groupoid C*-dynamical system

Definition 0.1. A C*-groupoid system or groupoid C*-dynamical system is a triple (A, G_lc,ρ), where: A is a C*-algebra, and 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 ρ : G_lc→Aut(A) defined by the assignment xρ_x(a) (from G_lc to A) which is continuous for any a ∈ A; moreover, one considers the norm topology on A in defining 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 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.).