A survey on nuclear groups (Q2762061)

The present article is part of the proceedings of a seminar on nuclear groups and related topics which took place at the Universidad Complutense de Madrid in 1999. Since their introduction by \textit{W. Banaszczyk} in [Lect. Notes Math. 1466 (1991; Zbl 0743.46002)] many researchers interested in topological groups have put their attention on nuclear groups. Locally compact Abelian groups and nuclear locally convex topological vector spaces are two very different classes of objects. However, they share some very nice properties of \(\mathbb{R}^n\) regarding convergence of sequences and series, compactness, summability, reflexivity etc. The class of nuclear groups is a Hausdorff variety of Abelian groups enjoying those properties and containing both classes mentioned above. The author uses this motivation to introduce the variety of nuclear groups and gives a very well written survey on them, where she outlines the methods of proof of some of the most remarkable properties of nuclear groups: NEWLINENEWLINENEWLINEProperties of the variety of nuclear groups.NEWLINENEWLINENEWLINECompletion of a nuclear group.NEWLINENEWLINENEWLINEDually closed and dually embedded subgroups.NEWLINENEWLINENEWLINELocal quasi-convexity of nuclear groups.NEWLINENEWLINENEWLINEPontryagin-van Kampen theorem in nuclear groups.NEWLINENEWLINENEWLINEGlicksberg theorem in nuclear groups.NEWLINENEWLINENEWLINEConvergence of sequences and series in nuclear groups.NEWLINENEWLINENEWLINELévy-Steinitz theorem on nuclear groups. Some of the main results of the paper were originally proved by \textit{W. Banaszczyk} (loc. cit.). The other ones are results by \textit{L. Außenhofer} [Diss. Math. 384, 113 p. (1999; Zbl 0953.22001)], \textit{J. Galindo} [Houston J. Math. 26, 315-334 (2000; Zbl 0978.22001)], \textit{W. Banaszczyk} and \textit{E. Martín-Peinador} [J. Pure Appl. Algebra 138, 99-106 (1999; Zbl 0935.22004)], \textit{X. Dominguez} and \textit{V. Tarieladze} [Acta Math. Hung. 88, 301-322 (2000; Zbl 0958.54042)]. Interested readers will find another article in the same proceedings, where other important results on nuclear groups as the Bochner or the Lévy theorems are treated by Banaszczyk himself.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00016].