The classification of injective Banach lattices
type: Item
instance of: scholarly article
title: The classification of injective Banach lattices
zbMATH Open document ID: 0579.46014
DOI: 10.1007/BF02760633
author: P. J. Mangheni
published in: Israel Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: A Banach lattice E is injective if there is a \(\lambda\geq 1\) such that for every Banach lattice G, every closed sublattice F of G and every positive linear operator S:\(F\to E\) there is a positive linear extension T:\(G\to E\) of S with \(\| T\| \leq \lambda \| S\|\). If \(\lambda =1\), E is 1-injective. Lotz has shown that AL-spaces are 1-injective. We obtain a representation for injective Banach lattices with order continuous norm: An order continuous Banach lattice is order isomorphic to an AL-space. Our result is deduced from the following characterisation of bounded linear operators on 1-injective Banach lattices: If E is a 1-injective Banach lattice, X a Banach space and T:\(E\to X\) a bounded linear operator then either T is an isomorphism on some \(\ell_{\infty}\)-subspace of E or for all \(\delta >0\) there is \(\phi\in E'\!_+\) with \(\| Tu\| \leq \phi (| u|)+\delta \| u\|\) for all \(u\in E\). The proof of this uses Haydon's characterisation of 1-injective Banach lattices as spaces of certain operators between C(K)-spaces, K Stonean. The representation of general injective Banach lattices seems to be still open.
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46M10
zbMATH DE Number: 3927643
zbMATH Keywords: positive linear extension
zbMATH Keywords: representation for injective Banach lattices with order continuous
zbMATH Keywords: norm
zbMATH Keywords: order isomorphic to an AL-space
zbMATH Keywords: Haydon's characterisation of 1-injective Banach lattices
zbMATH Keywords: representation for injective Banach lattices with order continuous norm
Wikidata QID: Q114233754
MaRDI profile type: Publication
cites work: Extensions of positive operators between Banach lattices
cites work: Injective Banach lattices
cites work: Q4136031
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full work available at URL: https://doi.org/10.1007/bf02760633
OpenAlex ID: W1967876737
description: scientific article; zbMATH DE number 3927643