Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness (Q2662330)

Summary: We study simplicity and pure infiniteness criteria for \(\mathrm{C}^*\)-algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over étale not necessarily Hausdorff groupoids. Inspired by recent work of \textit{R. Exel} and \textit{D. R. Pitts} [``Characterizing groupoid \(\mathrm{C}^*\)-algebras of non-Hausdorff étale groupoids'', Preprint (2019); \url{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 \(\mathrm{C}^*\)-algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic \(\mathrm{C}^*\)-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.