Ideal structure and pure infiniteness of inverse semigroup crossed products

Abstract

We give efficient conditions under which a $\mathrm{C}^*$-subalgebra $A\subseteq B$ separates ideals in a $\mathrm{C}^*$-algebra $B$, 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 $\mathrm{C}^*$-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.