Amenability for Fell bundles over groupoids

TL;DR

This paper identifies conditions where the universal and reduced C*-algebras of Fell bundles over groupoids are equal, especially focusing on measurewise amenable groupoids and those with T_0 orbit spaces.

Contribution

It establishes new criteria for the equality of full and reduced C*-algebras of Fell bundles over certain classes of groupoids, extending previous understanding.

Findings

Full and reduced C*-algebras coincide for measurewise amenable groupoids.

Equality holds when the orbit space is T_0 and the isotropy group restrictions' algebras coincide.

Provides conditions linking groupoid properties to C*-algebra equivalences.

Abstract

We establish conditions under which the universal and reduced norms coincide for a Fell bundle over a groupoid. Specifically, we prove that the full and reduced C*-algebras of any Fell bundle over a measurewise amenable groupoid coincide, and also that for a groupoid G whose orbit space is T_0, the full and reduced algebras of a Fell bundle over G coincide if the full and reduced algebras of the restriction of the bundle to each isotropy group coincide.