Exercise 28, § 4

Let $G_1, G_2$ be two locally compact groups, $\beta_1, \beta_2$ left Haar measures on $G_1, G_2$, and $A_1$ (resp. $A_2$) the topological algebra (over $\mathbf{R}$) $L^1(G_1, \beta_1)$ (resp. $L^1(G_2, \beta_2)$). Let $T$ be an algebra isomorphism of $A_1$ onto $A_2$, such that the relation $f \geq 0$ almost everywhere is equivalent to $T(f) \geq 0$ almost everywhere. a) Show that $T$ is an isomorphism of topological algebras. (First note that if a decreasing sequence $(f_n)$ of elements of $A_1$ tends to 0 almost everywhere, then the same is true of the sequence of the $T(f_n)$; deduce from this that for every sequence $(f_n)$ that is bounded above in $A_1$, $T(\sup(f_n)) = \sup(T(f_n))$, and conclude with the help of Fatou’s lemma.) b) Show that $T$ may be extended, in only one way, to an isomorphism of the topological algebra $\mathscr{M}^1(G_1)$ onto the topological algebra $\mathscr{M}^1(G_2)$, and that there exist a topological isomorphism $u$ of $G_1$ onto $G_2$ and a continuous homomorphism $\chi$ of $G_1$ into $\mathbf{R}_+^*$ such that $T(\varepsilon_s) = \chi(s)\varepsilon_{u(s)}$ (make use of Exer. 15 a) and Exer. 19 b); to prove that $u$ and $\chi$ are continuous, observe that $T$ defines an isomorphism of the algebra $\mathscr{L}(A_1)$ of continuous endomorphisms of the topological vector space $A_1$ onto the analogous algebra $\mathscr{L}(A_2)$, and that this isomorphism is bicontinuous for the topology of pointwise convergence; on the other hand, observe that $s \mapsto \delta(s^{-1})$ is an isomorphism of $G_1$ onto a multiplicative subgroup of $\mathscr{L}(A_1)$.

c) Deduce from b) that, for every $f \in A_1$,

for all $t \in G_2$.

Recall (A, III, §2, Exer. 9) that there exist finite groups $G_1, G_2$ such that the algebras $L^1(G_1, \beta_1)$ and $L^1(G_2, \beta_2)$ are isomorphic without $G_1$ and $G_2$ being so.