A Mackey-Arens theorem for topological Abelian groups
The attempt to generalize the Mackey-Arens theorem to topological Abelian groups has been undertaken only recently. As posed by \textit{M. J. Chasco}, \textit{E. Martín-Peinador} and \textit{V. Tarieladze} [Stud. Math. 132(3), 257--284 (1999; Zbl 0930.46006)], the problem can be formulated as follows: let \((G,X)\) be any duality of Abelian groups (i. e. \(X\) is a subgroup of the group of all characters on the Abelian group \(G\)), and let \({\mathcal T}_m\) be the supremum of all locally quasi-convex topologies on \(G\) which are compatible with \((G,X)\) (we say that a group topology \({\mathcal T}\) on \(G\) is compatible with \((G,X)\) if the topological dual of \((G,{\mathcal T})\) coincides with \(X\)). (i) Is \({\mathcal T}_m\) compatible with \((G,X)\)? (ii) If so, can it be expressed as the topology of uniform convergence on a family of subsets of \(X\), definable in terms of the duality? (Note that any locally quasi-convex topology \({\mathcal T}\) is the topology of uniform convergence on \(\mathcal T\)-equicontinuous subsets of the dual group.) Locally quasi-convex groups are a natural generalization of locally convex spaces, and from similar results in topological vector spaces it is not difficult to derive that the answer to (i) is negative if we replace ``locally quasi-convex topologies''{} with ``group topologies''{} above. In this paper the authors find a sufficient condition on the duality \((G,X)\), where \(G\) is a topological Abelian group and \(X\) is the group of all continuous characters on \(G\), which allows an affirmative answer to both (i) and (ii). Let \({\mathbb T}\) be the multiplicative group of all complex numbers with modulus 1. Let \(T_n=\exp( \pi i [-(2n)^{-1},(2n)^{-1}])\subset {\mathbb T}\), for every \(n\in {\mathbb N}\). Say that a subset \(U\) of \(G\) determines continuity if any character \(\chi:G\to {\mathbb T}\) which sends \(U\) into \(T_1\) is continuous. Let \({\mathcal M}\) be the family of all subsets \(A\subset X\) for which the set \(\{x \in G:\chi(x)\in T_n \;\forall \chi \in A\}\) determines continuity for every \(n\in {\mathbb N}.\) Theorem 3.11 says that if \({\mathcal M}\) is directed by inclusion, then \({\mathcal T}_m\) is compatible and coincides with the topology of uniform convergence on the sets in \({\mathcal M}.\) (The authors use a different notation.) The last part of the paper is devoted to provide several interesting examples which answer negatively some questions stated in the above mentioned paper by Chasco, Martín-Peinador and Tarieladze. It is worth to mention here that, using categorical methods, \textit{M. Barr} and \textit{H. Kleisli} [Theory Appl. Categ. 8, 54--62 (2001; Zbl 0965.22004)] have seemingly answered question (i) above in the negative; in fact they have found conditions on a category of topological Abelian groups which imply that the corresponding analogue of the Mackey topology within that category exists for every object in it.
scientific article; zbMATH DE number 2073857