We study strict u-ideals in Banach spaces. A Banach space X is a strict u-ideal in its bidual when the canonical decomposition $X*** = X* ⊕ X^{⊥}$ is unconditional. We characterize Banach spaces which are strict u-ideals in their bidual and show that if X is a strict u-ideal in a Banach space Y then X contains c₀. We also show that $ℓ_{∞}$ is not a u-ideal.
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We characterize the approximation property of Banach spaces and their dual spaces by the position of finite rank operators in the space of compact operators. In particular, we show that a Banach space E has the approximation property if and only if for all closed subspaces F of $c_0$, the space ℱ(F,E) of finite rank operators from F to E has the n-intersection property in the corresponding space K(F,E) of compact operators for all n, or equivalently, ℱ(F,E) is an ideal in K(F,E).
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We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space 𝔈 = {S ∘ T: S compact operator on X} is an ideal in 𝔉 = span(𝔈,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $(S_γ)$ of compact operators on X such that $sup_{γ}||S_{γ}T|| ≤ ||T||$ and $S_{γ} → I_{X}$ in the strong operator topology. Similar results for dual spaces are also proved.
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