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¨

• rnquist 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, NN, {0, 1}N = 2N, 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.

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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.

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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, II1, II∞, III, are Borel sets in vN(H). (Same for IIIλ, 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.

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Example: Ergodic measure preserving transformations

Fix a standard measure space (X, B, µ). Denote by MPT the set

• f 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 Gδ subset of MPT: Indeed fix a dense sequence {fi} in L2(X, B, µ), then EMPT =

• i
• j
• N

{T ∈ MPT : ||1/N

N

• n=1

T nfi −

• fidµ||2 < 1/j}

Thus EMPT is a Polish space.

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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) ∈ NN×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 Gδ and thus is Polish.

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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?

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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.

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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.”

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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

• f 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 toT −1} is complete analytic.

• Problem. Find another proof of this theorem.

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Isomorphism of II1 factors is complete analytic

Theorem: S.-Tornquist 2008.The isomorphism relation of II1 factors is complete analytic. i.e. if we denote by FII1(H) the (standard) space of II1 factors on H, then the set {(N, M) ∈ FII1(H) × FII1(H) : N is iso. M} is a complete analytic set. In fact this result is a corollary of a stronger statement about “Borel reducibility”.

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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.

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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.

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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.

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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.

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Examples of countable structures

Example 1: Graphs as a countable structure. GRAPHS = {f : N × N → {0, 1}; f (x, x) = 0; f (x, y) = f (y, x))} f1 ∼ f2 ⇐ ⇒ ∃φ : N → N bijection s.t. f1(x, y) = f2(φ(x), φ(y)). S∞ acts on GRAPHS as Θ ∈ S∞, Θf (x, y) = f (Θ−1(x), Θ−1(y)) Example 2: Countable groups as a countable structure. GROUPS = {(f , e) ∈ NN×N × N : f (f (i, j), k) = f (i, f (j, k)); f (i, e) = f (e, i) = i; ∀i ∃ l f (i, l) = e} (f1, e1) ∼ (f2, e2) ⇐ ⇒ ∃φ : N → N bijection s.t. φ(e1) = φ(e2), φ(f1(x, y)) = f2(φ(x), φ(y)). S∞ acts on GROUPS as Θ ∈ S∞, Θf (i, j) = Θf (Θ−1(i), Θ−1(j))

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Borel Complete for countable structures

• Definition. An equivalence relation E is Borel complete for

countable structures if for every S∞-space Y we have that: E Y

S∞ ≤B E.

Theorem (Mekler, ’81)

The isomorphism relation of (certain) discrete countable groups is Borel complete for countable structures. If we call these groups Mekler groups it follows that

Corollary

The isomorphism relation of Mekler groups is complete analytic

• Problem. Is the isomorphism relation of discrete torsion free

abelian groups Borel complete for countable structures? It is true though that it is complete analytic (Hjorth).

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Iso of II1 factors is Borel complete for countable structures

Denote by FII1(H) the (standard) space of II1 factors on H, and by ≃FII1(H) the isomorphism relation for factors of type II1 on H.

Theorem (1 S.-T¨

• rnquist, ’08)

If Y is an S∞ space, then E Y

S∞ ≤B≃FII1(H).

As an immediate corollary, we have:

Corollary

The isomorphism relation for factors of type II1 is complete analytic as a subset of FII1(H) × FII1(H). In particular it is not a Borel subset. Problem 1: We don’t know neither the theorem nor the corollary for factors of types II∞ and IIIλ. Problem 2: Same theorem but for conjugacy of EMPT.(of course this would imply the Theorem of FRW)

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How to prove the theorem

the strategy is the following: a) Find an equivalence relation E that is Borel complete for countable structures. b) Find a Borel reduction E ≤B≃FII1(H).

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A strong rigidity theorem for Bernoulli shifts

step b) is based on the following rigidity theorem of Popa, which shows that for a Bernoulli shift β coming from certain kind of group, the group can be recovered from the isomorphism type of the group measure space factor L∞(X G) ⋊β G:

Theorem (Popa, ’06)

Suppose G1 and G2 are countably infinite discrete groups, β1 and β2 are the corresponding Bernoulli shifts on X1 = [0, 1]G1 and X2 = [0, 1]G2, respectively, and M1 = L2(X1) ⋊β1 G1 and M2 = L2(X2) ⋊β2 G2 are the corresponding group-measure space II1 factors. Suppose further that G1 and G2 are ICC (infinite conjugacy class) groups having the relative property (T) over an infinite normal subgroup. Then M1 ≃ M2 iff G1 ≃ G2.

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Isomorphism of relative property (T) groups

An example of an ICC group with property (T) is SL(3, Z). Any group of the form H × SL(3, Z) has the relative property (T) (over SL(3, Z)). If H is ICC, then H × SL(3, Z) is ICC. Denote by wTICC the class of countable groups, having the relative property (T) over some infinite normal subgroup, and ≃wTICC the isomorphism relation in that class. After checking that the constructions are Borel, Popa’s Theorem reduces the problem of proving our theorem to proving step a):

Theorem (S.-T¨

• rnquist)

The equivalence relation ≃wTICC is Borel complete for countable structures

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More results

Theorem (2 S.-T¨

• rnquist, ’08)

The isomorphism relation for separable von Neumann factors of type II1, II∞ and IIIλ, λ ∈ [0, 1], are not classifiable by countable structures.

Corollary

The classification problem of II1 factors is not smooth.

Corollary

E Y

S∞ <B≃FII1(H).

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More results, II

Corollary

It is not possible in Zermelo-Fraenkel set theory without the Axiom

• f Choice to construct a function

f : FII1 → GROUPS such that M1 ≃FII1 M2 ⇐ ⇒ f (M1) ≃GROUPS f (M2).

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Even more results

Theorem (3 S.-T¨

• rnquist, ’09)

The isomorphism relation for ITPFI2 factors is not classifiable by countable structures.

Theorem (4 (Announcement) S.-T¨

• rnquist, ’11)

The isomorphism relation for free Araki-Woods factors is not classifiable by countable structures.

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