Separable lifting property and extensions of local reflexivity
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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 En** for every n (with c fixed), then (n En)0 has the separable lifting property. In particular, if En is a pn,-space for every n (1 < pn < , independent of n), an L or an L1 space, then (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)-1 u cn, where c is a numerical constant.