equivalent representations of groupoids

Definition 0.1   Two representations of groupoids $ (\mu_i, U_{{\mathsf{G}}} * \H , L_i)$ , for $ i=1,2$ are called equivalent if $ \mu_1 \sim \mu_2$ , and if there also exists a fiber-preserving isomorphism of analytical Hilbert space bundles $ v: (U_{{\mathsf{G}}}* \H _1)\vert _U \longrightarrow (U_{{\mathsf{G}}}* \H _2)\vert _U$ , where $ U$ is a measurable subset of $ U_{{\mathsf{G}}}$ of null complementarity; the isomorphism $ v$ also has the following property: $ \hat{v}[r(x)]\hat{L}_1(x) = \hat{L}_2 \hat{v}[d(x)]$ for $ x \in {\mathsf{G}}\vert _U $ .



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