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

@article{Kwaniewski2019EssentialCP,
  title={Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness},
  author={Bartosz Kosma Kwaśniewski and Ralf Meyer},
  journal={arXiv: Operator Algebras},
  year={2019}
}
We define "essential" crossed products for inverse semigroup actions by Hilbert bimodules on C*-algebras and for Fell bundles over etale, locally compact groupoids. If the underlying groupoid is non-Hausdorff, this is a quotient of the reduced crossed product by an ideal coming from a generalised expectation with values in the local multiplier algebra. We characterise when the essential and reduced crossed products coincide. We generalise the notion of aperiodicity or proper outerness from… 
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