Integrated operator of a continuous representation

Core idea

Let $G$ be a locally compact Hausdorff group with left Haar measure, and let $\pi:G\to\operatorname{GL}(E)$ be a strongly continuous representation on a Banach space. For $f\in C_c(G)$ , its integrated operator is

\pi(f)v=\int_G f(g)\pi(g)v\,dg .

This Bochner integral exists because $\{\pi(g):g\in\operatorname{supp}f\}$ is uniformly bounded. It defines a bounded operator on $E$ and satisfies $\pi(f*h)=\pi(f)\pi(h)$ . The same construction works on a complete Fréchet space using its locally-convex vector integral. Without a compatible Hilbert-space unitary structure, $f\mapsto\pi(f)$ is not asserted to preserve involution or to be a $*$ -representation.

Banach-space estimate

If $K=\operatorname{supp}f$ and $M_K=\sup_{g\in K}\lVert\pi(g)\rVert$ , then

\lVert\pi(f)v\rVert \leq M_K\lVert f\rVert_{L^1(G)}\lVert v\rVert .

Strong continuity and the uniform boundedness principle make $M_K$ finite. This estimate is local in $G$ : a continuous representation need not be uniformly bounded on the whole group, so its integrated action need not extend continuously from $C_c(G)$ to all of $L^1(G)$ .

Fréchet-space interpretation

Suppose $E$ is Fréchet and the action $G\times E\to E$ is continuous. For each compact $K\subseteq G$ , the operators $\pi(g)$ , $g\in K$ , form an equicontinuous family. Completeness permits integration of the compactly-supported continuous map $g\mapsto f(g)\pi(g)v$ . For every continuous seminorm $p$ on $E$ , equicontinuity supplies a continuous seminorm $q$ and $C_K>0$ such that

p(\pi(f)v)\leq C_K\lVert f\rVert_1q(v).

Hence $\pi(f)$ is a continuous linear endomorphism of $E$ .

Convolution and smoothing

For $f,h\in C_c(G)$ , Fubini's theorem and $\pi(xy)=\pi(x)\pi(y)$ give

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

When $G$ is a Lie group and $f\in C_c^\infty(G)$ , these operators are the basic smoothing operators used to form the Gårding subspace. Approximate identities supported near the identity recover vectors in the original topology under the usual continuity hypotheses.

Unitary specialization and warning

If $E$ is a Hilbert space and $\pi$ is unitary, the construction becomes the integrated form of a unitary representation. It then extends to $L^1(G)$ , is contractive, and respects the group-algebra involution.

For a general nonunitary representation, convolution multiplicativity still holds on $C_c(G)$ , but $\pi(f^*)=\pi(f)^*$ is unavailable or false. In particular, the integrated operator used for an admissible nonunitary representation must not be cited as though it were a unitary $*$ -representation.

References

Lars Gårding, “Note on Continuous Representations of Lie Groups,” Proceedings of the National Academy of Sciences 33 (1947), 331–332. DOI record. Relevant: integration of continuous representations against compactly supported smooth functions.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Elsevier book record. Relevant: vector-valued integration and continuous linear maps on complete locally convex spaces.
Joseph Bernstein and Bernhard Krötz, “Smooth Fréchet globalizations of Harish-Chandra modules,” Israel Journal of Mathematics 199 (2014), 45–111. DOI record. Relevant: §1 on continuous and smooth Fréchet representations.