The classification of injective Banach lattices
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.
representation for injective Banach lattices with order continuous
Haydon's characterisation of 1-injective Banach lattices
representation for injective Banach lattices with order continuous norm
scientific article; zbMATH DE number 3927643