# PlanetPhysics/Groupoid C Dynamical Systems

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A C*-groupoid system  or groupoid C*-dynamical system

is a triple $\displaystyle (A, \grp_{lc}, \rho )$ , where: ${\displaystyle A}$ is a C*-algebra, and $\displaystyle \grp_{lc}$ 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 $\displaystyle \rho: \grp_{lc} \longrightarrow Aut(A)$ defined by the assignment ${\displaystyle x\mapsto \rho _{x}(a)}$ (from $\displaystyle \grp_{lc}$ to ${\displaystyle A}$) which is continuous for any ${\displaystyle a\in A}$; moreover, one considers the norm topology on ${\displaystyle A}$ in defining $\displaystyle \grp_{lc}$ . (Definition introduced in ref. [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 $\displaystyle \grp_{lc}$ as a quantum groupoid; in the latter case, with additional conditions it or variable classical automata, depending on the added restrictions (ergodicity, etc.).

[1]

## References

1. T. Matsuda, Groupoid dynamical systems and crossed product, II-case of C*-systems., Publ. RIMS , Kyoto Univ., 20 : 959-976 (1984).