Weak metric approximation properties and nice projections

Abstract

We prove that a Banach space X has the weak MAP (the weak MCAP) [the very weak MCAP] if and only if there exists a norm one projection P on X^∗∗ with X⊂P(X^∗∗) such that P is in the weak^∗-closure of F(X,X) (K(X,X)) [K(X,X^∗∗)] in L(X^∗∗,X^∗∗).