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arXiv · 1103.1282

Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces

Vladimir Kadets·Miguel Martin·Javier Meri·Dirk Werner
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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.

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Vladimir Kadets, Miguel Martin, Javier Meri, Dirk Werner. 2011-03-07. Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces. https://arxiv.org/abs/1103.1282

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