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
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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].
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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).
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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 .
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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]. For some related results, see [44, 48]
102
A. G. Kusraev and S. S. Kutateladze
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