The Tracial Class Property for Crossed Products by Finite Group Actions

Abstract

We define the concept of tracial $𝒞$-algebra of $C^{*}$-algebras, which generalize the concept of local $𝒞$-algebra of $C^{*}$-algebras given by H. Osaka and N. C. Phillips. Let $𝒞$ be any class of separable unital $C^{*}$-algebras. Let $A$ be an infinite dimensional simple unital tracial $𝒞$-algebra with the (SP)-property, and let $\alpha :G\to \text{A}\text{u}\text{t}(A)$ 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 $𝒞$-algebra.