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The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra

Published online by Cambridge University Press:  20 November 2018

Ole A. Nielsen
Ole A. Nielsen*
Affiliation:
Queen's University, Kingston, Ontario
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The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 4 , August 1971 , pp. 598 - 607
Copyright
Copyright © Canadian Mathematical Society 1971

References

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