Stable rank of $\mathrm{C}(X)\rtimes\Gamma$

TL;DR

This paper proves that crossed product C*-algebras arising from free, minimal actions of countable discrete amenable groups on compact spaces have stable rank one, extending previous results and linking to properties like Jiang-Su absorption.

Contribution

It generalizes the stable rank one result to broader classes of group actions, including those with the uniform Rokhlin property and Cuntz comparison.

Findings

Crossed products from free, minimal $bZ^n$-actions have stable rank one.

Extension to actions of countable discrete amenable groups with specific properties.

Equivalence between Jiang-Su absorption and strict comparison in this context.

Abstract

It is shown that, for an arbitrary free and minimal $\mathbb Z^n$-action on a compact Hausdorff space $X$, the crossed product C*-algebra $\mathrm{C}(X)\rtimes\mathbb Z^n$ always has stable rank one, i.e., invertible elements are dense. This generalizes a result of Alboiu and Lutley on $\mathbb Z$-actions. In fact, for any free and minimal topological dynamical system $(X, \Gamma)$, where $\Gamma$ is a countable discrete amenable group, if it has the uniform Rokhlin property and Cuntz comparison of open sets, then the crossed product C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ has stable rank one. Moreover, in this case, the C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ absorbs the Jiang-Su algebra tensorially if, and only if, it has strict comparison of positive elements.