Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

Abstract

We study simplicity and pure infiniteness criteria for ${\mathrm{C}}^{\ast }$-algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over étale not necessarily Hausdorff groupoids. Inspired by recent work of R. Exel and D. R. Pitts ["Characterizing groupoid ${\mathrm{C}}^{\ast }$-algebras of non-Hausdorff étale groupoids", Preprint (2019); [arXiv: 1901.09683](arXiv: 1901.09683)], we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced ${\mathrm{C}}^{\ast }$-algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic ${\mathrm{C}}^{\ast }$-inclusions equipped with supportive generalised expectations. We thoroughly discuss the relationship between aperiodicity, detection of ideals, purely outer inverse semigroup actions, and non-triviality conditions for dual groupoids.