A note on derivations of Murray-von Neumann algebras
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
"A note on derivations of Murray-von Neumann algebras" (2014). Faculty Bibliography 2010s. 5540.