Exercise 26, § 1

For every compact commutative group H, we denote by $\mu_H$ the normalized Haar measure on H.

We suppose G compact. We denote by $E_G$ the set of complex measures $\mu$ on G such that the Fourier transform of $\mu$ has integer values. It is a commutative subgroup of $\mathscr{M}^1(G)$ which contains the Haar measure $\mu_G$. We denote by $F_G$ the subgroup of $E_G$ generated by the measures $\widehat{x} i_H(\mu_H)$, where H runs through the closed subgroups of G $(i_H$ denoting the canonical inclusion of H in G), and where $\widehat{x}$ runs through $\widehat{G}$.

The purpose of this exercise is to prove that $E_G= F_G$ (« Cohen's idempotent theorem »).

a) Let $\varphi : G_1\rightarrow G_2$ be a morphism of compact commutative groups. The mapping $\mu\mapsto \varphi (\mu)$ is a group homomorphism from $E_{G_1}$ into $E_{G_2}$.

b) For every $\mu\in E_G$ and every $\widehat{x}\in \widehat{G}$, one has $\widehat{x}\mu_G\in E_G$.

Let $\mu$ be a complex measure on G. We denote by $\theta$ the function on G such that $\mu=\theta |\mu|$. We denote by $A_\mu\subset \mathscr{M}^1(G)$ the set of measures of the form $\widehat{x} \mu$. Let $\nu$ be an adherent point to $A_\mu$ in $\mathscr{M}^1(G)$.

c) If $\nu$ does not belong to $A_\mu$, then $\nu$ is singular with respect to $\mu_G$.

d) Let $\varepsilon >0$. Let $f\in \mathscr{C}(G)$ of norm $\leqslant 1$ such that $\nu (f)>(1-\varepsilon )\|\mu\|$ and $\widehat{x}\in \widehat{G}$ such that $\mathscr{R}(\mu(\widehat{x} f))>(1-\varepsilon )\|\mu\|$. Prove that

e) Suppose that $\mu\in E_G$. Prove that $\|\nu \|\leqslant \|\mu\|-1/(36\|\mu\|)$. (Apply the preceding question to two characters $\widehat{x}_1$ and $\widehat{x}_2$ such that $\widehat{x}_1\mu\not=\widehat{x}_2\mu$, and note that $\|\widehat{x}_1\mu-\widehat{x}_2\mu\|\geqslant 1$.) f) Conclude that $E_G= F_G$ and more precisely that every measure $\mu$ in $E_G$ is written

where the set K is finite, $n_k\in \mathbf{Z},\widehat{x}_k\in \widehat{G}$, and the measures $i(\mu_{H_k})$ are mutually singular. (Let $\mu\in E_G$; prove that there exists a measure $\nu \not= 0$ in the closure of $\{\nu \in A_\mu|\nu (G)\not= 0\}$ whose norm is minimal; then prove using e) that $\{\widehat{x}\in \widehat{G}|\nu (\widehat{x})\not= 0\}$ is finite, and apply the preceding exercise.)