Upper Triangular Forms and Spectral Orderings in a II_1-factor
  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. 
Joseph Noles
math.OA