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Tytuł artykułu

On Mackey topology for groups

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EN
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EN
The present paper is a contribution to fill in a gap existing between the theory of topological vector spaces and that of topological abelian groups. Topological vector spaces have been extensively studied as part of Functional Analysis. It is natural to expect that some important and elegant theorems about topological vector spaces may have analogous versions for abelian topological groups. The main obstruction to get such versions is probably the lack of the notion of convexity in the framework of groups. However, the introduction of quasi-convex sets and locally quasi-convex groups by Vilenkin [26] and the work of Banaszczyk [1] have paved the way to obtain theorems of this nature. We study here the group topologies compatible with a given duality. We have obtained, among others, the following result: for a complete metrizable topological abelian group, there always exists a finest locally quasi-convex topology with the same set of continuous characters as the original topology. We also give a description of this topology as an S-topology and we prove that, for the additive group of a complete metrizable topological vector space, it coincides with the ordinary Mackey topology.
Twórcy
autor
  • Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain, EM_peinador@mat.ucm.es
Bibliografia
  • [1] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math. 1466, Springer, Berlin, 1991.
  • [2] W. Banaszczyk and E. Martín-Peinador, The Glicksberg theorem on weakly compact sets for nuclear groups, in: Ann. New York Acad. Sci. 788, 1996, 34-39.
  • [3] N. Bourbaki, Espaces vectoriels topologiques, Masson, Paris, 1981.
  • [4] M. Bruguera, Some properties of locally quasi-convex groups, Topology Appl. 77 (1997), 87-94.
  • [5] M. J. Chasco and E. Martín-Peinador, Pontryagin reflexive groups are not determined by their continuous characters, Rocky Mountain J. Math. 28 (1998), 155-160.
  • [6] W. W. Comfort and K. A. Ross, Topologies induced by groups of characters, Fund. Math. 55 (1964), 283-291.
  • [7] D. N. Dikranjan, I. R. Prodanov and L. N. Stoyanov, Topological Groups. Characters, Dualities and Minimal Group Topologies, Marcel Dekker, New York, 1990.
  • [8] I. Fleischer and T. Traynor, Continuity of homomorphisms on a Baire group, Proc. Amer. Math. Soc. 93 (1985), 367-368.
  • [9] I. Glicksberg, Uniform boundedness for groups, Canad. J. Math. 14 (1962), 269-276.
  • [10] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Grundlehren Math. Wiss. 115, Springer, 1963.
  • [11] H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981.
  • [12] J. Kąkol, Note on compatible vector topologies, Proc. Amer. Math. Soc. 99 (1987), 690-692.
  • [13] J. Kąkol, The Mackey-Arens theorem for non-locally convex spaces, Collect. Math. 41 (1990), 129-132.
  • [14] J. Kąkol, C. Pérez-García and W. Schikhof, Cardinality and Mackey topologies of non-Archimedian Banach and Fréchet spaces, Bull. Polish Acad. Sci. Math. 44 (1996), 131-141.
  • [15] G. Köthe, Topological Vector Spaces I, Springer, Berlin, 1969.
  • [16] I. Labuda and Z. Lipecki, On subseries convergent series and m-quasi-bases in topological linear spaces, Manuscripta Math. 38 (1982), 87-98.
  • [17] I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515-631.
  • [18] N. Noble, k-groups and duality, Trans. Amer. Math. Soc. 151 (1970), 551-561.
  • [19] B. J. Pettis, On continuity and openness of homomorphisms in topological groups, Ann. of Math. 52 (1950), 293-308.
  • [20] D. Remus and F. J. Trigos-Arrieta, Abelian groups which satisfy Pontryagin duality need not respect compactness, Proc. Amer. Math. Soc. 117 (1993), 1195-1200.
  • [21] W. Roelcke and S. Dierolf, On the three-space problem for topological vector spaces, Collect. Math. 32 (1981), 3-25.
  • [22] H. H. Schaefer, Topological Vector Spaces, Springer, 1971.
  • [23] M. F. Smith, The Pontryagin duality theorem in linear spaces, Ann. of Math. 56 (1952), 248-253.
  • [24] J. P. Troallic, Sequential criteria for equicontinuity and uniformities on topological groups, Topology Appl. 68 (1996), 83-95.
  • [25] N. T. Varopoulos, Studies in harmonic analysis, Proc. Cambridge Philos. Soc. 60 (1964), 467-516.
  • [26] N. Ya. Vilenkin, The theory of characters of topological Abelian groups with a given boundedness, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 439-462 (in Russian).
  • [27] J. H. Webb, Sequential convergence in locally convex spaces, Proc. Cambridge Philos. Soc. 64 (1968), 341-364.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv132i3p257bwm
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