A Rokhlin type theorem for C -algebras

Transcription

1 A Rokhlin type theorem for C -algebras Hung-Chang Liao Pennsylvania State University June 15, rd Canadian Operator Symposium University of Waterloo Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

2 What is a Rokhlin type theorem? Theorem (Rokhlin s Lemma) Let T : X X be an invertible ergodic m.p.t. of a probability space (X, µ). Then given n N and ε > 0, there exists a measurable set A X such that (i) A, T (A),..., T n 1 (A) are pairwise disjoint, and (ii) they cover X up to a set of measure less than ε. Rokhlin type theorem: approximate automorphism by shift. Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

3 Connes non-commutative Rokhlin theorem Theorem (Connes) Let R be the hyperfinite II 1 factor and let θ : Z Aut(R) be an outer action. Then given a finite subset F R, n N and ε > 0, there exists projections F 1, F 2,..., F n in R such that (i) θ(f k ) F k+1 2 < ε mod n (ii) [F k, x] 2 < ε (iii) n k=1 F k = 1. (1 k n, x F); (1 k n); Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

4 Rokhlin tower F n. F 1,..., F n are projections. θ F 3 θ cyclicly permuted by θ sum to 1 (hence mutually orthogonal) θ F 2 central F 1 Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

5 Connes theorem, again Theorem (Connes) Let R be the hyperfinite II 1 factor and let θ : Z Aut(R) be an outer action. Then given a finite subset F R, n N and ε > 0, there exists projections F 1, F 2,..., F n in R such that (i) θ(f k ) F k+1 2 < ε mod n (ii) [F k, x] 2 < ε (iii) n k=1 F k = 1. (1 k n, x F); (1 k n); A key step toward the classification of Z-actions on R up to outer conjugacy. C -version? Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

6 Kishimoto s Rokhlin type theorem for UHF algebras Theorem (Kishimoto) Let A be a UHF algebra, and let α : Z Aut(A) be a Z-action. Then TFAE: (1) α has the Rokhlin property. (2) α is strongly outer. Application: all strongly outer Z-actions on a UHF algebra are outer conjugate. Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

7 Rokhlin dimension F (0) n F (1) n F (d) n dim Rok (α) d... positive contractions α α F (0) 3 F (0) 2 F (0) 1 α α F (1) 3 F (1) 2 F (1) 1 α α F (d) 3 F (d) 2 F (d) 1 orthogonal within each color sum to 1 central Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

8 Rokhlin dimension and nuclear dimension Why is Rokhlin dimension useful? Theorem (Hirshberg, Winter and Zacharias) Let A be a unital separable C -algebra with dim nuc (A) < and α : Z Aut(A) an action with dim Rok (α) <. Then dim nuc (A α Z) <. Nuclear dimension is a non-commutative analogue of the topological covering dimension. dim(a) = 0 A is AF. dim nuc (A θ ) = 1 when θ R \ Q. dim nuc (O n ) = 1. Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

9 Kishimoto s theorem revisited Theorem (Kishimoto) Let A be a UHF algebra, and let α : Z Aut(A) be a Z-action. Then TFAE: (1) dim Rok (α) = 0. (2) α is strongly outer. Can we prove a similar result for a more general class of C -algebras, if we replace (1) by dim Rok (α) <? Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

10 Matui-Sato theorem Theorem (Matui and Sato) Let A be a unital separable simple stably finite C -algebra with dim nuc (A) <, and let α : Z Aut(A) be a Z-action. Assume that the extreme boundary T (A) is a finite set. Then TFAE (1) dim Rok (α) <. (2) α is strongly outer. Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

11 The main result Theorem Let A be a unital separable simple stably finite C -algebra with dim nuc (A) <, and let α : Z Aut(A) be a Z-action. Assume that T (A) is a finite dimensional compact set and that T(A) = T(A) α. Then TFAE: (1) dim Rok (α) <. (2) α is strongly outer. Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

12 Some ingredients Theorem...Then TFAE: (1) dim Rok (α) <. (2) α is strongly outer. (1) = (2): Almost the same as a classical result of Kishimoto. (2) = (1): Associate to each τ T (A) the von Neumann algebra π τ (A). We obtain a bundle over T (A). Connes Rokhlin theorem gives us Rokhlin towers in each fiber. Glue the towers together using a mutually orthogonal partition of unity. Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

13 References Alain Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup. (4), 8 No. 3, (1975) Akitaka Kishimoto, The Rohlin property for automorphisms of UHF algebras, J. Reine Angew. Math., 465, (1995) Ilan Hirshberg, Wilhelm Winter, and Joachim Zacharias, Rokhlin dimension and C dynamics, Comm. Math. Phys., 335 No. 2, (2015) Wilhelm Winter and Joachim Zacharias The nuclear dimension of C -algebras, Adv. Math., 224, No. 2, (2010) Hiroki Matui and Yasuhiko Sato, private communication. Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14

14 Thank you! Thank you! Hung-Chang Liao (Penn. State University) A Rokhlin type theorem for C -algebras June 15, / 14