The ideal property of tensor products of Banach lattices with applications to the local structure of spaces of absolutely summing operators

N. Nielsen

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We show that a Banach space with separable dual can be renormed to satisfy hereditarily an “almost” optimal uniform smoothness condition. The optimal condition occurs when the canonical decomposition $X * * * = X^{⊥} \oplus X *$ is unconditional. Motivated by this result, we define a subspace X of a Banach space Y to be an h-ideal (resp. a u-ideal) if there is an hermitian projection P (resp. a projection P with ∥I-2P∥ = 1) on Y* with kernel $X^{⊥}$ . We undertake a general study of h-ideals and u-ideals. For example...