A note on derivations of Murray-von Neumann algebras

作者：Kadison Richard V^*; Liu Zhe

来源：Proceedings of the National Academy of Sciences of the United States of America, 2014, 111(6): 2087-2093.

DOI：10.1073/pnas.1321358111

• quantum mechanics
• finite von Neumann algebra
• type II1 factor
• Murray-von Neumann algebra
• derivation

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 %26quot;extended derivations%26quot; 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%26apos;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.

• 出版日期2014-2-11