Some approximation properties of Banach spaces and Banach lattices

Abstract

The notion of the bounded approximation property = BAP (resp. the uniform approximation property = UAP) of a pair [Banach space, its subspace] is used to prove that if X is a ℒ ∞-space, Y a subspace with the BAP (resp. UAP), then the quotient X/Y has the BAP (resp. UAP). If Q: X → Z is a surjection, X is a ℒ 1-space and Z is a ℒ p -space (1 ≤ p ≤ ∞), then ker Q has the UAP. A complemented subspace of a weakly sequentially complete Banach lattice has the separable complementation property = SCP. A criterion for a space with GL-l.u.st. to have the SCP is given. Spaces which are quotients of weakly sequentially complete lattices and are uncomplemented in their second duals are studied. Examples are given of spaces with the SCP which have subspaces that fail the SCP. The results are applied to spaces of measures on a compact Abelian group orthogonal to a fixed Sidon set and to Sobolev spaces of functions of bounded variation on ℝ n .