PlanetPhysics/Representation of Locally Compact Groupoids

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Let Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}}
 be a locally compact ([[../CoIntersections/|topological]]) [[../EquivalenceRelation/|groupoid]] endowed with a [[../QuantumOperatorAlgebra5/|Haar system]] Failed to parse (unknown function "\grp"): {\displaystyle \nu = \nu^u, u \in U_{\grp_{lc}}}
. Then a [[../CategoricalGroupRepresentation/|representation]] of Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}}
 together with the

its associated Haar system $\nu$ is defined as a triple Failed to parse (unknown function "\grp"): {\displaystyle (\mu, U_{\grp_{lc}} * \mathbb{H}, L)} , where: $\mu$ is a quasi-invariant measure defined over Failed to parse (unknown function "\grp"): {\displaystyle U_{\grp_{lc}}} ,

Failed to parse (unknown function "\grp"): {\displaystyle U_{\grp_{lc}}*\mathbb{H}} is an analytical, fibered [[../NormInducedByInnerProduct/|Hilbert space]] or [[../HilbertBundle/|Hilbert bundle]] over Failed to parse (unknown function "\grp"): {\displaystyle U_{\grp_{lc}}} , and

Failed to parse (unknown function "\grp"): {\displaystyle L: U_{\grp_{lc}} \longrightarrow '''Iso''' (U_{\grp_{lc}}*\mathbb{H} )} is a Borelian groupoid [[../TrivialGroupoid/|morphism]] whose restriction on Failed to parse (unknown function "\grp"): {\displaystyle U_{\grp_{lc}}} is the identification map , that is, Failed to parse (unknown function "\grp"): {\displaystyle U_{'''Iso''' (U_{\grp_{lc}}*\mathbb{H})}} is being identified via $L$ with Failed to parse (unknown function "\grp"): {\displaystyle U_{\grp_{lc}}} . Thus,

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

where $\tilde{L}(x):ℍ(d(x))⟶ℍ(r(x))$ is a Hilbert space $ℍ$ [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]].