Talk:PlanetPhysics/Groupoid Representation Theorem

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\begin{document}

 \subsection{Groupoid representation theorem}

We shall briefly consider a main result due to Hahn (1978) that relates
\htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} and associated groupoid algebra representations:

\begin{theorem} {\rm (source: \cite{Hahn78, Hahn2}.)}~
Any \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of a groupoid $\grp$ with \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} $(\nu,
\mu)$ in a separable Hilbert space $\H$ induces a *-algebra
representation $f \mapsto X_f$ of the associated groupoid algebra
$ \Pi \grp, \nu)$ in $L^2 (U_{\grp} , \mu, \H )$ with the
following properties:
\begin{itemize}
\item[(1)]
For any $l,m \in H $ , one has that $\left|<X_f(u
\mapsto l), (u \mapsto m)>\right|\leq \left\|f_l\right\| \left\|l
\right\| \left\|m \right\|$ and

\item[(2)] $M_r (\alpha) X_f = X_{f \alpha \circ r}$, where
$M_r: L^\infty (U_{\grp}, \mu, \H) \longrightarrow \mathcal L (
L^2 (U_{\grp}, \mu, \H))$, with $M_r (\alpha)j = \alpha \cdot j$.
\end{itemize}
Conversely, any *-algebra representation with the above two
properties induces a \htmladdnormallink{groupoid representation}{http://planetphysics.us/encyclopedia/GroupRepresentations.html}, X, as follows:
\begin{equation}
\left\langle X_f , j, k\right\rangle ~ = ~ \int
f(x)[X(x)j(d(x)),k(r(x))d \nu (x)],
\end{equation}
(cf. p. 50 of Hahn, 1978).
\end{theorem}

\subsection{Remarks}

Furthermore, according to Seda (1986, on p.116) the continuity of a
\htmladdnormallink{Haar system}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} is equivalent to the continuity of the
convolution product $f*g$ for any pair $f, g$ of continuous \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} with
compact support. One may thus conjecture that similar results
could be obtained for functions with \textit{locally compact}
support in dealing with convolution products of either \htmladdnormallink{locally compact groupoids}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} or \htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}. Seda's result also implies
that the convolution algebra $C_{conv}(\grp)$ of a groupoid $\grp$ is
closed with respect to the convolution {*} if and only if the fixed Haar
system associated with the measured groupoid $\grp$ is
\textit{continuous} (Buneci, 2003).

In the case of groupoid algebras of transitive groupoids, Buneci
(2003) showed that representations of a measured groupoid
$({\grp, [\int \nu ^u d \tilde{ \lambda}(u)]=[\lambda]})$ on a
separable Hilbert space $\H$ induce \textit{non-degenerate}
$*$--representations $f \mapsto X_f$ of the associated groupoid
algebra $ \Pi (\grp, \nu,\tilde{\lambda})$ with properties
formally similar to (1) and (2) above. Moreover, as in the case
of \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \textit{there is a correspondence between the unitary
representations of a groupoid and its associated C*--convolution
algebra representations} (p.182 of Buneci, 2003), the latter
involving however fiber bundles of \htmladdnormallink{Hilbert spaces}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} instead of
single Hilbert spaces. Therefore, groupoid representations appear
as the natural construct for \htmladdnormallink{Algebraic Quantum Field Theories}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} (\htmladdnormallink{AQFT}{http://planetphysics.us/encyclopedia/MetaTheorems.html}) in
which nets of local \htmladdnormallink{observable}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} in Hilbert space fiber
bundles were introduced by Rovelli (1998).

\begin{thebibliography}{9}

\bibitem{GR02}
R. Gilmore: \emph{Lie Groups, Lie Algebras and Some of Their Applications.},
Dover Publs., Inc.: Mineola and New York, 2005.

\bibitem{Hahn78}
P. Hahn: Haar measure for measure groupoids., \textit{Trans. Amer. Math. Soc}. \textbf{242}: 1--33(1978).
(Theorem 3.4 on p. 50).

\bibitem{Hahn2}
P. Hahn: The regular representations of measure groupoids., \textit{Trans. Amer. Math. Soc}. \textbf{242}:34--72(1978).

\bibitem{HeynLifsctz}
R. Heynman and S. Lifschitz. 1958. \emph{Lie Groups and Lie Algebras}., New York and London: Nelson Press.

\bibitem{HLS2k8}
C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008);
arXiv:0709.4364v2 [quant--ph]

\end{thebibliography} 

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