Integrated form of a covariant representation

Definition

Let $(\pi,U)$ be a covariant representation of $(A,G,\alpha)$ on $H$ , and fix a left Haar measure on $G$ . Its integrated form, written $\pi\rtimes U$ , is the operator-valued map on the convolution $*$ -algebra $C_c(G,A)$ defined by

(\pi\rtimes U)(f) =\int_G\pi(f(s))U_s\,ds.

The integral is understood weakly, equivalently as a strong operator integral on compact support. Covariance implies that $\pi\rtimes U$ preserves convolution and involution, so it is a nondegenerate $*$ -representation and is bounded for the full crossed-product norm.

Representation correspondence

The integrated form therefore extends uniquely to a nondegenerate representation of the full crossed product $A\rtimes_\alpha G$ . Conversely, every nondegenerate representation of $A\rtimes_\alpha G$ determines a unique covariant pair, and reintegration recovers the original representation. This gives a bijection, compatible with unitary equivalence, between covariant representations of $(A,G,\alpha)$ and nondegenerate representations of the full crossed product Williams, Proposition 2.40.

How the formula encodes covariance

Formally, the crossed product contains compatible copies of $A$ and $G$ in its multiplier algebra. The integrated form sends these copies to $\pi(a)$ and $U_s$ . For $f\in C_c(G,A)$ , the factor $\pi(f(s))$ represents the coefficient at $s$ , while $U_s$ represents the group element; integration combines them into one bounded operator.

For a discrete group the integral becomes the finite sum

(\pi\rtimes U)(f)=\sum_{s\in G}\pi(f(s))U_s.

Omitting the unitary factors would generally lose the group action and would not represent the crossed-product convolution.

Conventions and scope

The order $\pi(f(s))U_s$ matches the covariance and convolution conventions used here. Sources using the opposite covariance identity or a right Haar measure may display a different-looking formula. The notation $\pi\rtimes U$ denotes the integrated representation, not the crossed-product algebra itself.

References

Dana P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: §2.3, Proposition 2.23 on integrated forms, and §2.4, Proposition 2.40 on the representation correspondence.