Separable lifting property and extensions of local reflexivity

Abstract

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 $E_1, E_2, \ldots$ has the joint uniform approximation property and $E_n$ is $c$-complemented in $E_n^{**}$ for every $n$ (with $c$ fixed), then $\eco$ has the separable lifting property. In particular, if $E_n$ is a ${\mathcal{L}}_{p_n, \lambda}$-space for every $n$ ($1 < p_n < \infty$, $\lambda$ independent of $n$), an $L_\infty$ or an $L_1$ space, then $\eco$ 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 $E \hra X^{**}$ such that if $u : X^{**} \raw X^{**}$ is an operator ($=$ bounded linear operator) such that $u (E) \subset X$, then $||(u|_E)^{-1}|| \cdot ||u|| \geq c \sqrt{n}$, where $c$ is a numerical constant.