Binz-Butzmann duality versus Pontryagin duality (Q1337779)

The set \(\Gamma G\) of all continuous characters on an abelian topological group \(G\), with addition defined pointwise, is also an abelian group. The ``standard'' dual of \(G\) (from the point of view of Pontryagin duality theory) is \(\Gamma G\) endowed with the compact open topology. The group \(G\) is called reflexive if it is topologically isomorphic to its bidual. Following the work of E. Binz and H. Butzmann, we endow \(\Gamma G\) with the ``continuous convergence structure'' and we call this object the BB- dual of \(G\), \(\Gamma_ c G\). It is a convergence group; if \(G\) is bicontinuously isomorphic to \(\Gamma_ c(\Gamma_ c G)\) then it is said to be BB-reflexive. In the present paper we prove that there are reflexive groups that are not BB-reflexive and conversely. We also prove that the class of BB-reflexive groups contains the countable products and sums of LCA groups. Nevertheless the BB-dual of a countable sum is not the product of the BB-duals of the summands as could be expected from its analogue in Pontryagin duality theory.