RepresentationOfLocallyCompactGroupoids

Definition 0.1 Let ${\mathsf{G}}_{lc}$ be a locally compact (topological) groupoid endowed with a Haar system $\nu = \nu^u, u \in U_{{\mathsf{G}}_{lc}}$ . Then a representation of ${\mathsf{G}}_{lc}$ together with the its associated Haar system $\nu$ is defined as a triple $(\mu, U_{{\mathsf{G}}_{lc}} * \H , L)$ , where: $\mu$ is a quasi-invariant measure defined over $U_{{\mathsf{G}}_{lc}}$ ,

$U_{{\mathsf{G}}_{lc}}*\H$ is an analytical, fibered Hilbert space or Hilbert bundle over $U_{{\mathsf{G}}_{lc}}$ , and

$L: U_{{\mathsf{G}}_{lc}} \longrightarrow \textbf{Iso} (U_{{\mathsf{G}}_{lc}}*\H )$ is a Borelian groupoid morphism whose restriction on $U_{{\mathsf{G}}_{lc}}$ is the identification map, that is, $U_{\textbf{Iso}(U_{{\mathsf{G}}_{lc}}*\H )}$ is being identified via $L$ with $U_{{\mathsf{G}}_{lc}}$ . Thus,

$L(x)= [r(x), \tilde{L}(x), d(x)]$ ,

where $\tilde{L}(x): \H (d(x)) \longrightarrow \H (r(x))$ is a Hilbert space $\H$ isomorphism.