Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Positive and negative results on the numerical index of Banach spaces and duality


Author: Miguel Martín
Journal: Proc. Amer. Math. Soc. 137 (2009), 3067-3075
MSC (2000): Primary 46B20, 46B04, 47A12
DOI: https://doi.org/10.1090/S0002-9939-09-09837-2
Published electronically: February 19, 2009
MathSciNet review: 2506465
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the numerical index of an $ L$-embedded space and that of its dual coincide. In particular, the numerical index of the predual of a real or complex von Neumann algebra or $ JBW^*$-triple coincides with the numerical index of the space. Also, we prove that when $ X$ is an $ M$-embedded Banach space with numerical index $ 1$, then every closed subspace of $ X^{**}$ containing $ X$ also has numerical index $ 1$ (in particular, $ X^*$ and $ X^{**}$ have numerical index $ 1$). Finally, we show that any Banach space $ X$ containing a complemented copy of $ c_0$ or a copy of $ \ell_\infty$ admits an equivalent norm for which the numerical index of its dual space is strictly less than the index of the space. In the special case of a separable space $ X$ containing $ c_0$, it is actually possible to renorm $ X$ with the maximum value of the numerical index (namely $ 1$) while the numerical index of the dual is as small as possible (namely, 0 in the real case, $ 1/\mathrm{e}$ in the complex case).


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

  • 1. Y. ABRAMOVICH AND C. ALIPRANTIS, An Invitation to Operator Theory, Graduate Texts in Math. 50, Amer. Math. Soc., Providence, RI, 2002. MR 1921782 (2003h:47072)
  • 2. Y. ABRAMOVICH AND C. ALIPRANTIS, Problems in Operator Theory, Graduate Texts in Math. 51, Amer. Math. Soc., Providence, RI, 2002. MR 1921783 (2003h:47073)
  • 3. 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)
  • 4. J. BECERRA, G. LóPEZ, A. M. PERALTA, AND A. RODRİGUEZ-PALACIOS, Relatively weakly open sets in closed balls of Banach spaces and real $ JB^*$-triples of finite rank, Math. Ann. 330 (2004), 45-58. MR 2091678 (2005f:46128)
  • 5. J. BECERRA GUERRERO AND M. MARTİN, The Daugavet property of $ C^*$-algebras, $ JB^*$-triples, and of their isometric preduals, J. Funct. Anal. 224 (2005), 316-337. MR 2146042 (2006g:46009)
  • 6. F. F. BONSALL AND J. DUNCAN, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, London Math. Soc. Lecture Note Ser. 2, Cambridge University Press, London-New York, 1971. MR 0288583 (44:5779)
  • 7. F. F. BONSALL AND J. DUNCAN, Numerical Ranges. II, London Math. Soc. Lecture Note Ser. 10, Cambridge University Press, London-New York, 1973. MR 0442682 (56:1063)
  • 8. K. BOYKO, V. KADETS, M. MARTİN, AND D. WERNER, Numerical index of Banach spaces and duality, Math. Proc. Cambridge Phil. Soc. 142 (2007), 93-102. MR 2296393
  • 9. K. BOYKO, V. KADETS, M. MARTİN, AND J. MERİ, Properties of lush spaces and applications to Banach spaces with numerical index $ 1$, Studia Math. 190 (2009), 117-133.
  • 10. J. DUNCAN, C. M. MCGREGOR, J. D. PRYCE, AND A. J. WHITE, The numerical index of a normed space, J. London Math. Soc. 2 (1970), 481-488. MR 0264371 (41:8967)
  • 11. C. FINET, M. MARTİN, AND R. PAYá, Numerical index and renorming, Proc. Amer. Math. Soc. 131 (2003), 871-877. MR 1937425 (2003h:46021)
  • 12. Y. FRIEDMAN AND B. RUSSO, Structure of the predual of a $ JBW^*$-triple, J. Reine Angew. Math. 356 (1985), 67-89. MR 779376 (86f:46073)
  • 13. P. HARMAND, D. WERNER, AND W. WERNER, $ M$-ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math. 1547, Springer-Verlag, Berlin, 1993. MR 1238713 (94k:46022)
  • 14. J. M. ISIDRO, W. KAUP, AND Á. RODRİGUEZ-PALACIOS, On real forms of $ JB^*$-triples, Manuscripta Math. 86 (1995), 311-335. MR 1323795 (96a:46121)
  • 15. V. KADETS. M. MARTİN, AND R. PAYá, Recent progress and open questions on the numerical index of Banach spaces, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Serie A Mat. 100 (2006), 155-182. MR 2267407 (2007h:46011)
  • 16. V. KADETS, R. SHVIDKOY, G. SIROTKIN, AND D. WERNER, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), 855-873. MR 1621757 (2000c:46023)
  • 17. A. KAIDI, A. MORALES, AND A. RODRIGUEZ PALACIOS, Geometrical properties of the product of a $ C^*$-algebra, Rocky Mountain J. Math. 31 (2001), 197-213. MR 1821377 (2002f:46140)
  • 18. G. LóPEZ, M. MARTİN, AND R. PAYá, Real Banach spaces with numerical index 1, Bull. London Math. Soc. 31 (1999), 207-212. MR 1664125 (99k:46024)
  • 19. M. MARTİN, The alternative Daugavet property of $ C^*$-algebras and $ JB^*$-triples, Math. Nachr. 281 (2008), 376-385. MR 2392119
  • 20. M. MARTİN AND T. OIKHBERG, An alternative Daugavet property, J. Math. Anal. Appl. 294 (2004), 158-180. MR 2059797 (2005b:46023)
  • 21. M. MARTİN AND R. PAYá, Numerical index of vector-valued function spaces, Studia Math. 142 (2000), 269-280. MR 1792610 (2001i:46017)
  • 22. H. PFITZNER, Separable $ L$-embedded Banach spaces are unique preduals, Bull. London Math. Soc. 39 (2007), 1039-1044. MR 2392827 (2009a:46030)
  • 23. B. RUSSO, Structure of $ JB^*$-triples, in: Jordan Algebras (Oberwolfach, 1992), 209-280, de Gruyter, Berlin, 1994. MR 1293321 (95h:46109)
  • 24. D. WERNER, Recent progress on the Daugavet property, Irish Math. Soc. Bull. 46 (2001), 77-97. MR 1856978 (2002i:46014)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B20, 46B04, 47A12

Retrieve articles in all journals with MSC (2000): 46B20, 46B04, 47A12


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-09-09837-2
Keywords: Numerical range, numerical index, duality, $L$-embedded, $M$-embedded
Received by editor(s): August 6, 2008
Received by editor(s) in revised form: November 20, 2008
Published electronically: February 19, 2009
Additional Notes: The author was supported by Spanish MEC project MTM2006-04837 and Junta de Andalucía grants FQM-185 and FQM-1438.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society