Set Theory and von Neumann algebras

Roman Sasyk

ENS Lyon & Universidad de Buenos Aires

Paris, May 25, 2011

Roman 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 theisomorphism relation for von Neumann factors, achieved byapplying tools from descriptive set theory.

This is joint work with Asger Tornquist from the University ofVienna and the University of Copenhagen.

Roman Sasyk Descriptive Set Theory and von Neumann algebras 2

Classical descriptive set theory

Recall that a Polish space is a completely metrizable separabletopological space. E.g. R, NN, {0, 1}N = 2N, C ([0, 1]).

Descriptive set theory is the study of definable subsets of Polishspaces, and definable functions on Polish spaces. Definable setsand functions include

I Borel sets and functions, i.e. those sets that occur in theσ-algebra generated by the open sets.

I Analytic sets: Those sets that are the image of a Polish spaceunder a continuous function.

Definition. A standard Borel space is a set X equipped with aσ-algebra B which is itself generated by the open sets of a Polishtopology on X .

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Motivation

Why is descriptive set theory relevant to classification problems?

I If the category or class of objects to be classified arethemselves countable or separable, then there often is anatural Polish space that “parametrizes” the class: I.e., aPolish space X where every element of the class is “coded” bysome x ∈ X .

I The relation of isomorphism among the objects in the classusually turns out to be Borel or analytic as a subset of X × X .

I Any reasonable solution to a classification problem has toassociate the invariants in the classifying category in acalculable (and therefore definable) way: Otherwise it doesnot provide a useful tool for distinguishing the isomorphismclasses.

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Example: The Effros-Borel space

There is a Borel structure in the space of separable von Neumannalgebras: Let H be a separable Hilbert. Let vN(H) denote the vonNeumann algebras acting on H The associated Borel structure isthe Effros Borel structure, i.e. the one generated by the sets

{N ∈ vN(H) : N ∩ U 6= ∅}

where U ranges over the weakly open subsets of B(H).

Theorem (Effros ’64): Sets of factors of types I, II1, II∞, III, areBorel sets in vN(H). (Same for IIIλ, 0 ≤ λ ≤ 1.)

To give a Borel structure to vN(H) was the first step in Effros’sattempt 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 setof measure preserving invertible transformations of X .MPT is a Polish space (via the Koopman representation).The set EMPT of ergodic measure preserving transformations is aGδ subset of MPT:Indeed fix a dense sequence {fi} in L2(X ,B, µ), then EMPT =

⋂i

⋂j

⋃N

{T ∈ MPT : ||1/NN∑

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 Polishspace.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 Nwith 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 butA itself was not Borel. (i.e. there exists sets that are analytic andnon Borel)Properties of Analytic sets. Unions, countable intersections,continuous images, continuous inverse images of analytic sets areanalytic. However the complement of an analytic set is usually notanalytic. In factTheorem (Suslin 1917): a set A and its complement are bothanalytic iff A is Borel. (This is the starting point of what is calledprojective hierarchy of sets)In concrete examples, “interesting” sets are obviously Analytic.The question is: How to show that a given set is analytic and notBorel?

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Complete analytic sets

Notation. Given two subsets A and B of two Polish spaces X andY , B ≤W A means that there exist a Borel function f : Y → Xsuch that f −1(A) = B.Definition. An analytic set A ⊂ X is complete analytic if for allB ⊂ 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 analyticand 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 setC ⊂ Z is analytic and not Borel, namely:a) Find a suitable Polish space X and a complete analytic subsetA ⊂ 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 isanalytic if E viewed as a subset of X × X is an analytic set.Theorem: Foreman-Rudolph-Weiss 2006-2010.The conjugacyrelation of ergodic measure preserving transformations is completeanalytic. 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 nomethod or protocol that involves a countable amount ofinformation and countable number of steps that reliablydistinguishes between non-isomorphic ergodic measure preservingtransformations. We view this as a rigorous way of saying that theclassification problem for ergodic measure preservingtransformations is intractable.”

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Proving the F.R.W. Theorem

The proof of this Theorem appeared in the last issue of Annals ofMathematics.The strategy of the proof is the one outlined in thesteps a) and b).for step a) They take X the space of countable trees and A the setof trees with an infinite branch.for step b) They construct a cont. function f: X → EMPT suchthat 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} iscomplete 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 II1factors 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 hasbeen developed as a tool to measure the relative complexity ofclassification problems that arise naturally in mathematics.

Its development has lead to what is now a large programme ofanalyzing the descriptive set theory of definable equivalencerelations. (Definable=Borel, analytic, etc.).Borel reducibility was first introduced in the late 80’s by Friedmanand Stanley in the context of model theory but it was quicklytaken over by descriptive set theorists. (Kechris, Louveau, Hjorth,etc.)The starting point comes from a generalization of theorems ofMackey, Glimm and Effros in operator algebras.

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Borel reducibility

Definition. Let E and F be equivalence relations on standard Borelspaces X and Y respectively.

E is Borel reducible to F , written E ≤B F , if there is a Borelf : X → Y such that

xEy ⇐⇒ f (x)Ff (y).

This means that the points of X can be classified up toE -equivalence by a Borel assignment of invariants that areF -equivalence classes.

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Borel reducibility

f is required to be Borel to make sure that the invariant f (x) has areasonable computation from x .

Without a requirement on f , the definition would only amount tostudying the cardinality of X/E vs. Y /F .

Observation. If E ≤B F , E is complete analytic and F is analyticthen 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 EYS∞

Where S∞ is the infinite symmetric group and EYS∞

denotes anequivalence relation induced by a continuous S∞-action on a Polishspace Y .

The motivation behind this “definition” is that isomorphism ofcountable structures, (countable groups, graphs, fields, etc.) canbe 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))Roman Sasyk Descriptive Set Theory and von Neumann algebras 17

Borel Complete for countable structures

Definition. An equivalence relation E is Borel complete forcountable structures if for every S∞-space Y we have that:EY

S∞≤B E .

Theorem (Mekler, ’81)

The isomorphism relation of (certain) discrete countable groups isBorel 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 freeabelian groups Borel complete for countable structures? It is truethough 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.-Tornquist, ’08)

If Y is an S∞ space, then EYS∞≤B'FII1

(H).

As an immediate corollary, we have:

Corollary

The isomorphism relation for factors of type II1 is completeanalytic as a subset of FII1(H)×FII1(H). In particular it is not aBorel subset.

Problem 1: We don’t know neither the theorem nor the corollaryfor factors of types II∞ and IIIλ.Problem 2: Same theorem but for conjugacy of EMPT.(of coursethis 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 forcountable 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, whichshows that for a Bernoulli shift β coming from certain kind ofgroup, the group can be recovered from the isomorphism type ofthe group measure space factor L∞(X G ) oβ G :

Theorem (Popa, ’06)

Suppose G1 and G2 are countably infinite discrete groups, β1 andβ2 are the corresponding Bernoulli shifts on X1 = [0, 1]G1 andX2 = [0, 1]G2 , respectively, and M1 = L2(X1) oβ1 G1 andM2 = L2(X2) oβ2 G2 are the corresponding group-measure spaceII1 factors. Suppose further that G1 and G2 are ICC (infiniteconjugacy class) groups having the relative property (T) over aninfinite 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). Anygroup of the form H × SL(3,Z) has the relative property (T) (overSL(3,Z)). If H is ICC, then H × SL(3,Z) is ICC.

Denote by wTICC the class of countable groups, having the relativeproperty (T) over some infinite normal subgroup, and 'wTICC theisomorphism relation in that class.

After checking that the constructions are Borel, Popa’s Theoremreduces the problem of proving our theorem to proving step a):

Theorem (S.-Tornquist)

The equivalence relation 'wTICC is Borel complete for countablestructures

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More results

Theorem (2 S.-Tornquist, ’08)

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

Corollary

The classification problem of II1 factors is not smooth.

Corollary

EYS∞

<B'FII1(H).

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More results, II

Corollary

It is not possible in Zermelo-Fraenkel set theory without the Axiomof 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.-Tornquist, ’09)

The isomorphism relation for ITPFI2 factors is not classifiable bycountable structures.

Theorem (4 (Announcement) S.-Tornquist, ’11)

The isomorphism relation for free Araki-Woods factors is notclassifiable by countable structures.

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