Exercise 26, § 4

We take up the notation of the preceding exercise. We denote by $\varphi_K$ the characteristic function of K. a) Let $\pi$ be a unitary representation of G in a complex Hilbert space E. The representation of $\mathscr{M}_c(G)$ in E defines, by passing to subspaces, a representation of the algebra $\mathscr{H}(G,K)$ in the space $E^K$ of K-invariant vectors of E.

b) If E admits a cyclic vector $x$ such that $x\in E^K$, then $\pi$ is irreducible. c) Suppose that $\pi$ is irreducible. If the space $E^K$ is not reduced to 0, the representation of $\mathscr{H}(G,K)$ in $E^K$ is irreducible.

d) Let $\varphi$ be a positive-type function on G and let $(\varrho , x)$ be a cyclic Hilbertian realization of $\varphi$. One has $\varphi \in \mathscr{H}(G,K)$ if and only if $x\in E^K$.