A disintegration theorem for non-second-countable etale groupoids

Abstract

We give a self-contained account of a version of Renault's disintegration theorem for (twisted) C${}^{\ast }$-algebras of not necessarily Hausdorff étale groupoids that can be covered by countably many open bisections. As an application we discuss the $I$-norm, crossed product decompositions by inverse semigroup actions, and KMS-states and weights for general étale groupoid C${}^{\ast }$-algebras.

27 pages; v3: minor changes; v2: a section on inverse semigroup crossed products and a discussion of the Steinberg algebras added, more references, minor fixes