Maximal ideals of disjointness preserving operators |
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Authors: | Fethi Benamor |
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Affiliation: | IPEST, University of Carthage, BP 51, 2070 La Marsa, Tunisia |
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Abstract: | Let L and M be vector lattices with M Dedekind complete, and let Lr(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 Lr(L,M) (briefly, maximal δ-ideals of Lr(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 δ-ideals. For instance, surprisingly enough, we prove that any maximal δ-ideal of Lr(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 δ-ideals on continuous function spaces. As an application, we furnish a characterization of lattice bimorphisms on such spaces in terms of weigthed composition operators. |
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Keywords: | Continuous functions spaces Disjointness preserving Lattice isomorphism Maximal order ideal Regular operator Vector lattice |
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