Weak approximation properties of subspaces

Abstract

The paper is concerned with weak approximation properties which are weaker than the classical approximation property. For $\lambda \geq 1$, we prove that a Banach space $X$ has the $\lambda$-bounded weak approximation property ($\lambda$-BWAP) if and only if every locally $1$-complemented subspace of $X$ has the $\lambda$-BWAP, and that if $X$ has the $\lambda$-BWAP and $Z$ is a locally $\mu$-complemented subspace of $X$, then $Z$ has the $(2\mu+4)\mu\lambda$-BWAP. It also follows that $X$ has the weak approximation property (WAP) if and only if every locally complemented subspace of $X$ has the WAP.