Exercise 54, § 1

We suppose that G is not compact. We denote by $\widehat{G}_d$ the dual group $\widehat{G}$ endowed with the discrete topology and by $\widetilde{G}$ the dual group of $\widehat{G}_d$; it is a compact topological group.

a) The mapping which associates with $x\in G$ the character $\chi \mapsto \chi (x)$ is a continuous injective homomorphism of G into $\widetilde{G}$; its image is dense in $\widetilde{G}$.

We shall henceforth identify G with a subset of $\widetilde{G}$.

b) Let K be a compact topological group and let $\varphi : G\rightarrow K$ be a continuous homomorphism. There exists a unique homomorphism $\widetilde{\varphi}$ from $\widetilde{G}$ into K which extends $\varphi$.

c) Let $f\in \mathscr{C}_b(G)$. The following conditions are equivalent:

(i) There exists a function $\widetilde{f}\in \mathscr{C}(\widetilde{G})$ which extends $f$;

(ii) The function $f$ belongs to the closure in $\mathscr{C}(G)$ of the subspace of $\mathscr{C}(G)$ generated by the characters $\chi \in \widehat{G}$;

(iii) The set of the $\boldsymbol{\gamma }(x)f$ for $x\in G$ is precompact in $\mathscr{C}(G)$.

(To verify that (iii) implies (i), prove that $f$ is uniformly continuous. Consider then the closure of the $\boldsymbol{\gamma }(x)f$; it is a compact metric space K. Prove that $\varrho :x\mapsto \boldsymbol{\gamma }(x)$ is a continuous homomorphism of G into the group of isometries of K; apply then question b) to extend $\varrho$ to $\widetilde{\varrho}$ and put $\widetilde{f}(x) = (\widetilde{\varrho}(x)f)(e)$.)

A function $f\in \mathscr{C}_b(G)$ satisfying these equivalent conditions is said to be almost periodic.

d) Let $f$ be an almost periodic function on G. For every $\varepsilon >0$ and for every compact subset F of G, there exists $x\in G$ - F such that $\|\boldsymbol{\gamma }(x)f-f\|\leqslant \varepsilon$.

e) The set of almost periodic functions on G is a star subalgebra of $\mathscr{C}_b(G)$.