# A note on commutators in algebras of unbounded operators

## TL;DR

This paper proves that the identity operator can be expressed as the sum of two commutators in the algebra of affiliated operators for a type II$_1$ von Neumann algebra, resolving a longstanding open question.

## Contribution

It demonstrates that the identity operator is the sum of two commutators in a specific algebra, answering a previously unresolved question.

## Key findings

- The identity operator can be written as a sum of two commutators.
- The result applies to the algebra of all operators affiliated with a type II$_1$ von Neumann algebra.
- The question about expressing the identity as a sum of commutators is settled negatively.

## Abstract

We show that the identity is the sum of two commutators in the algebra of all operators affiliated with a von Neumann algebra of type II$_1$, settling a question, in the negative, that had puzzled a number of us.