Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T11:12:55.463Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of orthomorphisms

Published online by Cambridge University Press:  09 April 2009

A. W. Wickstead
A. W. Wickstead
Affiliation:
Department of Pure Mathematics The Queen's University of BelfastBT7 INN Northern, Ireland
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.

We consider, on an Archimedean Riesz space, the spaces of all linear operators lying between two multiples of the identity (for the order), those leaving all ideals invariant and the order bounded orthomorphisms. We find, if E is uniformly complete, necessary and sufficient conditions for all such operators defined on sublattices of E to extend to the whole of E. Examples are given to show the role of uniform completeness. For the space of all orthomorphisms we give a sufficient condition on E for such an extension to exist.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 1 , February 1980 , pp. 87 - 98
Copyright
Copyright © Australian Mathematical Society 1980

References

Bigard, A. and Kejmel, K. (1969), ‘Sur les endomorphismes conservant les polaires d'un groupe réticulé Archimédien’, Bull. Soc. Math. France 97, 381–398.CrossRefGoogle Scholar
Conrad, P. F. and Diem, J. E. (1971), ‘The ring of polar preserving endomorphisms of an abelian lattice ordered group’, Illinois J. Math. 15, 224–240.CrossRefGoogle Scholar
Luxemburg, W. A. J. (1977), private communication.Google Scholar
Luxemburg, W. A. J. and Zaanen, A. C. (1971), Riesz spaces I (North-Holland, Amsterdam, London).Google Scholar
Meyer, M. (1977), ‘Quelques propriétés des homomorphismes d'espaces vectoriels réticulés’, preprint.Google Scholar
Wickstead, A. W. (1977), ‘Representation and duality of multiplication operator on Archimedean Riesz spaces’, Compositio Math. 35, 225–238.Google Scholar
Zaanen, A. C. (1975), ‘Examples of orthomorphisms’, J. Approximation Theory 13, 192–204.CrossRefGoogle Scholar