Exercise 15, § 1

Let $E\subset G$. One says that E is independent if, for every integer $k\geqslant 1$, for all elements $x_1, . . . , x_k$ pairwise distinct of E and for all integers $n_1, . . . , n_k$, the relation $x^{n_1}_1\cdots x^{n_k}_k=e$ implies $x^{n_1}_1=\cdots =x^{n_k}_k=e$. One says that E is a Kronecker set if every continuous complex function on E of absolute value 1 is the uniform limit on E of unitary characters of G. If $q$ is an integer $\geqslant 1$ and $G = (\mathbf{Z}/q\mathbf{Z})^{\mathbf{N}}$, one says that E is of type $K_q$ if every continuous mapping from E into the set of $q$-th roots of unity coincides on E with a unitary character of G. a) Let P be a compact Kronecker set. If $\mu\in \mathscr{M}^1(G)$ is concentrated on P, one has $\|\mathscr{F}(\mu)\|_{\infty}=\|\mu\|$. Deduce that every continuous complex function on P is the restriction of the Fourier transform of an element $f\in L^1(\widehat{G})$. b) Let $E\subset G$ be a finite independent set. Let $f$ be a complex function of absolute value 1 on E. Suppose that $f(x)^q$ = 1 when $x\in E$ is of order $q$. The function $f$ is the uniform limit on E of unitary characters of G. (Use exerc. 13.)

c) Every Kronecker set is independent and contains only elements of infinite order.

d) Every set of type $K_q$ is independent.

e) Let $k\geqslant 1$ be an integer and let $V_1, . . . ,V_k$ be non-empty disjoint open subsets of G. If every neighbourhood of $e$ in G contains an element of infinite order, there exist $x_i\in V_i$ for $1\leqslant i\leqslant k$ such that $\{x_1, . . . , x_k\}$ is a Kronecker set. If $G = (\mathbf{Z}/q\mathbf{Z})^{\mathbf{N}}$, there exist $x_i\in V_i$ such that $\{x_1, . . . , x_k\}$ is of type $K_q$.

f) Let $E\subset G$ be a compact independent set, $F = E\cup E^{-1}$. Let $\mu$ be a diffuse bounded measure on G, concentrated on F. Then the measures $\mu^n$, for $n\geqslant 0$, are pairwise mutually singular, where one puts $\mu^0=\varepsilon_e$ and $\mu^n=\mu*\mu^{n-1}$ for $n\geqslant 1$. (Now prove that, if $m < n$, the set of $(x_1, . . . , x_n)\in G^n$ such that $x_1. . . x_n\in F^m$ is negligible for $\mu\otimes \cdots \otimes \mu$). Deduce that, if $\mu\geqslant 0$, one has $\|\sum^n_{k=0}\alpha_k\mu^k\|=\sum^n_{k=0}|\alpha_k| \|\mu\|^k$ whatever the complex numbers $\alpha_0, . . . , \alpha_n$.

g) Let $r\geqslant 1$ be an integer. Let $E\subset G$ be a compact independent set, let $\mu_1, . . . , \mu_r$ be diffuse measures $\geqslant 0$ of mass 1, concentrated on disjoint parts $E_1, . . . ,E_r$ of E. Let $z_1, . . . , z_r\in \mathbf{C}$ with $|z_i|\leqslant 1$ for every $i$. There exists a character $\chi$ of $\mathscr{M}^1(G)$ such that $\chi (\mu_i) =z_i$ for $1\leqslant i\leqslant r$. (Now prove first, by reasoning as in f), that, if $(n_1, . . . , n_r)\not= (m_1, . . . , m_r)$, the measures $\mu^{m_1}_1*\cdots *\mu^{m_r}_r$ and $\mu^{n_1}_1*\cdots *\mu^{n_r}_r$ are mutually singular. Then, supposing $|z_i|= 1$ for every $i$, prove that the spectral radius of $\varepsilon_e+\overline{z}_1\mu_1+\cdots +\overline{z}_r\mu_r$ is $r+ 1$, whence a character $\chi$ of $\mathscr{M}^1(G)$ such that $\chi (\varepsilon_e+\overline{z}_1\mu_1+\cdots +\overline{z}_r\mu_r) =r+ 1$; this character answers the question. In the general case, write $z_i=\frac{1}{2}(z'_i+z''_i)$ with $|z'_i|=|z''_i|= 1$, and $\mu_i=\frac{1}{2}(\mu'_i+\mu''_i)$ where $\mu'_1, \mu''_1, . . . , \mu'_r, \mu''_r$ satisfy the same hypotheses as $\mu_1, . . . , \mu_r.)$

h) Let $\mathscr{M}_d(E)$ be the Banach space of diffuse measures on G concentrated on E. Every continuous linear form of norm $\leqslant 1$ on $\mathscr{M}_d(E)$ extends to a character of $\mathscr{M}^1(G)$. (Use g).)