{
  "id": "bicentralizers-of-type-iii1-factors",
  "title": "Bicentralizers of type III₁ factors",
  "description": "Bicentralizers of type III₁ factors and Connes' bicentralizer problem. Connes–Størmer transitivity makes the asymptotic centralizer of every faithful normal state a II₁ factor; the bicentralizer, its description in the Ocneanu ultrapower, its state-preserving expectation and the Houdayer–Isono self-bicentralizing property; Connes' isomorphism and the bicentralizer flow, built in a single ultrapower; modular averaging, bounded recovery and spectral intertwining rigidity (OpenAI, 2026), which make the flow trivial; Haagerup's criterion for a trivial bicentralizer; Marrakchi's strongly invariant states; and the theorem that every type III₁ factor with separable predual has trivial bicentralizer.",
  "units": [
    {
      "id": "asymptotic-centralizers-of-type-iii1-factors",
      "title": "Asymptotic centralizers of type III₁ factors",
      "reader": "asymptotic-centralizers-of-type-iii1-factors.html",
      "source": "src/asymptotic-centralizers-of-type-iii1-factors.md",
      "source_sha256": "15943c821f4dc82430e6425237988150d0911c1a9f6e8f1c79dced243d848565",
      "status": "author_self_checked"
    },
    {
      "id": "the-bicentralizer",
      "title": "The bicentralizer",
      "reader": "the-bicentralizer.html",
      "source": "src/the-bicentralizer.md",
      "source_sha256": "209098badd685ff52957cc1d2c7d853c3b146208dcf4a2ec00d760aded7f7383",
      "status": "author_self_checked"
    },
    {
      "id": "the-bicentralizer-flow",
      "title": "The bicentralizer flow",
      "reader": "the-bicentralizer-flow.html",
      "source": "src/the-bicentralizer-flow.md",
      "source_sha256": "fb398b933746df5464b3121964457937025c97164259652585d063d2a422f39f",
      "status": "author_self_checked"
    },
    {
      "id": "modular-averaging-and-bounded-recovery",
      "title": "Modular averaging and bounded recovery",
      "reader": "modular-averaging-and-bounded-recovery.html",
      "source": "src/modular-averaging-and-bounded-recovery.md",
      "source_sha256": "4136c269f1a03844f320497da0ab3d703687918ffc1dbb84e932d4ce89e4c1de",
      "status": "author_self_checked"
    },
    {
      "id": "spectral-intertwining-rigidity",
      "title": "Spectral intertwining rigidity",
      "reader": "spectral-intertwining-rigidity.html",
      "source": "src/spectral-intertwining-rigidity.md",
      "source_sha256": "c1a145a8a2dcf643c798fc77ad0667053e202bd811246c8d88e20bd07fc86dc7",
      "status": "author_self_checked"
    },
    {
      "id": "haagerups-criterion-for-a-trivial-bicentralizer",
      "title": "Haagerup's criterion for a trivial bicentralizer",
      "reader": "haagerups-criterion-for-a-trivial-bicentralizer.html",
      "source": "src/haagerups-criterion-for-a-trivial-bicentralizer.md",
      "source_sha256": "818cfc7fec2254da92ff44d9ed16fb28ab7b3c6ff89155f3c3e156987b833e31",
      "status": "author_self_checked"
    },
    {
      "id": "strongly-invariant-states",
      "title": "Strongly invariant states and approximate eigenstates",
      "reader": "strongly-invariant-states.html",
      "source": "src/strongly-invariant-states.md",
      "source_sha256": "7466b2cc018409d04e7fa6173da71d09246a7868c568d628cb14a05c53dbd665",
      "status": "author_self_checked"
    },
    {
      "id": "the-bicentralizer-is-trivial",
      "title": "The bicentralizer of a type III₁ factor is trivial",
      "reader": "the-bicentralizer-is-trivial.html",
      "source": "src/the-bicentralizer-is-trivial.md",
      "source_sha256": "e0bf52e1c95709f0dea734c020cffd41280d94d92b76b64d2d8b7d206b6803ef",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "type-iii-factors lesson ultraproducts-and-the-asymptotic-centralizer (Section 1, Lemmas 1.1, 1.3, 1.4, Theorem 5.1, Theorem 6.6, results (B1)-(B8))",
    "OA-APPROX lesson multiplier-ultraproducts-and-normal-embeddings (Ocneanu ultrapower, Theorem 3.2, Section 4)",
    "OA-FLOW lesson OA-FLOW-HC (Connes-Stormer transitivity, HC1)",
    "OA-MOD lesson OA-MOD-ME (conditional expectations from modular invariance, ME-01, ME-08)",
    "foundations-of-von-neumann-algebras lesson polar-decomposition-of-functionals-and-weak-compactness-in-preduals (Corollary 2.8)",
    "OA-FLOW lesson OA-FLOW-CR (standard-form corners CR-1, CR-3, CR-8; cone vectors CR-4; inequality CR18 of CR-6)",
    "OA-MOD lesson OA-MOD-NC (joint left/right spectral measure NC-20; estimate NC.4)",
    "OA-MOD lessons OA-MOD-FS (FS.1, FS.2), OA-MOD-SF (SF-05), OA-MOD-SK (spectral calculus with domains)",
    "OA-FLOW lessons OA-FLOW-CS (CS3, CS-4), OA-FLOW-MG (MG6, MG9, MG10, MG-5), OA-FLOW-MT (MT25)",
    "OA-APPROX lesson projection-comparison-and-finite-traces (normal center-valued trace)",
    "tensor-products-of-operator-algebras lesson tensor-products-of-c-star-algebras-and-the-minimal-norm (Theorems 2.2, 4.4, 5.1)",
    "analytic-elements-strips-and-kms lesson holomorphy-in-banach-spaces-stone-s-theorem-and-resolvent-convergence (Sections 3-4, Theorem 4.3)",
    "foundations-of-von-neumann-algebras lessons projections-and-types-of-von-neumann-algebras (Lemma 6.2), compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies, and weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian (separation theorem, Mazur)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}