### Impacto

Martín Peinador, Elena and Banaszczyk, W
(1999)
*Weakly pseudocompact subsets of nuclear groups.*
Journal of Pure and Applied Algebra , 138
(2).
pp. 99-106.
ISSN 0022-4049

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022404998000346

## Abstract

Let G be an Abelian topological group and G(+) the group G endowed with the weak topology induced by continuous characters. We say that G respects compactness (pseudocompactness, countable compactness, functional boundedness) if G and G+ have the same compact (pseudocompact, countably compact, functionally bounded) sets. The well-known theorem of Glicksberg that LCA groups respect compactness was extended by Trigos-Arrieta to pseudocompactness and functional boundedness. In this paper we generalize these results to arbitrary nuclear groups, a class of Abelian topological groups which contains LCA groups and nuclear locally convex spaces and is closed with respect to subgroups, separated quotients and arbitrary products.

Item Type: | Article |
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Uncontrolled Keywords: | locally compact Abelian group; nuclear groups; nuclear locally convex spaces; compactness; countable compactness; pseudocompactness; functional boundedness; Abelian groups |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 16691 |

References: | W. Banaszczyk. Additive subgroups of topological vector spaces. Lecture Notes in Mathematics, vol. 1466, Springer, Berlin (1991) W. Banaszczyk. Summable families in nuclear groups. Studia Math., 105 (1993), pp. 272–282 W. Banaszczyk, E. Martín-Peinador. The Glicksberg theorem on weakly compact sets for nuclear groups, Papers on General Topology and Applications. Ann. NY. Acad. Sci., 788 (1996), pp. 34–39 E. Hewitt, K.A. Ross. Abstract Harmonic Analysis, vol. ISpringer, Berlin (1963) D. Remus, F.J. Trigos-Arrieta. Abelian groups which satisfy Pontryagin duality need not respect compactness Proc. Amer. Math. Soc., 117 (1993), pp. 1195–1200 F.J. Trigos-Arrieta. Continuity, boundedness, connectedness and the Lindelöf property for topological groups J. Pure Appl. Algebra, 70 (1991), pp. 199–210 F.J. Trigos-Arrieta. Pseudocompactness on groups. Lecture Notes in Pure and Applied Mathematics, vol. 134Dekker, New York (1991), pp. 369–378 E. van Douwen. The maximal totally bounded group topology on G and the biggest minimal G-space, for Abelian groups G. Topology Appl., 34 (1990), pp. 69–91 |

Deposited On: | 11 Oct 2012 09:02 |

Last Modified: | 07 Feb 2014 09:34 |

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