Decomposibility and norm convergence properties in finite von Neumann algebras

TL;DR

This paper explores the properties of decomposability and norm convergence in finite von Neumann algebras, focusing on Schur-type forms, spectral properties, and introducing Borel decomposability, with applications to specific operators.

Contribution

It introduces Borel decomposability for finite von Neumann algebra elements and analyzes conditions for quasinilpotent parts in Schur-type decompositions.

Findings

The circular operator is Borel decomposable.

Existence of a thin-spectrum s.o.t.-quasinilpotent operator in the hyperfinite II_1-factor.

Connections between decomposability and norm convergence properties.

Abstract

We study Schur-type upper triangular forms for elements, T, of von Neumann algebras equipped with faithful, normal, tracial states. These were introduced in a paper of Dykema, Sukochev and Zanin; they are based on Haagerup-Schultz projections. We investigate when the s.o.t.-quasinilpotent part of this decomposition of T is actually quasinilpotent. We prove implications involving decomposability and strong decomposability of T. We show this is related to norm convergence properties of the sequence |T^n|^{1/n} which, by a result of Haagerup and Schultz, is known to converge in strong operator topology. We introduce a Borel decomposability, which is a property appropriate for elements of finite von Neumann algebras, and show that the circular operator is Borel decomposable. We also prove the existence of a thin-spectrum s.o.t.-quasinilpotent operator in the hyperfinite II_1-factor.