Rokhlin property for an inclusion of $C^*$-algebras

Hiroyuki Osaka  (Ritsumeikan University)

10:50 am to 11:40 am, June 3rd, 2015   Science Building A1414

Abstract:

We introduce the Rokhlin property for an inclusion $P \subset A$ of $C^*$-algebras with index finite in the sense of Pimsner-Popa and show that an action $\alpha$ from a finite group $G$ on a simple $C^*$-algebra $A$ has the Rokhlin property if and only if the canonical conditional expectation $E \colon A \rightarrow P$ has the Rokhlin property. We prove that a number of classes of separable $C^*$-algebras are closed under the assumption that $E \colon A \rightarrow P$ is of index finite with the Rokhlin property, including: (a) AF algebras, AI algebras, A$\mathbb{T}$ algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (b) Simple unital AH algebras with slow dimension growth and real rank zero. (c) $C^*$-algebras with real rank zero or stable rank one. (d) Simple $C^*$-algebras for which the order on projections is determined by traces, (e) Jiang-Su algebras absorption. (f) Strictly comparison property (g) $C^*$-algebras with finite nuclear dimension.