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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 Banach space with the Schur and the Daugavet property


Authors: Vladimir Kadets and Dirk Werner
Journal: Proc. Amer. Math. Soc. 132 (2004), 1765-1773
MSC (2000): Primary 46B04; Secondary 46B20, 46M07
Published electronically: October 24, 2003
MathSciNet review: 2051139
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Abstract: We show that a minor refinement of the Bourgain-Rosenthal construction of a Banach space without the Radon-Nikodým property which contains no bounded $\delta$-trees yields a space with the Daugavet property and the Schur property. Using this example we answer some open questions on the structure of such spaces; in particular, we show that the Daugavet property is not inherited by ultraproducts.


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Additional Information

Vladimir Kadets
Affiliation: Faculty of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine
Address at time of publication: Department of Mathematics, Freie Universität Berlin, Arnimallee 2–6, D-14195 Berlin, Germany
Email: vova1kadets@yahoo.com, kadets@math.fu-berlin.de

Dirk Werner
Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 2–6, D-14195 Berlin, Germany
Email: werner@math.fu-berlin.de

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07278-2
PII: S 0002-9939(03)07278-2
Keywords: Daugavet property, Schur property, ultraproducts of Banach spaces
Received by editor(s): February 13, 2003
Published electronically: October 24, 2003
Additional Notes: The work of the first author was supported by a fellowship from the Alexander-von-Humboldt Stiftung.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society