groupoid C*-dynamical system
Definition 0.1.
A C*-groupoid^{} system or groupoid C*-dynamical system is a triple $(A,{\U0001d5a6}_{lc},\rho )$, where: $A$ is a C*-algebra^{}, and ${\U0001d5a6}_{lc}$ 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^{} $\rho :{\U0001d5a6}_{lc}\u27f6Aut(A)$ defined by the assignment $x\mapsto {\rho}_{x}(a)$ (from ${\U0001d5a6}_{lc}$ to $A$) which is continuous for any $a\in A$; moreover, one considers the norm topology on $A$ in defining ${\U0001d5a6}_{lc}$. (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 ${\U0001d5a6}_{lc}$ 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 |