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This series of talks is an invitation to my recent joint work with Bartosz Kwasniewski. Actions of inverse semigroups contain actions of groups and etale groupoids and Fell bundles over them as special cases. The crossed products for such actions contain twisted groupoid C*-algebras of etale groupoids as a special case. Besides the full crossed product, I will explain the essential crossed product, which is defined through a conditional expectation with values in the local multiplier algebra of the coefficient algebra. If the inverse semigroup action is sufficiently non-trivial, then the inclusion of the coefficient algebra in the essential crossed product detects ideals, that is, a non-zero ideal in the crossed product has non-zero intersection with the coefficient algebra. The preferred definition of sufficiently non-trivial is a condition due to Kishimoto, which is called "aperiodic" in my joint articles with Kwasniewski; this property is often equivalent to other non-triviality properties such as the topological freeness of the dual groupoid, which is defined as the transformation groupoid of the induced inverse semigroup action on the space of equivalence classes of irreducible representations of the coefficient algebra. In particular, the essential groupoid C*-algebra of an etale, possibly non-Hausdorff, locally compact groupoid is simple if and only if the underlying groupoid is topologically free and minimal. I plan to start explaining the generalised intersection property for groupoid C*-algebras of etale groupoids, which does not yet require heavy notation. Then I will briefly discuss some of the classical results by Kishimoto and Olesen--Pedersen for group actions by automorphisms and modify Kishimoto's original condition to the definition of an aperiodic inclusion of C*-algebras. Then I formulate the main theorem on the generalised intersection property for aperiodic inclusions. I will sketch the elegant direct proof of this result. (In the original article, a stronger property related to pure infiniteness of crossed products is proven, which requires more work.) Then I turn to crossed products by inverse semigroup actions. I will define full and essential crossed products for them and discuss aperiodicity for inverse semigroup actions. |