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# Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces

Published:2012-04-12
Printed: Apr 2013
Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61022 Kharkov, Ukraine
• Miguel Martín,
• Javier Merí,
• Dirk Werner,
Department of Mathematics, Freie Universität Berlin, Arnimallee 6, D-14 195 Berlin, Germany
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## 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.
 Keywords: lush space, numerical index, Daugavet property, Köthe space, rearrangement invariant space
 MSC Classifications: 46B04 - Isometric theory of Banach spaces 46E30 - Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

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