groupoid C*-dynamical system

Definition 0.1.

A C*-groupoid system or groupoid C*-dynamical systemis a triple $(A,{\\mathsf{G}}_{lc},\\rho)$ , where: $A$ is a C*-algebra, and ${\\mathsf{G}}_{lc}$ is a locally compact (topological) groupoidwith a countable basis for which there exists an associated continuous Haar system and a continuousgroupoid (homo) morphism $\\rho:{\\mathsf{G}}_{lc}\\longrightarrow Aut(A)$ definedby 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 topologyon $A$ 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 conceptof dynamical system. Thus, it can also be utilized to represent a quantum dynamical systemupon 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).

单词	GroupoidCdynamicalSystem
释义	groupoid C-dynamical system Definition 0.1. A C-groupoid system or groupoid C-dynamical systemis a triple* $(A,{\\mathsf{G}}_{lc},\\rho)$ , where: $A$ is a C-algebra, and ${\\mathsf{G}}_{lc}$ is a locally compact (topological) groupoidwith a countable basis for which there exists an associated continuous Haar system and a continuousgroupoid (homo) morphism $\\rho:{\\mathsf{G}}_{lc}\\longrightarrow Aut(A)$ definedby 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 topologyon $A$ 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 conceptof dynamical system. Thus, it can also be utilized to represent a quantum dynamical systemupon 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).
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