The various central extensions of $(G\times G)$ by $T$. Let $G$ be a locally compact abelian group, isomorphic to $G^*$, its Pontryagin Dual. Let $T$ denote the unit circle in $\mathbb{C}$, where continuous morphisms $\chi: G\to T$ are the elements of $G^*$. How many different can $(G\times G^*)$ admit a central extension by $T$? I'm wondering how explicitly one may express such groups.
EDIT: I originally claimed this was the semi-direct product that I was interested in. I regret the mistake.

In general, given a group $H$ and an abelian group $Z$, the central extensions of $G$ by the central subgroup $Z$ are classified by the 2-cohomology group $H^2(H,Z)$ (2-cocycles modulo 2-coboundaries). The zero element of this group corresponds to the direct product extension. In your case $H=G\times G^*$ (no matter whether $G$ is isomorphic to $G^*$), there is a distinguished cocycle, given by $b((v,f),(v',f'))=f(v')-f'(v)$. There are probably other 2-cocycles, but it's unclear how they can be related to the decomposition $H=G\times G^*$. Also in the topological setting, you need to restrict to the cohomology group $H^2_m(G)$ based on measurable cochains, but the latter still probably contains irrelevant elements.
