Exercise 7, § 4

Let G be a locally compact group and $\beta$ a left Haar measure on G. Show that the algebra $L^1(G, \beta)$ has a unity element if and only if G is discrete. (Suppose G non-discrete, and let $f_0 \in L^1(G, \beta)$. There exists a compact neighborhood V of e such that

Let U be a symmetric compact neighborhood of e such that $U^2 \subset V$. Then, for almost every $x \in U$, thus $f_0$ is not a unity element for $L^1(G, \beta)$.)