## 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 E
_{1}, E
_{2},... has the joint uniform approximation property and E
_{n} is c-complemented in E
_{n}** for every n (with c fixed), then (∑
_{n} E
_{n})
_{0} has the separable lifting property. In particular, if E
_{n} is a ℒ
_{pn},λ-space for every n (1 < p
_{n} < ∞, λ independent of n), an L
_{∞} or an L
_{1} space, then (∑
_{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-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∥ ≥ c√n, where c is a numerical constant.

Original language | English (US) |
---|---|

Pages (from-to) | 123-137 |

Number of pages | 15 |

Journal | Illinois Journal of Mathematics |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - 2001 |

Externally published | Yes |