Hypergroupoids and C*-algebras

Let $G$ be a locally compact groupoid. If $X$ is a free and proper $G$-space, then $(X\ast X)\mathrm{/}G$ 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.