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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