A Stabilization Theorem for Fell Bundles over groupoids

TL;DR

This paper establishes a stabilization theorem linking Fell bundles over groupoids to dynamical systems, enabling analysis of their $C^*$-algebras through crossed products and applications to ideal structure and twisted $k$-graph algebras.

Contribution

It constructs a groupoid dynamical system equivalent to any given Fell bundle, generalizing the Packer--Raeburn stabilization trick to a broader class of bundles.

Findings

Full and reduced $C^*$-algebras are stably isomorphic to crossed products.

Describes the ideal lattice of the $C^*$-algebra of a Fell bundle.

Provides criteria for simplicity of the Fell-bundle $C^*$-algebra.

Abstract

We study the $C^*$-algebras associated to upper-semicontinuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer--Raeburn "Stabilization Trick," we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced $C^*$-algebras of any saturated upper-semicontinuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the $C^*$-algebra of a continuous Fell-bundle by applying Renault's results about the ideals of the $C^*$-algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle $C^*$-algebra of a bundle over $G$ in terms of an action, described by the first and last named authors, of $G$ on the primitive-ideal space of the $C^*$-algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted $k$-graph algebras, where the components of our results become more concrete.