A Dixmier–Douady theorem for Fell algebras

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 an analogue of the Dixmier–Douady invariant for Fell algebras, and to prove our classification theorem.