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'groupoid C*-dynamical system'
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| Title of object: |
groupoid C*-dynamical system |
| Canonical Name: |
GroupoidCDynamicalSystem |
| Type: |
Definition |
| Created on: |
2008-07-27 03:20:55 |
| Modified on: |
2008-09-07 22:16:04 |
| Classification: |
msc:22A22, msc:28C10, msc:22D25, msc:46L55, msc:37B45, msc:37-00, msc:18D05, msc:46L85, msc:18B30, msc:55U40, msc:55P10, msc:55N20, msc:55N33 |
| Keywords: |
groupoid C*-algebras, dynamical systems, dynamical systems as groupoid C*-algebras, C*-algebra, C*-groupoid system |
| Defines: |
C*-groupoid system, locally compact dynamical system, continuous groupoid automorphism, locally compact dynamical system with Haar measure, continuous groupoid homomorphism, dynamical system |
| Synonyms: |
groupoid C*-dynamical system=C*-groupoid system
groupoid C*-dynamical system=locally compact dynamical system with Haar measure |
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Content:
\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 C*-algebra, and $\grp_{lc}$ is a locally compact (topological) groupoid
with a countable basis for which there exists an associated continuous Haar system and a continuous
groupoid (homo) morphism $\rho: \grp_{lc} \longrightarrow Aut(A)$ 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 norm 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 concept
of dynamical system. Thus, it can also be utilized to represent a quantum dynamical system
upon further specification of the C*-algebra as a \PMlinkname{von Neumann algebra}{VonNeumannAlgebra}, and also of $\grp_{lc}$
as a quantum groupoid; in the latter case, with additional conditions it can also simulate either quantum automata,
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} |
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