PlanetPhysics/Equivalent Representations of Groupoids

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Two representations of groupoids $$(\mu_i, U_{\grp} * \mathbb{H}, L_i)$$, 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 $$v: (U_{\grp}* \mathbb{H}_1)|_U \longrightarrow (U_{\grp}* \mathbb{H}_2)|_U$$ , where $$U$$ is a measurable subset of $$U_{\grp}$$ 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 $$x \in \grp |_U $$.