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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 ${G}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}$, i.e., to the topological group of continuous homomorphisms of $G$ into the multiplicative group $𝕋=\left\{z\in ℂ:|z|=1\right\}$ with the topology of uniform convergence on compact subsets of $G$. The Pontryagin-van Kampen theorem establishes that the natural homomorphism ${\eta }_{G}:G\to {G}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}$ is a topological isomorphism for every locally compact abelian (LCA) group and thus that ${G}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}$ 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 ${\left({lim}_{\to }{G}_{n}\right)}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}\cong {lim}_{←}{G}_{n}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}$ and ${\left({lim}_{←}{G}_{n}\right)}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}\cong {lim}_{\to }{G}_{n}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}$ for sequences ${\left({G}_{n}\right)}_{n}$ 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 ${\left({lim}_{\to }{G}_{n}\right)}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}\cong {lim}_{←}{G}_{n}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}$ for every sequence ${\left({G}_{n}\right)}_{n}$ of topological groups that are ${k}_{\omega }$-spaces (a topological space $X$ is a ${k}_{\omega }$-space if there is an increasing sequence of compact spaces ${\left({K}_{m}\right)}_{m}$ with $X={\cup }_{m}{K}_{m}$ such that a subset $F\subset X$ is closed in $X$ if and only if $F\cap {K}_{m}$ 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 ${\left({lim}_{←}{G}_{n}\right)}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}\cong {lim}_{\to }{G}_{n}^{\phantom{\rule{0.277778em}{0ex}}\stackrel{^}{\phantom{\rule{0.166667em}{0ex}}}}$ 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 $\left\{{G}_{n},{f}_{m}^{n}\right\}$ (${f}_{m}^{n}:{G}_{m}\to {G}_{n}$) 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 ${lim}_{\to }{G}_{n}$ of a sequence $\left\{{G}_{n},{f}_{n}^{m}\right\}$ (${f}_{n}^{m}:{G}_{n}\to {G}_{m}$) of reflexive, ${k}_{\omega }$ groups with ${f}_{n}^{m}$ injective and ${f}_{n}^{m}\left({G}_{n}\right)$ dually embedded in ${G}_{m}$ is again reflexive. This applies in particular to the direct sum of any countable family of ${k}_{\omega }$, nuclear groups.

##### MSC:
 22A05 Structure of general topological groups 18A30 Limits; colimits 22D35 Duality theorems (locally compact groups) 54H11 Topological groups (topological aspects)