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1998 | 129 | 2 | 185-196
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An ideal characterization of when a subspace of certain Banach spaces has the metric compact approximation property

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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 $L^p$).
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  • Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
  • Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
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