Pontryagin duality relates a topological group to its dual group , i.e., to the topological group of continuous homomorphisms of into the multiplicative group with the topology of uniform convergence on compact subsets of . The Pontryagin-van Kampen theorem establishes that the natural homomorphism is a topological isomorphism for every locally compact abelian (LCA) group and thus that completely determines . A topological group with this property is usually called reflexive.
The Pontryagin-van Kampen theorem was extended by S. Kaplan [Duke Math. J. 17, 419-435 (1950; Zbl 0041.36101)] to direct and inverse sequential limits of LCA groups. Kaplan’s extension was based on the isomorphisms and for sequences of LCA groups. The object of the present paper is to extend these isomorphisms to other sequences of topological groups.
Following a categorical approach, the authors first prove that for every sequence of topological groups that are -spaces (a topological space is a -space if there is an increasing sequence of compact spaces with such that a subset is closed in if and only if is closed in for every ).
To obtain the other isomorphism the authors have to restrict their direct limits from the category of topological Abelian groups to the category of locally quasi-convex groups. The category of locally quasi-convex groups is a natural one for group duality, as the dual of an Abelian topological group is always locally quasi-convex. Regarding direct limits as defined in this category, the authors show that for every sequence of metrizable reflexive groups.
With these isomorphisms in hand the authors prove that the inverse limit of a sequence () of metrizable, reflexive groups such that all are onto is reflexive as well (this also follows from the results of the reviewer [Houston J. Math. 26, 315–334 (2000; Zbl 0978.22001)]) and that the direct limit of a sequence () of reflexive, groups with injective and dually embedded in is again reflexive. This applies in particular to the direct sum of any countable family of , nuclear groups.