On the bounded approximation property in Banach spaces

Abstract

We prove that the kernel of a quotient operator from an L 1-space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky-case l 1-and Figiel, Johnson and Pelczynski-case X* separable. Given a Banach space X, we show that if the kernel of a quotient map from some L 1-space onto X has the BAP, then every kernel of every quotient map from any L 1-space onto X has the BAP. The dual result for L ∞-spaces also holds: if for some L ∞-space E some quotient E/X has the BAP, then for every L ∞-space E every quotient E/X has the BAP.