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Paper IPM / M / 17267  


Abstract:  
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 they 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 thatunder a mild assumption(tracial) $\mathcal{Z}$absorption is preserved under crossed products by such automorphisms.
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