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The classification of flows on II1 factors and Connes' bicentralizer problem

  • Cyril Houdayer
  • Amine Marrakchi
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Abstract

We solve two long-standing open problems in von Neumann algebras. First, we classify all flows with full Connes spectrum on the hyperfinite II1 factor up to cocycle conjugacy. Every such flow is cocycle conjugate to an irrational rotation flow on a noncommutative torus. In particular, there is a unique outer flow with full Connes spectrum up to cocycle conjugacy. This settles Takesaki's classification problem for flows on the hyperfinite II1 factor. To prove this result, we draw on type III theory. Notably, we develop bicentralizer machinery for trace-preserving actions of locally compact groups. In the amenable case, we relate the bicentralizer conjecture to the Rokhlin property. For abelian groups, we prove an analog of the Connes--Størmer transitivity theorem and we generalize the Connes--Takesaki relative commutant theorem. A new resonance phenomenon is revealed which allows us to solve the bicentralizer conjecture for actions of ℝ. We then go back to the type III world and use this new resonance phenomenon to solve Connes' bicentralizer conjecture for all type III1 factors.

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Cite

BibTeXRIS
BibTeX
@misc{houdayer2026classification,
  title = {{The classification of flows on $\mathrmII_1$ factors and Connes' bicentralizer problem}},
  author = {Cyril Houdayer and Amine Marrakchi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.11462},
  eprint = {2609.11462},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11462},
}
RIS
TY  - UNPB
TI  - The classification of flows on II_1 factors and Connes' bicentralizer problem
AU  - Cyril Houdayer
AU  - Amine Marrakchi
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.11462
UR  - https://arxiv.org/abs/2609.11462
AB  - We solve two long-standing open problems in von Neumann algebras. First, we classify all flows with full Connes spectrum on the hyperfinite II_1 factor up to cocycle conjugacy. Every such flow is cocycle conjugate to an irrational rotation flow on a noncommutative torus. In particular, there is a unique outer flow with full Connes spectrum up to cocycle conjugacy. This settles Takesaki's classification problem for flows on the hyperfinite II_1 factor. To prove this result, we draw on type III theory. Notably, we develop bicentralizer machinery for trace-preserving actions of locally compact groups. In the amenable case, we relate the bicentralizer conjecture to the Rokhlin property. For abelian groups, we prove an analog of the Connes--Størmer transitivity theorem and we generalize the Connes--Takesaki relative commutant theorem. A new resonance phenomenon is revealed which allows us to solve the bicentralizer conjecture for actions of ℝ. We then go back to the type III world and use this new resonance phenomenon to solve Connes' bicentralizer conjecture for all type III_1 factors.
ER  -