On the bounded approximation property in Banach spaces

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 ? ₁-and Figiel, Johnson and Pe?czyński-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.