Ideal structure and pure infiniteness of inverse semigroup crossed products

TL;DR

This paper establishes conditions for ideal separation and pure infiniteness in $C^*$-inclusions, especially for crossed products by inverse semigroup actions, using concepts like essential crossed products and residual properties.

Contribution

It introduces new criteria for ideal separation and pure infiniteness in $C^*$-algebras involving inverse semigroup crossed products and develops the notions of essential crossed products, residual aperiodicity, and residual topological freeness.

Findings

Provided conditions for $A$ to separate ideals in $B$

Characterized pure infiniteness via positive elements in $A$

Extended theory to non-Hausdorff groupoid actions

Abstract

Let $A\subseteq B$ be a $C^*$-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 $C^*$-algebra of a Fell bundle over an \'etale, 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.