Stable rank for crossed products by actions of finite groups on C*-algebras

TL;DR

This paper proves that the stable rank of crossed product C*-algebras resulting from finite group actions on certain simple nuclear C*-algebras is equal to one under specific conditions, extending understanding of their algebraic structure.

Contribution

It establishes that the crossed product has stable rank one when the algebra absorbs the Jiang-Su algebra and satisfies certain trace space conditions, even with minimal assumptions on traces.

Findings

Stable rank of crossed products is one under given conditions.

Results apply to algebras with unique tracial states.

Extends known stable rank results to broader class of C*-algebras.

Abstract

Let $G$ be a finite group, $A$ a unital separable finite simple nuclear C*-algebra, and $\alpha$ an action of $G$ on $A$. Assume that $A$ absorbs the Jiang-Su algebra $\mathcal{Z}$, the extremal boundary of the trace space of $A$ is compact and finite dimensional and that $\alpha$ fixes any tracial state of $A$. Then tsr$(A \rtimes_\alpha G) = 1$. In particular, when $A$ has a unique tracial state, we conclude it without above conditons on a tracial state space of $A$.