Exercise 20, § 3

Let G be a locally compact group and $\mu$ a left Haar measure on G. We denote by Δ the module of G. For every function $f$ on G, we write $\widetilde{f}= \Delta f^*$.

A measure $\nu$ on G is said to be of positive type if $\langle \nu ,\widetilde{f}*f\rangle \geqslant 0$ for every function $f\in \mathscr{K}(G)$. A locally integrable function $\varphi$ on G is said to be of positive type if $\langle \varphi , f^**f\rangle \geqslant 0$ for every function $f\in \mathscr{K}(G)$.

a) Suppose that $\nu$ is bounded. The measure $\nu$ is of positive type if and only if $\boldsymbol{\gamma }(\nu )\geqslant 0$.

b) Let $\varphi \in \mathscr{K}(G)$. The function $\varphi$ is of positive type as a locally integrable function if and only if it is so according to Definition 4 of V, p. 442.

c) Let $\nu$ be a measure of positive type on G. The formula

defines a positive Hermitian form on $\mathscr{K}(G)$. We denote by $E_{\nu}$ the separated-completed Hilbert space of $\mathscr{K}(G)$ endowed with this Hermitian form.

d) For $f\in \mathscr{K}$ (G), the function $\sigma (g)f:g\mapsto (\varepsilon_g*f)\Delta^{1/2}$ belong to $\mathscr{K}(G)$ and $\|\sigma (g)f\|_{\nu}=\|f\|_{\nu}$. The mapping $g\mapsto \sigma (g)$ extends to a unitary representation of G in $E_{\nu}$.

e) Let U be an open neighbourhood of $e$ in G and $\nu$ a measure of positive type on G such that there exists a real number $c\geqslant 0$ such that

if $f\in \mathscr{K}(G)$ is positive and zero outside U. There exists a continuous function of positive type $\varphi$ on G such that $\nu =\varphi \Delta^{-1/2}\cdot \mu$. (Let $j$ be the canonical morphism of $\mathscr{K}(G)$ into $E_{\nu}$; if $\mathfrak{B}$ is a basis of the filter of neighbourhoods of $e$ whose elements are contained in U and $f_V\in \mathscr{K}_+(G)$ is a function with support in $V\in \mathfrak{B}$ and of integral 1, prove that $j(f_V)$ converges weakly according to $\mathfrak{B}$ to an element $y_0$ of $E_{\nu}$ such that

for $f\in \mathscr{K}(G)$; putting $\varphi (g) = (y_0|\sigma (g)y_0)$, verify that $\varphi$ has the required properties.)

f) Let $\varphi_1$ be a continuous function of positive type on G and $\varphi_2$ a locally integrable function of positive type. If $\varphi_1-\varphi_2$ is of positive type, then $\varphi_2$ is locally equal almost everywhere to a continuous function of positive type. (Verify that the preceding question can be applied.)