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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 II1 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 𝕂 of characteristic zero, the quotient division ring of the first Weyl algebra over 𝕂 embeds unitally into R. We also give a direct proof of this consequence.
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BibTeX
@misc{thom2026commutators,
title = {{Commutators in Continuous Regular Rings}},
author = {Andreas Thom and Jiaqi Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23510},
eprint = {2609.23510},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23510},
}
RIS
TY - UNPB
TI - Commutators in Continuous Regular Rings
AU - Andreas Thom
AU - Jiaqi Wang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.23510
UR - https://arxiv.org/abs/2609.23510
AB - 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 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 𝕂 of characteristic zero, the quotient division ring of the first Weyl algebra over 𝕂 embeds unitally into R. We also give a direct proof of this consequence.
ER -