81-00-RepresentationOfACcmathsfGTopologicalAlgebra.tex

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\pmcanonicalname{RepresentationOfACcmathsfGTopologicalAlgebra}
\pmcreated{2013-03-22 18:16:24}
\pmmodified{2013-03-22 18:16:24}
\pmowner{bci1}{20947}
\pmmodifier{bci1}{20947}
\pmtitle{representation of a  $C_c(\mathsf{G})  $ *- topological algebra}
\pmrecord{21}{40878}
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\pmauthor{bci1}{20947}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{81-00}
\pmclassification{msc}{18D05}
\pmclassification{msc}{55N33}
\pmclassification{msc}{55N20}
\pmclassification{msc}{55P10}
\pmclassification{msc}{55U40}
\pmsynonym{groupoid C*-algebra representations}{RepresentationOfACcmathsfGTopologicalAlgebra}
%\pmkeywords{representation of *-algebra on $B(\H)$}
%\pmkeywords{the $C^*$-algebra of bounded operators on the Hilbert space $\H$}
\pmrelated{C_cG}
\pmrelated{BoundedOperatorsOnAHilbertSpaceFormACAlgebra}
\pmrelated{GelfandNaimarkSegalConstruction}
\pmdefines{representation of a  topological *- algebra}

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\begin{document}
\begin{definition}
A \emph{representation of a \PMlinkname{$C_c(\grp)$}{C_cG} topological $*$--algebra} is defined as \\
a \emph{continuous $*$--morphism from $C_c(\grp)$ to $B(\H)$, where $\grp$ is a topological 
groupoid, (usually a locally compact groupoid, $\grp_{lc}$), $\H $ is a Hilbert space},
and $B(\H)$ is the $C^*$-algebra of bounded linear operators on the Hilbert space $\H$;
of course, one considers the inductive limit (strong) topology to be defined on $C_c(\grp)$, 
and then also an operator \emph{weak} topology to be defined on $B(\H)$.

\end{definition}

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