Positive and negative results on the numerical index of Banach spaces and duality
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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).References
- Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. MR 1921782, DOI 10.1090/gsm/050
- Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, Graduate Studies in Mathematics, vol. 51, American Mathematical Society, Providence, RI, 2002. MR 1921783, DOI 10.1090/gsm/051
- Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
- Julio Becerra Guerrero, Ginés López Pérez, Antonio M. Peralta, and A. Rodríguez-Palacios, Relatively weakly open sets in closed balls of Banach spaces, and real $\textrm {JB}^*$-triples of finite rank, Math. Ann. 330 (2004), no. 1, 45–58. MR 2091678, DOI 10.1007/s00208-004-0537-y
- Julio Becerra Guerrero and Miguel Martín, The Daugavet property of $C^*$-algebras, $JB^*$-triples, and of their isometric preduals, J. Funct. Anal. 224 (2005), no. 2, 316–337. MR 2146042, DOI 10.1016/j.jfa.2004.11.004
- F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Mathematical Society Lecture Note Series, vol. 2, Cambridge University Press, London-New York, 1971. MR 0288583
- F. F. Bonsall and J. Duncan, Numerical ranges. II, London Mathematical Society Lecture Note Series, No. 10, Cambridge University Press, New York-London, 1973. MR 0442682
- Kostyantyn Boyko, Vladimir Kadets, Miguel Martín, and Dirk Werner, Numerical index of Banach spaces and duality, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 93–102. MR 2296393, DOI 10.1017/S0305004106009650
- 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.
- J. Duncan, C. M. McGregor, J. D. Pryce, and A. J. White, The numerical index of a normed space, J. London Math. Soc. (2) 2 (1970), 481–488. MR 264371, DOI 10.1112/jlms/2.Part_{3}.481
- Catherine Finet, Miguel Martín, and Rafael Payá, Numerical index and renorming, Proc. Amer. Math. Soc. 131 (2003), no. 3, 871–877. MR 1937425, DOI 10.1090/S0002-9939-02-06576-0
- Yaakov Friedman and Bernard Russo, Structure of the predual of a $JBW^\ast$-triple, J. Reine Angew. Math. 356 (1985), 67–89. MR 779376, DOI 10.1515/crll.1985.356.67
- P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR 1238713, DOI 10.1007/BFb0084355
- José M. Isidro, W. Kaup, and Ángel Rodríguez-Palacios, On real forms of $\textrm {JB}^*$-triples, Manuscripta Math. 86 (1995), no. 3, 311–335. MR 1323795, DOI 10.1007/BF02567997
- Vladimir Kadets, Miguel Martín, and Rafael Payá, Recent progress and open questions on the numerical index of Banach spaces, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (2006), no. 1-2, 155–182 (English, with English and Spanish summaries). MR 2267407
- Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin, and Dirk Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), no. 2, 855–873. MR 1621757, DOI 10.1090/S0002-9947-99-02377-6
- Kaidi El Amin, Antonio Morales Campoy, and Angel Rodriguez Palacios, Geometrical properties of the product of a $C^\ast$-algebra, Rocky Mountain J. Math. 31 (2001), no. 1, 197–213. MR 1821377, DOI 10.1216/rmjm/1008959677
- Ginés López, Miguel Martín, and Rafael Payá, Real Banach spaces with numerical index 1, Bull. London Math. Soc. 31 (1999), no. 2, 207–212. MR 1664125, DOI 10.1112/S002460939800513X
- Miguel Martín, The alternative Daugavet property of $C^*$-algebras and $\textrm {JB^*}$-triples, Math. Nachr. 281 (2008), no. 3, 376–385. MR 2392119, DOI 10.1002/mana.200510608
- Miguel Martín and Timur Oikhberg, An alternative Daugavet property, J. Math. Anal. Appl. 294 (2004), no. 1, 158–180. MR 2059797, DOI 10.1016/j.jmaa.2004.02.006
- Miguel Martín and Rafael Payá, Numerical index of vector-valued function spaces, Studia Math. 142 (2000), no. 3, 269–280. MR 1792610, DOI 10.4064/sm-142-3-269-280
- Hermann Pfitzner, Separable L-embedded Banach spaces are unique preduals, Bull. Lond. Math. Soc. 39 (2007), no. 6, 1039–1044. MR 2392827, DOI 10.1112/blms/bdm077
- Bernard Russo, Structure of $\textrm {JB}^*$-triples, Jordan algebras (Oberwolfach, 1992) de Gruyter, Berlin, 1994, pp. 209–280. MR 1293321
- Dirk Werner, Recent progress on the Daugavet property, Irish Math. Soc. Bull. 46 (2001), 77–97. MR 1856978
Additional Information
- Miguel Martín
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain
- MR Author ID: 643000
- ORCID: 0000-0003-4502-798X
- Email: mmartins@ugr.es
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2506465