Abbreviated Journal Title
Proc. Natl. Acad. Sci. U. S. A.
quantum mechanics; finite von Neumann algebra; type II1 factor; Murray-von Neumann algebra; derivation; OPERATOR ALGEBRAS; RINGS; Multidisciplinary Sciences
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.
Proceedings of the National Academy of Sciences of the United States of America
Kadison, Richard V. and Liu, Zhe, "A note on derivations of Murray-von Neumann algebras" (2014). Faculty Bibliography 2010s. 5540.