Integrated form of a unitary representation - Imported Knowlpedia Content

Core idea

Let $G$ be a locally compact group with left Haar measure, and let $\pi:G\to\mathcal U(H)$ be a strongly continuous unitary representation. Its integrated form assigns to each $f\in L^1(G)$ the bounded operator

\pi(f)\xi=\int_G f(x)\pi(x)\xi\,dx,\qquad \xi\in H.

The integral is a Bochner integral in $H$ , equivalently characterized weakly by integrating matrix coefficients. It satisfies $\|\pi(f)\|\leq\|f\|_1$ , and the assignment $f\mapsto\pi(f)$ is a nondegenerate $*$ -representation of the Banach $*$ -algebra $L^1(G)$ .

Algebraic compatibility

For $f,h\in L^1(G)$ , Fubini's theorem and the representation identity give

\pi(f*h)=\pi(f)\pi(h).

With the group-algebra involution

f^*(x)=\Delta_G(x^{-1})\overline{f(x^{-1})},

where $\Delta_G$ is the modular function, one also has $\pi(f^*)=\pi(f)^*$ . The modular factor is indispensable for a general nonunimodular group; omitting it breaks the $*$ -identity.

Nondegeneracy and recovery of the group action

If $(u_i)$ is an approximate identity in $L^1(G)$ , then $\pi(u_i)\xi\to\xi$ for every $\xi\in H$ . Thus the integrated form is nondegenerate. Conversely, a nondegenerate $*$ -representation of $L^1(G)$ determines a unique strongly continuous unitary representation of $G$ . This is why integrated forms are the bridge between group representations and representations of group $C^*$ -algebras.

Smoothing and examples

When $G$ is a Lie group and $f$ is smooth and compactly supported, $\pi(f)$ often maps arbitrary vectors into smooth vectors. For the left regular representation, $\pi(f)$ is left convolution by $f$ on $L^2(G)$ . For a discrete group, Haar integration becomes summation and $\pi(f)=\sum_{x\in G}f(x)\pi(x)$ , with convergence in operator norm for $f\in\ell^1(G)$ .

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. Publisher record. Relevant: §§3.2 and 7.1 on integrated representations and group $C^*$ -algebras.
Dana P. Williams, Crossed Products of C-Algebras*, American Mathematical Society, 2007. DOI record. Relevant: §2 and Appendix B on integrated forms and vector-valued integration.