Abstract:We construct an equivariant E-theory and a Baum-Connes assembly map at the level of Fell bundles of inverse semigroups over separable C*-algebras. This generalizes previous work of several authors, and allows to discuss E-theoretic matters in the context of Cartan pairs; maximal and essential C*-algebras of non-Hausdorff groupoids; and Fell bundles over discrete groups and étale groupoids, among others. In order to do this we establish several functoriality properties for maximal, reduced and essential cross-sectional C*-algebras associated with a (saturated) Fell bundle of an inverse semigroup. This allows to discuss when these algebras give rise to short exact sequences, generalizing the classical case of discrete groups. We also introduce the adequate notion of ``proper'' Fell bundle, or ``proper'' action of an inverse semigroup, and prove a weak containment property for these. Using these functoriality properties and these proper actions we then introduce (maximal, reduced and/or essential) equivariant E-theory by means of adequately equivariant asymptotic morphisms, and construct a Baum-Connes assembly map that is both natural and reasonably well-behaved.
From: Diego Martínez [view email]
[v1]
Tue, 23 Jun 2026 12:31:21 UTC (67 KB)
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