Exercise 27, § 4

Let $G$ be a locally compact group, $\beta$ a left Haar measure on $G$. If $G$ is not reduced to $e$, then $L^1(G, \beta)$ is an algebra (for convolution) admitting divisors of zero other than 0. One may proceed as follows to form two nonzero elements $f, g$ of $L^1(G, \beta)$ such that $f * g = 0$: $1^\circ$ The case that $G$ contains a compact subgroup $H$, not reduced to $e$, in which $\Delta(x) = 1$. Take $f$ to be the characteristic function of a set $A$, and $g$ to be a difference $\varphi_{sB} - \varphi_B$ of two characteristic functions, with $A$ and $B$ suitably chosen. $2^\circ$ The case that $G = \mathbf{Z}$. Show that one can then take

for all $n \in \mathbf{Z}$, and $g(n) = f(-n)$. $3^\circ$ The general case. First prove that there exists an $a \neq e$ in $G$ such that $\Delta(a) = 1$. The closure $H$ in $G$ of the subgroup generated by $a$ is then either a compact subgroup or a subgroup isomorphic to $\mathbf{Z}$ (GT, V, §1, Exer. 2). In the first case, use the result of $1^\circ$; in the second, take suitably choosing $U$ and the sequences $(\alpha_n), (\beta_n)$ with the help of $2^\circ$.