The complexity of classification problem ofnuclear C*-algebras

Ilijas Farah(joint work with Andrew Toms and Asger Tornquist)

Nottingham, September 6, 2010

C*-algebras

H: a complex Hilbert space(B(H),+, ·,∗ , ‖ · ‖): the algebra of bounded linear operators on H

DefinitionA (concrete) C*-algebra is a norm-closed subalgebra of B(H).

Theorem (Gelfand–Naimark–Segal, 1942)

A Banach algebra with involution A is isomorphic to a concreteC*-algebra if and only if

‖aa∗‖ = ‖a‖2

for all a ∈ A.

Examples

Example

(1)B(H), Mn(C).

(2) If X is a compact metric space, C (X ).

C (X ) ∼= C (T ) ⇔ X ∼= Y .

(3) If (X , α) is a minimal dynamical system, C (X ) oα Z.

(X , α) ∼= (Y , β) ⇒ C (X ) oα Z ∼= C (Y ) oβ Z.

Classification results, I

UHF algebras are direct limits of full matrix algebras, Mn(C).

Theorem (Glimm, 1960)

Unital, separable UHF algebras are classified by the invariant∏p prime pnp in NN.

AF algebras are direct limits of finite-dimensional C*-algebras.

Theorem (Elliott, 1975)

Separable AF algebras are classified by the ordered group(K0(A),K0(A)+, 1).

Classification results, II

Theorem (Kirchberg–Phillips, 1995)

All purely infinite, nuclear, separable, simple, unital C*-algebraswith UCT are classified by their K-theoretic invariant.

Theorem (Elliott–Evans, 1993)

Irrational rotation algebras are are classified by their K-theoreticinvariant.

Elliott program as of 2003

All nuclear, separable, simple, unital C*-algebras are classified bythe Elliott invariant,

((K0(A),K0(A)+, 1),K1(A),T (A), ρA).

A //

��

Ell(A)

��B // Ell(B)

Counterexamples I

Theorem (Phillips, 2000)

There are 2ℵ0 nonisomorphic purely infinite, separable, simple,unital C*-algebras with the same (trivial) Elliott invariant.

Theorem (Rørdam, Toms, 2003)

There exist nonisomorphic separable, unital, simple, nuclearC*-algebras with the same Elliott invariant.

More counterexamples and some positive results

Theorem (Toms, 2006–2008)

There are 2ℵ0 nonisomorphic separable, unital, simple, nuclearC*-algebras with the same Elliott invariant and with the sameF -invariant, where F is any continuous homotopy invariant functor.

Theorem (Toms–Wnter, 2009)

C*-algebras C (X ) oα Z where X is infinite, compact,finite-dimensional metrizable space and α is a minimal uniquelyergodic homeomorphism are classified by the Elliott invariant.

New directionsClassification of nuclear, simple, unital, separable, Z-stableC*-algebras.Cuntz semigroup as an invariant.

Descriptive set theory: Abstract classification

Assume the collection X of objects we are trying to classify formsa ‘nice’ space, typically a Polish space or a standard Borel spaceand the equivalence relation E is a Borel or analytic subset of X 2.(Analytic set is a continuous image of a Borel set.)

The basic concept of abstract classification

DefinitionIf (X ,E ) and (Y ,F ) are equivalence relations, E is Borel-reducibleto F , in symbols

E ≤B F ,

if there is a Borel-measurable map f : X → Y such that

x E y ⇔ f (x) E f (y).

The intuitive meaning:(1) Classification problem represented by E is at most ascomplicated as that of F .(2) F -classes are complete invariants for E-classes.

Example

Spectral theorems.

Glimm–Effros Dichotomy

If E ≤B=R we say E is smooth.For x , y in 2N let

x E 0y iff (∃m)(∀n ≥ m)x(n) = y(n)

Theorem (Harrington–Kechris–Louveau, 1990)

If E is a Borel equivalence relation on a Polish space then either Eis smooth or E0 ≤B E .

Modelling classification problems I

Example (The Polish space of countable groups)

A countable group G is coded by(N, eG , xG ,

−1G ), for e ∈ N, ×G : N2 → N, −1

G : N→ N.

This is a closed subspace of the compact metric spaceP(N)× P(N3)× P(N2).

The isomorphism ∼=G is an S∞-orbit equivalence relation.

Modelling classification problems II

In general, a given concrete classification problem for category C ismodelled by a standard Borel space (X ,Σ) and F : X � C suchthat the relation E on X ,

x E y ⇔ F (x) ∼= F (y)

is analytic (i.e., a continuous image of a Borel set).

Classification by countable structures

An equivalence relation (X ,E ) is classified by countable structuresif there is a countable language L and a Borel map from X intocountable L-models such that

x E y iff F (x) ∼= F (y).

This is equivalent to being ≤B an S∞-orbit equivalence relation.

Lemma (Sasyk–Tornquist 2009, after Hjorth)

If G ( F are separable Banach spaces, G is dense in F , andid : G → F is bounded, then the coset equivalence F/G cannot beclassified by countable structures.

Example

c0/`2.

Examples

Theorem (Kechris–Sofronidis, 2001)

Unitary operators up to conjugacy are not classifiable by countablestructures.

Theorem (Sasyk–Tornquist, 2009)

Type II1 factors are not classifiable by countable structures. Thesame result applies to II∞ factors and IIIλ factors for 0 ≤ λ ≤ 1, toinjective III0 factors and to ITPFI factors.

The space of irreducible representations

On A define π0 ∼A π1 if

B(H0)

Ad π0(u)

��A

π0 55llllllllπ1

))RRRRRRRR

B(H1)

commutes for some u ∈ A.

TheoremA is type I implies ∼A is smooth.

A strong dichotomy

Theorem (Glimm, 1960)

A is non-type I iff E0 ≤B ∼A

Theorem (Kerr–Li–Pichot, Farah, 2008)

A is non-type I iff ∼A is not classifiable by countable structures.(More precisely, if A is non-type I then RN/`2 ≤B ∼A.)

There is no dichotomy for equivalence relations not classifiable bycountable structures analogous to the Glimm–Effros dichotomy fornon-smooth equivalence relations.

Effros Borel space

For a Polish space X let X ∗ be the space of closed subsets of X .The σ-algebra Σ on X ∗ is generated by sets

{A ∈ X ∗ : A ⊆ U}

where U ranges over open subsets of X .

Proposition

(X ∗,Σ) is a standard Borel space. If X is a separable C*-algebrathen

S(X ) = {B ∈ X ∗ : B is a subalgebra of X}

is a Borel subspace of X ∗.

Examples

Theorem (Kirchberg, 1994)

S(O2) is the space of all exact separable C*-algebras.

Theorem (Pisier–Junge, 1995)

S(A) is not the space of all separable C*-algebras for any separableC*-algebra A.

Borel space of separable C*-algebras

Definition (Kechris, 1996)

Let Γ be B(`2)N, with respect to the weak operator topology. Then

Γ 3 γ 7→ C ∗(γ)

maps Γ onto the space of all separable C*-algebras represented onH, and

γ0 E γ1 ⇔ C ∗(γ0) ∼= C ∗(γ1)

is analytic.

There is also a space ∆ of abstract separable C*-algebras.Two representations are equivalent.

Classification problem of C*-algebras

Proposition (Farah–Toms–Tornquist)

Computation of the Elliott invariant is Borel.

Theorem (Farah–Toms–Tornquist)

The isomorphism of separable, simple, unital, nuclear C*-algebrasis not classifiable by countable structures.

Actually we can do this for AI algebras.

Classifiable C*-algebras are not classifiable. . .by countable structures

Theorem (Elliott, 199?)

AI algebras are classified by the Elliott invariant.

Theorem (Farah–Toms–Tornquist)

If L is a countable language, then the isomorphism of countableL-models is ≤B to the isomorphism of AT algebras.

The top

Theorem (Louveau–Rosendal, 2005)

Biembeddability of separable Banach spaces is the ≤B -maximalanalytic equivalence relation.

Theorem (Ferenczi–Louveau–Rosendal, 2009)

Isomorphism of separable Banach spaces is the ≤B -maximalanalytic equivalence relation.

In particular, separable Banach spaces cannot be classified byorbits of a polish group action.

Theorem (Kechris–Solecki, 200?)

Homeomorphism of compact metric spaces is ≤B a Polish groupaction.

Below group action

Proposition (Farah–Toms–Tornquist)

The isomorphism of simple separable nuclear C*-algebras is ≤B toan orbit equivalence relation of a Polish group action.

Pf. (The unital case.) Consider the Effros Borel space ofsubalgebras of C ∼= O2 and the natural action of Aut(C ) on it.For nuclear, separable, simple, unital A we have C ∼= A⊗O2

(Kirchberg).Borel map: A 7→ F (A)

where F (A) is a subalgebra of C isomorphic to A such thatC = F (A)⊗O2.Then A ∼= B if and only if there is an automorphism of C sendingF (A) to F (B).

Relative complexity of some isomorphism relations

separable Banach spaces

AI algebras

OO

abelian C*-algebras

55jjjjjjjjjjjjj___________ compact metriz-

able spaces

iiSSSSSSSSS

AF algebras

OO 22fffffffffffffffffffffffffff// Kirchbergalgebras

66mmmmmmmm

hhRRRRRRRRRRR

any countablestructures

llXXXXXXXXXXXXXXXXXXXXXXXXXXX

OO

UHF algebras

OO 55jjjjjjjjjj

22eeeeeeeeeeeeeeeeeeeeeeeeeee______ =R

iiTTTTTTTTTTTTTTTT

OO 55kkkkkkkkkkkkk

finite simple groups

jjUUUUUUUUUUUU

OO

Theorem (Farah–Toms–Tornquist)

Biembeddability relation Ebi of separable AF algebras is notclassifiable by countable strucutures. Moreover, any Kσ

equivalence relation is ≤B Ebi .

Some problems

QuestionIs the isomorphism of all separable, unital C*-algebras ≤B orbitequivalence relation?What about the not necessarily simple nuclear C*-algebras? ExactC*-algebras? Arbitrary C*-algebras?

QuestionIs biembeddability of separable AF algebras a ≤B -maximal analyticequivalence relation?

QuestionIs isomorphism of AF algebras a ≤B -maximal S∞ orbit equivalencerelation? (I.e., If L is a countable language, is the isomorphism ofcountable L-models ≤B to the isomorphism of AF algebras?)

ProblemDevelop set-theoretic framework for Elliott’s functorialclassification.