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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.

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

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