Definition 0.1
A
C*-groupoid system or
groupoid C*-dynamical system
is a
triple

, where:

is a
C*-algebra, and

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

defined by the assignment

(from

to

)
which is continuous for any

; moreover, one considers the
norm
topology
on

in defining

. (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

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