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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

Topics: Mathematics - Functional Analysis, Primary 46B04. Secondary 46E30
Year: 2011
OAI identifier: oai:arXiv.org:1103.1282
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