{
  "id": "generators-of-ii1-factors",
  "title": "Generators of II₁ factors",
  "description": "Every II₁ factor with separable predual is generated by one operator, equivalently by two self-adjoint operators (OpenAI, 2026). Haar unitaries and freeness of a unitary from a C*-algebra in tracial von Neumann algebras; tracial ultraproducts; Popa's free independence theorem for irreducible inclusions, proved by patching partial isometries with local quantization in corners; Haagerup's inequality for words of length one and a perturbation of a unitary whose correlation with a fixed vector stays uniform along long words; relative generation W*(P,u) = M for a dense G_δ set of unitaries u; irreducible hyperfinite subfactors, two self-adjoint generators of the hyperfinite factor and a 2×2 matrix encoding. Finally the generator problem in full: every von Neumann algebra with separable predual is singly generated, through properly infinite algebras (Wogen), finite factors and direct integrals with a Borel choice of fibre generators (Willig); and the vanishing of the generator invariant of Dykema, Sinclair, Smith and White.",
  "units": [
    {
      "id": "haar-unitaries-freeness-and-ultraproducts",
      "title": "Haar unitaries, freeness and tracial ultraproducts",
      "reader": "haar-unitaries-freeness-and-ultraproducts.html",
      "source": "src/haar-unitaries-freeness-and-ultraproducts.md",
      "source_sha256": "1b656f7f8761b2f2af8e75539a2b9a2fea55aa7c03d101c738e399a65bd5b1ba",
      "status": "author_self_checked"
    },
    {
      "id": "free-independence-in-ultraproducts",
      "title": "Free independence in ultraproducts",
      "reader": "free-independence-in-ultraproducts.html",
      "source": "src/free-independence-in-ultraproducts.md",
      "source_sha256": "bc004a6c4a6fb6d1aa8667e349eb097f2f5cf713b08c704fc4c1169147ec7b28",
      "status": "author_self_checked"
    },
    {
      "id": "relative-generation-by-one-unitary",
      "title": "Relative generation by one unitary",
      "reader": "relative-generation-by-one-unitary.html",
      "source": "src/relative-generation-by-one-unitary.md",
      "source_sha256": "f6c9e4fd229fad4c8a2c5f95e6bebc449f103fe009bf8457e3b304cfe5f63e84",
      "status": "author_self_checked"
    },
    {
      "id": "single-generation-of-ii1-factors",
      "title": "Single generation of II₁ factors",
      "reader": "single-generation-of-ii1-factors.html",
      "source": "src/single-generation-of-ii1-factors.md",
      "source_sha256": "95a906bc1cb31206bd11ad450cc6cd6f3dffe7c719ba5d4a2a896f94b6a79d6e",
      "status": "author_self_checked"
    },
    {
      "id": "the-generator-problem",
      "title": "The generator problem",
      "reader": "the-generator-problem.html",
      "source": "src/the-generator-problem.md",
      "source_sha256": "17e70d6f3f2e4f2b654a366aaff5c90a02d9481f03931b0ffc6eb2e5453741d1",
      "status": "author_self_checked"
    },
    {
      "id": "the-generator-invariant",
      "title": "The generator invariant",
      "reader": "the-generator-invariant.html",
      "source": "src/the-generator-invariant.md",
      "source_sha256": "54017b4b585055903aed439e24837ed62bc449ed28c80a88855548428c3ed868",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "Course foundations-of-von-neumann-algebras (double commutant and Kaplansky density theorems, polar decomposition, comparison of projections, spectral projections, preduals, Baire and Banach–Alaoglu)",
    "Course type-iii-factors, lesson on ultraproducts (Theorems 3.2, 3.3, Lemma 1.4: tracial ultraproducts)",
    "OA-SUBFACTORS lesson finite-phase-local-quantization (Lemma 55.1, Theorem 55.7: local quantization)",
    "OA-APPROX lessons abelian-pinching-and-relative-commutants (Lemma 3.1), masas-in-expected-semifinite-subfactors (Theorem 2.1) and infinite-tensor-products (Corollary 5.3)",
    "Course traces-and-noncommutative-integration (Theorem 9.1: trace-preserving conditional expectations)",
    "foundations-of-von-neumann-algebras lessons projections-and-types-of-von-neumann-algebras (Theorem 7.2, Proposition 8.4, Proposition 15.3(3)), abelian-operator-algebras (Theorem 8.2), kaplansky-s-density-theorem-and-its-consequences (Theorem 7.1), compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies (Theorems 6.2(d), 9.4(a)); measurable-fields-and-direct-integrals lesson the-central-decomposition-and-the-types-of-the-fibres (Facts 1.1, 1.2, 1.7(a), Theorem 4.1, Lemma 5.1, Theorem 5.3); OA-FOUND-REMAINDER reader base-changes-and-disintegration (Lemma 4.2)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}