Lemma 4

Let $\varphi \in$ Pos(G). The function $\varphi$ is bounded on G and one has $\|\varphi \|_{\infty}=\varphi (e)$. Moreover, one has

for every $(g, h)\in G\times G$.

Let $(\varrho , x)$ be a Hilbert realization of $\varphi$. One has $\varphi (e) =\|x\|^2$. For every $g\in G$, one therefore has $|\varphi (g)|=|\langle x|\varrho (g)x\rangle |\leqslant \varphi (e)$ by the Cauchy-Schwarz inequality. This proves the first assertion.

For every $(g, h)\in G\times G$, one has

since $\varrho$ is unitary, whence

We conclude by observing that