Weakly tracially approximately representable actions

TL;DR

This paper introduces a new concept called weak tracial approximate representability for finite group actions on C*-algebras, linking it to the weak tracial Rokhlin property of dual actions, generalizing earlier results.

Contribution

It defines weak tracial approximate representability and establishes its equivalence with the weak tracial Rokhlin property of dual actions for finite abelian group actions on simple unital C*-algebras.

Findings

Weak tracial Rokhlin property characterizes dual actions.

Weak tracial approximate representability is equivalent to the dual action having the Rokhlin property.

Generalizes results by Izumi and Phillips on dual 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.