A Dixmier-Douady classification for Fell algebras

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.

37 pages. v2: updated to correct an oversight in the proof of Lemma 3.2

References in corpus (7)

• T-duality for torus bundles via noncommutative topology
• Nonassociative tori and applications to T-duality
• Renault's Equivalence Theorem for Reduced Groupoid C*-algebras
• Non-Hausdorff groupoids
• Tensor products of maximal abelian subalgebras of C*-algebras
• The transformation groups whose C^*-algebras are Fell algebras
• Properties preserved under Morita equivalence of C*-algebras

Cited by in corpus (11)

• Reconstruction of groupoids and C*-rigidity of dynamical systems
• Étale groupoids and their $C^{\ast }$-algebras
• Cartan subalgebras in C*-algebras of Hausdorff etale groupoids
• Reconstruction of twisted Steinberg algebras
• $K$-theory and homotopies of 2-cocycles on higher-rank graphs
• Iterated crossed products for groupoid fibrations
• The equivalence relations of local homeomorphisms and Fell algebras
• Simplicity of twisted C*-algebras of Deaconu--Renault groupoids
• The local bisection hypothesis for twisted groupoid C*-algebras
• Topological freeness for $C^{\ast }$-correspondences
• The stable algebra of a Wieler solenoid: inductive limits and K-theory