Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-15T18:58:07.817Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiplication in Vector Lattices

Published online by Cambridge University Press:  20 November 2018

Norman M. Rice
Norman M. Rice*
Affiliation:
Queen's University, Kingston, Ontario
Rights & Permissions [Opens in a new window]

Extract

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.

B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1136 - 1149
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

The contents of this paper are largely derived from the author's Ph.D. thesis, written under the supervision of Professor W. A. J. Luxemburg and presented to the California Institute of Technology in 1966.

References

1. Amemiya, I., A generalization of the Riesz-Fischer theorem, J. Math. Soc. Japan 5 (1953), 353–354.Google Scholar
2. Bourbaki, N., Éléments de mathématique. XIII. Première partie: Les structures fondamentales de l'analyse. Livre VI: Intégration, Actualités Sci. Indust., no. 1175 (Hermann, Paris, 1952).Google Scholar
3. Freudenthal, H., Teilweise geordnete Moduln, Proc. Acad. Sci. Amsterdam 39 (1936), 641–651.Google Scholar
4. Hewitt, E. (Review of a paper of B. Z. Vulih), The product in linear partially ordered spaces and its applications to the theory of operators. I and II, Math. Reviews 10 (1949), 46.Google Scholar
5. Johnson, D. G. and Kist, J. E., Prime ideals in vector lattices, Can. J. Math. 14 (1962), 517–528.Google Scholar
6. Luxemburg, W. A. J. and Zaanen, A. C., Notes on Banach function spaces, Proc. Acad. Sci. Amsterdam, Note II, A66 (1963), 148-153, Note VI, ibid., 655-668; Note VII, ibid., 669-681; Note VIII, A67 (1964), 104-119; Note IX, ibid., 360–376.Google Scholar
7. Nakano, H., Modern spectral theory (Maruzen, Tokyo, 1950).Google Scholar
8. Pinsker, A. G., Sur l'extension des espaces semi-or donnés, Dokl. Akad. Nauk SSSR 21 1938), 6–9.Google Scholar
9. Pinsker, A. G., Sur certaines propriétés des K-espaces étendus, Dokl. Akad. Nauk SSSR 22 (1939), 216–219.Google Scholar
10. Pinsker, A. G., On representations of a K-space as a ring of self-adjoint operators, Dokl. Akad. Nauk SSSR 106 (1956), 195–198. (Russian)Google Scholar
11. Rice, N. M., Multiplication in Riesz spaces (Thesis, California Institute of Technology, 1966).Google Scholar
12. Segal, I. E., Equivalences of measure spaces, Amer. J. Math. 73 (1951), 275–313.Google Scholar
13. Vulih, B. Z., The product in linear partially ordered spaces and its applications to the theory of operators, Mat. Sb. (N.S.) 22 (52) (1948); I, 27-78; II, 267–317. (Russian)Google Scholar
14. Zaanen, A. C., An introduction to the theory of intgreation (North-Holland, Amsterdam, 1961).Google Scholar
15. Zaanen, A. C., The Radon-Nikodym theorem, Proc. Acad. Sci. Amsterdam, A64 (1961); I, 157-170; II, 171–187.10.1016/S1385-7258(61)50017-0CrossRefGoogle Scholar