Normal Limits of Nilpotent Operators in von Neumann Algebras

摘要

We will investigate the intersection of the normal operators with the norm closure of the nilpotent operators in von Neumann algebras. A complete description of said intersection will be given for type I and type III von Neumann algebras with separable predual whereas difficulties and restrictions in the type II setting will be discussed. In addition, the closure of the span of the nilpotent operators and the distance from a projection to the nilpotent operators will be examined in the context of von Neumann algebras.