinite measure, 338 — weight, 337 Semiprime f -algebra, 63 Separable extension, 241 Separably injective, 354 Separated Boolean valued universe, 9 — module, 252 Separately band preserving, 253 Separating mapping, 188 Sequentially order continuous operator, 121 Shadow, 158 Subject Index — of an operator, 191 Shift by a Boolean automorphism, 326 — of a disjoint preserving operator, 160 — operator in a vector lattice, 160 Signature, 21 Sikorski extension theorem, 34 Simple element, 308 Space with mixed norm, 309 Special axiom, 5 Spectral operator, 114 — integral, 89 — measure, 88 — — of an element, 94 — system, 83 — — of an element, 70 Splitting property, 319 Square of a vector lattice, 126 Square-mean closed, 235 Stabilisateur, 258 Standard cardinal, 28 — name, 13 — ordinal, 28 Steinitz theorem, 240 Stone theorem, 137 Stratum of an operator, 137 Strictly positive operator, 274 Strong Freudenthal property, 83 — homomorphism, 23 — order unit, 57 — unit, 57 Strongly diagonal operator, 233 — generating set, 164 Subdiﬀerential, 137 Sublattice, 236 397 Sublinear operator, 121, 137 Submorphism, 32 Summable in order, 67 Supermorphism, 32 Support of a quantum set, 43 Symmetric bilinear operator, 126, 171, 254 — diﬀerence, 21 Tautology, 23 Trace, 336 — of an element, 69 Transcendence basis, 240 — degree, 264 Transcendental extension, 240 Transfer principle, 12 Transfinite number, 26 Transitive set, 26 Trivial automorphism, 238 — involution, 248 Two-point relation, 137 Type I AW ∗ -algebra, 257 Underlying set of an algebraic system, 22 Uniformity generated by a Λ-metric, 102 Uniformly order continuous mapping, 344 Unit of a vector lattice, 57 — — an AM -space, 303 — — an algebra, 334 Unital f -module, 96 Universal completion, 99 Universally complete, 58 Universe, 2, 22 Up-down theorem, 351 Subject Index Valid formula, 23 Vector semimetric, 100 — lattice, 54 — measure algebra, 340 — norm, 305 — order, 54 — sublattice, 57 Von Neumann universe, 6 M -module, 305 M -projection, 304 T -saturated band, 277 Y -perfect f -module, 272 Y -valued measure, 338 B-JBW -algebra, 336 B-JBW -factor, 336 B-complete Banach lattice, 305 B-complete vector lattice, 96 Weak Freudenthal property, 83 — order unit, 57 — orthomorphism, 195 Weakly orthomodular ortholattice, 51 Weight on a JB-algebra, 336 — system, 161 Weighted conditional expectation operator, 286 Weight-shift-weight factorization, 160 Wickstead problem, 194 — — for bilinear operators, 253 B-isometry, 310 Zermelo–Fraenkel set theory, 2 B-σ-distributive Boolean algebra, 252 A-linear operator, 268 A-module homomorphisms, 268 AL-space, 303 AM -space, 303 AW ∗ -algebra, 256 AW ∗ -factor, 256 G-space, 141 JB-algebra, 334 JB-factor, 335 KB-space, 302 K-space, 58 Kσ -space, 58 398 B-cyclic Banach lattice, 305 B-cyclic Banach space, 310 B-dense subset, 313, 327 B-dual, 310 B-embeddable C ∗ -algebra, 257 B-isomorphism, 310 B-lattice, 96 B-linear operator, 310 B-metric, 20 B-reflexive, 316 B-set, 20 B-summing operator, 329 B-valued interpretation, 22 — semimetric, 30 — set, 7 P-automorphism, 202 — of an algebra, 238 P-derivative, 202 — of an algebra, 238 P-endomorphism of an algebra, 238 P(Y )-permutation, 151, 162 B-complete Λ-metric space, 102 B-cyclic base, 102 Subject Index B-decomposable Λ-metric space, 102 Q-valued universe, 43 Λ-metric space, 100 Λ-uniformly complete, 308 — convergent sequence, 308 Λ-valued norm, 305 — semimetric, 305 — state, 336 Φ-ample sublattice, 286 α-concavification, 187 α-convexification, 187 (γ, Y )-homogeneous, 164 (κ, λ)-distributive Boolean algebra, 48 λ-injective Banach lattice, 353 λ-uniformly Cauchy sequence, 308 — convergent sequence, 308 µ-atom, 167 µ-integrable element, 339 π-atom, 181 (ρ, e, π)-net, 109 ρo-fundamental net, 101 ρo-convergence, 101 ρr-convergence, 101 399 ρr-fundamental net, 101 σ-distributive Boolean algebra, 29 σ-inductive Boolean algebra, 225 d-basis, 260 d-decomposable norm, 306 d-expansion, 260 d-independence, 260 e-uniform convergence, 67 f -algebra, 63 f -module, 95 h-modular measure, 340 n-ary operation, 21 — predicate, 21 n-disjoint bilinear operator, 191 — linear operator, 146 — set of operators, 176 o-bounded operator, 119 — set, 57 o-convergence, 67 o-ideal, 57 o-limit, 67 o-sum, 67 o-summable family, 67 r-convergence, 101 r-limit, 67 Symbol Index :=, 1 |=, 1 ⇐⇒, 1 =⇒, 1 ⇔, 1 ⇒, 1 ↔, 1 ⊢, 1 →, 1 ZFC, 2 ZF, 2 AC, 2 (∃ x ∈ y) ϕ(x), 3 (∀ x ∈ y) ϕ(x), 3 ∃ !z, 3 P, 3 ∅, 3 {x, y}, 3 (x, y), 3 CL, 4 dom(f ), 4 im(f ), 4 Fnc, 4 f : x → y, 4 V, 6 V(B) , 6 [[ϕ]], 7, 40 V(B) ϕ, 8 a ⇒ b, 8 [[x = y]], 8 [[x ∈ y]], 8 π ∗ , 10 b ∧ u, 11 cyc(A), 13 mixξ∈Ξ (bξ xξ ), 13 x∧ , 13 im(Φ), 14 dom(Φ), 14 fin(X), 15 Pfin (X), 15 x↓, 15 Φ↓, 16 f ↓, 16 P ↓, 17 Ψ↓, 17 g↓, 17 Φ↑, 18 x↑, 18 mix(X), 19 f ↑, 19 Φ↑, 20 f ↑, 20 b1 △ b2 , 21 χ, 24 On, 26 Ord(x), 26 Tr(x), 26 ω, 27 ω0 , 27 ωα , 27 lim(x), 27 N, 27 Card(α), 28 U(A∧ ), 33 Hom(A, B), 33 x ⇒ y, 35 x∗ , 35 R(Ω), 36 x ◦| y, 37 Q, 37, 209 IL, 38 ⊥ ⊥(A), 38 Ex, 38 ZFI , 38 ∃˙ x, 39 ∀˙ x, 39 V(Ω) , 40 QL, 41 ZFQ , 41 ∨(A), 43 L(u), 43 V(Q) , 43 V(Γ) , 47 stab(x), 47 B ⊗ D, 49 V(L) , 51 N, 53 Z, 53 Q, 53 R, 53 C, 53 X+ , 54 x+ , 54 x− , 54 |x|, 54 x1 ∨ · · · ∨ xn , 54 x1 ∧ · · · ∧ xn , 54 M ⊥ N , 55 [a, b], 55 401 Symbol Index x ⊥ y, 55 M ⊥ , 56 PB , 56 [B], 56 B(X), 56 C(u), 56 X(u), 57 [u] := [u⊥⊥ ], 57 X δ , 58 X u , 58 R, 59 XC , 64 C , 65 b ∧ R, 66 o-lim, 67 r-lim, 67 ex , 69 exλ , 70 S(B), 83 {f < λ}, 86 {f λ}, 86 C∞ (Q), 87 σ(f, β), 88 σ(f, β), 88 Iµ , 89 Clop(Q), 92, 223 Clopσ (Q), 92 Bor(Q), 93 x̂(f ), 94 f , 94 mixξ∈Ξ bξ xξ , 96 U (ρ, e), 102 B-mix, 102 PrtN (B), 108 L(X, Y ), 119 Lr (X, Y ), 119 L∼ (X, Y ), 119 L+ (X, Y ), 119 L∼ c (X, Y ), 121 L∼ n (X, Y ), 121 Hom(X, Y ), 123 Orth(X), 123 BLr (X, Y ; Z), 124 BLbv (X, Y ; Z), 124 BL+ (X, Y ; Z), 124 X ⊗ Y , 125 (X ⊙ , ⊙), 126 BLor (X, Z), 126 X ∧∼ , 128 L∼ P (X, Y ), 128 L∼ a (X, Y ), 130 L∼ d (X, Y ), 130 (X ∧ )∼ a , 130 (X ∧ )∼ d , 130 L∼ dp (X, Y ), 130 ker(f ), 132 ∂p, 137 RT , 151 D(ϕ), 155 x/e, 158 1/e, 158 ∆↓ , 164 D↑ , 164 Orth(X, Y ), 176 ba(A ), 180 ba(A , Y ), 180 FP , 199 F0 (E , P), 199 Lbp (X), 203 Lbp (X, Y ), 203 L∼ bp (X), 203 L∼ bp (X, Y ), 203 LR∧ (X , Y ), 203 L∼ R∧ (X , Y ), 203 RR , 203 End(RR ), 205 (←, u], 209 [u, →), 209 1U , 224 P[x], 232 ϕT , 232 σp (T ), 232 XC , 235 TC , 236 L∼ (XC , YC ), 237 Lbp (XC ), 239 End(CC ), 239 K(E ), 240 D(C ↓), 245 DC∧ (C ), 245 MN (C ↓), 245 MC∧ (C ), 245 Z (A), 256, 335 L∼ A (X, Y ), 268 L∼ n,A (X, Y ), 268 LA (X, Y ), 268 XT , 274 YT , 274 CT , 274 Dm (T ), 274 NT , 274 BT (X), 277 E(·|Z0 ), 285 E (·|Σ0 ), 286 X ϕ , 289 U ↑ , 290 U ↓ , 290 U ↑↓ , 290 U ↿ , 290 U ⇃ , 290 U ↿⇃ , 290 (X̄, T̄ ), 295 L1 (T ), 296 S (T ), 298 S (X̄), 298 πG , 298 πe , 298 π x , 299 K, 299 A (T ), 299 C (T ), 299 C (X̄), 299 L (X, Y ), 304 ∼ (X, Y ), 304 Ln,A LA (X, Y ), 304 · , 305 402 ⊥ M⊥ , 306 ⊥ ⊥, 306 X # , 310 LB (X, Y ), 310 Xn# , 314 Mu , 321 L1 (Φ), 324 Ln,B (Z (X), Λ), 328 Symbol Index SB (X, Y ), 329 C∞ (Q, X), 329 Prtσ := Prtσ (B), 329 σB (T ), 329 C# (Q, X), 330 P(A), 335 Pc (A), 335 L 1 (µ), 339 L ∞ (µ), 339 L1 (µ), 340 C0 (G), 345 Cc (G), 345 Lm (X, E), 345 D(G, YC ), 345 Lp (Φ), 349 CONTENTS Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Chapter 1. Boolean Valued Requisites . . . . . . . . . . . . . . . . . . . . . 1 1.1. Zermelo–Fraenkel Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Boolean Valued Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3. Transformations of the Boolean Valued Universe . . . . . . . . . . . . . 9 1.4. Principles of Boolean Valued Set Theory . . . . . . . . . . . . . . . . . . . 12 1.5. Descents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6. Ascents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7. Algebraic B-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.8. Boolean Valued Algebraic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.9. Boolean Valued Ordinals and Cardinals . . . . . . . . . . . . . . . . . . . . 26 1.10. Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.11. Applications to Boolean Homomorphisms . . . . . . . . . . . . . . . . . 31 1.12. Variations on the Theme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.13. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Chapter 2. Boolean Valued Numbers . . . . . . . . . . . . . . . . . . . . 2.1. Vector Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Gordon’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Gordon’s Theorem Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Boolean Valued Reals Translated . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Vector Lattices Within Boolean Valued Reals . . . . . . . . . . . . . . 2.6. Order Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Freudenthal Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Representation of Vector Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. Spectral Measure and Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 53 58 63 67 70 75 78 83 89 404 Contents 2.10. Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.11. Boolean Valued Vector Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.12. Variations on the Theme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.13. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Chapter 3. Order Bounded Operators . . . . . . . . . . . . . . . . . . . 119 3.1. Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.2. Bilinear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.3. Boolean Valued Positive Functionals . . . . . . . . . . . . . . . . . . . . . . 128 3.4. Disjointness Preserving Operators . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.5. Diﬀerences of Lattice Homomorphisms . . . . . . . . . . . . . . . . . . . . 137 3.6. Sums of Lattice Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.7. Polydisjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.8. Sums of Disjointness Preserving Operators . . . . . . . . . . . . . . . . 152 3.9. Representation of Disjointness Preserving Operators . . . . . . . 157 3.10. Pseudoembedding Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.11. Diﬀuse operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 3.12. Variations on the Theme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.13. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Chapter 4. Band Preserving Operators . . . . . . . . . . . . . . . . . 196 4.1. Orthomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 4.2. The Cauchy Functional Equation . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.3. Representation of Band Preserving Operators . . . . . . . . . . . . . 205 4.4. Dedekind Cuts and Continued Fractions . . . . . . . . . . . . . . . . . . 210 4.5. Hamel Bases in Boolean Valued Models . . . . . . . . . . . . . . . . . . . 215 4.6. Locally One-Dimensional Vector Lattices . . . . . . . . . . . . . . . . . . 219 4.7. σ-Distributive Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 224 4.8. Band Preserving Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 4.9. Algebraic Band Preserving Operators . . . . . . . . . . . . . . . . . . . . . 234 4.10. Band Preserving Operators on Complex Vector Lattices . . 238 4.11. Automorphisms and Derivations on the Complexes . . . . . . . 242 4.12. Automorphisms and Derivations on Complex f -Algebras . 246 Contents 405 4.13. Involutions and Complex Structures . . . . . . . . . . . . . . . . . . . . . 250 4.14. Variations on the Theme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 4.15. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Chapter 5. Order Continuous Operators . . . . . . . . . . . . . . . . 272 5.1. Order Bounded Module Homomorphisms . . . . . . . . . . . . . . . . . 272 5.2. Maharam Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 5.3. Representation of Order Continuous Operators . . . . . . . . . . . . 284 5.4. Conditional Expectation Type Operators . . . . . . . . . . . . . . . . . 289 5.5. Maharam Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 5.6. Properties of Maharam Extension . . . . . . . . . . . . . . . . . . . . . . . . . 300 5.7. Banach Lattices and Banach f -Modules . . . . . . . . . . . . . . . . . . . 306 5.8. Lattice Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 5.9. Boolean Valued Banach Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 316 5.10. Injective Banach Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 5.11. Injectives: M -Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 5.12. Representation of Injective Banach Lattices . . . . . . . . . . . . . . 328 5.13. Operators Factorable Through Injective Banach Lattices . 333 5.14. Variations on the Theme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 5.15. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 Научное издание Серия МАТЕМАТИЧЕСКАЯ МОНОГРАФИЯ Выпуск 6 Кусраев А. Г., Кутателадзе С. С. БУЛЕВОЗНАЧНЫЙ АНАЛИЗ: ИЗБРАННЫЕ ТЕМЫ Ответственный редактор А. Е. Гутман Редакторы серии: Ю. Ф. Коробейник, А. Г. Кусраев Утверждено к печати Ученым советом Южного математического института Владикавказского научного центра Российской академии наук Компьютерная верстка В. В. Кибизова Подписано в печать 19.12.2014. Формат бумаги 60×841/16 . Усл. п. л. 23,48. Тираж 200 экз. Заказ є 123. Отпечатано ИП Цопановой А. Ю. 362000, г. Владикавказ, пер. Павловский, 3.

RELATED PAPERS

RELATED TOPICS

Log In

Boolean Valued Analysis: Selected Topics

Boolean Valued Analysis: Selected Topics

Related Papers

RELATED PAPERS

RELATED TOPICS