Example

Let $\varrho$ be a unitary representation of G in a Hilbert space E and let $x\in E$. Let $\varphi$ be the diagonal matrix coefficient defined by $\varphi (g) =\langle x|\varrho (g)x\rangle$ for all $g\in G$; the function $\varphi$ is continuous. We have

for all $(g, h)\in G\times G$. Consequently $\varphi \in$ Pos(G) (Theorem 1 of V, p. 432, (iv)). One has $\varphi \in$ Pos$_1(G)$ if and only if $\|x\|= 1$.