Let $T$ be a locally compact abelian (LCA) group. For any other LCA group $G$, let 
$\hom(G,T)$ be the set of continuous homomorphisms $G\to T$. With the compact-open 
topology, $\hom(G,T)$ is certainly a topological group, but is not in general locally 
compact, even if $T$ is compact. In any case there is an obvious homomorphism 
$$
 \alpha\_G : G \to \hom(\hom(G,T),T) 
$$
sending $x\in G$ to the functional $\chi\mapsto \chi(x)$. 


I have three questions: 


1. For what LCA groups $T$ is $\hom(G,T)$ locally compact for all locally compact 
 $G$?
2. For what groups $T$ is $\alpha\_G$ always a (topological) isomorphism?
 - More generally, is there any full subcategory $\mathcal L$ of the category of LCA 
 groups for which $\hom(G,H)$ is in $\mathcal L$ whenever $G$ and $H$ are in 
 $\mathcal L$?


(**Edit**: I'm most interested in 1 and 2. If there is some $\mathcal L$ satisfying 3 
where the objects in $\mathcal L$ can be characterized by some interesting topological 
criterion, that would be fantastic. My question is not whether there are "ad-hoc" ways of constructing $\mathcal L$.)



When does Pontryagin duality generalize?
I gave an answer to the second question already on math.stackexchange at <https://math.stackexchange.com/questions/124379/why-unitary-characters-for-the-dual-group-in-pontryagin-duality-if-g-is-not-co>, but I'll put it here too: if $\alpha\_G$ is an isomorphism of topological groups for all locally compact abelian groups $G$ then $T$ *must* be the unit circle. This explains the importance of the unit circle as a target group. It is discussed in Pontryagin's book on topological groups (see Example 72) and the first volume on abstract harmonic analysis by Hewitt and Ross (p. 424). 


