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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Banach spaces having the Radon-Nikodym property and numerical index 1

Author: Miguel Martín
Journal: Proc. Amer. Math. Soc. 131 (2003), 3407-3410
MSC (2000): Primary 46B20, 47A12
Published electronically: June 19, 2003
MathSciNet review: 1990629
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Abstract: Let $X$ be a Banach space with the Radon-Nikodym property. Then, the following are equivalent.

(i) $X$ has numerical index 1.

(ii) $\vert x^{**}(x^*)\vert=1$ for all $x^*\in \mathrm{ex}(B_{X^*})$ and $x^{**}\in \mathrm{ex}(B_{X^{**}})$. (iii) $X$ is an almost-CL-space.

(iv) There are a compact Hausdorff space $K$ and a linear isometry $J:X \to C(K)$ such that $\vert x^{**}(J^*\delta_s)\vert=1$ for all $s\in K$ and $x^{**}\in\mathrm{ex}(B_{X^{**}})$.

If $X$ is a real space, the above conditions are equivalent to being semi-nicely embedded in some space $C(K)$.

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Additional Information

Miguel Martín
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

PII: S 0002-9939(03)07176-4
Received by editor(s): November 20, 2001
Published electronically: June 19, 2003
Additional Notes: This research was partially supported by Spanish MCYT projects no. BFM2000-1467 and BFM2002-00061
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society