Separable lifting property and extensions of local reflexivity

A Banach space X is said to have the {\it separable lifting property} if for every subspace Y of X∗∗ containing X and such that Y/X is separable there exists a bounded linear lifting from Y/X to Y. We show that if a sequence of Banach spaces E1,E2,… has the joint uniform approximation property and En is c-complemented in E∗∗n for every n (with c fixed), then \eco has the separable lifting property. In particular, if En is a Lpn,λ-space for every n (1