Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-12-01T00:54:50.627Z Has data issue: false hasContentIssue false
  • English
  • Français

Extension of Riesz homomorphisms. III

Part of: Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Gerard Buskes
Gerard Buskes
Affiliation:
Department of Mathematics, University of Mississippi, University, Mississippi 38677, U.S.A.
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.

In this paper we prove an analogue of the separable version of Nachbin's characterization of injective Banach spaces in the setting of Banach lattices. The mappings involved are continuous Riesz homomorphisms defined on ideals of separable Banach lattices which can be extended to Riesz homomorphisms on the whole Banach lattice. We discuss applications to simultaneous extension operators and to extension of continuous mappings between certain topological spaces.

MSC classification

Secondary: 46A40: Ordered topological linear spaces, vector lattices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 1 , August 1987 , pp. 35 - 46
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Arens, R., ‘Extensions of functions on fully normal spaces’, Pacific J. Math. 2 (1952), 11–22.CrossRefGoogle Scholar
[2]Buskes, Gerard, ‘On extension of Riesz homomorphisms II’, to appear as a technical report in the report series of the Catholic University of Nijmegen, the Netherlands.Google Scholar
[3]Buskes, Gerard, Extension of Riesz homomorphisms, Thesis 1983, Nijmegen.Google Scholar
[4]Buskes, G. J. H. M., ‘Disjoint sequences and completeness properties’, Nedevl. Akad. Wetensch. Proc. Ser. A 88 (1), (1985), 11–19.Google Scholar
[5]Buskes, Gerard, ‘On an extension theorem by Sikorski’, Report 8523, 07 1985, KUN, Nijmegen.Google Scholar
[6]Cohen, H. B., ‘The k-norm extension property for Banach spaces’, Proc. Amer. Math. Soc. 15 (1964), 797–802.Google Scholar
[7]Gillman, L. and Jerison, M., Rings of continuous functions (Springer-Verlag, New York-Heidelberg-Berlin, 1976).Google Scholar
[8]de Jonge, E. and van Rooij, A. C. M., Introduction to Riesz spaces (Math. Centre Tracts 78, Math. Centrum, Amsterdam, 1977).Google Scholar
[9]Kakutani, S., ‘Simultaneous extension of continuous functions considered as a positive linear operation’, Japan Math. 17 (1940), 1–4.CrossRefGoogle Scholar
[10]Lindenstrauss, J., ‘On the extension of operators with range in a C(K)-space’, Proc. Amer. Math. Soc. 15 (1964), 218–224.Google Scholar
[11]Luxemburg, W. A. J. and Zaanen, A. C., Riesz spaces I (North-Holland Publishing Company, Amsterdam-London, 1971).Google Scholar
[12]Negrepontis, S., ‘The Stone space of the saturated Boolean algebras’, Trans. Amer. Math. Soc. 141 (1969), 515–527.CrossRefGoogle Scholar
[13]Rudin, W., Functional analysis (McGraw-Hill, New York, 1973).Google Scholar
[14]Semadeni, Z., Banach spaces of continuous functions, Vol. I (PWN, Tom 55, 1971).Google Scholar
[15]Walker, R., The Stone-Čech compactification (Ergebnisse der Math. 83, Springer Verlag, Berlin-Heidelberg-New York, 1974).CrossRefGoogle Scholar
[16]Willard, Stephen, General topology (Addison-Wesley, Reading, Massachussets, Menlo Park, California, London, Don Mills, Ontario, 1970).Google Scholar
[17]Zaanen, A. C., Riesz spaces II (North-Holland, Amsterdam-New York-Oxford, 1983).Google Scholar