Exercise 2, § 3

Let G be a locally compact group. For every function of positive type $\varphi$ on G, let $(\pi_{\varphi}, x_{\varphi})$ denote a cyclic Hilbert representation of $\varphi$.

a) Let $\varphi_1,\varphi_2$ and $\varphi_3$ be functions of positive type on G. If $\varphi_3=\varphi_1+\varphi_2$, then there exists an injective G-morphism of $\pi_{\varphi_1}$ into $\pi_{\varphi_3}$.

b) If $\varphi$ is a constant non-zero function of positive type, then $\pi_{\varphi}$ is trivial of dimension 1.

c) For every unitary representation $\pi$ of G in a Hilbert space E, and for every $x\in E$, there exists an injective G-morphism of $\pi_{\varphi}$ into $\pi$, where $\varphi (g) =\langle x|\pi (g)x\rangle$.

d) Let H be a closed subgroup of G and $\varphi$ a function of positive type on G. Let $\psi$ denote the restriction of $\varphi$ to H; it is a function of positive type on H, and there exists an injective H-morphism of $\pi_{\psi}$ into Res$^H_H(\pi )$.