Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-10-05038-5
Published electronically: April 8, 2010
MathSciNet review: 2645054
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. F. ALBIAC AND N. J. KALTON, Topics in Banach space theory, Graduate Texts in Mathematics 233, Springer-Verlag, New York, 2006. MR 2192298 (2006h:46005)
  • 2. S. ARGYROS, E. ODELL, AND H. ROSENTHAL, On certain convex subsets of $ c\sb 0$, Functional analysis (Austin, TX, 1986-87), 80-111, Lecture Notes in Math., 1332, Springer, Berlin, 1988. MR 967090 (89k:46030)
  • 3. Y. BENYAMINI AND J. LINDENSTRAUSS, Geometric nonlinear functional analysis, vol. 1, American Mathematical Society Colloquium Publications 48, AMS, Providence, RI, 2000. MR 1727673 (2001b:46001)
  • 4. J. BOURGAIN, La propriété de Radon-Nikodym, Publ. Math. de l'Univ. Pierre et Marie Curie 36, 1979.
  • 5. J. BOURGAIN, Dentability and finite-dimensional decompositions, Studia Math. 67 (1980), 135-148. MR 583294 (81m:46034)
  • 6. R. R. BOURGIN, Geometric Aspects of Convex Sets with the Radon-Nikodym Property, Lecture Notes in Math. 993, Springer-Verlag, Berlin 1983. MR 704815 (85d:46023)
  • 7. K. BOYKO, V. KADETS, M. MARTíN, AND J. MERí, Properties of lush spaces and applications to Banach spaces with numerical index $ 1$, Studia Math. 190 (2009), 117-133. MR 2461290
  • 8. K. BOYKO, V. KADETS, M. MARTíN, AND D. WERNER, Numerical index of Banach spaces and duality, Math. Proc. Cambridge Phil. Soc. 142 (2007), 93-102. MR 2296393
  • 9. J. M. F. CASTILLO, AND M. GONZALEZ, Three-space problem in Banach space theory, Lecture Notes in Mathematics 1667, Springer-Verlag, Berlin 1997. MR 1482801 (99a:46034)
  • 10. G. CHOQUET, Lectures on Analysis. Volume II: Representation Theory, W. A. Benjamin, Inc., London, 1969. MR 0250012 (40:3253)
  • 11. R. DEVILLE, G. GODEFROY AND V. ZIZLER, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific & Tecnical, London, 1993. MR 1211634 (94d:46012)
  • 12. D. VAN DULST, Characterizations of Banach spaces not containig $ \ell_1$, CWI Tract 59, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 1002733 (90h:46037)
  • 13. J. DUNCAN, C. MCGREGOR, J. PRYCE, AND A. WHITE, The numerical index of a normed space, J. London Math. Soc. 2 (1970), 481-488. MR 0264371 (41:8967)
  • 14. N. GHOUSSOUB, G. GODEFROY, B. MAUREY, AND W. SCHACHERMAYER, Some topological and geometrical structures in Banach spaces, Memoirs of the AMS, Providence, RI, 1987. MR 912637 (89h:46024)
  • 15. Y. IVAKNHO, V. KADETS, AND D. WERNER, The Daugavet property for spaces of Lipschitz functions, Math. Scand. 101 (2007), 261-279. MR 2379289 (2009c:46014)
  • 16. V. KADETS, M. MARTíN, J. MERí, AND R. PAYá, Convexity and smoothness of Banach spaces with numerical index one, Illinois J. Math. 53 (2009), 163-182.
  • 17. V. KADETS, M. MARTíN, J. MERí, AND V. SHEPELSKA, Lushness, numerical index one and duality, J. Math. Anal. Appl. 357 (2009), no. 1, 15-24. MR 2526802
  • 18. V. KADETS. M. MARTíN, AND R. PAYá, Recent progress and open questions on the numerical index of Banach spaces, Rev. R. Acad. Cien. Serie A. Mat. 100 (2006), 155-182. MR 2267407 (2007h:46011)
  • 19. V. M. KADETS, R. V. SHVIDKOY, G. G. SIROTKIN, AND D. WERNER, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), 855-873. MR 1621757 (2000c:46023)
  • 20. V. M. KADETS, R. V. SHVIDKOY, AND D. WERNER, Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math. 147 (2001), 269-298. MR 1853772 (2002f:46018)
  • 21. V. KADETS AND D. WERNER, A Banach space with the Schur and the Daugavet property, Proc. Amer. Math. Soc. 132 (2004), 1765-1773. MR 2051139 (2005b:46021)
  • 22. J. LINDENSTRAUSS AND L. TZAFRIRI, Classical Banach Spaces. I, Springer-Verlag, Berlin, 1977. MR 0500056 (58:17766)
  • 23. G. LóPEZ, M. MARTíN, AND R. PAYá, Real Banach spaces with numerical index 1, Bull. London Math. Soc. 31 (1999), 207-212. MR 1664125 (99k:46024)
  • 24. M. MARTíN, The alternative Daugavet property for $ C^*$-algebras and $ JB^*$-triples, Math. Nachr. 281 (2008), 376-385. MR 2392119
  • 25. M. MARTíN AND T. OIKHBERG, An alternative Daugavet property, J. Math. Anal. Appl. 294 (2004), 158-180. MR 2059797 (2005b:46023)
  • 26. H. P. ROSENTHAL, On the structure of nondentable closed bounded convex sets, Adv. in Math. 70 (1988), 1-58. MR 947756 (89g:46041)
  • 27. R. A. RYAN, Introduction to Tensor Products of Banach spaces, Springer-Verlag, London, 2002. MR 1888309 (2003f:46030)
  • 28. W. SCHACHERMAYER, An example concerning strong regularity and points of continuity in Banach spaces. Functional analysis (Austin, TX, 1986-87), 64-79, Lecture Notes in Math., 1332, Springer, Berlin, 1988. MR 967089 (90a:46039)
  • 29. R. V. SHVIDKOY, Geometric aspects of the Daugavet property, J. Funct. Anal. 176 (2000), 198-212. MR 1784413 (2001h:46019)
  • 30. S. TODORČEVIĆ, Compact subsets of the first Baire class, J. Amer. Math. Soc. 12 (1999), 1179-1212. MR 1685782 (2000d:54028)
  • 31. L. WEIS, On the surjective (injective) envelope of strictly (co-) singular operators, Studia Math. 54 (1976), 285-290. MR 0399908 (53:3749)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46B20, 46B03, 46B04, 46B22, 47A12

Retrieve articles in all journals with MSC (2010): 46B20, 46B03, 46B04, 46B22, 47A12


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: https://doi.org/10.1090/S0002-9947-10-05038-5
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

American Mathematical Society