### Impacto

Martín Peinador, Elena and Chasco, M.J.
(1998)
*Pontryagin reflexive groups are not determined by their continuous characters.*
Rocky Mountain Journal of Mathematics, 28
(1).
pp. 155-160.
ISSN 0035-7596

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Official URL: http://projecteuclid.org/euclid.rmjm/1181071826

## Abstract

A theorem of Glicksberg states that, for an abelian group G, two locally compact topologies with the same set of continuous characters must coincide. In [12] it is asserted that this fact also holds for two Pontryagin reflexive topologies. We prove here that this statement is not correct, and we give some additional conditions under which it is true. We provide some examples of classes of groups determined by their continuous characters.

Item Type: | Article |
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Uncontrolled Keywords: | Continuous character; reflexive space; compact-open topology; Pontryagin duality; Glicksberg theorem; Montel space |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 16699 |

References: | W. Banaszczyk, Additive subgroups of topological vector spaces, Lecture Notes in Math. 1466 1991. W. Banaszczyk and E. Martí n-Peinador, The Glicksberg theorem on weakly compact sets for nuclear groups, Ann. N.Y. Acad. Sci. 788 (1996), 34-39. J. Diestel, Sequences and series in Banach spaces, Grad. Texts in Math. 92 1984. D.N. Dikranjan, I.R. Prodanov and L.N. Stoyanov, Topological groups, Marcel Dekker, New York, 1990. I. Glicksberg, Uniform boundedness for groups, Canad. J. Math. 14 (1962), 269-276. G. Köthe, Topological vector spaces I, Springer Verlag, New York, 1969. E. Martí n-Peinador, A reflexive admissible group must be locally compact, Proc. Amer. Math. Soc. 123 (1995), 3563-3566. D. Remus and F.J. Trigos, Abelian groups which satisfy Pontryagin duality need not respect compactness, Proc. Amer. Math. Soc. 117 (1993), 1195-1200. M.F. Smith, The Pontryajin duality theorem in linear spaces, Ann. of Math. 56 (1952), 248-253. V. Tarieladze, personal, communication. N.Th. Varopoulos, Studies in harmonic analysis, Proc. Camb. Phil. Soc. 60 (1964), 465-516. R. Venkataramann, Compactness in abelian topological groups, Pacific J. Math. 57 (1975), 591-595. |

Deposited On: | 11 Oct 2012 08:54 |

Last Modified: | 07 Feb 2014 09:34 |

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