关于 Murray-von Neumann 代数的推导的注释。
A note on derivations of Murray-von Neumann algebras.
机构信息
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104.
出版信息
Proc Natl Acad Sci U S A. 2014 Feb 11;111(6):2087-93. doi: 10.1073/pnas.1321358111. Epub 2014 Jan 27.
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.
Murray-von Neumann 代数是与有限 von Neumann 代数相关联的算子代数。在本文中，我们首先从物理和数学的角度简要介绍算子代数的导数理论。然后，我们描述了我们最近在 Murray-von Neumann 代数导数方面的工作。我们证明了 Murray-von Neumann 代数的“扩展导数”，即那些将相关有限 von Neumann 代数映射到自身的导数，是内导的。特别地，我们证明了将与 II1 型因子相关联的 Murray-von Neumann 代数映射到该因子的唯一导数是 0。这些结果是 Singer 对 Kaplansky 问题的开创性回答的扩展，适用于 von Neumann 代数：该代数可以是非交换的，甚至可以包含无界元素。
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