Groupoid actions and Koopman representations

We study the $C^{\ast }$-algebra $C^{\ast }(\kappa )$ generated by the Koopman representation $\kappa ={\kappa }^{\mu }$ of a locally compact groupoid $G$ acting on a measure space $(X,\mu )$, where $\mu$ is quasi-invariant for the action. We interpret $\kappa$ as an induced representation and we prove that if the groupoid $G⋉X$ is amenable, then $\kappa$ is weakly contained in the regular representation $\rho ={\rho }^{\mu }$ associated to $\mu$, so we have a surjective homomorphism $C_r^{\ast }(G)\to C^{\ast }(\kappa )$. We consider the particular case of Renault-Deaconu groupoids $G=G(X,T)$ acting on their unit space $X$ and show that in some cases $C^{\ast }(\kappa )\cong C^{\ast }(G)$.