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

By B. Kwaśniewski, Ralf Meyer. Documenta Mathematica, 2019

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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 group actions to general C*-inclusions. For inverse semigroup actions, aperiodicity implies that the coefficient algebra supports the essential crossed product and detects ideals in it. It is equivalent to the topological freeness of the dual groupoid for actions of inverse semigroups on C*-algebras that are separable or of Type I. This gives efficient criteria for the essential crossed product to be purely infinite and simple. In particular, for a possibly non-Hausdorff groupoid, the commutative C*-algebra of functions on the object space detects ideals in the essential groupoid C*-algebra if and only if it supports the essential groupoid C*-algebra, if and only if the groupoid is topologically free.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c168b84002844856d5ac4fe2b610dd815a2bb0c9