An ideal characterization of when a subspace of certain Banach spaces has the metric compact approximation property

Volume 129 / 1998

J. C. Cabello Studia Mathematica 129 (1998), 185-196 DOI: 10.4064/sm-129-2-185-196


C.-M. Cho and W. B. Johnson showed that if a subspace E of $ℓ_p$, 1 < p < ∞, has the compact approximation property, then K(E) is an M-ideal in ℒ(E). We prove that for every r,s ∈ ]0,1] with $r^2 + s^2 < 1$, the James space can be provided with an equivalent norm such that an arbitrary subspace E has the metric compact approximation property iff there is a norm one projection P on ℒ(E)* with Ker P = K(E)^{⊥} satisfying ∥⨍∥ ≥ r∥Pf∥ + s∥φ - Pf∥ ∀⨍ ∈ ℒ(E)*. A similar result is proved for subspaces of upper p-spaces (e.g. Lorentz sequence spaces d(w, p) and certain renormings of $L^p$).


  • J. C. Cabello

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