2016/12/21 by Ionescu, Marius, Kumjian, Alex
#22A22 #46L05 #46L35 #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1612.07257
Given a short exact sequence of locally compact abelian groups 0 → A → B → C → 0 and a continuous C-valued 1-cocycle ϕ on a locally compact Hausdorff groupoid Γ we construct a twist of Γ by A that is trivial if and only if ϕ lifts. The cocycle determines a strongly continuous action of \widehatC into Aut C^*(Γ) and we prove that the C^*-algebra of the twist is isomorphic to the induced algebra of this action if Γ is amenable. We apply our results to a groupoid determined by a locally finite cover of a space X and a cocycle provided by a Čech 1-cocycle with coefficients in the sheaf of germs of continuous \mathbbT-valued functions. We prove that the C^*-algebra of the resulting twist is continuous trace and we compute its Dixmier-Douady invariant.