Boolean Valued Analysis: Background and Results A. G. Kusraev and S. S. Kutateladze Abstract The paper provides a brief overview of the origins, methods and results of Boolean valued analysis. Boolean valued representations of some mathematical structures and mappings are given in tabular form. A list of some problems arising outside the theory of Boolean valued models, but solved using the Boolean valued approach, is given. The relationship between the Kantorovich’s heuristic principle and the Boolean valued transfer principle is also discussed. Keywords Vector lattice · Kantorovich’s principle · Gordon’s theorem · Boolean valued analysis · Boolean valued representation Mathematics Subject Classification (2000) 06F25, 46A40. 1 Introduction In 1977, Eugene Gordon, a young teacher of Lobachevsky Nizhny Novgorod State University, published the short note [13] which begins with the words: This article establishes that the set whose elements are the objects representing reals in a Boolean valued model of set theory V(B) , can be endowed with the structure of a vector A. G. Kusraev () Southern Mathematical Institute, Vladikavkaz Scientific Center of the RAS, Vladikavkaz, Russia e-mail: kusraev@smath.ru S. S. Kutateladze Sobolev Institute of Mathematics, Siberian Branch of the RAS, Novosibirsk, Russia e-mail: sskut@math.nsc.ru © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. G. Kusraev, Z. D. Totieva (eds.), Operator Theory and Differential Equations, Trends in Mathematics, https://doi.org/10.1007/978-3-030-49763-7_9 91 92 A. G. Kusraev and S. S. Kutateladze space and an order relation so that it becomes an extended K-space1 with base2 B. It is shown that in some cases this fact can be used to generalize the theorems about reals to extended K-spaces. His note has become the bridge between various areas of mathematics which helps, in particular, to solve numerous problems of functional analysis in “semiordered vector spaces” [36] by using the techniques of Boolean valued models of set theory [6]. In the same year, at the Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis (Durham, July 9–11, 1977), Gaisi Takeuti, a renowned expert in proof theory, observed that if B is a complete Boolean algebra of orthogonal projections in a Hilbert space H , then the set whose elements represent reals in the Boolean valued model V(B) can be identified with the vector lattice of selfadjoint operators in H whose spectral resolutions take values in B; see [93]. These two events marked the birth of a new section of functional analysis, which Takeuti designated by the term Boolean valued analysis. The history and achievements of Boolean valued analysis are reflected in [56–58]. It should be mentioned that in 1969 Dana Scott foresaw that the new nonstandard models must be of mathematical interest beyond the independence proof, but he was unable to give a really good evidence of this; see [87]. In fact Takeuti found a narrow path whereas Gordon paved a turnpike to the core of mathematics, which justifies the vision of Scott. Boolean valued analysis signifies the technique of studying the properties of an arbitrary mathematical object by comparison between its representations in two different Boolean valued models of set theory. As the models, we usually take the von Neumann universe V (the mundane embodiment of the classical Cantorian paradise) and the Boolean valued universe V(B) (a specially-trimmed universe whose construction utilizes a complete Boolean algebra B). The principal difference between V and V(B) is the way of verification of statements: There is a natural way of assigning to each statement φ about x1 , . . . , xn ∈ V(B) the Boolean truthvalue [[φ(x1 , . . . , xn )]] ∈ B. The sentence φ(x1 , . . . , xn ) is called true in V(B) if [[φ(x1 , . . . , xn )]] = 1. All theorems of Zermelo–Fraenkel set theory with the axiom of choice are true in V(B) for every complete Boolean algebra B. There is a smooth and powerful mathematical technique for revealing interplay between the interpretations of one and the same fact in the two models V and V(B) . The relevant ascending-and-descending machinery rests on the functors of canonical embedding X → X∧ , descent X → X↓, and ascent X → X↑ acting between V and V(B) , see [56, 57]. Everywhere below B is a complete Boolean algebra and V(B) the corresponding Boolean valued model of set  theory; see [6, 99]. A partition of unity in B is a family (bξ )ξ ∈ ⊂ B such that ξ ∈ bξ = 1 and bξ ∧ bη = O 1A K-space or a Kantorovich space is a Dedekind complete vector lattice. An extended K-space is a universally complete vector lattice, cp. [4] and [104]. 2 The base of a vector lattice is the inclusion ordered set of all of its bands (that forms a complete Boolean algebra) [36, 104]. Boolean Valued Analysis: Background and Results 93 whenever ξ = η. The unexplained terms of vector lattice theory can be found in [4, 70, 71, 85, 104]. 2 Kantorovich’s Heuristic Principle Definition 1 A vector lattice or a Riesz space is a real vector space X equipped with a partial order ≤ for which the join x ∨ y and the meet x ∧ y exist for all x, y ∈ X, and such that the positive cone X+ := {x ∈ X : 0 ≤ x} is closed under addition and multiplication by positive reals and for any x, y ∈ X the relations x ≤ y and 0 ≤ y − x are equivalent. A band in a vector lattice X is the disjoint complement Y ′ of any set Y ⊂ X where Y ′ := {x ∈ X : (∀y ∈ Y ) |x| ∧ |y| = 0}. Let P(X) stand for the complete Boolean algebra of all band projections in X. Definition 2 A subset U ⊂ X is order bounded if U lies in an order interval [a, b] := {x ∈ X : a ≤ x ≤ b} for some a, b ∈ X. A vector lattice X is Dedekind complete (respectively, laterally complete) if each nonempty order bounded set (respectively, each nonempty set of pairwise disjoint positive vectors) U in X has a least upper bound sup(U ) ∈ X. The vector lattice that is laterally complete and Dedekind complete simultaneously is referred to as universally complete. Definition 3 An f -algebra is a vector lattice X equipped with a distributive multiplication such that if x, y ∈ X+ then xy ∈ X+ , and if x ∧ y = 0 then (ax) ∧ y = (xa) ∧ y = 0 for all a ∈ X+ . An f -algebra is semiprime provided that xy = 0 implies x ⊥ y for all x and y. A complex vector lattice XC is the complexification XC := X ⊕ iX (with i standing for the imaginary unity) of a real vector lattice X. Leonid Kantorovich was among the first who studied operators in ordered vector spaces. He distinguished an important instance of ordered vector spaces, a Dedekind complete vector lattice, often called a Kantorovich space or a K-space. This notion appeared in Kantorovich’s first fundamental article [35] on this topic where he wrote: In this note, I define a 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. Here Kantorovich stated an important methodological principle, the heuristic transfer principle for K-spaces, claiming that the elements of a K-space can be considered as generalized reals. Essentially, this principle turned out to be one of those profound ideas that, playing an active and leading role in the formation of a new branch of analysis, led eventually to a deep and elegant theory of Kspace rich in various applications. At the very beginning of the development of the theory, attempts were made at formalizing the above heuristic argument. In this direction, there appeared the so-called identity preservation theorems which claimed that if some proposition involving finitely many relations is proven for the reals then an analogous fact remains valid automatically for the elements of 94 A. G. Kusraev and S. S. Kutateladze every K-space (see [36, 71, 104]). The depth and universality of Kantorovich’s principle were demonstrated within Boolean valued analysis. See more about the Kantorovich’s universal heuristics and innate integrity of his methodology in [67]. The contemporary forms of above mentioned relation preservation theorems, basing on Boolean valued models, may be found in Gordon [15, 18, 21] and Jech [30]. 3 Boolean Valued Reals Boolean valued analysis stems from the fact that each internal field of reals of a Boolean valued model descends into a universally complete vector lattice. Thus, a remarkable opportunity opens up to expand and enrich the mathematical knowledge by translating information about the reals to the language of other branches of functional analysis. According to the principles of Boolean valued set theory there exists an internal field of reals R in a model V(B) which is unique up to isomorphism. In other words, there exists R ∈ V(B) for which [[ R is a field of reals ]] = 1. Moreover, if [[ R ′ is a field of reals ]] = 1 for some R ′ ∈ V(B) then [[ the ordered fields R and R ′ are isomorphic ]] = 1. By the same reasons there exists an internal field of complex numbers C ∈ V(B) which is unique up to isomorphism. Moreover, V(B) | C = R ⊕ iR. We call R and C the internal reals and internal complexes in V(B) . The fundamental result of Boolean valued analysis is Gordon’s Theorem [13] which reads as follows: Each universally complete vector lattice is an interpretation of the reals in an appropriate Boolean valued model. Formally: Gordon Theorem Let B be a complete Boolean algebra, R be a field of reals within V(B) . Endow R := R↓ with the descended operations and order. Then (1) The algebraic structure R is a universally complete vector lattice. (2) The field R ∈ V(B) can be chosen so that [[ R∧ is a dense subfield of R ]] = 1. (3) There is a Boolean isomorphism χ from B onto P(R) such that χ (b)x = χ (b)y ⇐⇒ b ≤ [[ x = y ]], χ (b)x ≤ χ (b)y ⇐⇒ b ≤ [[ x ≤ y ]] (x, y ∈ R; b ∈ B). For a detailed proof of the Gordon Theorem, see [45, 56, 58]. Observe also some additional properties of Boolean valued reals, namely multiplicative structure and complexification: Boolean Valued Analysis: Background and Results 95 Corollary 1 The universally complete vector lattice R↓ with the descended multiplication is a semiprime f -algebra with the ring unity 1 := 1∧ . Moreover, for every b ∈ B the band projection χ (b) ∈ P(R) acts as multiplication by χ (b)1. Corollary 2 Let C be the field of complex numbers within V(B) . Then the algebraic system C ↓ is a universally complete complex f -algebra. Moreover, C ↓ is the complexification of the universally complete real f -algebra R↓; i.e., C ↓ = R↓ ⊕ iR↓. Example 1 Assume that a measure space (, , μ) is semi-finite; i.e., if A ∈  and μ(A) = ∞ then there exists B ∈  with B ⊂ A and 0 < μ(B) < ∞. The vector lattice L0 (μ) := L0 (, , μ) (of cosets) of μ-measurable functions on  is universally complete if and only if (, , μ) is localizable (≡ Maharam). In this event Lp (, , μ) is Dedekind complete; see [11, 241G]. Observe that P(L0 (, , μ)) ≃ /μ−1 (0). Example 2 Given a complete Boolean algebra B of orthogonal 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 if and only if A = R λ dEλ and Eλ ∈ B for all λ ∈ R. Define the partial order in B by putting A ≥ B whenever Ax, x ≥ Bx, x holds for all x ∈ D(A) ∩ D(B), where D(A) ⊂ H stands for the domain of A. Then B is a universally complete vector lattice and P(B) ≃ B. Example 3 Let = R⇓ stands for the bounded part of the universally complete vector lattice R↓, that is, := {x ∈ R↓ : |x| ≤ C ∧ for some C ∈ R}. Then is a Dedekind complete vector lattice and ¯ := ⊕ i is a complex Dedekind complete vector lattice. Moreover, can be endowed with a norm x∞ := inf{α > 0 : |x| ≤ α1}. If μ is a Maharam measure and B in the Gordon Theorem is the algebra of all μmeasurable sets modulo μ-negligible sets, then R↓ is lattice isomorphic to L0 (μ); see Example 1. If B is a complete Boolean algebra of projections in a Hilbert space H then R↓ is isomorphic to B; see Example 2. The two indicated particular cases of Gordon’s Theorem were intensively and fruitfully exploited by Takeuti [92–95]. The object R↓ for general Boolean algebras was also studied by Jech [30, 31], and [32] who in fact rediscovered Gordon’s Theorem. The difference is that in [30] a (complex) universally complete vector lattice with unit is defined by another system of axioms and is referred to as a complete Stone algebra. By selecting special B’s, it is possible to obtain some properties of R. Remark 1 In 1963 P. Cohen discovered his method of ‘forcing’ and also proved the independence of the Continuum Hypothesis. A comprehensive presentation of the Cohen forcing method gave rise to the Boolean valued models of set theory, which were first introduced by D. Scott and R. Solovay (see Scott [87]) and P. Vopěnka [103]. A systematic account of the theory of Boolean valued models and its applications to independence proofs can be found in [6, 33, 91, 99]. 96 A. G. Kusraev and S. S. Kutateladze Remark 2 Gordon came to his theorem, while trying to solve the Solovay’s famous problem. Assuming the consistency with ZFC of the existence of inaccessible cardinal, R. Solovay established the following result: The statement “Every subset of R is Lebesgue measurable” is consistent with ZF+DC (Dependent choice), see [90]. The Solovay’s problem asks whether or not this result remains true without assumption of consistency of existence of inaccessible cardinal? Gordon failed to solve this problem but proved the following weaker assertion: The statement “The Lebesgue measure on R can be extended to a σ -additive invariant measure on the σ -algebra of sets definable by a countable sequence of ordinals” is consistent with ZFC,3 see [13, Theorem 7] and [16]. In order to prove this result he needed to examine a Boolean algebra B with a measure μ : B → R inside V(B) and identify the descent μ↓ : B↓ → R↓ of μ in V. Thus, he discovered that the algebraic structure of R↓ is a well-known object, and it is K-space, which he learned from the book [101]. Remark 3 Many delicate properties of the objects inside 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 [6, 33, 99]. Here it is worth mentioning two deep independence results in analysis: The sentences SH4 (Souslin’s Hypothesis) and NDH5 (No Discontinuous Homomorphisms) are independent of ZFC, see [29, 91] and [10], respectively. 4 Boolean Valued Representation of Structures Every Boolean valued universe has the collection of mathematical objects in full supply. Available in plenty are all sets with extra structure: groups, rings, algebras, normed spaces, operators etc. Applying the descent functor to these internal algebraic systems of a Boolean valued model, we distinguish some bizarre entities or recognize old acquaintances, which leads to revealing the new facts of their life and structure. 3 Earlier G. Saks [88] without assumption of existence of inaccessible cardinal proved that the statement “The Lebesgue measure on R can be extended to the σ -additive invariant measure defined on all subsets of R” is consistent with ZF + DC. 4 H: Every order complete order dense linearly ordered set having no first or last element is order isomorphic to the ordered set of reals R, provided that every collection of mutually disjoint nonempty open intervals in it is countable. 5 NDH: For each compact space X, each homomorphism from C(X, C) into a Banach algebra is continuous. Boolean Valued Analysis: Background and Results 97 It thus stands to reason to raise the following question: What structures significant for mathematical practice are obtainable by the Boolean values interpretation of the most typical algebraic systems? The answer is given in terms of Boolean sets. 1. A Boolean set or, more precisely, a B-set is by definition a pair (X, d), where X ∈ V, X = ∅, and d is a mapping from X×X to B satisfying for all x, y, z ∈ X the conditions: (1) d(x, y) = O if and only if x = y; (2) d(x, y) = d(y, x); (3) d(x, y) ≤ d(x, z) ∨ d(z, y). Each nonempty subset ∅ = X ⊂ V(B) provides an example of a B-set on assuming that d(x, y) := [[x = y]] = [[x = y]]∗ for all x, y ∈ X. Another example arises if we furnish a nonempty set X with the “discrete B-metric” d; i. e., on letting d(x, y) = 1 in case x = y and d(x, y) = O in case x = y. 2. For every B-set (X, d) there are a member X of V(B) and an injection ı : X → X′ := X ↓ such that d(x, y) = [[ı(x) = ı(y)]] for all x, y ∈ X and every x ′ ∈ X′ admits the representation x ′ = mixξ ∈ (bξ ı(xξ )), with (xξ )ξ ∈ ⊂ X and (bξ )ξ ∈ a partition of unity in B. The element X of V(B) is said to be the Boolean valued representation of the B-set X. If X is a discrete B-set then X = X∧ and ı(x) = x ∧ for all x ∈ X. If X ⊂ V(B) then ı↑ is an injection from X↑ to X within V(B) . Say that X is B-complete (or B-cyclic), whenever ι(X) = X′ . 3. A mapping f from a B-set (X, d) to a B-set (X′ , d ′ ) is contractive provided that d ′ (f (x), f (y)) ≤ d(x, y) for all x, y ∈ X. Assume that X and Y are some Bsets. Assume further that X and Y are the Boolean valued representations of X and Y , while ı : X → X ↓ and j : Y → Y ↓ are the corresponding injections. If f : X → Y is a contractive mapping then there is a unique member g of V(B) such that [[g : X → Y ]] = 1 and f = j −1 ◦ g↓ ◦ ı. 4. In case a B-set X has some a priori structure we may try to furnish the Boolean valued representation of X with an analogous structure, so as to apply the technique of ascending and descending to the study of the original structure of X. Consequently, the above questions may be treated as instances of the unique problem of searching a well-qualified Boolean valued representation of a B-set with some additional structure, algebraic B-systems. 5. Thus an algebraic B-system refers to a B-set endowed with a few contractive operations and B-predicates, the latter meaning B-valued contractive mappings. The Boolean valued representation of an algebraic B-system appears to be a conventional two valued algebraic system of the same type. This means that an appropriate B-completion of each algebraic B-system coincides with the descent of some two valued algebraic system. 6. The following table shows Boolean valued representations of some structures. Of course, all these representation results are applied to the study of their properties by means of Boolean valued analysis. For details, we refer to the sources indicated in the third column of the table (Table 1). 98 A. G. Kusraev and S. S. Kutateladze Table 1 Structures Algebraic structure with order, norm, etc. Complete Boolean algebra with a complete subalgebra Amalgated free product of Boolean algebras over B Universally complete Kantorovich space Boolean extension of a uniform space Rationally complete semiprime abelian ring Complete ring of fractions of a semiprime abelian ring Unital separated injective module Continuous geometrya Von Neumann algebra Kaplansky–Hilbert module B-complete C ∗ -algebra Type I AW ∗ -algebra AW ∗ -module Embeddable AW ∗ -algebra Banach–Kantorovich space Operator caps and faces B-simple groups and B-simple rings B-complete Banach space B-compactification (or cyclic compactification) B-Dedekind domainb B-complete Lie algebra over a Stone algebra AL∗ -algebrac Boolean valued representation Complete Boolean algebra Free product of Boolean algebras Field of reals Complete uniform space Arbitrary field The field of fractions of an integral domain Vector space Author [·], year Solovay and Tennenbaum [91] Can be extracted from [91] Gordon [13] Gordon and Lyubetskiı̆ [22–24] Gordon [19] Gordon [19] Gordon [20] Irreducible CGb Von Neumann factor Hilbert space C ∗ -algebra W ∗ -algebra End(H ) for a Hilbert space H Hilbert space Von Neumann algebra Banach space Caps and faces of sets of functionals Simple groups and Simple rings Banach space Stone-Čech compactification Nishimura [73] Ozawa [78], Takeuti [96] Takeuti [96], Ozawa [79, 80] Takeuti [97] Ozawa [80] Dedekind domainb Lie algebra Nishimura [75] Nishimura [76] L∗ -algebrac Nishimura [77] Ozawa [80] Ozawa [81] Kusraev [41] Kutateladze [64, 65] Takeuti [98] Kusraev [41, 42], Ozawa [84] Abasov and Kusraev [1] Boolean Valued Analysis: Background and Results B-complete J B-algebra B-complete B-dual J B-algebra Injective Banach lattice Kaplansky–Hilbert latticed Ordered preduals to injective Banach lattices 99 J B-algebra Dual J B-algebra Kusraev [43] Kusraev [43] AL-space (L1 space) Hilbert lattice L1 -preduals Kusraev [50, 54]e Kusraev [51] Kusraev, Kutateladze [59] aA continuous geometry (= CG) is a complete complemented modular lattice L satisfying the axioms: supα∈A (xα ∧ z) = (supα∈A xα ) ∧ z and infα∈A (yα ∨ z) = (infα∈A yα ) ∨ z for all z ∈ L, increasing family (xα )α∈A , and decreasing family (yα )α∈A in L. A continuous geometry with a trivial center is called irreducible, Neuman [102] b A B-Dedekind domain is a B-integral domain that is B-hereditary. A B-integral domain is a Bcomplete ring R in which every B-ideal of R is B-projective and for all a, b ∈ R with ab = 0 there exist e, f ∈ B such that ef = 0, e + f = 1, ea = 0, and f b = 0; see [75, p. 69]. A Dedekind domain is an integral domain in which every ideal is projective or, equivalently, each nonzero ideal is a product of prime ideals [7, Chap. 7, § 2] c An AL∗ -algebra is an AW ∗ -module L over a commutative von Neumann algebra A endowed with an A-bilinear operation [·, ·] : L × L → L and a unary ∗-operation (·)∗ : L → L such that for all u, v, w ∈ L we have: (1) [u, u] = 0; (2) [[u, v], w] + [[v, w]u] + [[w, u]v] = 0; (3) [u, v], w = v, [u∗ , w]; see [77, p. 245]. An L∗ -algebra is a complex Lie algebra L that is simultaneously a Hilbert space endowed with a ∗-operation satisfying [u, v], w = v, [u∗ , w] for all u, v, w ∈ L ; see [86] d A Kaplansky–Hilbert lattice over is a real √Banach lattice X such that X ⊕ iX is a Kaplansky–Hilbert module over ¯ and x :=  x, x ∞ for all x ∈ X, see Example 3. A Kaplansky–Hilbert lattice over = R is called a Hilbert lattice, see [71]. The norm x + iy := √ x, x + y, y∞ is given incorrectly in [51] e Some related results can be found in [51, 59, 60] 5 Boolean Valued Representation of Operators 1. Let X be a normed space and let E be a vector lattice. Say that a linear operator T : X → E has an abstract norm or is dominated if the image T (BX ) of the unit ball BX of X is order bounded in E. Assume now that X is a multinormed space and E has an order unit 1. An operator T is called piecewise bounded if there is a partition of unity (πα ) in P(E) and a family of continuous seminorms (pα ) such that |πα T x| ≤ 1pα (x) for all α and x ∈ X 2. An operator T : E → F between two vector lattices is said to be interval preserving whenever T is a positive operator and T [0, x] = [0, T x] holds for each x ∈ E+ . A Maharam operator is an order continuous interval preserving operator. An operator T : E → E on vector lattice is said to be band preserving if x ⊥ y implies T x ⊥ y for all x, y ∈ E or, equivalently, whenever T keeps all bands of E invariant, i. e., T (B) ⊂ B holds for each band B of E. 3. Consider a B-complete Banach space Y . Denote by Prtσ (B) the set of all countable partitions of unity in B. Say that a sequence (yn )n∈N B-approximates y ∈ Y if, for each k ∈ N, we have inf{supnk πn (yn − y) : (πn )n≥k ∈ Prtσ (B)} = 0. Call a set K ⊂ Y B-compact if K is B-complete and every sequence (yn )n∈N ⊂ K B-approximates some y ∈ K. An operator from a normed space X to Y is called B-compact or cyclically compact if the image of every norm bounded subset of X lies in some B-compact subset of Y . 100 A. G. Kusraev and S. S. Kutateladze 4. Suppose E is a Banach lattice. A linear operator T : E → Y is cone B-summing if and only if there exists a positive constant C such that for every finite collection x1 , . . . xn ∈ E there is a countable partition of unity (πk )k∈N in B such that the inequality   n   n   sup πk T xi  ≤ C  |x | i   k∈N i=1 i=1 holds, see [50]. Observe that if B = {0, IY } then a cone B-summing operator is a cone absolutely summing operator; cp. [85, Ch. 4]. 5. Let P = R or P = C. Given an algebra A over the field P, we call a P-linear operator D : A → A a derivation provided that D(uv) = D(u)v + uD(v) for all u, v ∈ A. It can be easily seen that an order bounded derivation of a universally complete f -algebra is trivial (Table 2). Table 2 Operators Operator, representation homomorphism, etc. Unitary representation of an LCA group Ordinary differential operator with parameters in Ba B-valued Fourier transform on LCA groups Linear operator with abstract norm Conditional expectation B-Compact operator Maharam operator Piecewise bounded linear operator Differential polynomial on D ′ (Rn , C) or S ′ (Rn , C) with coefficients in Cb Unitary representation of a locally compact group Band preserving operator Derivation on a universally complete f -algebra over C Cone B-summing operator Weighted conditional expectation type operator a See Boolean valued representation Author [·], year Character of an LCA group Takeuti [93] Ordinary differential operator Takeuti [93] Fourier transform on LCA groups Norm bounded linear functional Lebesgue integral Compact operator Order continuous positive functional Continuous linear functional Constant coefficients differential polynomial on D ′ (Rn ), S ′ (Rn ) Irreducible unitary representation R∧ -linear function on Boolean valued reals Derivation on the complex plane Cone absolutely summing operator Weighted conditional expectation operator Takeuti [94] Gordon [14, 17] Gordon [17] Kusraev [39] Kusraev [40] Sikorskiı̆ [89] Sikorskiı̆ [88, 89] Nishimura [74] Kusraev [46] Kusraev [47] Kusraev [49] Kusraev, Kutateladze [58] Example 2 in § 3 (resp. S ′ (Rn , C)) is the space of all piecewise bounded operators from D (Rn ) (resp. n S (R ) to C), where C := C ↓ = R ⊕ iR is a complex universally complete vector lattice, see Corollary 2 b D ′ (Rn , C) Boolean Valued Analysis: Background and Results 101 6 Problems and Solutions Boolean valued analysis sheds new light on some old problems and generates a large number of new ones. We now give a small list of problems that arose independently of the theory of Boolean valued models, but which were solved by means of Boolean valued analysis. Details as well as many other aspects of Boolean valued analysis may be found in the books [10, 21, 56–58, 92] and the survey papers [23, 34, 61] (Table 3). Table 3 Problems and solutions Problem Stems from Intrinsic characterization of subdifferentials General disintegration in Kantorovich spaces Kutateladze [63] Kaplansky Problem: Homogeneity of a type I AW ∗ -algebra The trace problem for finite AW ∗ -algebra Wickstead problem: Order boundedness of all band preserving operators Maharam extension of a positive operator Goodearl problem 18 in [12] B-Atomic decomposition of vector measures (into a sum of spectral measures) Classification of AJ W -algebrasa Kaplansky [38] Description of operators T with |T | a sum of two lattice homomorphisms Classification of injective Banach lattices Grothendieck [25] a An Ioffe, Levin [28]; Neumann [72] Kaplansky [37] Wickstead [105] Reduced to (by means of BA) Weakly compact convex sets of functionals Hahn–Banach and Radon–Nikodým theorems Homogeneity of End(H ) with H a Hilbert space The trace problem for a W ∗ -factor Solvability of Cauchy type functional equations Solved Kusraev and Kutateladze [55] Kusraev [40] Ozawa [80] Ozawa [82, 83] Gutman [26] and Kusraev [47] Luxemburg and Schep [69] Goodearl [12] Daniel extension of an elementary integral Theorem 12.16 in [12] Akilov, Kolesnikov, and Kusraev [2, 3] Chupin [9] Hoffman-Jørgenson [27] Hammer–Sobczyc decomposition theorem Kusraev and Malyugin [62] Topping [100] Classification of predual JB-factors (J BW -factors) Description of functionals with the same property Kusraev [52, 53] Classification of AL-spaces (L1 spaces) Kusraev [52, 53] Cartwright[8] and Lotz [68] Kutateladze [66] AJ W -algebra is a J B-algebra with a Jordan counterpart of Baire condition (= annihilators are generated by projections), see [5]. 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