Exercise 18, § 3

Let G be a topological commutative discrete ordered group. We write the law of the group of G additively, and we denote its unit by 0. We denote by $G^+$ the set of elements $x\geqslant 0$ of G.

a) The space $L^1_+$ of functions $f\in L^1(G)$ vanishing outside $G^+$ is a closed subalgebra of $L^1(G)$.

b) Suppose that G is not Archimedean (TG, V, p. 16, exercise 1 of §3). The algebra $L^1_+$ is not maximal in $L^1(G)$.

Suppose now that G is Archimedean. Let A be a closed subalgebra of $L^1(G)$ containing $L^1_+$. For every $x\in G$, we denote by $\varphi_x\in L^1(G)$ the characteristic function of ${x}$. It is invertible in $L^1(G)$ and its inverse is $\varphi_{-x}$.

c) There exists $y\in G_+$ such that $\varphi_y\in A$ and $\varphi_{-y}=\varphi^{-1}_y\notin A$ and there exists a character $\chi \not= 0$ of A such that $\chi (y) = 0$.

d) For every $x >0$ in G, one has $\chi (x) = 0$. (There exists an integer $k\geqslant 1$ such that $-y+kx >0$.)

e) There exists a measure $\mu\in \mathscr{M}^1(G)$ of norm 1 such that $\chi$ is the restriction of $\mu$ to the algebra A viewed as a subspace of $\mathscr{C}(G)$. (For $f\in A$, one has $|\chi (f)|\leqslant \varrho (f)$ and moreover $\varrho (f) =|f|_{\infty}.$)

f) Let $\widehat{\mu}: G\rightarrow \mathbf{C}$ be the Fourier cotransform of $\mu$. One has $\widehat{\mu}(x) = 0$ if $x >0$.

g) The measure $\mu$ is the counting measure on G.

h) The subalgebra $L^1_+$ is maximal in $L^1(G)$.