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

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.