{
  "id": "bounded-hochschild-cohomology-of-von-neumann-algebras",
  "title": "Bounded Hochschild cohomology of von Neumann algebras",
  "description": "Every bounded Hochschild cocycle of degree at least two on a von Neumann algebra is the coboundary of a bounded cochain, after OpenAI's 2026 proof; with the Kadison-Sakai theorem this settles the Kadison-Ringrose cohomology problem. A rigidity theorem for first-letter identities on free Haar families; a random walk on the unitary group whose long products are asymptotically free; a Liouville theorem for its harmonic maps; bounded primitives by averaging, removal of separability and an explicit central homotopy.",
  "units": [
    {
      "id": "the-cohomology-problem",
      "title": "The cohomology problem",
      "reader": "the-cohomology-problem.html",
      "source": "src/the-cohomology-problem.md",
      "source_sha256": "e662c25f1cda5a521807e5fa1f89e69dec1fabd8b492ac8d7f112e9128fa6781",
      "status": "author_self_checked"
    },
    {
      "id": "rigidity-of-first-letter-identities",
      "title": "Rigidity of first-letter identities",
      "reader": "rigidity-of-first-letter-identities.html",
      "source": "src/rigidity-of-first-letter-identities.md",
      "source_sha256": "606da536b47f92e369a954357203f45a0be2b65175b39508c30d4f586f555095",
      "status": "author_self_checked"
    },
    {
      "id": "a-random-walk-with-asymptotically-free-products",
      "title": "A random walk with asymptotically free products",
      "reader": "a-random-walk-with-asymptotically-free-products.html",
      "source": "src/a-random-walk-with-asymptotically-free-products.md",
      "source_sha256": "7ba71d9e3fd0c44c6337e977556bb35a10a5607beb514d0d135c2904786842eb",
      "status": "author_self_checked"
    },
    {
      "id": "the-liouville-theorem",
      "title": "The Liouville theorem",
      "reader": "the-liouville-theorem.html",
      "source": "src/the-liouville-theorem.md",
      "source_sha256": "4a555439e7bcb1400d0617a3ad4edea9c2449825b978f0cdbfe7eb26dcce3fb8",
      "status": "author_self_checked"
    },
    {
      "id": "bounded-primitives",
      "title": "Bounded primitives",
      "reader": "bounded-primitives.html",
      "source": "src/bounded-primitives.md",
      "source_sha256": "057880b134cde49500b20f1e6120a5dc518c8667da63d823118d274b90eb1ff2",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "von Neumann algebras: the double commutant theorem, operator topologies and preduals, Kaplansky's density theorem, the spectral theorem, projections and types, W*-algebras (course Foundations of von Neumann algebras)",
    "Haar unitaries, freeness and Popa's free independence theorem in tracial ultraproducts (course Generators of II1 factors, lessons 1 and 2)",
    "the central decomposition and the types of the fibres (course Measurable fields and direct integrals)",
    "trace-preserving conditional expectations (course Traces and noncommutative integration, Theorem 9.1)",
    "tracial ultraproducts (course Type III factors, Ultraproducts and the asymptotic centralizer, Theorem 3.2)",
    "products of probability spaces (core course D10 Measure and Integration, Fremlin, Measure Theory, Theorem 254F)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}