A Dixmier-Douady classification for Fell algebras

TL;DR

This paper extends the Dixmier-Douady classification from continuous-trace C*-algebras to Fell algebras by introducing a new invariant based on twisted groupoid C*-algebras and sheaf cohomology.

Contribution

It introduces a classification framework for Fell algebras using a Dixmier-Douady invariant, generalizing existing theory for continuous-trace C*-algebras.

Findings

Fell algebras are Morita equivalent to those with a diagonal subalgebra.

C*-diagonals in Fell algebras are exactly abelian subalgebras with the extension property.

A new invariant for Fell algebras is defined via twisted groupoid C*-algebras and sheaf cohomology.

Abstract

We generalise the Dixmier-Douady classification of continuous-trace C*-algebras to Fell algebras. To do so, we show that C*-diagonals in Fell algebras are precisely abelian subalgebras with the extension property, and use this to prove that every Fell algebra is Morita equivalent to one containing a diagonal subalgebra. We then use the machinery of twisted groupoid C*-algebras and equivariant sheaf cohomology to define the Dixmier-Douady invariant of a Fell algebra A, and to prove our classification theorem.