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 xÎy, with x and y in , are defined by recursion. The sign Þ 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 right-hand 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 j(×) is a formula of ZFC, then the phrase “x satisfies j inside a Boolean valued universe” or, briefly, “j(x) is true inside ” means that the Boolean truth-value of j(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.