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%%% Primary Title: groupoid C*-dynamical system
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%%% Owner: bci1
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\begin{document}
\begin{definition}
A \emph{C*-groupoid system} or \emph{groupoid C*-dynamical system}
is a \emph{triple} $(A, \grp_{lc}, \rho )$, where:
$A$ is a \htmladdnormallink{C*-algebra}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html}, and $\grp_{lc}$ is a locally compact (\htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html}) \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} with a countable basis for which there exists an associated continuous \htmladdnormallink{Haar system}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} and a continuous groupoid (homo) \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\rho: \grp_{lc} \longrightarrow Aut(A)$ defined by the assignment $x \mapsto \rho_x(a)$ (from $\grp_{lc}$ to $A$)
which is continuous for any $a \in A$; moreover, one considers the \htmladdnormallink{norm}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} topology
on $A$ in defining $\grp_{lc}$. (Definition introduced in ref. \cite{MT1984}.)
\end{definition}
\begin{remark}
A \emph{groupoid C*-dynamical system} can be regarded as an extension of the ordinary \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of dynamical system. Thus, it can also be utilized to represent a quantum dynamical system
upon further specification of the C*-algebra as a \emph{\htmladdnormallink{von Neumann algebra}{http://planetphysics.us/encyclopedia/CoordinateSpace.html}}, and also of $\grp_{lc}$ as a \emph{\htmladdnormallink{quantum groupoid}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}}; in the latter case, with additional conditions it or variable classical automata, depending on the added restrictions (ergodicity, etc.).
\end{remark}
\begin{thebibliography}{9}
\bibitem{MT1984}
T. Matsuda, Groupoid dynamical systems and crossed product, II-case of C*-systems.,
\emph{Publ. RIMS}, Kyoto Univ., \textbf{20}: 959-976 (1984).
\end{thebibliography}
\end{document}
