PlanetPhysics/Representation of Locally Compact Groupoids

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Let $$\grp_{lc}$$ be a locally compact (topological) groupoid endowed with a Haar system $$\nu = \nu^u, u \in U_{\grp_{lc}}$$. Then a representation of $$\grp_{lc}$$ together with the its associated Haar system $$\nu$$ is defined as a triple $$(\mu, U_{\grp_{lc}} * \mathbb{H}, L)$$, where: $$\mu$$ is a quasi-invariant measure defined over $$U_{\grp_{lc}}$$,

$$U_{\grp_{lc}}*\mathbb{H}$$ is an analytical, fibered Hilbert space or Hilbert bundle over $$U_{\grp_{lc}}$$, and

$$L: U_{\grp_{lc}} \longrightarrow Iso (U_{\grp_{lc}}*\mathbb{H} )$$ is a Borelian groupoid morphism whose restriction on $$U_{\grp_{lc}}$$ is the identification map, that is, $$U_{Iso (U_{\grp_{lc}}*\mathbb{H})}$$ is being identified via $$L$$ with $$U_{\grp_{lc}}$$. Thus,

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

where $$ \tilde{L}(x): \mathbb{H} (d(x)) \longrightarrow \mathbb{H} (r(x))$$ is a Hilbert space $$ \mathbb{H} $$ isomorphism.