Covariant representation of a C*-dynamical system

Definition

Let $(A,G,\alpha)$ be a $C^*$ -dynamical system. A covariant representation of $(A,G,\alpha)$ on a Hilbert space $H$ is a pair $(\pi,U)$ in which $\pi:A\to\mathcal B(H)$ is a nondegenerate $*$ -representation, $U:G\to\mathcal U(H)$ is a strongly continuous unitary representation, and

\pi(\alpha_s(a))=U_s\pi(a)U_s^*

for every $s\in G$ and $a\in A$ . The displayed covariance identity is the compatibility axiom relating the algebra representation to the group action.

Integrated compatibility

The covariance identity is precisely what makes

f\longmapsto\int_G\pi(f(s))U_s\,ds

a $*$ -representation of the crossed-product convolution algebra. Completion then turns covariant pairs into nondegenerate representations of the full crossed product. This correspondence is the universal role of covariant representations Williams, §§2.2–2.4.

Examples and non-examples

For the trivial action of $G$ on $A$ , covariance says that every $U_s$ commutes with $\pi(A)$ . The one-dimensional pair $(\operatorname{id}_{\mathbb C},U)$ is covariant for the trivial action on $\mathbb C$ for any continuous unitary character $U$ .

A representation $\pi$ and a unitary representation $U$ on the same Hilbert space do not form a covariant pair merely by coexisting: if conjugation by $U_s$ fails to implement $\alpha_s$ through $\pi$ , the covariance axiom fails.

Conventions and scope

Some treatments permit degenerate $\pi$ , but the standard crossed-product representation correspondence uses nondegenerate representations. The phrase “covariant representation” is setting-dependent: representations of correspondences, operator systems, and dynamical systems have different covariance relations. Here it always refers to the displayed $C^*$ -dynamical-system identity.

References

Dana P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: §2.2 on covariant representations and §2.3 on integrated forms.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §7.6 on covariant representations.