Authors

R. V. Kadison;Z. Liu

Comments

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Abbreviated Journal Title

Proc. Natl. Acad. Sci. U. S. A.

Keywords

quantum mechanics; finite von Neumann algebra; type II1 factor; Murray-von Neumann algebra; derivation; OPERATOR ALGEBRAS; RINGS; Multidisciplinary Sciences

Abstract

A Murray-von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray-von Neumann algebras. We show that the "extended derivations" of a Murray-von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray-von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer's seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements.

Journal Title

Proceedings of the National Academy of Sciences of the United States of America

Volume

111

Issue/Number

6

Publication Date

1-1-2014

Document Type

Article

Language

English

First Page

2087

Last Page

2093

WOS Identifier

WOS:000330999600021

ISSN

0027-8424

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