The tracial class property for crossed products by finite group actions (Q1938292)

Summary: We define the concept of tracial \(\mathcal C\)-algebra of \(C^\ast\)-algebras, which generalizes the concept of local \(\mathcal C\)-algebra of \(C^\ast\)-algebras given by \textit{H. Osaka} and \textit{N. C. Phillips} [Math. Z. 270, No.~1--2, 19--42 (2012; Zbl 1244.46032)]. Let \(\mathcal C\) be any class of separable unital \(C^\ast\)-algebras. Let \(A\) be an infinite dimensional simple unital tracial \(\mathcal C\)-algebra with the (SP)-property, and let \(\alpha : G \rightarrow \text{Aut}(A)\) be an action of a finite group \(G\) on \(A\) which has the tracial Rokhlin property. Then \(A \times_\alpha G\) is a simple unital tracial \(\mathcal C\)-algebra.