# Journal of Operator Theory

Volume 91, Issue 1, Winter 2024 pp. 3-25.

Weakly tracially approximately representable actions

**Authors**:
M. Ali Asadi-Vasfi

**Author institution:** Department of Mathematics, Univ. of Toronto, Toronto, Ontario, M5S~2E4, Canada

**Summary: ** We describe a weak tracial analog of approximate representability under the name
\textit{weak tracial approximate representability}
for finite group actions.
We then investigate the dual actions on the crossed products by this class of group actions.
Namely, 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.

**DOI: **http://dx.doi.org/10.7900/jot.2021dec07.2430

**Keywords: **weak tracial approximate representability, duality, simple $C^*$-algebras, crossed product

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