Group actions on simple tracially $\mathcal{Z}$-absorbing C*-algebras

Abstract

We show that if $A$ is a simple (not necessarily unital) tracially $\mathcal{Z}$-absorbing C*-algebra and $\alpha :G\to \mathrm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then the crossed product $C^{\ast }(G,A,\alpha )$ 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 $A\subseteq C^{\ast }(G,A,\alpha )$. 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.

42 pages, a new main result is added (Corollary C)