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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Slicely countably determined Banach spaces


Authors: Antonio Avilés, Vladimir Kadets, Miguel Martín, Javier Merí and Varvara Shepelska
Journal: Trans. Amer. Math. Soc. 362 (2010), 4871-4900
MSC (2010): Primary 46B20; Secondary 46B03, 46B04, 46B22, 47A12
Published electronically: April 8, 2010
MathSciNet review: 2645054
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Abstract: We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the Radon-Nikodým property and all spaces without copies of $ \ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index $ 1$. In particular, we show that the dual of a real infinite-dimensional Banach space with the alternative Daugavet property contains $ \ell_1$ and that operators which do not fix copies of $ \ell_1$ on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.


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Additional Information

Antonio Avilés
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain
Email: avileslo@um.es

Vladimir Kadets
Affiliation: Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine
Email: vova1kadets@yahoo.com

Miguel Martín
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: mmartins@ugr.es

Javier Merí
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: jmeri@ugr.es

Varvara Shepelska
Affiliation: Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine
Email: shepelskaya@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05038-5
PII: S 0002-9947(10)05038-5
Keywords: Numerical radius, numerical index, Daugavet equation, Radon-Nikod\'{y}m property, Asplund spaces, containing of $\ell _1$, narrow operators
Received by editor(s): September 15, 2008
Received by editor(s) in revised form: March 2, 2009
Published electronically: April 8, 2010
Additional Notes: The first-named author was supported by the Marie Curie Intra-European Fellowship MCEIF-CT2006-038768 (EU) and the Spanish research project MTM2005-08379 (MEC and FEDER). The second-named author was supported by Junta de Andalucía and FEDER grant P06-FQM-01438. The third- and fourth-named authors were partially supported by the Spanish MEC and FEDER project no. MTM2006-04837 and Junta de Andalucía and FEDER grants FQM-185 and P06-FQM-01438. The fifth-named author was partially supported by the N. I. Akhiezer Foundation
Article copyright: © Copyright 2010 American Mathematical Society