C*-Algebras of extensions of groupoids by group bundles

Abstract

Given a normal subgroup bundle $\mathcal{A}$ of the isotropy bundle of a groupoid $\Sigma$, we obtain a twisted action of the quotient groupoid $\Sigma /\mathcal{A}$ on the bundle of group $C^{\ast }$-algebras determined by $\mathcal{A}$ whose twisted crossed product recovers the groupoid $C^{\ast }$-algebra $C^{\ast }(\Sigma )$. Restricting to the case where $\mathcal{A}$ is abelian, we describe $C^{\ast }(\Sigma )$ as the $C^{\ast }$-algebra associated to a $\mathbf{T}$-groupoid over the tranformation groupoid obtained from the canonical action of $\Sigma /\mathcal{A}$ on the Pontryagin dual space of $\mathcal{A}$. We give some illustrative examples of this result.

28 Pages. This is a substantially revised version of the original with a new title. Section 4 of the old version will appear separately