On dualities of actions

TL;DR

This paper establishes a duality between two properties of group actions on certain $C^*$-algebras, linking the weak tracial Rokhlin property and weak tracial approximate representability, especially for finite abelian groups.

Contribution

It introduces the concept of weak tracial approximate representability for actions on projectionless $C^*$-algebras and proves a duality with the weak tracial Rokhlin property for finite abelian groups.

Findings

Duality between weak tracial Rokhlin property and weak tracial approximate representability.

Equivalence of properties under dual actions for finite abelian groups.

Applicable to unital simple infinite dimensional $C^*$-algebras, including Jiang-Su algebra.

Abstract

We introduce the notion of the weak tracial approximate representability of a discrete group action on a unital $C^*$-algebra which could have no projections like the Jiang-Su algebra $\mathcal{Z}$. Then we show a duality between the weak tracial Rokhlin property and the weak tracial approximate representability. More precisely, when $G$ is a finite abelian group and $\alpha:G\curvearrowright A$ is a group action on a unital simple infinite dimensional $C^*$-algebra, we prove that 1. $\alpha$ has the weak tracial Rokhlin property if and only if $\hat{\alpha}$ has the weak tracial approximate representability. 2. $\alpha$ has the weak tracial approximate representability if and only if $\hat{\alpha}$ has the weak tracial Rokhlin property.