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

Source abstract

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 , ℓ 1 andℓ∞. 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.