Maximal ideals of disjointness preserving operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:322

Article

Benamor, Fethi ; Boulabiar, Karim

关键词: continuous functions spaces;    disjointness preserving;    lattice isomorphism;    maximal order ideal;    regular operator;    vector lattice;   

DOI  :  10.1016/j.jmaa.2005.09.038

来源: Elsevier

【 摘 要 】

Let L and M be vector lattices with M Dedekind complete, and let L-r (L, M) be the vector lattice of all regular operators from L into M. We introduce the notion of maximal order ideals of disjointness preserving operators in L-r (L, M) (briefly, maximal delta-ideals of L-r (L, M)) as a generalization of the classical concept of orthomorphisms and we investigate some aspects of this 'new' structure. In this regard, various standard facts on orthomorphisms are extended to maximal delta-ideals. For instance, surprisingly enough, we prove that any maximal delta-ideal of L-r (L, M) is a vector lattice copy of M, when L, in addition, has an order unit. Moreover, we pay a special attention to maximal delta-ideals on continuous function spaces. As an application, we furnish a characterization of lattice bimorphisms on such spaces in terms of weigthed composition operators. (c) 2005 Elsevier Inc. All rights reserved.

【 授权许可】

Free