How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ becomes a LCA group?

Question

How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ itself becomes a LCA group ?

I would really really appreciate if I can get a step by step and clear answer, because I have not been able to find a good resource online to understand this fact.

Thank you in advance,

Answer 1

We put a topology on $\Gamma$ as follows :

1. Let $K \subseteq G$ be a compact subset and $U$ be an open subset of the circle group $\mathbb{T}$. Then the set $P(K,U) \subseteq \Gamma$ is defined to be $$ \big\{ \phi :~~ \phi \in \Gamma \text{ and } \phi(K) \subseteq U\big\}.$$

2. Let the sets $P(K,U)$ be a subbasis for the topology of $\Gamma$ that is every open is a union of sets of the form $$ P(K_1, U_1)\cap P(K_2,U_2)\cap ... \cap P(K_n,U_n),$$ where each $K_i$ is compact in $G$ and each $U_i$ is open in $\mathbb{T}$.

This topology is called the compact-open topology and you can verify that $\Gamma$ is Hausdorff using the fact $\mathbb{T}$ is Hausdorff. Showing that this topology is locally compact is non-trival but you can find a proof here : http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/loccptascoli.pdf

Now that the abelian group $\Gamma$ has been given a locally compact Hausdorff topology, it remains to check that the algebraic and topological structures mesh well so as to make $\Gamma$ a locally compact abelian group. For this, you can check Section 1.2.6 of Rudin's Fourier Analysis on Groups :

http://www.math.rochester.edu/people/faculty/iosevich/rudingroupsbook.pdf