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

TL;DR

This paper proves that under certain conditions, the crossed product of a simple unital C*-algebra by a finite group action with the weak tracial Rokhlin property retains key structural properties like property (TM) and stable rank one.

Contribution

It establishes that the weak tracial Rokhlin property ensures the preservation of property (TM) and stable rank one in crossed products of C*-algebras.

Findings

Crossed product preserves property (TM) under weak tracial Rokhlin property.

Stable rank one is maintained in crossed products with the weak tracial Rokhlin property.

Results apply to infinite-dimensional simple unital C*-algebras with strict comparison.

Abstract

Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, let $G$ be a finite group, and let $\alpha\colon G\rightarrow \mathrm{Aut}(A)$ 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 $A\rtimes_\alpha G$ 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, $\alpha\colon G\rightarrow \mathrm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then $A\rtimes_\alpha G$ has stable rank one.