Exercise 7, § 3

Let I be a finite set and G = F(I) the free group constructed on I (A, I, p. 84). Let A be the involutive subalgebra of $\mathscr{L}(\ell^2(G))$ generated by the image of the left regular representation $\boldsymbol{\lambda }$.

a) For each $g\in G$, let $|g|$ denote the length of the unique reduced decomposition of $g$ (A, I, p. 146, exercise 19). The mapping $d(g, h) =|h^{-1}g|$ is a metric on G.

b) Let $\lambda \in \mathbf{R}^*_+$. The function on G defined by $\psi_{\lambda}(g) =$ exp($-\lambda |g|$) is a function of positive type. (Now prove that the function $g\mapsto |g|$ is a function of negative type, cf. exercise 5.)

c) For each integer $k\in \mathbf{N}$, let $G_k$ be the set of $g\in G$ such that $|g|=k$, and let $\varphi_k$ be the characteristic function of $G_k$. Let $f$ and $g$ be functions on G with support in $G_k$ and $G_l$, respectively. For each $m\in \mathbf{N}$, one has $(f*g)\varphi_m= 0$ unless $k+l-m$ is even and $|k-l|\leqslant m\leqslant k+l$; in this case, one has

d) Let $f$ be a function on G with finite support. One has

and

e) Let $\psi$ be a function on G such that M = sup$_{g\in G}|\psi (g)|(1 +|g|)^2$ is finite. There exists a unique linear mapping $u_{\psi}$ of A into A such that $u_{\psi}(\boldsymbol{\lambda }(g)) =$ $\psi (g)\boldsymbol{\lambda }(g)$ for each $g\in G$. One has $\|u_{\psi}\|\leqslant 2M$.

f ) Let X be the set of functions $\psi$ on G with finite support such that $\|u_{\psi}\|\leqslant 1$. There exists a filter $\mathfrak{F}$ on X such that

lim$_{\psi ,\mathfrak{F}}u_{\psi}= 1_A$

in the space $\mathscr{L}(A)$ endowed with the topology of simple convergence. In particular, the underlying Banach space of A has the approximation property. (Consider the functions $\psi_{\lambda ,n}(g) = (\varphi_0+\cdots +\varphi_n)\psi_{\lambda}$ for $\lambda \in \mathbf{R}_+^*$ and $n\in \mathbf{N}$, and use the preceding exercise.)