{
  "abstract": "We solve two long-standing open problems in von Neumann algebras. First, we classify all flows with full Connes spectrum on the hyperfinite $\\mathrm{II}_1$ 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 $\\mathrm{II}_1$ factor. To prove this result, we draw on type $\\mathrm{III}$ 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 $\\mathbb{R}$. We then go back to the type $\\mathrm{III}$ world and use this new resonance phenomenon to solve Connes' bicentralizer conjecture for all type $\\mathrm{III}_1$ factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11462",
  "author_total": 2,
  "authors": [
    {
      "name": "Cyril Houdayer"
    },
    {
      "name": "Amine Marrakchi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6n0ffg487ykx0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11462",
  "fulltext_url": null,
  "id": "6n0ffg487ykx0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11462",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7212406397",
  "pdf_url": "https://arxiv.org/pdf/2609.11462",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6n0ffg487ykx0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.11462",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11462"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The classification of flows on $\\mathrm{II}_1$ factors and Connes' bicentralizer problem",
  "title_text": "The classification of flows on II_1 factors and Connes' bicentralizer problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/6n0ffg487ykx0",
  "venue": "arXiv",
  "versions": []
}