Separable Lifting Property and Extensions of Local Reflexivity

March 7, 2000 Abstract. A Banach space X is said to have the 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 P n En 0 has the separable lifting property. In particular, if En is a Lpn,λ-space for every n (1 < pn < ∞, λ independent of n), an L∞ or an L1 space, then P n En 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-dimensional subspace E →֒ X such that if u : X → X is an operator (= bounded linear operator) such that u(E) ⊂ X, then ||(u|E)|| · ||u|| ≥ c √ n, where c is a numerical constant.