Set Theory and von Neumann algebras Rom an Sasyk ENS Lyon & - PowerPoint PPT Presentation

Set Theory and von Neumann algebras Rom´ an Sasyk ENS Lyon & Universidad de Buenos Aires Paris, May 25, 2011 Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 1

Set Theory and von Neumann algebras The purpose of this talk is to discuss some recent results about the isomorphism relation for von Neumann factors, achieved by applying tools from descriptive set theory. This is joint work with Asger T¨ ornquist from the University of Vienna and the University of Copenhagen. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 2

Classical descriptive set theory Recall that a Polish space is a completely metrizable separable topological space. E.g. R , N N , { 0 , 1 } N = 2 N , C ([0 , 1]). Descriptive set theory is the study of definable subsets of Polish spaces, and definable functions on Polish spaces. Definable sets and functions include ◮ Borel sets and functions , i.e. those sets that occur in the σ -algebra generated by the open sets. ◮ Analytic sets : Those sets that are the image of a Polish space under a continuous function. Definition. A standard Borel space is a set X equipped with a σ -algebra B which is itself generated by the open sets of a Polish topology on X . Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 3

Motivation Why is descriptive set theory relevant to classification problems? ◮ If the category or class of objects to be classified are themselves countable or separable, then there often is a natural Polish space that “parametrizes” the class: I.e., a Polish space X where every element of the class is “coded” by some x ∈ X . ◮ The relation of isomorphism among the objects in the class usually turns out to be Borel or analytic as a subset of X × X . ◮ Any reasonable solution to a classification problem has to associate the invariants in the classifying category in a calculable (and therefore definable) way: Otherwise it does not provide a useful tool for distinguishing the isomorphism classes. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 4

Example: The Effros-Borel space There is a Borel structure in the space of separable von Neumann algebras: Let H be a separable Hilbert. Let vN( H ) denote the von Neumann algebras acting on H The associated Borel structure is the Effros Borel structure , i.e. the one generated by the sets { N ∈ vN( H ) : N ∩ U � = ∅} where U ranges over the weakly open subsets of B ( H ). Theorem (Effros ’64): Sets of factors of types I, II 1 , II ∞ , III, are Borel sets in vN( H ). (Same for III λ , 0 ≤ λ ≤ 1.) To give a Borel structure to vN( H ) was the first step in Effros’s attempt to show that there exists uncountably many factors. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 5

Example: Ergodic measure preserving transformations Fix a standard measure space ( X , B , µ ). Denote by MPT the set of measure preserving invertible transformations of X . MPT is a Polish space (via the Koopman representation). The set EMPT of ergodic measure preserving transformations is a G δ subset of MPT : Indeed fix a dense sequence { f i } in L 2 ( X , B , µ ), then EMPT = N � � � � � T n f i − { T ∈ MPT : || 1 / N f i d µ || 2 < 1 / j } i j N n =1 Thus EMPT is a Polish space. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 6

Example: Countable groups The set of countable groups with underlying sets N is also a Polish space.It may be identified with the set GP = { ( f , e ) ∈ N N × N × N :The operation n · f m = f ( n , m ) defines a group operation on N with identity e } This set is easily seen to be G δ and thus is Polish. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 7

Borel vs Analytic sets Historically, descriptive set theory started with an example of Luzin (1917) of a set A that was the continuous image of a Borel set but A itself was not Borel. (i.e. there exists sets that are analytic and non Borel) Properties of Analytic sets. Unions, countable intersections, continuous images, continuous inverse images of analytic sets are analytic. However the complement of an analytic set is usually not analytic. In fact Theorem (Suslin 1917): a set A and its complement are both analytic iff A is Borel. (This is the starting point of what is called projective hierarchy of sets) In concrete examples, “interesting” sets are obviously Analytic. The question is: How to show that a given set is analytic and not Borel? Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 8

Complete analytic sets Notation. Given two subsets A and B of two Polish spaces X and Y , B ≤ W A means that there exist a Borel function f : Y → X such that f − 1 ( A ) = B . Definition. An analytic set A ⊂ X is complete analytic if for all B ⊂ Y analytic, B ≤ W A . Observation 1: If A is complete analytic, then A is not Borel. Observation 2: If A ⊂ X is complete analytic, C ⊂ Z is analytic and A ≤ W C then C is also complete analytic. Observation 3: There exists complete analytic sets. These observations provide a strategy to show that a given set C ⊂ Z is analytic and not Borel, namely: a) Find a suitable Polish space X and a complete analytic subset A ⊂ X . b) Find f : X → Z Borel such that A ≤ W C . Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 9

Analytic equivalence relations Definition. An equivalence relation E on a Polish space X is analytic if E viewed as a subset of X × X is an analytic set. Theorem: Foreman-Rudolph-Weiss 2006-2010.The conjugacy relation of ergodic measure preserving transformations is complete analytic. i.e. the set { ( S , T ) ∈ EMPT × EMPT : S is conjugate to T } is a complete analytic set. Quoting from their paper: “This [theorem] can be interpreted as saying that there is no method or protocol that involves a countable amount of information and countable number of steps that reliably distinguishes between non-isomorphic ergodic measure preserving transformations. We view this as a rigorous way of saying that the classification problem for ergodic measure preserving transformations is intractable.” Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 10

Proving the F.R.W. Theorem The proof of this Theorem appeared in the last issue of Annals of Mathematics.The strategy of the proof is the one outlined in the steps a) and b). for step a) They take X the space of countable trees and A the set of trees with an infinite branch. for step b) They construct a cont. function f: X → EMPT such that f ( t ) has an infinite branch iff f ( t ) is conjugate to f ( a ) − 1 . This implies that the set { T ∈ EMPT : T is conjugate to T − 1 } is complete analytic. Problem. Find another proof of this theorem. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 11

Isomorphism of II 1 factors is complete analytic Theorem: S.-Tornquist 2008.The isomorphism relation of II 1 factors is complete analytic. i.e. if we denote by F II 1 ( H ) the (standard) space of II 1 factors on H , then the set { ( N , M ) ∈ F II 1 ( H ) × F II 1 ( H ) : N is iso. M } is a complete analytic set. In fact this result is a corollary of a stronger statement about “Borel reducibility”. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 12

Borel reducibility: Less classical set theory Borel reducibility is a theory within descriptive set theory which has been developed as a tool to measure the relative complexity of classification problems that arise naturally in mathematics. Its development has lead to what is now a large programme of analyzing the descriptive set theory of definable equivalence relations . (Definable=Borel, analytic, etc.). Borel reducibility was first introduced in the late 80’s by Friedman and Stanley in the context of model theory but it was quickly taken over by descriptive set theorists. (Kechris, Louveau, Hjorth, etc.)The starting point comes from a generalization of theorems of Mackey, Glimm and Effros in operator algebras. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 13

Borel reducibility Definition. Let E and F be equivalence relations on standard Borel spaces X and Y respectively. E is Borel reducible to F , written E ≤ B F , if there is a Borel f : X → Y such that xEy ⇐ ⇒ f ( x ) Ff ( y ) . This means that the points of X can be classified up to E -equivalence by a Borel assignment of invariants that are F -equivalence classes. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 14

Borel reducibility f is required to be Borel to make sure that the invariant f ( x ) has a reasonable computation from x . Without a requirement on f , the definition would only amount to studying the cardinality of X / E vs. Y / F . Observation. If E ≤ B F , E is complete analytic and F is analytic then F is also complete analytic. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 15

Classification by countable structures Definition. Let E be an equivalence relation on a Polish space X . E is classifiable by countable structures if E ≤ B E Y S ∞ Where S ∞ is the infinite symmetric group and E Y S ∞ denotes an equivalence relation induced by a continuous S ∞ -action on a Polish space Y . The motivation behind this “definition” is that isomorphism of countable structures, (countable groups, graphs, fields, etc.) can be realized as S ∞ -actions on appropriate Polish spaces. Rom´ an Sasyk Descriptive Set Theory and von Neumann algebras 16