C*-dynamical system - Imported Knowlpedia Content

Definition

A $C^*$ -dynamical system is a triple $(A,G,\alpha)$ consisting of a $C^*$ -algebra $A$ , a locally compact group $G$ , and a group homomorphism

\alpha:G\longrightarrow\operatorname{Aut}(A),\qquad s\longmapsto\alpha_s,

such that the action is strongly continuous in the $C^*$ -algebraic sense: for every $a\in A$ , the orbit map $s\mapsto\alpha_s(a)$ is continuous in the norm of $A$ . Equivalently, $\alpha$ is continuous when $\operatorname{Aut}(A)$ has the point-norm topology. Neither $A$ nor $G$ is required to be unital or discrete.

Why the continuity axiom matters

Point-norm continuity ensures that $s\mapsto f(s)\alpha_s(a)$ has the continuity needed in the convolution and integration constructions underlying crossed products. It is stronger than requiring continuity after applying only selected representations or functionals. When $G$ is discrete the condition is automatic, so an action by $*$ -automorphisms is already a $C^*$ -dynamical system.

Standard examples

The trivial action $\alpha_s=\operatorname{id}_A$ gives a $C^*$ -dynamical system for every $A$ and $G$ . If $G$ acts continuously on a locally compact Hausdorff space $X$ , then

(\alpha_s f)(x)=f(s^{-1}x)

defines a point-norm continuous action on $C_0(X)$ . For a non-example, a homomorphism $G\to\operatorname{Aut}(A)$ whose orbit map is discontinuous for some $a$ is an algebraic group action but not a $C^*$ -dynamical system in this sense.

Conventions and scope

Many sources define a $C^*$ -dynamical system for an arbitrary topological group. Crossed-product theory typically assumes that $G$ is locally compact Hausdorff so that Haar integration is available; that convention is built into this knowl. “Strong continuity” here means point-norm continuity of automorphisms, not continuity in the strong operator topology of a particular Hilbert-space representation Williams, §2.1, Definition 2.6.

References

Dana P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: §2.1, especially Definition 2.6 on $C^*$ -dynamical systems.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 7 on automorphism groups and covariant representations.