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Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces
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- Journal:
- Canadian Journal of Mathematics / Volume 65 / Issue 2 / 01 April 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 331-348
- Print publication:
- 01 April 2013
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- Article
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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 
$\left[ 0,1 \right]$ is 
${{L}_{1}}\left[ 0,1 \right]$; the same space is the only r.i. separable function space on $\left[ 0,1 \right]$ with the Daugavet property over the reals.
