发表机构
TU Dresden; Institute of Geometry, TU Dresden; State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences（德累斯顿工业大学; 德累斯顿工业大学几何研究所; 中国科学院数学与系统科学研究院数学科学重点实验室; 中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明 $\IIone$ 型冯·诺依曼代数的从属算子代数中单位元是单个换位子，回答 Kadison-Liu 问题并加强双换位子定理，同时证明第一外尔代数商除环的单位嵌入性质。
AI 中文摘要
设 $\M$ 为 $\IIone$ 型冯·诺依曼代数，$\U(\M)$ 为其从属算子代数。我们证明单位元在 $\U(\M)$ 中是单个换位子。这回答了 Kadison 和 Liu 提出的一个问题，并加强了 Kadison、Liu 和 Thom 的双换位子定理。这两个算子可被选取为从属于一个单位超有限子因子，且无需对 $\M$ 作可分性假设。我们还证明第一外尔代数的商除环可单位地嵌入 $\U(\M)$，且每个非零元素具有全支撑。
英文摘要
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 López-Álvarez, 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.
Comments11 pages. Strengthened main theorem: every element of a continuous regular ring with no nonzero abelian idempotents is a single additive commutator
