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%%% Primary Title: groupoid C*-convolution algebra
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\begin{document}
\subsubsection{Background and Data for the Definition of a Groupoid $C^*$--Convolution Algebra}
Jean Renault introduced in ref. \cite{JR80} the \emph{$C^*$--algebra of a \htmladdnormallink{locally compact groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} $\grp$} as follows: the space of continuous \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} with compact support on a \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} $\grp$ is made into a *-algebra whose multiplication is the \emph{\htmladdnormallink{convolution}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html}}, and that is also endowed with the smallest $C^*$--norm which makes its \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} continuous, as shown in ref.\cite{MAB2k3}. Furthermore, for this convolution to be defined, one needs also to have a \htmladdnormallink{Haar system}{http://planetphysics.us/encyclopedia/GroupoidRepresentationsInducedByMeasure.html}
associated to the locally compact groupoids $\grp$
that are then called \emph{measured groupoids} because they are endowed with an associated \htmladdnormallink{Haar system}{http://planetphysics.us/encyclopedia/Groupoid.html} which involves the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of measure, as introduced in ref. \cite{Hahn1} by P. Hahn.
With these concepts one can now sum up the definition (or construction) of the \emph{groupoid $C^*$-convolution algebra}, or \htmladdnormallink{groupoid $C^*$-algebra}{http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf}, as follows.
\begin{definition} a {\em groupoid C*--convolution algebra}, $G_{CA}$, is defined for \emph{measured groupoids}
as a \emph{*--algebra with ``$*$'' being defined by convolution so that it has a smallest $C^*$--norm which makes its representations continuous}.
\end{definition}
\begin{remark}
One can also produce a functorial construction of $G_{CA}$ that has additional interesting properties.
\end{remark}
Next we recall a result due to P. Hahn \cite{PH78} which shows how \htmladdnormallink{groupoid representations}{http://planetphysics.us/encyclopedia/GroupRepresentations.html} relate to
induced \htmladdnormallink{*-algebra representations}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} and also how--under certain conditions-- the former can be derived from
the appropriate *-algebra representations.
\begin{theorem}
(source: ref. \cite{PH78}). Any representation of a groupoid $(\grp,C)$ with \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} $(\nu, \mu)$ in a \htmladdnormallink{separable Hilbert space}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} $\H$ 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:
(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
\med
(2) $M_r (\alpha) X_f = X_{f \alpha \circ r}$, where
\med
$M_r: L^\infty (U_{\grp}, \mu \longrightarrow L[L^2 (U_{\grp}, \mu, \H]$, with
$M_r (\alpha)j = \alpha \cdot j$.
\textit{Conversely, any *- algebra representation with the above two properties induces a groupoid representation, X, as follows:}
\med
$<X_f , j, k> ~ = ~ \displaystyle{\int} f(x)[X(x)j(d(x)),k(r(x))d \nu (x)].$
(viz. p. 50 of ref. \cite{PH78}).
\end{theorem}
Furthermore, according to Seda (ref. \cite {Seda86,Seda2k8}), the continuity of a Haar system is equivalent to the continuity of the convolution product $f*g$ for any pair $f$, $g$ of continuous functions 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 locally compact groupoids or \htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}. Seda's result also implies that the convolution algebra $C_c (\G)$ of a groupoid $\G$ is closed with respect to convolution if and only if the fixed Haar system associated with the measured groupoid $\G$ is \textit{continuous} (see ref. \cite{MAB2k3}).
Thus, in the case of groupoid algebras of transitive groupoids, it was shown in \cite{MAB2k3} that any representation of a measured groupoid $(\G, [\displaystyle{\int} \nu ^u d \tilde{\lambda}(u)] = [\lambda])$ on a separable Hilbert space $\H$ induces a \textit{non-degenerate} *-representation $f \mapsto X_f$ of the associated groupoid algebra
$\Pi (\G, \nu,\tilde{\lambda})$ with properties formally similar to (1) and (2) above.
Moreover, as in the case of \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \emph{there is a correspondence between the unitary representations of a groupoid and its associated C*-convolution algebra representations} (p. 182 of \cite{MAB2k3}), the latter involving however fiber bundles of \htmladdnormallink{Hilbert spaces}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} instead of single Hilbert spaces.
\begin{thebibliography}{9}
\bibitem{Hahn1}
P. Hahn: Haar measure for measure groupoids., \textit{Trans. Amer. Math. Soc}. \textbf{242}: 1--33(1978).
\bibitem{PH78}
P. Hahn: The regular representations of measure groupoids., \textit{Trans. Amer. Math. Soc}. \textbf{242}:35--72(1978).
Theorem 3.4 on p. 50.
\bibitem{MAB2k3}
M. R. Buneci. \emph{Groupoid Representations}, Ed. Mirton: Timishoara (2003).
\bibitem{MRB2k6}
M.R. Buneci. 2006.,
\htmladdnormallink{Groupoid C*-Algebras.}{http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf},
{\em Surveys in Mathematics and its Applications}, Volume 1: 71--98.
\bibitem{MAB2k5}
M. R. Buneci. Isomorphic groupoid C*-algebras associated with
different Haar systems., {\em New York J. Math.}, \textbf{11} (2005):225--245.
\bibitem{JR80}
J. Renault. A groupoid approach to C*-algebras, \emph{Lecture Notes in Math}., 793, Springer,
Berlin, (1980).
\bibitem{JR97}
J. Renault. 1997. The Fourier Algebra of a Measured Groupoid and Its Multipliers,
{\em Journal of Functional Analysis}, \textbf{145}, Number 2, April 1997, pp. 455--490.
\bibitem{Seda76}
A. K. Seda: Haar measures for groupoids, \emph{Proc. Roy. Irish Acad.
Sect. A} \textbf{76} No. 5, 25--36 (1976).
\bibitem{Seda82}
A. K. Seda: Banach bundles of continuous functions and an integral
representation theorem, \emph{Trans. Amer. Math. Soc.} \textbf{270} No.1 : 327-332(1982).
\bibitem{Seda86}
A. K. Seda: On the Continuity of Haar measures on topological groupoids, \emph{Proc. Amer Math. Soc.} \textbf{96}: 115--120 (1986).
\bibitem{Seda2k8}
A. K. Seda. 2008. \emph{Personal communication}, and also Seda (1986, on p.116).
\end{thebibliography}
\end{document}
