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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A property of Dunford-Pettis type in topological groups


Authors: E. Martín-Peinador and V. Tarieladze
Journal: Proc. Amer. Math. Soc. 132 (2004), 1827-1837
MSC (2000): Primary 22A05; Secondary 46A16
Published electronically: October 21, 2003
MathSciNet review: 2051147
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Abstract: The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory.

In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the Dunford-Pettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups.

For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties.


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Additional Information

E. Martín-Peinador
Affiliation: Departamento de Geometría y Topología, Universidad Complutense de Madrid, Av. Complutense s/n, Madrid 28040, España
Email: peinador@mat.ucm.es

V. Tarieladze
Affiliation: Muskhelishvili Institute of Computational Mathematics, Georgian Academy of Sciences, Akuri str. 8. Tbilisi 93, Georgia
Email: tar@gw.acnet.ge

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07249-6
PII: S 0002-9939(03)07249-6
Keywords: Dunford-Pettis property, Schur property, Bohr topology, dual group, Pontryagin reflexive, locally convex space.
Received by editor(s): June 26, 2001
Received by editor(s) in revised form: April 20, 2002, and January 28, 2003
Published electronically: October 21, 2003
Additional Notes: This paper was written while the second author was enjoying a Sabático at the Complutense University of Madrid. The first author was partially supported by D.G.I.C.Y.T. BFM 2000-0804-C03-01.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society