Talk:PlanetPhysics/Representation of Locally Compact Groupoids

Original TeX Content from PlanetPhysics Archive

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%%% Primary Title: representation of locally compact groupoids
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\begin{document}

 \begin{definition}
Let $\grp_{lc}$ be a locally compact (\htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html}) \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/EquivalenceRelation.html} endowed with a \htmladdnormallink{Haar system}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} $\nu = \nu^u, u \in U_{\grp_{lc}}$. Then a \emph{\htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}} of $\grp_{lc}$ together with the
its associated Haar system $\nu$ is defined as a \emph{triple} $(\mu, U_{\grp_{lc}} * \H, L)$,
where:
$\mu$ is a \emph{quasi-invariant measure} defined over $U_{\grp_{lc}}$,

$U_{\grp_{lc}}*\H$ is an analytical, fibered \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} or \htmladdnormallink{Hilbert bundle}{http://planetphysics.us/encyclopedia/HilbertBundle.html} over
$U_{\grp_{lc}}$, and

$L: U_{\grp_{lc}} \longrightarrow \textbf{Iso} (U_{\grp_{lc}}*\H )$ is a
Borelian groupoid \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} whose restriction on $U_{\grp_{lc}}$
is the \emph{identification map}, that is, $U_{\textbf{Iso}(U_{\grp_{lc}}*\H)}$ is
being identified \emph{via} $L$ with $U_{\grp_{lc}}$. Thus,

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

where $ \tilde{L}(x): \H (d(x)) \longrightarrow \H (r(x))$ is a Hilbert space $ \H $
\htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}.

\end{definition}

\end{document}