A universal property for groupoid C*-algebras. II. Fell bundles

TL;DR

This paper introduces a universal property for representations of full section C*-algebras of Fell bundles over groupoids, extending key theorems and establishing functoriality and exactness.

Contribution

It defines a universal property for Fell bundle C*-algebras and extends Renault's theorems to a broader class of bundles.

Findings

Proves the full section C*-algebra is functorial and exact.

Defines a quasi-orbit space and map for Fell bundles.

Extends Renault's Integration and Disintegration Theorems.

Abstract

We define possibly unsaturated, upper semicontinuous Fell bundles over Hausdorff, locally compact groupoids and establish a universal property for representations of their full section C*-algebras on Hilbert modules over arbitrary C*-algebras. Based on this, we prove that the full section C*-algebra is functorial and exact, and we define a quasi-orbit space and a quasi-orbit map. We deduce and extend Renault's Integration and Disintegration Theorems to general Fell bundles using our universal property.