Stable rank of $\mathrm{C}(X)⋊\Gamma$

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 $\mathrm{C}(X)⋊{\mathbb{Z}}^n$ 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 $\mathrm{C}(X)⋊\Gamma$ has stable rank one. Moreover, in this case, the C*-algebra $\mathrm{C}(X)⋊\Gamma$ absorbs the Jiang-Su algebra tensorially if, and only if, it has strict comparison of positive elements.

The previous version is revised, and an error in the proof of Lemma 7.2 is fixed

References in corpus (2)

• Almost finiteness for general etale groupoids and its applications to stable rank of crossed products
• Comparison radius and mean topological dimension: ${\mathbb{Z}}^d$-actions