Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.

Question

Let $ G $ be a locally compact abelian (LCA) group and $ \widehat{G} $ the Pontryagin dual of $ G $, i.e., the set of all continuous homomorphisms $ G \to \mathbb{R} / \mathbb{Z} $. Clearly, $ \widehat{G} $ is an abelian group. The topology on $ \widehat{G} $ is generated by the sub-basic sets $$ U_{K,\xi_{0},\epsilon} \stackrel{\text{def}}{=} \left\{ \xi \in \widehat{G} ~ \Big| ~ |\xi(x) - {\xi_{0}}(x)| < \epsilon ~ \text{for all} ~ x \in K \right\} $$ for compact subsets $ K \subseteq G $, $ \xi_{0} \in \widehat{G} $ and $ \epsilon > 0 $. Show that $ \widehat{G} $ equipped with this topology is an LCA group.

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Anyone is welcome to write a proof or to give a link to a proof. Thanks!

Answer 1

Added in edit: Keith Conrad points out that the proof can be found in Hewitt and Ross's Abstract Harmonic Analysis, volume I, Theorem 23.15 (p. 361). The proof is essentially as I describe below, except Hewitt and Ross do not bother with the C$^*$-completion of $L^1(G)$, they just define a topology directly on $\widehat{G}$ agreeing with the weak-* topology on the spectrum of $L^1(G)$ as a Banach algebra (Theorem 23.13).

My original post follows:

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I don't know if this is the standard proof or not. I wasn't able to find a proof in the standard reference, Hewitt and Ross, and someone worried me a bit elsewhere by claiming that $\widehat{G}$ wasn't necessarily locally compact if $G$ was. Nonetheless, when I had looked into this previously I came up with this argument.

If $G$ is locally compact abelian, the convolution algebra $L^1(G)$ is commutative and can be completed to a commutative (non-unital unless G is discrete) C$^*$-algebra $C^*(G)$. The spectrum of $C^*(G)$ is locally compact (Gelfand duality for nonunital commutative C$^*$-algebras), and it is isomorphic to the spectrum of $L^1(G)$. There is an isomorphism (as sets) between the spectrum of $L^1(G)$ and the group $\widehat{G}$. However, $\mathrm{Spec}(C^*(G))$ has the weak-* topology, which is not prima facie the same as the usual Pontryagin topology on $\widehat{G}$.

If we examine Dixmier's C$^*$-algebras, Theorem 13.5.2, we have that the weak-* topology for states coincides with uniform convergence on compact subsets (of $G$) for the corresponding positive-definite functions, since the set of states is bounded. (Multiplicative) Characters of a group live a double life as representations and positive definite functions (just as characters of a C$^*$-algebra live a double life as representations and pure states), so uniform convergence on compact subsets of $G$ is the Pontryagin topology, as was required.