Hypergroupoids and C*-algebras

Let $G$ be a locally compact groupoid. If $X$ is a free and proper $G$-space, then $(X*X)/G$ is a groupoid equivalent to $G$. We consider the situation where $X$ is proper but no longer free. The formalism of groupoid C*-algebras and their representations is suitable to attach C*-algebras to this new object.Comment: This is the authors' English version of a work that was published in Comptes rendus-Math\'ematique [Ser. I 351 (2013) 911-914]. References [5,6,10,12] have been added since publicatio

C∗^*-algebras of Fell bundles over \'etale groupoids

We describe a construction for the full C$^*$-algebra of a possibly unsaturated Fell bundle over a possibly non-Hausdorff locally compact \'etale groupoid without appealing to Renault's disintegration theorem. This construction generalises the standard construction given by Muhly and Williams.Comment: 15 pages. Initial version, comments are welcom

Topological fundamental groupoid. III. Haar systems on the fundamental groupoid

Let $X$ be a path connected, locally path connected and semilocally simply connected space; let $\tilde{X}$ be its universal cover. We discuss the existence and description of a Haar system on the fundamental groupoid $\Pi_1(X)$ of $X$. The existence of a Haar system on $\Pi_1(X)$ is justified when $X$ is a second countable, locally compact and Hausdorff. We provide equivalent criteria for the existence of the Haar system on a locally compact (locally Hausdorff) fundamental groupoid in terms of certain measures on $X$ and $\tilde{X}$. $\mathrm{C}^*(\Pi_1(X))$ is described using a result of Muhly, Renault and Williams. Finally, two formulae for the Haar system on $\Pi_1(X)$ in terms of measures on $X$ or $\tilde{X}$ are given.Comment: 14 pages, 2 figures. arXiv admin note: text overlap with arXiv:2305.0466