PlanetPhysics/Groupoid C Dynamical Systems

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A C*-groupoid system  or groupoid C*-dynamical system

is a triple Failed to parse (unknown function "\grp"): {\displaystyle (A, \grp_{lc}, \rho )} , where: $A$ is a [[../VonNeumannAlgebra2/|C*-algebra]], and Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is a locally compact ([[../CoIntersections/|topological]]) [[../QuantumOperatorAlgebra5/|groupoid]] with a countable basis for which there exists an associated continuous [[../QuantumOperatorAlgebra5/|Haar system]] and a continuous groupoid (homo) [[../TrivialGroupoid/|morphism]] Failed to parse (unknown function "\grp"): {\displaystyle \rho: \grp_{lc} \longrightarrow Aut(A)} defined by the assignment $x\mapsto {\rho }_x(a)$ (from Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} to $A$) which is continuous for any $a\in A$; moreover, one considers the [[../NormInducedByInnerProduct/|norm]] topology on $A$ in defining Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} . (Definition introduced in ref. .)

A groupoid C*-dynamical system can be regarded as an extension of the ordinary [[../PreciseIdea/|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 [[../CoordinateSpace/|von Neumann algebra]], and also of Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} as a [[../WeakHopfAlgebra/|quantum groupoid]]; in the latter case, with additional conditions it or variable classical automata, depending on the added restrictions (ergodicity, etc.).