Multiplicative representation of disjointness preserving operators

Abstract

AbstractLet X and Y be vector lattices and let T: X→Y be a disjointness preserving linear operator, i.e., |Tx1|∧|Tx2|=0 if |x1|∧|x2|=0; x1,x2 ϵ X. Necessary and sufficient conditions are obtained under which T can be written as a multiplication, i.e., Tx(·)=e(·)x(τ(·)), where τ is a continuous mapping from a topological space on which Y can be represented as a vector lattice of extended real valued continuous functions into a topological space on which X can be similarly represented. If X and Y are normed lattices and T is continuous, these conditions are satisfied and hence T can be represented as such a multiplication. With the author's permission, several remarks made by C.B. Huijsmans and B. de Pagter are incorporated in the text