Invertibility in Groupoid C*-algebras

Abstract

Given a second-countable, Hausdorff, étale, amenable groupoid $\mathcal{G}$ with compact unit space, we show that an element a in $C^{\star}(\mathcal{G})$ is invertible if and only if $\lambda_{x}(a)$ is invertible for every x in the unit space of $\mathcal{G}$,where $\lambda_{x}$ refers to the regular representation of $C^{\star}(\mathcal{G})$ on $l_{2}(\mathcal{G}_{x})$. We also prove that, for every a in $C^{\star}(\mathcal{G})$, there exists some $x\;\in\;\mathcal{G}^{(0)}$ such that $\parallel a \parallel\,=\,\parallel \lambda_{x}(a) \parallel$.

Mathematics Subject Classification (2010). Primary 22A22; Secondary 46L05, 46L55.