Show that: - $\hat{\mathbb{Z}} \cong \mathbb{S^1}$ , $\hat{\mathbb{S^1}} \cong \mathbb{Z}$ , $\hat{\mathbb{R}} \cong \mathbb{R}$

Question

For a locally compact abelian (LCA) group $G$ let $\hat{G}$ denotes the dual group (the group of characters on $G$). Now with the compact open topology $\hat{G}$ becomes a LCA group. I need to prove the following -

1) $\hat{\mathbb{Z}} \cong \mathbb{S}^1$

2) $\hat{\mathbb{S}}^1 \cong \mathbb{Z}$

3) $\hat{\mathbb{R}} \cong \mathbb{R}$

as LCA groups. I have absolutely no idea how to do this. Help is appreciated..

Answer 1

This is very basic. The first isomorphism is almost tautological: $\hat{\mathbb{Z}}$ is the group of (continuous) homomorphisms $\mathbb{Z}\to\mathbb{S}^1$ and this is algebraically and topologically isomorphic to $\mathbb{S}^1$.

If you already know Pontryagin duality, then the second statement is again obvious. Without Pontryagin duality, you have to prove

1. $\widehat{\mathbb{S}^1}$ is discrete;
2. A continuous endomorphism of $\widehat{\mathbb{S}^1}$ is the multiplication by an integer.

For $\hat{\mathbb{R}}$, it's a bit more difficult. For this it is convenient to consider $\mathbb{S}^1=\mathbb{R}/\mathbb{Z}$, so we have the canonical map $\pi\colon \mathbb{R}\to\mathbb{R}/\mathbb{Z}$.

For $r\in\mathbb{R}$, define $\alpha_r\colon\mathbb{R}\to\mathbb{R}/\mathbb{Z}$ by $$ \alpha_r(x)=\pi(rx) $$ It is clear that $\alpha_r\in\hat{\mathbb{R}}$ and it's not really difficult to show that $r\mapsto\alpha_r$ is a topological isomorphism.

Answer 2

The characters groups of elementary groups should be described in any book about Pontrjagin duality. For instance, in [DPS, Ex. 3.1.7] or [Pon, § 36]. The following quotations are from [DPS]

Below are referenced claims. I remark that since 2.4.5 is not a corollary, but a theorem claiming that there exists a Haar integral in every locally compact abelian group, probably there is a misprint and Corollary 2.5.5 is meant instead. Also probably referenced Proposition 1.2, Theorem 1.5, and Lemma 1.6 are in the Chapter 3.

References

[DPS] D. Dikranjan, I. Prodanov, L. Stoyanov, Topological groups. Characters, dualities and minimal group topologies, Marcel Dekker, Inc., New York, 1990.

[Pon] Lev S. Pontrjagin, Continuous groups, 2nd ed., M., (1954) (in Russian). (An English translation of some its edition is “Topological groups”).