Ideal structure and pure infiniteness of ample groupoid C∗-algebras

Abstract

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