On single commutators in II_1-factors

TL;DR

This paper explores whether every trace-zero element in a II_1-factor can be expressed as a single commutator, establishing this for nilpotent and certain normal elements with discrete spectral measures.

Contribution

It proves that all nilpotent elements and specific normal trace-zero elements with discrete spectral measures are single commutators in II_1-factors.

Findings

Nilpotent elements are single commutators.

Normal elements with discrete spectral measures are single commutators.

Some additional classes of elements are discussed.

Abstract

We investigate the question of whether all elements of trace zero in a II_1-factor are single commutators. We show that all nilpotent elements are single commutators, as are all normal elements of trace zero whose spectral distributions are discrete measures. Some other classes of examples are considered.