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The Pontryagin duality of sequential limits of topological Abelian groups. (English) Zbl 1090.22001

Pontryagin duality relates a topological group G to its dual group G ^ , i.e., to the topological group of continuous homomorphisms of G into the multiplicative group 𝕋={z:|z|=1} with the topology of uniform convergence on compact subsets of G. The Pontryagin-van Kampen theorem establishes that the natural homomorphism η G :GG ^ ^ is a topological isomorphism for every locally compact abelian (LCA) group and thus that G ^ completely determines G. A topological group G 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 (lim G n ) ^ lim G n ^ and (lim G n ) ^ lim G n ^ for sequences (G n ) n 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 (lim G n ) ^ lim G n ^ for every sequence (G n ) n of topological groups that are k ω -spaces (a topological space X is a k ω -space if there is an increasing sequence of compact spaces (K m ) m with X= m K m such that a subset FX is closed in X if and only if FK m is closed in K m for every m).

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 (lim G n ) ^ lim G n ^ for every sequence of metrizable reflexive groups.

With these isomorphisms in hand the authors prove that the inverse limit lim G n of a sequence {G n ,f m n } (f m n :G m G n ) of metrizable, reflexive groups such that all f m n 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 lim G n of a sequence {G n ,f n m } (f n m :G n G m ) of reflexive, k ω groups with f n m injective and f n m (G n ) dually embedded in G m is again reflexive. This applies in particular to the direct sum of any countable family of k ω , nuclear groups.


MSC:
22A05Structure of general topological groups
18A30Limits; colimits
22D35Duality theorems (locally compact groups)
54H11Topological groups (topological aspects)