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On the Radon-Nikodym Derivative with a Chain Rule in a Von Neumann Algebra

Published online by Cambridge University Press:  20 November 2018

George A. Elliott
George A. Elliott*
Affiliation:
Mathematics Institute University of Copenhagen, Denmark
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The purpose of this paper is to show that by a reorganization of the proofs of the main results concerning Radon-Nikodym derivatives in a von Neumann algebra of Pedersen and Takesaki in [12] and of Connes in paragraphs 1.1 and 1.2 of [5], considerable technical simplification can be achieved. Roughly speaking, the analytic vector techniques developed by these authors for the study of weights on a von Neumann algebra can be replaced, to a large extent, by the tensor product methods introduced by Connes, which are essentially algebraic in nature. In the exposition which follows, analytic vectors are not used at all (see, however, 4.4).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 5 , 01 December 1975 , pp. 661 - 669
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Araki, H., Relative Hamiltonian for faithful states of von Neumann algebras, Publ. Res. Inst. Math. Sci. 9 (1973), 165-209.Google Scholar
2. Araki, H., One-parameter family of Radon-Nikodym theorems for a von Neumann algebra, Publ. Res. Inst. Math. Sci., 10 (1974), 1-10.Google Scholar
3. Combes, F., Poids sur une C*-algèbre, J. Math. Pures Appl. 47 (1968), 57-100.Google Scholar
4. Combes, F., Poids associÉ à une algèbre hilbertienne à gauche, Compositio Math. 23 (1971), 49-77.Google Scholar
5. Connes, A., Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. 6 (1973), 133-252.Google Scholar
6. Connes, A., Applications of Tomita-Takesaki theory to classification of factors of type III, Proceedings of the International School of Physics "Enrico Fermi", Varenna, July 23 to August 4, 1973, to appear in Nuovo Cimento,Google Scholar
7. Connes, A., Sur le thÉorème de Radon-Nikodym pour les poids normaux fidèles semi-finis, Bull. Sci. Math. 97 (1973), 253-258.Google Scholar
8. Connes, A., CaractÉrisation des espaces vectoriels ordonnÉs sous-jacents aux algèbres de von Neumann, Ann. Inst. Fourier (Grenoble) 24 (1974), 121-155.Google Scholar
9. Dixmier, J., Algèbres quasi-unitaires, Comment, Math. Helv. 26 (1952), 275-322.Google Scholar
10. Haagerup, U., Tomita's theory for von Neumann algebras with a cyclic separating vector, preprint, University of Copenhagen.Google Scholar
11. Haagerup, U., Normal weights on W*-algebras, J. Functional Analysis 19 (1975), 302-317.Google Scholar
12. Pedersen, G. K. and Takesaki, M., The Radon-Nikodym theorem for von Neumann algebras, Acta Math. 130 (1973), 53-87.Google Scholar
13. Takesaki, M., Tomita's theory of modular Hilbert algebras, Lecture Notes in Mathematics 128, Springer, New York, 1970.Google Scholar
14. Van Daele, A., A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras, J. Functional Analysis 15 (1974), 378-392.Google Scholar