Journal of Operator Theory

Volume 92, Issue 2, Autumn 2024 pp. 505-547.

Finite group and integer actions on simple tracially $\mathcal{Z}$-absorbing $C^*$-algebras

Authors: Massoud Amini (1), Nasser Golestani (2), Saeid Jamali (3), and N. Christopher Phillips (4)
Author institution: (1) Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran
(2) Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran
(3) Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran
(4) Department of Mathematics, University of Oregon, Eugene OR 97403-1222, U.S.A.

Summary: We show that if $A$ is a simple (not necessarily unital) tracially $\mathcal{Z}$-absorbing $C^*$-algebra and $\alpha \colon G \to \mathrm{Aut} (A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then the crossed product $C^*(G, A,\alpha)$ and the fixed point algebra $A^{\alpha}$ are simple and tracially $\mathcal{Z}$-absorbing, and are $\mathcal{Z}$-stable if, in addition, $A$ is separable and nuclear. The same conclusion holds for all intermediate $C^*$-algebras of the inclusions $A^{\alpha} \subseteq A$ and $A \subseteq C^*(G, A,\alpha)$. We prove that if $A$ is a simple tracially $\mathcal{Z}$-absorbing $C^*$-algebra, then, under a finiteness condition, the permutation action of the symmetric group $S_m$ on the minimal $m$-fold tensor product of $A$ has the weak tracial Rokhlin property. We define the weak tracial Rokhlin property for automorphisms of simple $C^*$-algebras and we show that - under a mild assumption - (tracial) $\mathcal{Z}$-absorption is preserved under crossed products by such automorphisms.

DOI: http://dx.doi.org/10.7900/jot.2022nov02.2417
Keywords: $C^*$-algebra, crossed product, group action, tracial $\mathcal{Z}$-absorption, weak tracial Rokhlin property

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