Exercise 22, § 3

The notation of the preceding exercise is retained.

Let $f$ be a continuous function of positive type belonging to $\mathscr{L}^2(G)$.

a) There exists an increasing sequence $(p_n)_{n\in\mathbf{N}}$ of polynomial functions on $[0\surd ,1]$ which converges uniformly on [0,1] to the function $t\mapsto t$. b) Assume that $f$ is bounded and that $0\leqslant f\leqslant 1$ in $\mathscr{L}(L^2(G))$. There exists a sequence $(h_n)$ in $\mathscr{L}^2(G)$ satisfying the following conditions :

(i) Each $h_n$ is bounded and of positive type ;

(ii) The function $h_{n+1}-h_n$ is of positive type for all $n\in \mathbf{N}$;

(iii) The endomorphisms $\varrho (h_n)$ commute pairwise ;

(iv) One has $\varrho (h_n)^2\leqslant \varrho (f)$ for all $n$.

c) The sequence $(h_n)_{n\in\mathbf{N}}$ converges in $\mathscr{L}^2(G)$; if $h$ is a limit of $(h_n)$, then $h$ is of positive type and bounded, and satisfies $\varrho (h)^2=\varrho (f)$; one has $f=h*h$. d) No longer assume that $f$ is bounded. For $t\in \mathbf{R}$, let $p_t$ denote the spectral projector of the self-adjoint partial operator $\varrho (f)$ defined by $]-\infty , t]$. One has $p_t\circ \boldsymbol{\gamma }(g) =\boldsymbol{\gamma }(g)\circ p_t$ for all $t\in \mathbf{R}$ and $g\in G$.

e) For $t\in \mathbf{R}$, put $f_t=p_t(f)$; it is the class in L^2(G) of a continuous bounded function of positive type in $\mathscr{L}^2(G)$. (Verify that $\boldsymbol{\gamma }(\varphi )f_t=\varrho (f)p_t(g)$ for $g\in \mathscr{K}$ (G), and deduce that $\varrho (f_t) =\varrho (f)\circ p_t$, then apply the results of Exercise 20.)

f ) There exists $h\in \mathscr{L}^2(G)$ such that $f=h*h$. (Consider $f_n=p_n(f)$ for $n\in \mathbf{N}$ and apply part c).)