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

TL;DR

This paper demonstrates that for simple tracially $ ext{Z}$-absorbing C*-algebras with finite group actions having the weak tracial Rokhlin property, the resulting crossed products and fixed point algebras retain simplicity and tracial $ ext{Z}$-absorption, extending to $ ext{Z}$-stability under certain conditions.

Contribution

It establishes the preservation of simplicity and tracial $ ext{Z}$-absorption under crossed products and fixed point algebras for finite group actions with the weak tracial Rokhlin property, and introduces this property for automorphisms.

Findings

Crossed products and fixed point algebras are simple and tracially $ ext{Z}$-absorbing.

The weak tracial Rokhlin property is defined for automorphisms of simple C*-algebras.

Tracial $ ext{Z}$-absorption is preserved under crossed products by automorphisms with the weak tracial Rokhlin property.

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 that -- under a mild assumption -- (tracial) $\mathcal{Z}$-absorption is preserved under crossed products by such automorphisms.