Regular covariant representation - Imported Knowlpedia Content

Definition

Let $(A,G,\alpha)$ be a $C^*$ -dynamical system and let $\pi:A\to\mathcal B(H)$ be a nondegenerate representation. The regular covariant representation induced by $\pi$ acts on $L^2(G,H)$ by

(\widetilde\pi(a)\xi)(s)=\pi(\alpha_{s^{-1}}(a))\xi(s), \qquad (\lambda_t\xi)(s)=\xi(t^{-1}s).

Here $L^2(G,H)$ uses a left Haar measure, and $\lambda$ is the left regular representation. The pair $(\widetilde\pi,\lambda)$ is covariant because

\widetilde\pi(\alpha_t(a))=\lambda_t\widetilde\pi(a)\lambda_t^*.

Both representations are nondegenerate, and $\lambda$ is strongly continuous. Their integrated form is consequently a nondegenerate representation of the crossed-product convolution algebra; when $\pi$ is faithful, its operator norm defines the reduced crossed-product norm.

Integrated regular representation

Its integrated form is

((\widetilde\pi\rtimes\lambda)(f)\xi)(s) =\int_G\pi(\alpha_{s^{-1}}(f(t)))\xi(t^{-1}s)\,dt.

The operator norm of this representation supplies the reduced crossed-product norm. Thus “regular” refers to the use of translation on the group variable, not to a regularity condition on vectors or coefficients.

Dependence on the coefficient representation

If $\pi$ is faithful and nondegenerate, the resulting reduced norm and reduced crossed product are independent of $\pi$ up to canonical isomorphism Williams, §7.2, Definition 7.7. Faithfulness matters: starting from a representation with nonzero kernel can discard coefficient-algebra information and produce a smaller image.

The construction can also be viewed as inducing $\pi$ from the identity subgroup. This perspective explains the common notation $\operatorname{Ind}\pi$ for the integrated regular representation.

Standard cases

When $A=\mathbb C$ with the trivial action and $\pi$ is the scalar representation, $\widetilde\pi$ acts by scalars and the integrated form is the usual regular representation of $C_c(G)$ . For a discrete $G$ , $L^2(G,H)=\ell^2(G,H)$ , and the formulas become pointwise coefficient action together with shifts of the $G$ -coordinate.

References

Dana P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: §2.2, Example 2.14 on regular covariant representations, and §7.2 on reduced crossed products.