This is an announcement for the paper "Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces" by Vladimir Kadets, Miguel Martin, Javier Meri, and Dirk Werner.

Abstract: We show that for spaces with 1-unconditional bases lushness, the alternative Daugavet property and numerical index~1 are equivalent. In the class of rearrangement invariant (r.i.)\ sequence spaces the only examples of spaces with these properties are $c_0$, $\ell_1$ and $\ell_\infty$. The only lush r.i.\ separable function space on $[0,1]$ is $L_1[0,1]$; the same space is the only r.i.\ separable function space on $[0,1]$ with the Daugavet property over the reals.

Archive classification: math.FA

Mathematics Subject Classification: Primary 46B04. Secondary 46E30

Submitted from: werner@math.fu-berlin.de

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

http://front.math.ucdavis.edu/1103.1282

or

http://arXiv.org/abs/1103.1282