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%%% Primary Title: uniform continuity over locally compact quantum groupoids
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%%% Filename: UcLocallyCompactQuantumGroupoids.tex
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\begin{document}
\subsection{Uniform Continuity over Locally Compact Quantum Groupoids}
Let us consider \emph{locally compact quantum groupoids} ($LCQGn$) defined as {\em \htmladdnormallink{locally compact groupoids}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} endowed with a \htmladdnormallink{Haar system}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}}, $\nu$, $(\G,\nu):= ([\grp, G_2, \mu], \nu)$, or as derived from a {\em (\htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}) \htmladdnormallink{weak Hopf algebra}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}} (WHA), with the additional condition of \emph{uniform continuity} over $\grp$ defined as follows . Let us also consider a space $LUC(\grp)$ of left uniformly continuous elements in $L^{\infty}(\grp)$ defined over $G_2$, which is endowed with the induced product topology from the subset $G^2$ of composable pairs in the \emph{\htmladdnormallink{topological groupoid}{http://planetphysics.us/encyclopedia/EquivalenceRelation.html}} $\grp$. This step completes the construction of uniform continuity over $LCQGn$ that can be then compared with the results obtained from `\htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}' derived from a weak Hopf algebra.
\subsubsection{C*-algebra Comparison and Example}
Consider $LCG$ to be a \htmladdnormallink{locally compact quantum group}{http://planetphysics.us/encyclopedia/LocallyCompactQuantumGroup.html}. Then consider the space $LUC(G)$ of left uniformly continuous elements in $L^{\infty}(G)$ introduced in ref. \cite{VRunde2k8}. (The definition according to {\em V. Runde} (\em loc. cit.) covers both the space of left uniformly continuous \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} on a locally compact \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and (Granirer's) uniformly continuous functionals on the Fourier algebra.) Also consider $LUC(G)$ which is then an {\em \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} containing the \htmladdnormallink{C*-algebra}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html} $C_o(G)$}. One may compare the \htmladdnormallink{groupoid C*-convolution algebra}{http://planetphysics.us/encyclopedia/GroupoidCConvolutionAlgebra.html}, $G_{CA}$ -- obtained in the general case-- with the C*-algebra $C_o(G)$ obtained from $LUC(G)$ in the particular case of uniform continuity over a locally compact group.
\begin{thebibliography}{9}
\bibitem{Buneci2k3}
M. Buneci. 2003. {\em Groupoid Representations}, Publs: Ed. Mirton, Timishoara.
\bibitem{VRunde2k8}
V. Runde. 2008. Uniform continuity over locally compact quantum groups.
\htmladdnormallink{(math.OA -arxiv/0802.2053v4).}{http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.2053v4.pdf}
\end{thebibliography}
\end{document}
