An upper triangular decomposition theorem for some unbounded operators affiliated to II_1-factors

TL;DR

This paper extends invariant subspace results to unbounded operators affiliated with II_1-factors, enabling a decomposition into normal and negligible parts, and shows traces are spectral under certain conditions.

Contribution

It generalizes Haagerup and Schultz's invariant subspace theorem to a broader class of unbounded operators in tracial von Neumann algebras.

Findings

Operators can be decomposed into normal and negligible parts.

Traces on certain bimodules depend only on the Brown measure.

The results apply to a large class of unbounded operators in II_1-factors.

Abstract

Results of Haagerup and Schultz (2009) about existence of invariant subspaces that decompose the Brown measure are extended to a large class of unbounded operators affiliated to a tracial von Neumann algebra. These subspaces are used to decompose an arbitrary operator in this class into the sum of a normal operator and a spectrally negligible operator. This latter result is used to prove that, on a bimodule over a tracial von Neumann algebra that is closed with respect to logarithmic submajorization, every trace is spectral, in the sense that the trace value on an operator depends only on the Brown measure of the operator.