Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-20T19:28:04.274Z Has data issue: false hasContentIssue false
  • English
  • Français

Extension and inversion of extended orthomorphisms on Riesz spaces

Part of: Nonlinear operators and their properties Topological linear spaces and related structures Ordered 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Aliprantis, C. D. and Burkinshaw, O., Locally solid Riesz spaces (Academic Press, New York-San Francisco-London, 1978).Google Scholar
[2]Duhoux, M. and Meyer, M., ‘A new proof of the lattice structure of orthomorphisms’, J. London Math. Soc. (2) 25 (1982), 375–378.CrossRefGoogle Scholar
[3]Duhoux, M. and Meyer, M., ‘Extended orthomorphisms on Archimedean Riesz spaces’, Annali di Matematica (4) 133 (1983), 193–226.CrossRefGoogle Scholar
[4]Duhoux, M. and Meyer, M., ‘Extended orthomorphisms and lateral completions of Archimedean Riesz spaces’, to appear.Google Scholar
[5]Huijmans, C. B. and de Pagter, B., ‘Ideal theory in f-algebras’, Trans. Amer. Math. Soc. 269 (1982), 225–245.Google Scholar
[6]Luxemburg, W. A. J., Some aspects of the theory of Riesz spaces (Lecture Notes in Mathematics 4 University of Arkansas, Fayetteville, 1979).Google Scholar
[7]Luxemburg, W. A. J. and Schep, A. R., ‘A Radon-Nikodym type theorem for positive operators and a dual’, Nederl. Akad. Wetensch. Proc. Ser. A 81 (1978), 357–375.CrossRefGoogle Scholar
[8]Luxemburg, W. A. J. and Zaanen, A. C., Riesz spaces I (North-Holland, Amsterdam-London, 1971).Google Scholar
[9]Meyer, M., ‘Richesses du centre d'un espace vectoriel réticulé’, Math. Ann. 236 (1978) 147–169.CrossRefGoogle Scholar
[10]Meyer, M., ‘Les algèbres réticulées archimédiennes’, Rev. Roumaine Math. Pures et Appl., to appear.Google Scholar
[11]Nakano, H., ‘Teilweise geordnete Algebra’, Japan J. Math. 17 (1941), 425–511.CrossRefGoogle Scholar
[12]de Pagter, B., f-algebras and orthomorphisms (Thesis, University of Leiden, 1981).Google Scholar
[13]de Pagter, B., ‘The space of extended orthomorphisms in a Riesz space’, to appear.Google Scholar
[14]Schaefer, H. H., Banach lattices and positive operators (Springer-Verlag, Berlin-Heidelberg-New York, 1974).CrossRefGoogle Scholar
[15]Vulikh, B. Z., Introduction to the theory of partially ordered spaces (Wolters-Noordhoff Scientific Publ. Ltd, Groningen, 1967).Google Scholar
[16]Wickstead, A. W., ‘Extensions of orthomorphisms’, J. Austral. Math. Soc. Ser. A 29 (1980), 87–98.CrossRefGoogle Scholar