M-complete approximate identities in operator spaces

A. Arias; H. Rosenthal

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Unconditional ideals in Banach spaces

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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...

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On subspaces of Banach spaces where every functional has a unique norm-preserving extension

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Let X be a Banach space and Y a closed subspace. We obtain simple geometric characterizations of Phelps' property U for Y in X (that every continuous linear functional g ∈ Y* has a unique norm-preserving extension f ∈ X*), which do not use the dual space X*. This enables us to give an intrinsic geometric characterization of preduals of strictly convex spaces close to the Beauzamy-Maurey-Lima-Uttersrud criterion of smoothness. This also enables us to prove that the U-property of the subspace...

Unconditional ideals of finite rank operators

Trond A. Abrahamsen, Asvald Lima, Vegard Lima (2008)

Czechoslovak Mathematical Journal

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Let $X$ be a Banach space. We give characterizations of when $ℱ (Y, X)$ is a $u$ -ideal in $𝒲 (Y, X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when $ℱ (X, Y)$ is a $u$ -ideal in $𝒲 (X, Y)$ for every Banach space $Y$ , when $ℱ (Y, X)$ is a $u$ -ideal in $𝒲 (Y, X^{* *})$ for every Banach space $Y$ , and when $ℱ (Y, X)$ is a $u$ -ideal in $𝒦 (Y, X^{* *})$ for every Banach space $Y$ .