Upper Triangular Forms and Spectral Orderings in a II_1-factor

TL;DR

This paper extends the methods for decomposing operators in a II_1-factor into normal and quasinilpotent parts by using broader classes of functions to order the Brown measure support.

Contribution

It generalizes previous results by allowing a wider class of measurable functions for ordering the Brown measure support, enabling new decompositions.

Findings

Broader class of functions can order the Brown measure support.

Operators can be decomposed into normal plus quasinilpotent parts.

Generalization of previous decomposition techniques.

Abstract

Dykema, Sukochev and Zanin used a Peano curve covering the support of the Brown measure of an operator T in a diffuse, finite von Neumann algebra to give an ordering to the support of the Brown measure, and create a decomposition T = N + Q, where N is normal and Q is s.o.t.-quasinilpotent. In this paper we prove a broader class of measurable functions can be used to order the support of the Brown measure, giving normal plus s.o.t.-quasinilpotent decompositions.