Aperiodicity and related properties for crossed product inclusions

Abstract: In recent work with Bartosz Kwa?niewski, we have vastly generalised the condition that was introduced by Kishimoto in order to prove that reduced crossed products for outer group actions on simple C*-algebras are again simple. We call this condition aperiodicity, and it applies to arbitrary inclusions of C*-algebras, without requiring a crossed product structure. We relate this to topological non-triviality conditions in the special case of actions of inverse semigroups or étale groupoids (which are possibly non-Hausdorff). In that generality, we define an essential crossed product, which is a quotient of the reduced crossed product. If the action satisfies Kishimoto's condition, then the coefficient algebra detects ideals in this essential crossed product. And in the simple case, we also get criteria for the essential crossed product to be simple. We also relate aperiodicity to other properties that have been used to study the ideal structure of crossed products. This includes unique pseudo-expectations and the almost extension property, which assume that the set of pure states on the coefficient algebra that extend uniquely to the crossed product is dense.

operator algebrasrings and algebras

Audience: researchers in the topic

Western Sydney University Abend Seminars

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