Transformation-group C*-algebra - Imported Knowlpedia Content

Definition

Let $X$ be a locally compact Hausdorff space and let a locally compact group $G$ act continuously on $X$ . The induced action $\alpha:G\to\operatorname{Aut}(C_0(X))$ is

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

The full transformation-group $C^*$ -algebra of $(G,X)$ is the full crossed product $C_0(X)\rtimes_\alpha G$ . Its reduced version is $C_0(X)\rtimes_{\alpha,r}G$ . Thus the object records both the topology of $X$ and the dynamics of the action; the acting group and the action are part of the defining data.

Covariant description

A nondegenerate representation of $C_0(X)\rtimes_\alpha G$ is determined by a covariant pair $(\pi,U)$ : a nondegenerate representation $\pi$ of $C_0(X)$ , a strongly continuous unitary representation $U$ of $G$ , and the covariance equations

U_s\pi(f)U_s^*=\pi(\alpha_s(f)).

The full algebra is universal for these pairs. The reduced algebra instead uses regular covariant representations. This distinction is essential when the action is not amenable; there is always a canonical quotient from the full algebra to the reduced one Williams, Chapters 2–3.

Standard examples

If $X$ is a one-point space with the trivial action, the full and reduced transformation-group algebras are respectively the full and reduced group $C^*$ -algebras of $G$ . If $G$ acts on itself by left translation, then $C_0(G)\rtimes_r G$ is naturally isomorphic to the compact operators on $L^2(G)$ . More generally, transformation-group algebras are the operator algebras attached to dynamical quotients even when the orbit space $X/G$ has poor separation properties.

Conventions and scope

This construction concerns an action on a space. A general $C^*$ -dynamical system need not arise from a commutative coefficient algebra.

References

Dana P. Williams, Crossed Products of C-Algebras*, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: Chapters 2–3 on crossed products, covariant representations, and transformation groups.