Ideal structure and pure infiniteness of inverse semigroup crossed products

Abstract

Let $A\subseteq B$ be a $C^{\ast }$-inclusion. We give efficient conditions under which $A$ separates ideals in $B$, and $B$ is purely infinite if every positive element in $A$ is properly infinite in $B$. We specialise to the case when $B$ is a crossed product for an inverse semigroup action by Hilbert bimodules or a section $C^{\ast }$-algebra of a Fell bundle over an étale, possibly non-Hausdorff, groupoid. Then our theory works provided $B$ is the recently introduced essential crossed product and the action is essentially exact and residually aperiodic or residually topologically free. These last notions are developed in the article.

An error in the earlier version of Proposition 4.2 is fixed (we thank Alcides Buss, Diego Martínez and Jonathan Taylor for pointing it to us). To appear in Journal of Noncommutative Geometry