Convex Sets Associated to C -Algebras Scott Atkinson University of Virginia Joint Mathematics Meetings 2016
Overview Goal: Define and investigate invariants for a unital separable tracial C*-algebra A. These invariants will be convex sets that in some sense generalize the trace space of A. They are built out of equivalence classes of -homomorphisms from A into certain II 1 -factors. This is an adaptation of a 2011 contruction of Nate Brown involving ultrapowers. Our construction uses no ultrapowers, and these invariants are always separable Brown s are either nonseparable or trivial. A will denote a unital separable tracial C*-algebra, and R will denote the separable hyperfinite II 1 -factor throughout.
Classical Situation: BDF Theory and Ext(A) BDF theory and (more generally) the theory of Ext(A) are classical examples of placing a nice structure on equivalence classes of -homomorphisms. Ext(A) is given by the set of unital -monomorphisms π : A B(H)/K(H) modulo B(H)-unitary equivalence. A semigroup structure on Ext(A) is described by the following picture. [π] + [ρ] = [( )] π 0 0 ρ
Definition For a separable, unital C -algebra A, and a separable II 1 -factor N, we define Hom w (A, N) to be the space of unital -homomorphisms π : A N modulo the equivalence relation of weak approximate unitary equivalence (w.a.u.e.). That is, [π] = [ρ] if there is a sequence {u n } of unitaries in N such that for every a A we have lim n π(a) u n ρ(a)u n 2 = 0 where x 2 2 = τ N(x x) for τ N the unique tracial state on N.
Convex Structure If the target algebra is McDuff (M = M R) then yields a convex structure, rather than a semigroup structure as in Ext(A). Picture: t[π] + (1 t)[ρ] ( pπp 0 0 p ρp ), τ(p) = t Definition Let M be a separable McDuff II 1 -factor. For t [0, 1] and [π], [ρ], we define t[π] + (1 t)[ρ] := [σ M (π p + ρ p )] where σ M : M R M is an isomorphism satisfying σ M (id M 1 R ) id M and p is a projection in R with τ R (p) = t.
Non-McDuff? No Problem! Theorem Let N be a separable II 1 -factor. Given π, ρ : A N, consider π 1 R, ρ 1 R : A N R. If π 1 R ρ 1 R then π ρ. Consequence: For N non-mcduff, Hom w (A, N) Hom w (A, N R) via [π] [π 1 R ]
Extreme Points By a characterization due to Brown and Capraro-Fritz, we may consider as a closed, bounded (sometimes compact but not always), separable, convex subset of a Banach space. So what about extreme points?
Extreme Points Let π : A M be given. Theorem (A.) If [π] is extreme then W (π(a)) is a factor, but the converse fails in general. Hom w (A, M U ) via constant sequence embedding M M U. Theorem (Brown, A.) If π U (A) M U is a factor then [π] is extreme in.
Extreme Points in Amenable Cases Theorem (A.) If either A or M is amenable, then given π : A M, TFAE: [π] is extreme; W (π(a)) is a factor; π U (A) M U is a factor.
Quotients Faces Theorem (A.) If J A is a closed two-sided ideal of A, then Hom w (A/J, M) is naturally embedded as a(n exposed) face of. With the observation that any separable unital C*-algebra is a quotient of C (F ), we get the following corollary. Corollary (A.) For any separable unital A, may be embedded as a face of Hom w (C (F ), M).
Connection to Given [π] we get a tracial state on A given by τ M π. The map [π] τ M π is well-defined, continuous, and affine. Natural question: For a fixed M, how much data does share with T (A)?
Nuclear Case Theorem (A., Ding-Hadwin) If A is nuclear then for any McDuff M, is affinely homeomorphic to T (A) via [π] τ M π. English Version: All traces of a separable unital nuclear algebra lift through any fixed McDuff factor; and the traces remember their homomorphisms up to w.a.u.e.
Alternative Characterization of Hyperfiniteness Let N be a separable, finite, tracial, R U -embeddable von Neumann algebra. Theorem (Jung) N is hyperfinite if and only if any two embeddings π, ρ : N R U are unitarily conjugate. (embedding = unital, injective, normal, trace-preserving -homomorphism) Corollary (A.) N is hyperfinite if and only if for any separable McDuff II 1 -factor M, any two embeddings π, ρ : N M are weakly approximately unitarily equivalent.
Table of Property A M compact nuclear not compact has non extreme point with factorial closure [π] τ M π not injective [π] τ M π not surjective C (Γ) for Γ a non-amenable, residually finite, discrete group any McDuff Dense in Z L(F 2 ) Z L(F 2 ) C r (F 2 ) C (F ) R L(F 2 ) R any McDuff
Minimal Faces in Hom(N, R U ) Theorem (A.) Let π : N R U be an embedding such that the center of π(n) R U has dimension n <. Then the minimal face in Hom(N, R U ) containing [π] is an n-vertex simplex. Conjecture (A.) If [π] is a nontrivial average of n(< ) extreme points in Hom(N, R U ) then the center of π(n) R U has dimension n. (And thus the minimal face containing [π] is an n-vertex simplex) So, for example, the hull of four extreme points in Hom(N, R U ) cannot be a square it has to be a tetrahedron.
Thanks! Preprint: arxiv:1509.00822