An alternative proof of a Radon-Nikodým theorem for lattice homomorphisms
Let \(E\) be an Archimedean and \(F\) a Dedekind complete vector lattice. Then the space \(L_ b(E,F)\) of order bounded linear operators from \(E\) to \(F\) is also a Dedekind complete lattice, but in general the evaluation mapping \(T\to Tu\) \((u>0)\) is not a lattice homomorphism. So the crucial result of this paper is of great interest: Let \(T\) be a lattice homomorphism from \(E\) to \(F\), and let \(0\leq U\), \(V\) be arbitrary elements of the band generated by \(T\) in \(L_ b(E,F)\). Then \((U\wedge V)(u)= U(u)\wedge V(u)\) for all \(u\geq 0\). As an application the authors get a new proof of the following result due to the second author and \textit{A. R. Schep} [Indag. Math. 40, 357-375 (1978; Zbl 0389.47018)]: Let again \(T\) be a lattice homomorphism from \(E\) to \(F\). Then the following assertions are equivalent: (i) \(S\in \{T\}^{dd}\). (ii) \(Su\in \{Tu\}^{dd}\) for all \(u\geq 0\). Here \(\{x\}^{dd}\) denotes the band generated by \(x\).
scientific article; zbMATH DE number 417084