Ideal structure and pure infiniteness of inverse semigroup crossed products

  • Bartosz Kosma Kwaśniewski

    University of Białystok, Poland
  • Ralf Meyer

    Georg-August-Universität Göttingen, Germany
Ideal structure and pure infiniteness of inverse semigroup crossed products cover
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Abstract

We give efficient conditions under which a C\mathrm{C}^*-subalgebra ABA\subseteq B separates ideals in a C\mathrm{C}^*-algebra BB, and BB is purely infinite if every positive element in AA is properly infinite in BB. We specialise to the case when BB is a crossed product for an inverse semigroup action by Hilbert bimodules or a section C\mathrm{C}^*-algebra of a Fell bundle over an étale, possibly non-Hausdorff, groupoid. Then our theory works provided BB is the recently introduced essential crossed product and the action is essentially exact and residually aperiodic or residually topologically free. These last notions are developed in the article.

Cite this article

Bartosz Kosma Kwaśniewski, Ralf Meyer, Ideal structure and pure infiniteness of inverse semigroup crossed products. J. Noncommut. Geom. 17 (2023), no. 3, pp. 999–1043

DOI 10.4171/JNCG/506