Geodesic
Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness
Documenta mathematica, Tome 26 (2021), pp. 271-335.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 46L55, 46L05, 20M18, 22A22
Keywords: Fell bundles, inverse semigroups, étale groupoids, simplicity, pure infiniteness 
@article{DOCMA_2021__26__a49,
     author = {Kwa\'sniewski, Bartosz Kosma and Meyer, Ralf},
     title = {Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness},
     journal = {Documenta mathematica},
     pages = {271--335},
     publisher = {mathdoc},
     volume = {26},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a49/}
}
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Kwaśniewski, Bartosz Kosma; Meyer, Ralf. Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness. Documenta mathematica, Tome 26 (2021), pp. 271-335. http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a49/