Commutators in Continuous Regular Rings

Abstract

Let $R$ be a continuous von Neumann regular ring with no nonzero abelian idempotents. We prove that every element of $R$ is a single additive commutator. In particular, every operator affiliated with a type $\mathrm{II}_1$ von Neumann algebra is a commutator of operators affiliated with the same algebra. Our commutator theorem, together with a theorem of Jaikin-Zapirain and L\'opez-\'Alvarez, implies that, if $R$ is a nonzero unital algebra over a field $\mathbb{K}$ of characteristic zero, the quotient division ring of the first Weyl algebra over $\mathbb{K}$ embeds unitally into $R$. We also give a direct proof of this consequence.