Weakly tracially approximately representable actions

Abstract

We describe a weak tracial analog of approximate representability under the name "weak tracial approximate representability" for finite group actions. Let $G$ be a finite abelian group, let $A$ be an infinite-dimensional simple unital C*-algebra, and let $\alpha \colon G \to \operatorname{Aut} (A)$ be an action of $G$ on $A$ which is pointwise outer. Then $\alpha$ has the weak tracial Rokhlin property if and only if the dual action $\widehat{\alpha}$ of the Pontryagin dual $\widehat{G}$ on the crossed product $C^*(G, A, \alpha)$ is weakly tracially approximately representable, and $\alpha$ is weakly tracially approximately representable if and only if the dual action $\widehat{\alpha}$ has the weak tracial Rokhlin property. This generalizes the results of Izumi in 2004 and Phillips in 2011 on the dual actions of finite abelian groups on unital simple C*-algebras.