Separable lifting property and extensions of local reflexivity

William B. Johnson, Timur Oikhberg

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)123-137
Number of pages15
JournalIllinois Journal of Mathematics
Volume45
Issue number1
DOIs
StatePublished - 2001
Externally publishedYes

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