PlanetPhysics/Representation of Locally Compact Groupoids
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Let Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} be a locally compact (topological) groupoid endowed with a Haar system Failed to parse (unknown function "\grp"): {\displaystyle \nu = \nu^u, u \in U_{\grp_{lc}}} . Then a representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \grp_{lc}} together with the
its associated Haar system is defined as a triple Failed to parse (unknown function "\grp"): {\displaystyle (\mu, U_{\grp_{lc}} * \mathbb{H}, L)} , where: is a quasi-invariant measure defined over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle U_{\grp_{lc}}} ,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle U_{\grp_{lc}}*\mathbb{H}} is an analytical, fibered Hilbert space or 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 morphism whose restriction on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle U_{\grp_{lc}}} . Thus,
,
where is a Hilbert space isomorphism.