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Proceedings of the American Mathematical Society
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
Published electronically: February 19, 2009
MathSciNet review: 2506465
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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).


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09837-2
PII: S 0002-9939(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.