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Denseness of norm-attaining operators into strictly convex spaces

Published online by Cambridge University Press:  14 November 2011

M. D. Acosta
M. D. Acosta
Affiliation:
Departamento de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain ([email protected])
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Extract

We show that no infinite-dimensional Banach space provided with a strictly convex norm satisfies Lindenstrauss's property B. This is a generalization of previous results by Lindenstrauss for rotund spaces isomorphic to C0 and by Gowers for ℓp (1 < p < ∞). Also, there is an appropriate complex version of the announced result that works for all the C-strictly convex spaces. As a consequence, the Hardy space H1, any infinite-dimensional complex L1(μ), and, in general, any infinite-dimensional predual of a von Neumann algebra lacks Lindenstrauss's property B.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 6 , 1999 , pp. 1107 - 1114
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Aguirre, F. J.. Norm attaining operators into strictly convex Banach spaces. J. Math. Anal. Appl. 222 (1998), 431–437.CrossRefGoogle Scholar
2Akemann, C. A. and Russo, B.. Geometry of the unit sphere of a C*-algebra and its dual. Pacific J. Math. 32 (1970), 575–585.CrossRefGoogle Scholar
3Bishop, E. and Phelps, R. R.. A proof that every Banach space is subrefiexive. Bull. Am. Math. Soc. 67 (1961), 97–98.CrossRefGoogle Scholar
4Bourgain, J.. On dentability and the Bishop–Phelps property. Israel J. Math. 28 (1977), 265–271.CrossRefGoogle Scholar
5Deville, R., Godefroy, G. and Zizler, V.. Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64 (New York: Longman, 1993).Google Scholar
6Diestel, J., Jarchow, H. and Tonge, A.. Absolutely summing operators. Cambridge Studies in Advanced Mathematics, vol. 43 (Cambridge University Press, 1995).CrossRefGoogle Scholar
7Gowers, W. T.. Symmetric block bases of sequences with large average growth. Israel J. Math. 69 (1990), 129–149.CrossRefGoogle Scholar
8Istratescu, V. I.. Strict convexity and complex strict complexity. Lecture Notes in Pure and Applied Mathematics, vol. 89 (New York: Dekker, 1984).Google Scholar
9Iwanik, A.. Norm attaining operators on Lebesgue spaces. Pacific J. Math. 83 (1979), 381–386.CrossRefGoogle Scholar
10Johnson, J. and Wolfe, J.. Norm attaining operators. Studia Math. 65 (1979), 7–19.CrossRefGoogle Scholar
11Johnson, J. and Wolfe, J.. Norm attaining operators and simultaneously continuous retractions. Proc. Am. Math. Soc. 86 (1982), 609–612.CrossRefGoogle Scholar
12Lindenstrauss, J.. On operators which attain their norm. Israel J. Math. 1 (1963), 139–148.CrossRefGoogle Scholar
13Partington, J. R.. Norm attaining operators. Israel J. Math. 43 (1982), 273–276.CrossRefGoogle Scholar
14Schachermayer, W.. Norm attaining operators on some classical Banach spaces. Pacific J. Math. 105 (1983), 427–438.CrossRefGoogle Scholar
15Uhl, J.. Norm attaining operators in L1[0,1] and the R.N.P. Pacific J. Math. 63 (1976), 293–300.CrossRefGoogle Scholar