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

  Published:2012-04-12
 Printed: Apr 2013
  • Vladimir Kadets,
    Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61022 Kharkov, Ukraine
  • Miguel Martín,
    Departamento de Análisis Matematico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
  • Javier Merí,
    Departamento de Análisis Matematico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
  • 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 lush space, numerical index, Daugavet property, Köthe space, rearrangement invariant space
MSC Classifications: 46B04, 46E30 show english descriptions Isometric theory of Banach spaces
Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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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