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groupoid C*-dynamical system (Definition)
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, and also of ${\mathsf{G}}_{lc}$ as a quantum groupoid; in the latter case, with additional conditions it or variable classical automata, depending on the added restrictions (ergodicity, etc.).

Bibliography

1
T. Matsuda, Groupoid dynamical systems and crossed product, II-case of C*-systems., Publ. RIMS, Kyoto Univ., 20: 959-976 (1984).



"groupoid C*-dynamical system" is owned by bci1.
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See Also: C*-algebra, general dynamic systems, differential logic, Fundamental physical concepts

Other names:  groupoid C*-dynamic system, C*-groupoid system, locally compact dynamical system with Haar measure
Also defines:  C*-groupoid system, locally compact dynamical system, continuous groupoid automorphism, locally compact dynamical system with Haar measure, continuous groupoid homomorphism, dynamical system
Keywords:  C*-groupoid system, locally compact dynamical system, continuous groupoid automorphism, locally compact dynamical system with Haar measure, continuous groupoid homomorphism, dynamical system

Cross-references: quantum groupoid, von Neumann algebra, concept, norm, morphism, Haar system, groupoid, topological, C*-algebra
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This is version 4 of groupoid C*-dynamical system, born on 2009-03-03, modified 2009-05-28.
Object id is 561, canonical name is GroupoidCDynamicalSystems.
Accessed 2616 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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