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# 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 can also simulate either quantum automata, or variable classical automata, depending on the added restrictions (ergodicity, etc.).

# References

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

## Mathematics Subject Classification

55N33*no label found*55N20

*no label found*55P10

*no label found*55U40

*no label found*18B30

*no label found*46L85

*no label found*18D05

*no label found*37-00

*no label found*37B45

*no label found*46L55

*no label found*22D25

*no label found*28C10

*no label found*22A22

*no label found*

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