Ideal structure and pure infiniteness of ample groupoid $C^*$-algebras

Abstract

In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to étale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in $C_r^*(G)$ and the open invariant subsets of the unit space $G^{(0)}$ of $G$. As a consequence, we show that if $G$ is an inner exact, essentially principal, ample groupoid, then $C_r^*(G)$ is (strongly) purely infinite if and only if every non-zero projection in $C_0(G^{(0)})$ is properly infinite in $C_r^*(G)$. We also establish a sufficient condition on the ample groupoid $G$ that ensures pure infiniteness of $C_r^*(G)$ in terms of paradoxicality of compact open subsets of the unit space $G^{(0)}$. Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: Let $G$ be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then $C_r^*(G)$ is a simple $C^*$-algebra which is either stably finite or strongly purely infinite.

25 pages To appear in Ergodic Theory and Dynamical Systems