Exercise 22, § 4

Let $r\geqslant 1$ be an integer and let $H\subset \mathbf{G}\mathbf{L}(r,\mathbf{C})$ be a closed subgroup.

a) The set of functions on H of the form $g\mapsto \langle \varrho (g)x, \lambda \rangle$, where $\varrho$ is a continuous linear representation of H in a finite-dimensional complex vector space E, $x\in E$ and $\lambda \in E'$, is a dense involutive subalgebra of $\mathscr{C}(H)$.

b) For every nonzero function $f\in \mathscr{K}(H)$, there exists a continuous linear representation $\varrho$ of H in a finite-dimensional complex vector space such that $\varrho (f)$ is nonzero.

c) Suppose that H is a connected semisimple real Lie group. For every nonzero function $f\in \mathscr{K}(H)$, there exists an irreducible continuous linear representation $\varrho$ of H in a finite-dimensional complex vector space such that $\varrho (f)$ is nonzero.