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, 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).



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