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

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.

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

• What is your definition of locally compact? Jun 18, 2014 at 0:14
• I guess the least restrictive definition ‘a locally compact space is one in which every point has a compact neighborhood’ would be appropriate. Actually, this matters only if $G$ is not Hausdorff. Jun 18, 2014 at 0:20

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:

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.

• There is a proof in "the standard reference, Hewitt and Ross" Volume I. See Theorem 23.15 (p. 361). It uses the maximal ideal space of the Banach algebra $L^1(G)$ and involves showing two differently define topologies on $\widehat{G}$ agree, so it is essentially the argument you describe.
– KCd
Mar 22, 2019 at 16:33
• @KCd Thank you, Keith. I must have just failed to read to the end of those 5 lines the first time I skimmed through. Mar 23, 2019 at 20:46
• A proof of local compactness that was inspired by Corollary 23.16 in Hewitt & Ross vol. 1 is at kconrad.math.uconn.edu/blurbs/gradnumthy/loccptascoli.pdf. It uses Ascoli's theorem to get local compactness and does not directly use Banach algebra methods.
– KCd
Mar 26, 2019 at 11:52