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Extension and inversion of extended orthomorphisms on Riesz spaces

Part of: Nonlinear operators and their properties Ordered structures Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Michel Duhoux  and
Mathieu Meyer
Michel Duhoux
Affiliation:
Université Catholique de Louvain Institut de Mathématique Pure etAppliquée Chemin du Chclotron 1348 Louvain-la-Neuve Belgique
Mathieu Meyer
Affiliation:
Equipe d'Analyse Université ParisVI Tour 46 4emeetage 4, Place Jussieu 75230 Paris-Cedex 05, France
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Abstract

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Let E be an Archimedean Riesz space and let Orth∞(E) be the f-algebra consisting of all extended orthomorphisms on E, that is, of all order bounded linear operators T:D→E, with D an order dense ideal in E, such that T(B∩D) ⊆ B for every band B in E. We give conditions on E and on a Riesz subspace F of E insuring that every T ∈ Orth∞(F) can be extended to some ∈ Orth∞(E), and we also consider the problem of inversing an extended orthomorphism on its support. The same problems are also studied in the case of σ-orthomorphisms, that is, extended orthomorphisms with a super order dense domain. Furthermore, some applications are given.

Keywords

Archimedean Riesz spacesorder dense idealsorthomorphismsf-algebras

MSC classification

Secondary: 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces 46A40: Ordered topological linear spaces, vector lattices 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 37 , Issue 2 , October 1984 , pp. 223 - 242
Copyright
Copyright © Australian Mathematical Society 1984

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