Obstructions to lifting cocycles on groupoids and the associated $C^{\ast }$-algebras

Abstract

Given a short exact sequence of locally compact abelian groups $0\to A\to B\to C\to 0$ and a continuous $C$-valued $1$-cocycle $\varphi$ on a locally compact Hausdorff groupoid $\Gamma$ we construct a twist of $\Gamma$ by $A$ that is trivial if and only if $\varphi$ lifts. The cocycle determines a strongly continuous action of $\hat{C}$ into $\mathrm{Aut}{C}^{\ast }(\Gamma )$ and we prove that the $C^{\ast }$-algebra of the twist is isomorphic to the induced algebra of this action if $\Gamma$ 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 $\mathbb{T}$-valued functions. We prove that the $C^{\ast }$-algebra of the resulting twist is continuous trace and we compute its Dixmier-Douady invariant.

We made some changes indicated by the referee. The paper is accepted for publication in the Münster Journal of Mathematics

References in corpus (1)

• Renault's Equivalence Theorem for Groupoid Crossed Products