Ideal structure and pure infiniteness of ample groupoid -algebras

Abstract

In this paper, we study the ideal structure of reduced$C^{\ast }$-algebras$C_{r}^{\ast }(G)$associated to étale groupoids$G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in$C_{r}^{\ast }(G)$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$C_{r}^{\ast }(G)$is (strongly) purely infinite if and only if every non-zero projection in$C_{0}(G^{(0)})$is properly infinite in$C_{r}^{\ast }(G)$. We also establish a sufficient condition on the ample groupoid$G$that ensures pure infiniteness of$C_{r}^{\ast }(G)$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$C_{r}^{\ast }(G)$is a simple$C^{\ast }$-algebra which is either stably finite or strongly purely infinite.