Abstract
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 R. Exel and D. R. Pitts ["Characterizing groupoid \mathrm{C}^* -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}^* -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.
Published Version (Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have