Regularity for Subfactors — Draconitic Research

Abstract

We study three naturally associated notions of regularity for a unital inclusion of von Neumann algebras ℬ ⊆ ℳ: regularity, groupoid regularity, and unitary regularity. We first observe that regularity and groupoid regularity are unconditionally equivalent by showing that the standard normaliser and the groupoid normaliser of ℬ in ℳ generate the same linear span. The central question addressed in this work is whether regularity implies unitary regularity. For arbitrary von Neumann subalgebras, we demonstrate that this implication fails dramatically; we construct explicit examples to show that every factor (of type I_∞, II₁, II_∞, and III) contains a regular von Neumann subalgebra that is not unitarily regular. In stark contrast, we prove that the behaviour changes completely in the setting of subfactors. For subfactor inclusions across all these types, regularity always implies unitary regularity, and thus all three notions of regularity coincide.

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BibTeX RIS

BibTeX

@misc{bakshi2026regularity,
  title = {{Regularity for Subfactors}},
  author = {Keshab Chandra Bakshi and Indrajit Ghosh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.10118},
  eprint = {2609.10118},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10118},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Regularity for Subfactors
AU  - Keshab Chandra Bakshi
AU  - Indrajit Ghosh
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.10118
UR  - https://arxiv.org/abs/2609.10118
AB  - We study three naturally associated notions of regularity for a unital inclusion of von Neumann algebras ℬ ⊆ ℳ: regularity, groupoid regularity, and unitary regularity. We first observe that regularity and groupoid regularity are unconditionally equivalent by showing that the standard normaliser and the groupoid normaliser of ℬ in ℳ generate the same linear span. The central question addressed in this work is whether regularity implies unitary regularity. For arbitrary von Neumann subalgebras, we demonstrate that this implication fails dramatically; we construct explicit examples to show that every factor (of type I_∞, II_1, II_∞, and III) contains a regular von Neumann subalgebra that is not unitarily regular. In stark contrast, we prove that the behaviour changes completely in the setting of subfactors. For subfactor inclusions across all these types, regularity always implies unitary regularity, and thus all three notions of regularity coincide.
ER  -

Abstract

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