Forough, Marzieh; Golestani, Nasser

The Weak Tracial Rokhlin Property for Finite Group Actions on Simple C*-Algebras

Doc. Math. 25, 2507-2552 (2020)
DOI: 10.25537/dm.2020v25.2507-2552
Communicated by Wilhelm Winter

Summary

We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*-algebras, minimal tensor products, and direct limits of actions. Some of these results are new even in the unital case and answer open questions asked by N. C. Phillips in full generality. We present several examples of finite group actions with the weak tracial Rokhlin property on simple stably projectionless C*-algebras. We prove that if \(\alpha \colon G \rightarrow \text{Aut}(A)\) is an action of a finite group \(G\) on a simple C*-algebra \(A\) with tracial rank zero and \(\alpha\) has the weak tracial Rokhlin property, then the crossed product \(A \rtimes_{\alpha} G\) and the fixed point algebra \(A^{\alpha}\) are simple with tracial rank zero. This extends a result of N. C. Phillips to the nonunital case. We use the machinery of Cuntz subequivalence to work in this nonunital setting.

Mathematics Subject Classification

46L55, 46L40, 46L05

Keywords/Phrases

C*-algebra, weak tracial Rokhlin property, group action, crossed product

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Affiliation

Forough, Marzieh
Institute of Mathematics, Czech Academy of Sciences, 115 67 Praha 1, Czech Republic
Golestani, Nasser
Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Iran

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