groupoid C*-dynamical system

Definition 0.1.

A C*-groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath system or groupoid C*-dynamical system is a triple (A,𝖦lc,ρ), where: A is a C*-algebraPlanetmathPlanetmath, and 𝖦lc is a locally compact (topological) groupoid with a countable basis for which there exists an associated continuousPlanetmathPlanetmath Haar systemPlanetmathPlanetmath and a continuous groupoid (homo) morphismMathworldPlanetmathPlanetmath ρ:𝖦lcAut(A) defined by the assignment xρx(a) (from 𝖦lc to A) which is continuous for any aA; moreover, one considers the norm topology on A in defining 𝖦lc. (Definition introduced in ref. [1].)

Remark 0.1.

A groupoid C*-dynamical system can be regarded as an extensionPlanetmathPlanetmathPlanetmath 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 algebraMathworldPlanetmathPlanetmath (, and also of 𝖦lc as a quantum groupoidPlanetmathPlanetmath (; in the latter case, with additional conditions it can also simulate either quantum automata (, or variable classical automata, depending on the added restrictionsPlanetmathPlanetmath (ergodicity, etc.).


  • 1 T. Matsuda, Groupoid dynamical systems and crossed product, II-case of C*-systems., Publ. RIMS, Kyoto Univ., 20: 959-976 (1984).
Title groupoid C*-dynamical system
Canonical name GroupoidCdynamicalSystem
Date of creation 2013-03-22 18:16:33
Last modified on 2013-03-22 18:16:33
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 23
Author bci1 (20947)
Entry type Definition
Classification msc 55N33
Classification msc 55N20
Classification msc 55P10
Classification msc 55U40
Classification msc 18B30
Classification msc 46L85
Classification msc 18D05
Classification msc 37-00
Classification msc 37B45
Classification msc 46L55
Classification msc 22D25
Classification msc 28C10
Classification msc 22A22
Synonym C*-groupoid system
Synonym locally compact dynamical system with Haar measure
Related topic CAlgebra
Related topic CAlgebra3
Related topic VonNeumannAlgebra
Related topic DynamicalSystem
Related topic NuclearCAlgebra
Related topic SystemDefinitions
Related topic SimilarityAndAnalogousSystemsDynamicAdjointness2
Related topic QuantumAutomataAndQuantumComputation2
Related topic VariableTopology3
Related topic QuantumGroupoids2
Related topic OrganismicSupercategoriesAndComplexS
Defines C*-groupoid system
Defines locally compact dynamical system
Defines continuous groupoid automorphism
Defines locally compact dynamical system with Haar measure
Defines continuous groupoid homomorphism
Defines dynamical system