# PlanetPhysics/Groupoid C Dynamical Systems

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AC*-groupoid systemorgroupoid C*-dynamical system

is a *triple* **Failed to parse (unknown function "\grp"): {\displaystyle (A, \grp_{lc}, \rho )}**
, where:
is a C*-algebra, and **Failed to parse (unknown function "\grp"): {\displaystyle \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 **Failed to parse (unknown function "\grp"): {\displaystyle \rho: \grp_{lc} \longrightarrow Aut(A)}**
defined by the assignment (from **Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}}**
to )
which is continuous for any ; moreover, one considers the norm topology
on in defining **Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}}**
. (Definition introduced in ref. ^{[1]}.)

A *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 *von Neumann algebra*, and also of **Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}}**
as a *quantum groupoid*; in the latter case, with additional conditions it or variable classical automata, depending on the added restrictions (ergodicity, etc.).

## All SourcesEdit

^{[1]}

## ReferencesEdit

- ↑
^{1.0}^{1.1}T. Matsuda, Groupoid dynamical systems and crossed product, II-case of C*-systems.,*Publ. RIMS*, Kyoto Univ.,**20**: 959-976 (1984).