Geodesic
Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness
Documenta mathematica, Tome 26 (2021), pp. 271-335 Cet article a éte moissonné depuis la source EMS Press

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We study simplicity and pure infiniteness criteria for 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 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 C∗-algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic 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.
DOI : 10.4171/dm/815
Classification : 20M18, 22A22, 46L05, 46L55
Mots-clés : Fell bundles, inverse semigroups, étale groupoids, simplicity, pure infiniteness 
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     author = {Bartosz Kosma Kwa\'sniewski and Ralf Meyer},
     title = {Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness},
     journal = {Documenta mathematica},
     pages = {271--335},
     year = {2021},
     volume = {26},
     doi = {10.4171/dm/815},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/815/}
}
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Bartosz Kosma Kwaśniewski; Ralf Meyer. Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness. Documenta mathematica, Tome 26 (2021), pp. 271-335. doi: 10.4171/dm/815

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