PlanetMath: groupoid C*-dynamical system

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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:

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

) which is continuous for any $a \in A$ ; moreover, one considers the norm topology on

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 can also simulate either quantum automata, 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:

-algebra, C*-algebras and quantum compact groupoids, von Neumann algebra, dynamical system, nuclear C*-algebra, general system definitions, similarity and analogous systems: dynamic adjointness and topological equivalence, quantum automata and quantum computation, variable network topology, quantum groupoids, organismic supercategories and super-complex systems biodynamics-1, groupoids

Other names:

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:

groupoid C*-algebras, dynamical systems, dynamical systems as groupoid C*-algebras, C*-algebra, C*-groupoid system

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Cross-references: ergodicity, restrictions, automata, variable, specification, represent, extension, topology, norm, morphism, Haar system, continuous, countable basis, groupoid, locally compact, C*-algebra
There are 14 references to this entry.

This is version 19 of groupoid C*-dynamical system, born on 2008-07-27, modified 2008-09-07.
Object id is 10881, canonical name is GroupoidCDynamicalSystem.
Accessed 778 times total.

Classification:

AMS MSC:	22A22 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Topological groupoids )
	28C10 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures on topological groups, Haar measures, invariant measures)
	22D25 (Topological groups, Lie groups :: Locally compact groups and their algebras :: $C^$-algebras and $W$-algebras in relation to group representations)
	46L55 (Functional analysis :: Selfadjoint operator algebras :: Noncommutative dynamical systems)
	37B45 (Dynamical systems and ergodic theory :: Topological dynamics :: Continua theory in dynamics)
	37-00 (Dynamical systems and ergodic theory :: General reference works )
	18D05 (Category theory; homological algebra :: Categories with structure :: Double categories, $2$-categories, bicategories and generalizations)
	46L85 (Functional analysis :: Selfadjoint operator algebras :: Noncommutative topology)
	18B30 (Category theory; homological algebra :: Special categories :: Categories of topological spaces and continuous mappings)
	55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory)
	55P10 (Algebraic topology :: Homotopy theory :: Homotopy equivalences)
	55N20 (Algebraic topology :: Homology and cohomology theories :: Generalized homology and cohomology theories)
	55N33 (Algebraic topology :: Homology and cohomology theories :: Intersection homology and cohomology)

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