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