GROUPOID C*-ALGEBRAS INVERTIBILITY

In this work given a second-countable, Hausdorff, etale, amenable groupoid $ G$ with compact unit space, we show that an element in ${C^{*}(G)} $ is invertible if and only if $\lambda_{x}(a) $ is invertible for every $ x$ in the unit space of $ G$, where $ \lambda_{x} $ refers to the regular representation of $C^{*}(G)$ on $\textit{l}_{2}(G_{x})$. We also prove that, for every $a$ in $C^{*}(G)$, there exists some $ x\in G^{(0)} $ such that $\parallel x \parallel=\parallel \lambda_{x}(a)\parallel $.