Boolean Valued Analysis and Positivity
A. G. Kusraev and S. S. Kutateladze
Slide 1. Boolean Valued Analysis and Positivity by Kusraev A. G. (Vladikavkaz) and
Kutateladze S. S. (Novosibirsk).
This talk overviews Boolean valued analysis in its interaction with positivity.
The term Boolean valued analysis minted by Gaisi Takeuti signifies a general mathematical
method that rests on a special model-theoretic technique. This technique consists primarily in
studying the properties of an arbitrary mathematical objects by means of comparison between its
representations in two different set-theoretic models whose constructions utilizes principally distinct
Boolean algebras. The comparative analysis presumes that there is some close interconnection
between two models under consideration.
Our starting point is a brief description of the best Cantorian paradise in shape of the von
Neumann universe and a specially-trimmed Boolean valued universe that are usually taken as these
two models. Then we present a special ascending and descending machinery for interplay between
the models.
We consider the reals and complexes inside a Boolean valued model by using the celebrated
Gordon's Theorem which we reads as follows: Every universally complete vector lattice is an
interpretation of the reals in an appropriate Boolean-valued model.
We proceed with demonstrating the Boolean valued approach to the two familiar problems:
1. When is a band preserving operator order bounded?
2. When is an order bounded operators a sum or difference of two lattice homomorphisms?
In conclusion we overview some typical spaces and operators together with their Boolean
valued representations.
Slide 2. Kusraev A. G. and Kutateladze S. S., Introduction to Boolean Valued Analysis [in
Russian], Moscow: Nauka, 2005.
Professor M. Weber had invited us to this Positivity Conference at the end of 2004 when we
were completing this book . The book was recently published and so this talk is a kind of
presentation.
Slide 3. Saunders Mac Lane (August 4, 1909 – April 14, 2005), Cofather of Category Theory.
Another recent event of relevance to this talk is grievous. Saunders Mac Lane passed away in
San Francisco on April 14, 2005. We reverently dedicate this talk to the memory of this great
master and servant of mathematics.
The power of mathematics rests heavily on the trick of socializing the objects and problems
under consideration. The understanding of the social medium of set-theoretic models belongs to
category theory.
A category is called an elementary topos provided that it is cartesian closed and has a subobject
classifier. A. Grothendieck and F.W. Lawvere, the followers of Saunders Mac Lane, created topos
theory in the course of “point elimination” and in the dream of invariance of the objects we operate
in mathematics. It is on this road that we met the conception of varying sets, underlying the notion
of topos and bringing about the understanding of the social medium of set-theoretic models. The
Boolean valued models belong happily in the family of Boolean toposes enjoying the classical
Aristotelian logic.
Slide 4. The von Neumann Universe.
The von Neumann universe  results by recursion with On for the class of all ordinals and
presents the unions of transfinite collections of sets. As the initial object 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 these steps, we exhaust the universe .
Slide 5. A Boolean Valued Universe.
We start with recalling some auxiliary facts about the construction and treatment of Boolean
valued models. Let  be a complete Boolean algebra. Just as in the case of the von Neumann
universe a Boolean valued universe  is introduced by recursion and presents to the union of
transfinite collections of sets. Again, the starting set is empty and the case of a limit ordinal yields
the union of preceding levels, while in the case of a nonlimit ordinal we take the set of all functions
from subsets of the preceding level to the Boolean algebra under consideration .
Observe that every Boolean valued universe consists only of functions. In particular, the empty
set is the function whose domain and image are empty.
The internal equality of two elements (inside ) implies in no way that these elements
coincide as functions (members of ). This circumstance involves inconveniences. Therefore, we
pass to a separated Boolean-valued universe. To define it, we declare two elements equivalent if
they are equal inside . Choosing an element (a representative of least rank) in each class of
equivalent functions, we arrive at a separated Boolean valued universe. We still denote it .
Slide 6. Boolean Valued Truth. Principles of Analysis.
The Boolean truth-values of the atomic formulas x=y and xОy, with x and y in , are defined by
recursion. The sign  symbolizes the implication in  and a* is the complement of a. The intuitive
idea is in viewing a  -valued set y is a “fuzzy set,” i.e., a “set that contains an element z in domain
y with probability y(z).”
The Boolean truth value of an arbitrary formula of ZFC is introduced by induction on the length
of a formula on naturally interpreting the propositional connectives and quantifiers in . The righthand sides of these definitions involve Boolean operations corresponding to the logical connectives
and quantifiers on the left-hand sides.
Now we are able to define the sense in which a set-theoretic proposition with constants in .
We say that the formula is valid inside  if the Boolean truth-value of this formula is equal to
unity: If x is an element of a Boolean valued universe and () is a formula of ZFC, then the phrase
“x satisfies  inside a Boolean valued universe” or, briefly, “(x) is true inside ” means that the
Boolean truth-value of (x) is equal to unity.
The Transfer Principle. Boolean valued universe with the Boolean truth-value of a formula is
a model of set theory in the sense that every theorem of ZFC is true inside .
The Maximum Principle. The least upper bound is attained on the right-hand side of the
formula for the Boolean truth-value of the existentential quantifier.
A partition of unity in a Boolean algebra  is an arbitrary family of pairwise disjoint elements
whose least upper bound is equal to the unity. If x is a -valued set and b is an arbitrary element of
 then bx denotes a -valued set defined as the infimum of x and the constant singleton {b}
function.
The Mixing Principle. Every Boolean valued universe is rich in mixings.
Proliferation of Boolean valued models is due to P.Cohen’s final breakthrough in Hilbert's
Problem Number One (Continuum Problem). His method of forcing was rather intricate and the
inevitable attempts at simplification gave rise to the Boolean valued models by Dana Scott, Robert
Solovay, and Peter Vopenka.
Slide 7. Ascending and Descending: The Escher Rules.
Boolean valued analysis is impossible to carry out without some dialog between the von
Neumann and Boolean valued universes. In other words, we need a smooth mathematical toolkit
for interplay. The relevant ascending-and-descending technique rests on the functors of canonical
embedding, descent, and ascent.
We start with the canonical embedding of the von Neumann universe . Given a set, we define
its standard name; i.e., the element in  defined by recursion.
The Standard Name Functor implements an embedding of  into . Moreover, the standard
name sends  onto the subclass of two-valued sets.
Thus, the standard name can be considered as a covariant functor from the category of sets in 
onto the subcategory of two-valued sets in . In other words, the standard name of a function is a
function from the standard name of the domain into the standard name of the image inside .
Composites and identity mappings are preserved by the standard name functor.
The Descent Functor. Given an arbitrary element X of  , we define the descent of X as the set
of all Boolean valued sets in X inside . The descent of a function is an extensional function. The
descent of the internal composite of functions is the composite of their descents. Moreover, the
descent operation is a functor from the category of Boolean valued sets and mappings to the
category of the von Neumann sets and mappings. The descent of an internal relation is an external
relation. Therefore, the descent of an internal algebraic structure is also an external algebraic
structure of a similar type.
The Ascent Functor. Let X be a set (= an element of ) composed of Boolean valued sets.
Then there is a Boolean valued set defined as the function with domain X and range the unity of .
This element is called the ascent of X.
The ascent of an extensional function is an internal function. The composite of extensional
fuctions is extensional. Moreover, the ascent of a composite is equal to the internal composite of the
ascents. Thus, the ascent operation can be considered as a functor from the category of subsets of
^ and extensional functions into the category of Boolean valued sets and mappings.
The Escher rules. Given a set X of Boolean valued sets, we denote by mix(X) the set of all
mixings of subfamilies of X by arbitrary partitions of unity in . The subsequent application of
ascent and descent obeys some simple rules referred to as the arrow cancellation rules or
ascending-and-descending rules. There are many good reasons to call them simply the Escher
rules.
Slide 8. Maurits Cornelis Escher. Ascending and Descending, 1960. Lithograph, 35.5x28.5.
Maurits Cornelis Escher (1889–1972), an outstanding Dutch graphic artist. The graphic works
of Maurits Escher are neither figments of imagination, nor proof of scientific theories, nor
documents of an eyewitness. What the artist shows is not the reality empirically known by No
staircase exists on which you could go down as you are going up as in the lithograph Ascending and
Descending (1960), showing a castle full of soldiers who are doomed for ever to go in circles within
the inclosed spatial whirlpool.
Slide 9. Boolean Valued Numbers.
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 enjoys the axiom of completeness. The same object
can be defined as a one-dimensional K-space.
It is provable in Zermelo–Fraenkel set theory that there exists a field of reals that is unique up to
isomorphism. By the transfer principle, the same assertion is true inside any Boolean valued model.
By the maximum principles, we find an element R in the Boolean valued universe that is an internal
field of reals. Moreover, if there is one more field R’ of internal reals then R and R ’ are
isomorphic.
By the same reasons there exists an internal field of complex numbers C which is unique up to
isomorphism. Moreover, C is the complexification of R. We call R and C the internal reals and
internal complexes.
Now, the standard name of the field of usual reals is also an internal Archimedean ordered field.
In Zermelo–Fraenkel set theory the latter can be considered as a dense subfield of the field of reals;
thus by transfer the standard name of the usual (external) reals can be viewed as an internal dense
subfield of the internal reals. Analogously, the standard name of the usual (external) complexes can
be considered as an internal dense subfield of the internal complexes.
Look now at the descent of the internal reals R that is the descent of the algebraic system R. In
other words, consider the descent of the underlying set of R together with descended operations and
order. For simplicity, we denote the internal operations and order by the encircled symbols.
The fundamental result of Boolean valued analysis is Gordon's Theorem which reads as follows:
The descent of the internal reals (with the descended operations and order) is a universally
complete Kantorovich space. In addition, there is a Boolean isomorphism from  onto the Boolean
algebra of all band projections which gives an interconnection between internal and external
equality and inequality relation.
By the same reasons the descent of the internal complexes C (with the descended operations
and order) is a universally complete complex Kantorovich space, which is the complexification of
the descent of R.
Thus, each universally complete Kantorovich space is an interpretation of the reals (or
complexes) in an appropriate Boolean valued model. This fact opens up a remarkable opportunity to
expand and enrich the treasure-trove of mathematical knowledge by translating information about
the reals (and complexes) to the language of other noble families of functional analysis.
Applications of Boolean valued models to functional analysis stem from the works by E.I.
Gordon and G. Takeuti. The Gordon Theorem of was first established in 1977 and rediscovered in
by T. Jech in 1985. If  is the algebra of measurable sets modulo negligible sets then the descent of
R is isomorphic to the universally complete K-space of measurable functions. If  is a complete
Boolean algebra of projections in a Hilbert space then the descent of R is isomorphic to the space of
all (densely defined) selfadjoint operators whose spectral resolutions take values in . These two
particular cases of Gordon's Theorem were intensively and fruitfully exploited by G.Takeuti.
Slide 10. Leonid Kantorovich (January 19, 1912–April 7, 1986). Cofather of the Theory of
Vector Lattices.
The theory of vector lattices and positive operators with a vast field of applications is
thoroughly covered in many monographs. The credit for finding the most important instance among
ordered vector spaces, an order complete vector lattice or K-space, is due to L.V. Kantorovich. This
notion appeared in Kantorovich's first article on this topic (1935) 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.”
Thus, the heuristic transfer principle was stated for Kantorovich spaces which becomes the
Ariadna thread of many subsequent studies. The depth and universality of Kantorovich’s principle
are explained within Boolean valued analysis. Indeed, each theorem about the reals within
Zermelo--Fraenkel set theory has an analog in the original universally complete K-space.
Translation of theorems is carried out by general operations of Boolean valued analysis. This makes
precise the Kantorovich motto: “The members of every K-space are generalized reals.”
Slide 11. Band Preserving Operators.
Now we give some examples demonstrating the Boolean valued machinery in settling the
problems of operator theory. First, we deal with the class of band preserving operators. Simplicity
of these operators notwithstanding, the question about their order boundedness is far from trivial.
Recall that a linear operator in a Kantorovich space is band preserving if every band is its
invariant subspace. A linear operator D in a lattice ordered algebra is said to be a derivation if it
obeys the Leibnitz rule D(fg)=D(f)g+fD(g) for all f,g. It can be easily checked that every derivation
is band preserving.
From any two partitions of unity in an arbitrary Boolean algebra one can refine a partition of
unity by taking infimum of any pair of members of the partitions. The same is true for a finite set of
partitions of unity. A -complete Boolean algebra  is called -distributive if from every sequence
of countable partitions of unity in , it is possible to refine a (possibly, uncountable) partition of
unity.
We now address the problem, which is often referred to in the literature as Wickstead's problem:
Characterize the universally complete vector lattices in which every band preserving linear operator
is order bounded. We restrict exposition to the case of complex Kantorovich space.
By the Gordon theorem we can regard the Kantorovich space under consideration as the descent
C of the internal complexes C. It is easy to prove that a linear operator in the K-space is band
preserving if and only if it is extensional. Since each extensional mapping has an ascent, we arrive
at the assertion: The modules of all linear band preserving operators End(C) in the complex Kspace C and the descent of the internal space End(C,P) of P-linear functions in the internal
complexes C (considered as a vector space over P) are isomorphic by sending each band preserving
operator to its ascent.
Moreover we arrived at an internal Cauchy type functional equation: find an additive function in
the internal complexes that is P-homogeneous for some dense subfield P. If P coincides with the
field of rationals then we obtain exactly the Cauchy functional equation inside Boolean valued
model.
Slide 12. Band Preserving Operators (continued).
Let P be a dense subfield of the internal complexes C. Therefore, Wickstead’s problem is
reduced to the existence of P-linear functions in the internal complexes. The corresponding scalar
result reads as follows.
Theorem. Let P be a dense subfield of the field of complexes C. The following are equivalent:
(1) P coincides with the complexes;
(2) Every P-linear function on the complexes is order bounded;
(3) There are no nontrivial P-derivations on the complexes;
(4) Each P-linear endomorphism on the complexes is the zero or identity function;
(5) There is no P-linear automorphism on the complexes other than the identity.
The equivalence (1) - (2) is checked by using a Hamel basis of the vector space C over P. The
remaining equivalences rest on replacing a Hamel basis with a transcendence degree of C over P.
By transfer this fact is true for an arbitrary dense subfield P of the internal complexes inside
any Boolean valued model. Let P be the standard name of the external complexes. Then Booleanvalued interpretation leads to the following two results. Let  be a complete Boolean algebra, C be
the field of complexes inside the corresponding Boolean valued model, and C be the descent of C.
Theorem (A.E. Gutman, 1995). The following are equivalent:
(1) The standard name of the external complexes coincides with the internal complexes;
(2) Every band preserving linear operator is order bounded in C;
(3)  is -distributive.
Theorem (A.G.Kusraev, 2005). The following are equivalent:

The standard name of the external complexes coincides with the internal complexes;

There is no nontrivial derivations in C;

Each band preserving endomorphism is a band projection in C;

There is no band preserving automorphism other than the identity in C;

 is -distributive.
Slide 13. Boolean Valued Positive Functionals.
Let E be a vector lattice, and let F be a K-space with the complete Boolean algebra of band
projections . By the Gordon Theorem, we may assume that F is a nonzero space embedded as an
order dense ideal in the universally complete Kantorovich space, which is the descent of the internal
reals R.
The standard name of a vector lattice is an internal vector lattice over the standard name of the
external reals. Therefore, if T is a linear operator from E into F then its ascent is an internal linear
functional from the standard name of E into the internal reals R. Moreover, a linear operator and its
ascent are order bounded (or positive) simultaneously. Thus, the ascent and descent operations
implement a lattice isomorphism of the vector lattice of all linear order bounded operators from E to
F and the descent of the internal space of all linear order bounded functionals in the standard name
of E.
The Boolean valued representation does not preserve order continuity and is not suitable for the
study of order continuous operators. But it may reduce some problems on general order bounded
and positive operators to those on functionals and provide a rather promising approach. Consider an
instance of this approach.
A linear functional on a vector space is determined up to a scalar from its zero hyperplane. In
contrast, a linear operator is recovered from its kernel up to a simple multiplier on a rather special
occasion. Fortunately, Boolean valued analysis prompts us that some operator analog of the
functional case is valid for each operator with target a Kantorovich space.
Theorem. Let S and T be linear operators from E to F. Then ker(bS) is contained in ker(bT) for
any band projections b  if and only if there is an orthomorphism  of F such that S=T.
We see that a linear operator T is, in a sense, determined up to an orthomorphism from the
family of the kernels of the strata bT of T. This remark opens a possibility of studying some
properties of a linear operator in terms of the kernels of its strata.
Theorem (S.S. Kutateladze, 2005). An order bounded operator from a vector lattice to a Kspace may be presented as the difference of two lattice homomorphisms if only if the kernel of each
its stratum is a vector sublattice of the ambient vector lattice.
Recall that a subspace of a vector lattice is a Grothendieck subspace provided that it contains
sup{x, y, 0}+inf{x, y, 0} for every two elements x and y.
Theorem (S.S. Kutateladze, 2005). The modulus of an order bounded operator from a vector
lattice to a K-space is the sum of some pair of lattice homomorphisms if and only if the kernel of
each strata of this operator is a Grothendieck subspace of the ambient vector lattice.
Slide 14. Different Classes of Spaces and Operators.
1. A lattice normed space (LNS) is a vector space endowed with a norm taking values in some
vector lattice. A lattice normed space is decomposable if its every element admits a decomposition
into the sum of two elements with given norms provided that the sum of these norms is equal to the
norm of an initial element. A Banach–Kantorovich space (BKS) is a decomposable LNS which is
complete in the sense that each norm order fundamental net has a norm order limit.
2. The disjointness in LNS can be introduced by setting two elements disjoint if their norms are
disjoint. With this in mind now a band preserving operators acting between two BKS with the same
norm lattice is defined just as in the case of vector lattices. A linear operator is said to be order
bounded if it sends every norm order bounded set into a norm order bounded set.
3. A Maharam operator is an order continuous order interval preserving linear operator acting
in a Kantorovich space. We also say that a linear operator enjoys the Maharam property if it is order
interval preserving. The domain of a Maharam operator T is a decomposable LNS if it is endowed
with the vector valued norm T(|x|).
Slide 15. Different Classes of Spaces and Operators (continued).
4. A Banach space is said to be -cyclic if there is a complete Boolean algebra of norm one
projections  in it, we may find a unique mixing of each norm bounded family by each partition of
unity in , and the unit ball is closed under mixings. Given a partition of unity (b()), we refer to an
element x satisfying the condition b()x()=b()x for all  as a mixing of (x())) by (b()).
5. Consider an order complete complex AM-space with fixed order unity. It can be furnished by
a uniquely defined multiplicative structure and then it is also called a Stone algebra. Replacing in
the definition of a Hilbert space the complex valued inner product by a Stone algebra valued inner
product we arrive at the definition of a Kaplansky–Hilbert module. Given a module over Stone
algebra and a Stone algebra valued inner product we can define a vector valued norm as square root
of the inner square. Now a Kaplansky–Hilbert module is a BKS whose norm is generated by a
Stone algebra valued inner product.
6. An AW*-algebra is a C*-algebra presenting a Baer *-algebra. More explicitly, an AWalgebra is a C*-algebra whose every right annihilator is generated by a projection. (A projection p
in an AW*-algebra is a Hermitian (p=p*) idempotent (p p= p) element).
An element is said to be central if it commutes with every member of an aalgebra. The center of
an AW*-algebra is the set of all central elements. Clearly, the center is a commutative AW*subalgebra and the set of all central projections is a complete Boolean algebra. If the center is onedimensional then a AW*-algebra is called a AW*-factor.
7. JB-algebra is a real nonassociative analog of C*-algebra. More precisely, a JB-algebra is a
real Banach space which is a unital Jordan algebra with the norm satisfying three additional
conditions. The set of all squares in JB-algebra is a proper convex cone and determines in it the
structure of an ordered vector space so that the unit of the algebra serves as a strong order unit, and
the order interval [-1, 1] serves as a unit ball.
The intersection of all maximal associative subalgebras is called a center. If the center is onedimensional then a JB-algebra is called a JB-factor. Idempotent elements are called projections. The
set of all projections containing in the center of a JB-algebra is a Boolean algebra . If a JB-algebra
with the Boolean algebra of central projections  is -cyclic then it is called -JB-algebra.
Slide 16. Table.
1. An arbitrary Banach space inside a Boolean-valued model may be interpreted as an external
universally complete Banach–Kantorovich space and vice versa any external Banach–Kantorovich
space may be represented as a Banach space in an appropriate Boolean valued model.
2. An internal bounded linear operator in a Banach spaces is interpreted as an external order
bounded band preserving linear operator in a Banach–Kantorovich space. Conversely, each order
bounded band preserving operator in a Banach–Kantorovich space can be represented as a linear
bounded operator in an appropriate Boolean valued model.
3. An internal order continuous positive linear functional in a Kantorovich space may be
interpreted as an external Maharam operators. Conversely, each essentially positive Maharam
operator is represented as an internal order continuous positive linear functional in a Kantorovich
space in an appropriate Boolean valued model.
This fact enables us to claim that each fact about order continuous positive linear functionals in
K-spaces has a parallel version for the Maharam operators which can be revealed by Boolean
valued representation.
4. A Boolean valued representation of a -cyclic Banach space is an internal Banach space. A
Banach space is linearly isometric to the bounded descent of some internal Banach space if and only
if it is -cyclic for some complete Boolean algebra of norm one projections .
5. The bounded descent of an internal Hilbert space in is a Kaplansky–Hilbert module.
Conversely, any Kaplansky–Hilbert module can be represented as a Hilbert space in an appropriate
Boolean valued model.
The Boolean valued transfer implies that the scope of the formal theory of the initial
Kaplansky–Hilbert module is the same as that of internal Hilbert space, its Boolean valued
representation.
6. The bounded descent of an AW*-factor inside  is an AW*-algebra whose Boolean algebra
of central projections is isomorphic with . Conversely, each AW*-algebra is represented as an
AW*-factor in an appropriate Boolean valued model.
An AW*-algebra is embeddable if and only if its Boolean valued representation is an internal
von Neumann algebra.
7. The bounded descent of a JB-factor inside  is a -JB-algebra. Conversely, each -JBalgebra can be represented as an internal JB-factor in .