Normal Limits of Nilpotents in C*-Algebras

TL;DR

This paper characterizes the limits of nilpotent and quasinilpotent operators in various C*-algebras, revealing structural restrictions and specific behaviors in different algebra types.

Contribution

It provides a complete characterization of normal operators as limits of nilpotent operators in certain von Neumann and C*-algebras, including type I, type III, and purely infinite cases.

Findings

Normal operators are limits of nilpotent operators in specific algebra types.

Restrictions prevent some normal operators from being approximated by quasinilpotent operators.

The paper examines the span of nilpotent operators and distances from projections in various C*-algebras.

Abstract

We will investigate the intersection of the normal operators with the closure of the nilpotent and quasinilpotent operators in various C*-algebras. A complete characterization will be given for type I and type III von Neumann algebras with separable predual and for unital, simple, purely infinite C*-algebras. Some restrictions that prevent normal operators from being limits of quasinilpotent operators will be given. The case of AF C*-algebras will be of particular interest. In addition, the closure of the span of the nilpotent operators and the distance from a projection to the nilpotent operators of various C*-algebras will be examined.