PlanetPhysics/Groupoid C Dynamical Systems

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A C*-groupoid system or groupoid C*-dynamical system is a triple $$(A, \grp_{lc}, \rho )$$, where: $$A$$ is a C*-algebra, and $$\grp_{lc}$$ 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 $$\rho: \grp_{lc} \longrightarrow Aut(A)$$ defined by the assignment $$x \mapsto \rho_x(a)$$ (from $$\grp_{lc}$$ to $$A$$) which is continuous for any $$a \in A$$; moreover, one considers the norm topology on $$A$$ in defining $$\grp_{lc}$$. (Definition introduced in ref. .)

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 $$\grp_{lc}$$ as a quantum groupoid; in the latter case, with additional conditions it or variable classical automata, depending on the added restrictions (ergodicity, etc.).