Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

Transcription

1 Tracial Rokhlin property for actions of amenable group on C*-algebras Qingyun Wang University of Toronto June 8, 2015 Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

2 Notation In this talk, we assume that all C*-algebras are simple, unital and separable, all groups are discrete, countable, and amenable. Notation Let A be a C*-algebra, G be a group. A l (N, A)/c 0 (N, A) A A A For K G finite and ε > 0, we say a finite set T G is (K, ε)-invariant or (K, ε)-følner if T g K gt (1 ε) T Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

3 Motivation (Rokhlin s Lemma) Let (X, µ) be a probability space and α: Z (X, µ) be a free measure preserving action. Then for any ε > 0 and any n N, there exists some Y X such that 1 α i (Y ) α j (Y ) =. 2 µ( i n α i (Y )) > 1 ε. The sets α 1 (Y ), α 2 (Y ), is called a Rokhlin tower. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

4 Motivation (Weiss-Ornstein) Let G be a countable discrete amenable group. Let (X, µ) be a probability space and let α: G (X, µ) be a free measure preserving action. Then for any ε > 0, any finite K G, there exists (K, ε)-invariant subsets T 1, T 2, T n and measurable subsets Y i X such that 1 α g (Y i ) α h (Y j ) =, for any (g, i) (h, j) 2 µ( g Ti,1 i nα g (Y i )) > 1 ε. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

5 Motivation The idea has been adapted to actions on von Neumann algebras: Fact free action outer action subsets projections measure trace Connes (integer group), Jones (finite group), Ocneanu (amenable group) classified outer actions on von Neumann algebras. The starting point is that the existence of the Rokhlin towers. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

6 Motivation How about actions on C*-algebras? It turns out that outerness is probably too weak to work with. Definition Let α: G A be an action on C*-algebra. We say α has the tracial Rokhlin property, if for any finite subset K of G, any ε > 0, and any z A + \{0}, there exist (K, ε)-invariant finite subsets T 1, T 2,, T n and projections {e i 1 i n} such that, (1) α g (e i )α h (e j ) = 0, for any g T i, h T j such that g h or i j. (2) With e = g T i,1 i n α g (e i ), 1 e p.w. z. If f as above is weakened to be a positive contraction, then we say that α has the weak tracial Rokhlin property. Definition Let f (A ) + and a be an element of A +. We say f is pointwisely Cuntz subequivalent to a and write f p.w. a, if f has a representative (f n ) n N l (N, A), such that each f n is positive and f n a in A, for all n N. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

7 Motivation Let α: G A be an action. Let g G and τ be an α g -invariant trace. Then there is an induced automorphism ᾱ g on π τ (A), where π τ is thegns representation associated to τ. We say α is strongly outer, if for any g 1 G and any α g -invariant trace τ, the induced automorphism ᾱ g is outer. If the C*-algebra A has only finitely many extreme tracial states. Then the action has the weak tracial Rokhlin property if and only if it is strongly outer. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

8 Motivation What s good about Rokhlin towers? The idea is that, using the Rokhlin towers, we can build a family of large subalgebras of the crossed product with good properties. We can therefore deduce good properties on the crossed product. More precisely, we have Proposition Let α Act G (A) be an action with tracial Rokhlin property, where A has real rank zero and has strict comparison for projections. Then for any finite F A α G, any ε > 0, and any z A + \{0}, there is subalgebra D A α G which is isomorphic to i M ni (C) e i Ae i, where e i are projections in A, and a projection p D such that (1) pa ε D, ap ε D, a D. (2) 1 p z. If G is finite, the assumptions on A can be dropped, and we can have p almost commute with F. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

9 real rank zero and stable rank one (Phillips) Let A have tracial rank zero, let G be a finite group. Let α: G A be an action with the tracial Rokhlin property. Then A α G has tracial rank zero. (Phillips and Osaka) Let A be a simple unital C*-algebra with real rank zero, stable rank one and has strict comparison for projections. Let α Aut(A) be an automorphism with the tracial Rokhlin property. Then A α Z has the same properties. (W) The same is true if we replace Z by a general amenable group. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

10 pureness of crossed product Definition We say the Cuntz semigroup W (A) is almost divisible, if for any positive elment a M (A), any ε > 0 and any n N, there is some positive b M (A), such that n[b] [a] [(a ε) + ] (n + 1)[b] (OPW) Let A be a C*-algebra whose Cuntz semigroup has strict comparison and is almost divisible. Let α: G A be an action with the weak tracial Rokhlin property. Then A α G has the same properties. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

11 Z-stability (Mutui & Sato) Let A be a nuclear Z-stable C*-algebra with finitely many extremal tracial states. Let G be an elementary amenable group. Let α: G A be an action with the weak tracial Rokhlin property. Then A α G has the same properties. (Hirshberg & Orovitz) Let A be a nuclear Z-stable C*-algebra. Let α: G A be an action with the weak tracial Rokhlin property. Suppose either G is finite or G = Z and A has only finitely many extremal tracial states. Then A α G is also nuclear Z-stable. (OPW) Let A be a nuclear Z-stable C*-algebra. Let α: G A be an action with the weak tracial Rokhlin property. Then A α G is also nuclear Z-stable. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

12 Application to classification Corollary Let A be a simple Z-stable AH C*-algebra. Suppose the extreme boundary of T (A) is compact. Suppose also that the projections in A separate traces. Let G = Z n. Let α Act G (A) be an action with the weak tracial Rokhlin property. Then A α G is rationally tracial rank zero, and thus classifiable. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

13 Application to classification Proof. The assumptions indicates that A UHF has real rank zero and is Z-stable. Our results indicate that A α G UHF has real rank zero and is Z-stable Lin s result implies that every trace on the crossed product is quasidiagonal. BNSTWW s result implies that the crossed product rationally has finite decomposition rank. Winter s result says that real rank zero + finite decomposition rank implies tracial rank zero. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

14 Examples Let A be a separble unital C*-algebra and G be a countable discrete group. Let G A be the minimal tensor product of countably many copies of A indexed by the elements of G. The left multiplication of G on itself induces an action on G A (permuting the indices), which we shall call the Bernoulli shift on G A. Proposition The Bernoulli shift on G Z has the tracial Rokhlin property. More generally, we have Let α Act G (A), where G is amenable. Let X be a compact metrizable space with a Borel probability measure µ. Let β : G (X, µ) be a free and measure-preserving action which is also a topological action (acts on X by homeomorphisms). It induces an action on C(X ). Let τ be a tracial state on A. Suppose that there are approximate equivariant central unital homomorphisms ι i : C(X ) A with µ i the measure induced by τ ι i. If µ is the weak* limit of (µ i ) i N, then α has the weak tracial Rokhlin property Qingyun with Wang respect (University to of τ. Toronto) Furthermore, Tracial Rokhlinifproperty X isfor totally actions of amenable disconnected, group C*-algebras then α has June 8, / 17

15 Examples Definition Let A = i=1 B(H i), where H i is a finite dimensional Hilbert space for each i. An action α: G Aut(A) is called a product-type action if and only if for each i, there exists a unitary representation π i : G B(H i ), which induces an inner action α i : g Ad(π i (g)), such that α = i=1 α i. (W)Let α: G Aut(A) be a product-type action where A is UHF. Let H i, π i and α i be defined as above. Let d i be the dimension of H i and χ i be the character of π i. We will use the same notations if we do a telescope to the action. Define χ: G C to be the characteristic function on 1 G. Then the action α has the tracial Rokhlin property if and only if there exists a telescope, such that for any n N, the infinite product n i< 1 d i χ i = χ. (1) Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

16 Examples Let θ be a non-degenerate anti-symmetric bicharacter on Z d. We identify it with its matrix under the canonical basis of Z d. Then the associated non-commutative tori A θ is simple, unital AT algebra with a unique trace. A θ is generated by unitaries U x x Z d subject to the relation U y U x = exp(πi < x, θy >)U x+y, x, y Z d. Proposition For any T M n (Z), the map U x U Tx give rises to an endomorphism α T of A θ if and only if 1 2 (T t θt θ) M d (Z). It is an automorphism if and only if T is invertible. For any T I, α T is not weakly inner. Corollary Let G be any amenable subgroup of {T GL d (Z) 1 2 (T t θt θ) M d (Z)}. Then the action α: G Aut(A θ ), defined by T α T, has the tracial Rokhlin property. Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17

17 Permanance properties Let α: G A be an action with the (weak) tracial Rokhlin property. Let β : G A B be an arbitrary action. Then α β : G A min B has the (weak) tracial Rokhlin property. This shows that for any Z-stable C*-algebra A and any group G, there is at least one action with the weak tracial Rokhlin property. One can in fact generalize Phillips result that the set of actions with weak tracial Rokhlin property is generic. Question Let α: G A be an action with the weak tracial Rokhlin property. Let H be a subgroup. Does α H : H A has the weak tracial Rokhlin property? Tracial Rokhlin property for actions of amenable group on C*-algebras June 8, / 17