Groupoid actions and Koopman representations

TL;DR

This paper investigates the $C^*$-algebra generated by Koopman representations of groupoids acting on measure spaces, establishing conditions for their equivalence and connections to groupoid $C^*$-algebras.

Contribution

It provides a new interpretation of Koopman representations as induced representations and characterizes when these algebras are isomorphic to groupoid $C^*$-algebras.

Findings

If the groupoid action is amenable, the Koopman $C^*$-algebra is a quotient of the reduced groupoid $C^*$-algebra.

Under certain conditions, the Koopman $C^*$-algebra is isomorphic to the groupoid $C^*$-algebra.

The paper applies these results to Renault-Deaconu groupoids, identifying cases of isomorphism.

Abstract

We study the $C^*$-algebra $C^*(\kappa)$ 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 $G\ltimes X$ is amenable, then $\kappa$ is weakly contained in the regular representation $\rho=\rho^\mu$ associated to $\mu$, so we have a surjective homomorphism $C^*_r(G)\to C^*(\kappa)$. 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 $C^*(\kappa)\cong C^*(G)$.