Inclusions of Fell bundles ${\mathrm{C}}^{\ast }$-algebras and coaction crossed products

Abstract

Let $p:\mathcal{A}\to G$ be a Fell bundle over a locally compact Hausdorff second countable groupoid $G$ equipped with a Haar system, and let $\Gamma$ be a discrete group. Given a continuous $1$-cocycle $c:G\to \Gamma$, we show that the ${\mathrm{C}}^{\ast }$-algebra of the restricted Fell bundle $\mathcal{A}{|}_{G_e}$ embeds isometrically into ${\mathrm{C}}^{\ast }(G;\mathcal{A})$, where $G_e=c^{-1}(e)$ is the clopen subgroupoid corresponding to the identity element. We exploit this embedding to show that ${\mathrm{C}}^{\ast }(G;\mathcal{A})$ admits a natural structure of a topologically graded ${\mathrm{C}}^{\ast }$-algebra in the sense of Exel. As a consequence, we obtain a canonical coaction $\delta$ of $\Gamma$ on ${\mathrm{C}}^{\ast }(G;\mathcal{A})$. We further show that the associated coaction crossed product ${\mathrm{C}}^{\ast }(G;\mathcal{A}){⋊}_{\delta }\Gamma$ is naturally isomorphic to the ${\mathrm{C}}^{\ast }$-algebra of a Fell bundle constructed from the cocycle data.

19 pages