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arXiv · 2609.23510

The Identity as a Single Commutator of Affiliated Operators

Jiaqi Wang
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Abstract

Let $\M$ be a von Neumann algebra of type $\IIone$, and let $\U(\M)$ be its algebra of affiliated operators. We prove that the identity is a single commutator in $\U(\M)$. This answers a question stated by Kadison and Liu and strengthens the two-commutator theorem of Kadison, Liu and Thom. The two operators may be chosen affiliated with a unital hyperfinite subfactor, without any separability assumption on $\M$. We also show that the quotient division ring of the first Weyl algebra embeds unitally into $\U(\M)$, with every nonzero element having full support.

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Jiaqi Wang. 2026-09-20. The Identity as a Single Commutator of Affiliated Operators. https://arxiv.org/abs/2609.23510

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