Stable rank for crossed products by finite group actions with the weak tracial Rokhlin property

Abstract

Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, let $G$ be a finite group, and let $\alpha :G\to \mathrm{Aut}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. We prove that if $A$ has property (TM), then the crossed product $A{⋊}_{\alpha }G$ has property (TM). As a corollary, if $A$ is an infinite-dimensional separable simple unital C*-algebra which has stable rank one and strict comparison, $\alpha :G\to \mathrm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then $A{⋊}_{\alpha }G$ has stable rank one.

20 pages