groupoid C*-dynamical system

Definition 0.1.

A C*-groupoid system or groupoid C*-dynamical system is a triple $(A,{\mathsf{G}}_{lc},\rho)$, where: $A$ is a C*-algebra, and ${\mathsf{G}}_{lc}$ 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 $\rho:{\mathsf{G}}_{lc}\longrightarrow Aut(A)$ defined by the assignment $x\mapsto\rho_{x}(a)$ (from ${\mathsf{G}}_{lc}$ to $A$) which is continuous for any $a\in A$; moreover, one considers the norm topology on $A$ in defining ${\mathsf{G}}_{lc}$. (Definition introduced in ref. [1].)

Remark 0.1.

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 (http://planetmath.org/VonNeumannAlgebra), and also of ${\mathsf{G}}_{lc}$ as a quantum groupoid (http://planetmath.org/QuantumGroupoids2); in the latter case, with additional conditions it can also simulate either quantum automata (http://planetmath.org/QuantumAutomataAndQuantumComputation2), or variable classical automata, depending on the added restrictions (ergodicity, etc.).

References

• 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