Exercise 27, § 3

Let $G$ be a locally compact group, $\beta$ a left Haar measure on $G$. If $G$ is not reduced to $e$, $L^1(G, \beta)$ is an algebra (for convolution) admitting zero divisors other than $0$. One can proceed as follows to form two non-zero elements $f, g$ of $L^1(G, \beta)$ such that $f * g = 0$ :

1° Case where $G$ contains a compact subgroup $H$ not reduced to $e$. Take then for $f$ a characteristic function of a set $A$, for $g$ a difference $\varphi_{sB} - \varphi_B$ of two characteristic functions of sets, $A$ and $B$ being suitably chosen.

2° Case where $G = \mathbf{Z}$. Now prove that one can take

for every $n \in \mathbf{Z}$ and $g(n) = f(-n)$.

3° General case. First prove that there exists $a \neq e$ in $G$ such that $\Delta(a) = 1$. The closure $H$ in $G$ of the subgroup generated by $a$ is then a compact subgroup or a subgroup isomorphic to $\mathbf{Z}$ (General Top., Chapter V, § 1, exerc. 2). In the first case, use the result of the

1° ; in the second, take $f(t) = \sum_{n=-\infty}^{+\infty} \alpha_n \varphi_{Ua^{-n}}(t), g(t) = \sum_{n=-\infty}^{+\infty} \beta_n \varphi_{a^n U}(t)$, choosing suitably $U$ and the sequences $(\alpha_n), (\beta_n)$ with the aid of the 2°.