PlanetPhysics/Equivalent Representations of Groupoids

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 Two representations of groupoids Failed to parse (unknown function "\grp"): {\displaystyle (\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 Failed to parse (unknown function "\grp"): {\displaystyle v: (U_{\grp}* \mathbb{H}_1)|_U \longrightarrow (U_{\grp}* \mathbb{H}_2)|_U} , where $U$ is a measurable subset of Failed to parse (unknown function "\grp"): {\displaystyle 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 Failed to parse (unknown function "\grp"): {\displaystyle x \in \grp |_U } .