The complexity of classification problem of nuclear C*-algebras

Transcription

1 The complexity of classification problem of nuclear C*-algebras Ilijas Farah (joint work with Andrew Toms and Asger Törnquist) Nottingham, September 6, 2010

2 C*-algebras H: a complex Hilbert space (B(H), +,,, ): the algebra of bounded linear operators on H Definition A (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 concrete C*-algebra if and only if for all a A. aa = a 2

3 Examples Example (1)B(H), M n (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 ) α Z. (X, α) = (Y, β) C(X ) α Z = C(Y ) β Z.

4 Classification results, I UHF algebras are direct limits of full matrix algebras, M n (C). Theorem (Glimm, 1960) Unital, separable UHF algebras are classified by the invariant p prime pnp in N N. AF algebras are direct limits of finite-dimensional C*-algebras. Theorem (Elliott, 1975) Separable AF algebras are classified by the ordered group (K 0 (A), K 0 (A) +, 1).

5 Classification results, II Theorem (Kirchberg Phillips, 1995) All purely infinite, nuclear, separable, simple, unital C*-algebras with UCT are classified by their K-theoretic invariant. Theorem (Elliott Evans, 1993) Irrational rotation algebras are are classified by their K-theoretic invariant.

6 Elliott program as of 2003 All nuclear, separable, simple, unital C*-algebras are classified by the Elliott invariant, ((K 0 (A), K 0 (A) +, 1), K 1 (A), T (A), ρ A ). A B Ell(A) Ell(B)

7 Counterexamples I Theorem (Phillips, 2000) There are 2 ℵ 0 nonisomorphic purely infinite, separable, simple, unital C*-algebras with the same (trivial) Elliott invariant. Theorem (Rørdam, Toms, 2003) There exist nonisomorphic separable, unital, simple, nuclear C*-algebras with the same Elliott invariant.

8 More counterexamples and some positive results Theorem (Toms, ) There are 2 ℵ 0 nonisomorphic separable, unital, simple, nuclear C*-algebras with the same Elliott invariant and with the same F -invariant, where F is any continuous homotopy invariant functor. Theorem (Toms Wnter, 2009) C*-algebras C(X ) α Z where X is infinite, compact, finite-dimensional metrizable space and α is a minimal uniquely ergodic homeomorphism are classified by the Elliott invariant. New directions Classification of nuclear, simple, unital, separable, Z-stable C*-algebras. Cuntz semigroup as an invariant.

9 Descriptive set theory: Abstract classification Assume the collection X of objects we are trying to classify forms a nice space, typically a Polish space or a standard Borel space and the equivalence relation E is a Borel or analytic subset of X 2. (Analytic set is a continuous image of a Borel set.)

10 The basic concept of abstract classification Definition If (X, E) and (Y, F ) are equivalence relations, E is Borel-reducible to 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 as complicated as that of F. (2) F -classes are complete invariants for E-classes. Example Spectral theorems.

11 Glimm Effros Dichotomy If E B = R we say E is smooth. For x, y in 2 N let x E 0 y 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 E is smooth or E 0 B E.

12 Modelling classification problems I Example (The Polish space of countable groups) A countable group G is coded by (N, e G, x G, 1 G ), for e N, G : N 2 N, 1 G : N N. This is a closed subspace of the compact metric space P(N) P(N 3 ) P(N 2 ). The isomorphism = G is an S -orbit equivalence relation.

13 Modelling classification problems II In general, a given concrete classification problem for category C is modelled by a standard Borel space (X, Σ) and F : X C such that the relation E on X, x E y F (x) = F (y) is analytic (i.e., a continuous image of a Borel set).

14 Classification by countable structures An equivalence relation (X, E) is classified by countable structures if there is a countable language L and a Borel map from X into countable 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 Törnquist 2009, after Hjorth) If G F are separable Banach spaces, G is dense in F, and id: G F is bounded, then the coset equivalence F /G cannot be classified by countable structures. Example c 0 /l 2.

15 Examples Theorem (Kechris Sofronidis, 2001) Unitary operators up to conjugacy are not classifiable by countable structures. Theorem (Sasyk Törnquist, 2009) Type II 1 factors are not classifiable by countable structures. The same result applies to II factors and III λ factors for 0 λ 1, to injective III 0 factors and to ITPFI factors.

16 The space of irreducible representations On  define π 0 A π 1 if commutes for some u A. B(H 0 ) π 0 A π 1 B(H 1 ) Theorem A is type I implies A is smooth. Ad π 0 (u)

17 A strong dichotomy Theorem (Glimm, 1960) A is non-type I iff E 0 B A Theorem (Kerr Li Pichót, Farah, 2008) A is non-type I iff A is not classifiable by countable structures. (More precisely, if A is non-type I then R N /l 2 B A.) There is no dichotomy for equivalence relations not classifiable by countable structures analogous to the Glimm Effros dichotomy for non-smooth equivalence relations.

18 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*-algebra then S(X ) = {B X : B is a subalgebra of X } is a Borel subspace of X.

19 Examples Theorem (Kirchberg, 1994) S(O 2 ) 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 separable C*-algebra A.

20 Borel space of separable C*-algebras Definition (Kechris, 1996) Let Γ be B(l 2 ) N, with respect to the weak operator topology. Then Γ γ C (γ) maps Γ onto the space of all separable C*-algebras represented on H, and γ 0 E γ 1 C (γ 0 ) = C (γ 1 ) is analytic. There is also a space of abstract separable C*-algebras. Two representations are equivalent.

21 Classification problem of C*-algebras Proposition (Farah Toms Törnquist) Computation of the Elliott invariant is Borel. Theorem (Farah Toms Törnquist) The isomorphism of separable, simple, unital, nuclear C*-algebras is not classifiable by countable structures. Actually we can do this for AI algebras.

22 Classifiable C*-algebras are not classifiable... by countable structures Theorem (Elliott, 199?) AI algebras are classified by the Elliott invariant. Theorem (Farah Toms Törnquist) If L is a countable language, then the isomorphism of countable L-models is B to the isomorphism of AT algebras.

23 The top Theorem (Louveau Rosendal, 2005) Biembeddability of separable Banach spaces is the B -maximal analytic equivalence relation. Theorem (Ferenczi Louveau Rosendal, 2009) Isomorphism of separable Banach spaces is the B -maximal analytic equivalence relation. In particular, separable Banach spaces cannot be classified by orbits of a polish group action. Theorem (Kechris Solecki, 200?) Homeomorphism of compact metric spaces is B a Polish group action.

24 Below group action Proposition (Farah Toms Törnquist) The isomorphism of simple separable nuclear C*-algebras is B to an orbit equivalence relation of a Polish group action. Pf. (The unital case.) Consider the Effros Borel space of subalgebras of C = O 2 and the natural action of Aut(C) on it. For nuclear, separable, simple, unital A we have C = A O 2 (Kirchberg). Borel map: A F (A) where F (A) is a subalgebra of C isomorphic to A such that C = F (A) O 2. Then A = B if and only if there is an automorphism of C sending F (A) to F (B).

25 Relative complexity of some isomorphism relations separable Banach spaces AI algebras abelian C*-algebras compact metrizable spaces AF algebras Kirchberg any countable algebras structures UHF algebras = R finite simple groups

26 Theorem (Farah Toms Törnquist) Biembeddability relation E bi of separable AF algebras is not classifiable by countable strucutures. Moreover, any K σ equivalence relation is B E bi.

27 Some problems Question Is the isomorphism of all separable, unital C*-algebras B orbit equivalence relation? What about the not necessarily simple nuclear C*-algebras? Exact C*-algebras? Arbitrary C*-algebras? Question Is biembeddability of separable AF algebras a B -maximal analytic equivalence relation? Question Is isomorphism of AF algebras a B -maximal S orbit equivalence relation? (I.e., If L is a countable language, is the isomorphism of countable L-models B to the isomorphism of AF algebras?) Problem Develop set-theoretic framework for Elliott s functorial classification.