This is an announcement for the paper "The Bishop-Phelps-Bollob'as moduli of a Banach space" by Mario Chica, Vladimir Kadets, Miguel Martin, Soledad Moreno and Fernando Rambla.

Abstract: We introduce two Bishop-Phelps-Bollob'as moduli of a Banach space which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollob'as theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of these moduli and an inequality with respect to duality. We calculate the two moduli for Hilbert spaces and also present many examples for which the moduli have the maximum possible value (among them, there are $C(K)$ spaces and $L_1(\mu)$ spaces). Finally, we show that if a Banach space has the maximum possible value of any of the moduli, then it contains almost isometric copies of the real space $\ell_\infty^{(2)}$ and present an example showing that this condition is not sufficient.

Archive classification: math.FA

Mathematics Subject Classification: 46B04

Remarks: 26 pages, 5 figures

Submitted from: mmartins@ugr.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1304.0376

or

http://arXiv.org/abs/1304.0376