Separable lifting property and extensions of local reflexivity

A Banach space X is said to have the separable lifting property if for ever y subspace Y of X** containing X and such that Y/X is separable there exist s a bounded linear lifting from Y/X to Y. We show that if a sequence of Ban ach spaces E-1, E-2,E- . . . has the joint uniform approximation property a nd E is c-complemented in E**(n) for every n (with c fixed), then (Sigma (n ) E-n)(0) has the separable lifting property. In particular, if E-n, is a L -pn,L-lambda-Space for every n (1 < p, < infinity, lambda independent of n) , an L-infinity or an L-1 space, then (Sigma (n) E-n)(0) has the separable lifting property. We also show that there exists a Banach space X which is not extendably locally reflexive; moreover, for every n there exists an n-d imensional subspace E hooked right arrow X** such that if u : X** --> X** i s an operator (= bounded linear operator) such that u(E) C X, then parallel to (u/E)(-1) parallel to.parallel tou parallel to greater than or equal to c rootn, where c is a numerical constant.