| Version 6 |
Version 5 |
| \begin{definition} |
\begin{definition} |
| Let $\grp_{lc}$ be a locally compact (topological) groupoid endowed with a Haar system |
Let $\grp_{lc}$ be a locally compact (topological) groupoid endowed with a Haar system |
| $\nu = \nu^u, u \in U_{\grp_{lc}}$. Then a \emph{representation) of $\grp_{lc}$ together with the |
$\nu = \nu^u, u \in U_{\grp_{lc}}$. Then a \emph{representation) of $\grp_{lc}$ together with the |
| its associated Haar system $\nu$ is defined as a \emph{triple} $(\mu, U_{\grp_{lc}} * \H, L)$, |
its associated Haar system $\nu$ is defined as a \emph{triple} $(\mu, U_{\grp_{lc}} * \H, L)$, |
| where: \\ |
where: \\ |
| $\mu$ is a \emph{quasi-invariant measure} defined over $U_{\grp_{lc}}$, \\ |
$\mu$ is a \emph{quasi-invariant measure} defined over $U_{\grp_{lc}}$, \\ |
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| $U_{\grp_{lc}}*\H$ is an analytical, fibered Hilbert space or Hilbert bundle over |
$U_{\grp_{lc}}*\H$ is an analytical, fibered Hilbert space or Hilbert bundle over |
| $U_{\grp_{lc}}$, and \\ |
$U_{\grp_{lc}}$, and \\ |
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| $L: $U_{\grp_{lc}} \longrightarrow \textbf{Iso} (U_{\grp_{lc}}*\H )$ is a |
$L: $U_{\grp_{lc}} \longrightarrow \textbf{Iso} (U_{\grp_{lc}}*\H )$ is a |
| Borelian (or \emph{borelian}) groupoid morphism whose restriction on $U_{\grp_{lc}}$ |
Borelian (or \emph{borelian}) groupoid morphism whose restriction on $U_{\grp_{lc}}$ |
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is the \emph{identification map}, that is, $U_{\textbf{Iso} (U_{\grp_{lc}}*\H)}}$ is
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is the \emph{identification map}, that is, $U_{\textbf{Iso} (U_{\grp_{lc}}*\H)}}$ is
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| being identified \emph{via} $L$ with $U_{\grp_{lc}}$. Thus, \\ |
being identified \emph{via} $L$ with $U_{\grp_{lc}}$. Thus, \\ |
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| $L(x)= [r(x), \tilde{L}(x), d(x)]$, \\ |
$L(x)= [r(x), \tilde{L}(x), d(x)]$, \\ |
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| where $ \tilde{L}(x): \H (d(x)) \longrightarrow \H (r(x))$ is a Hilbert space $ \H $ |
where $ \tilde{L}(x): \H (d(x)) \longrightarrow \H (r(x))$ is a Hilbert space $ \H $ |
| isomorphism. |
isomorphism. |
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| \end{definition} |
\end{definition} |
