{
  "id": "irrational-trace-and-baum-connes",
  "title": "A projection of irrational trace and the Baum–Connes conjecture",
  "description": "The reduced group C*-algebra of a countable group carries a canonical trace, and the Baum-Connes conjecture predicts its K-theory from proper actions. By a theorem of Lück every predicted class has a rational trace. OpenAI constructed in 2026 a finitely generated group with a projection in its reduced group C*-algebra whose trace is the irrational number sum p^(-k_i), so the reduced Baum-Connes conjecture without coefficients fails. The group is a semidirect product of a graphical small cancellation group, whose Cayley graph contains labelled expanders isometrically, with a module over F_p defined by linear codes on the copies of the expanders. A Fourier transform turns a Laplacian-type element into random subgraph Laplacians with a uniform spectral gap, and its kernel projection counts the copies whose random pattern is the prototype. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "traces-of-projections-and-the-baum-connes-conjecture",
      "title": "Traces of projections and the Baum–Connes conjecture",
      "reader": "traces-of-projections-and-the-baum-connes-conjecture.html",
      "source": "src/traces-of-projections-and-the-baum-connes-conjecture.md",
      "source_sha256": "399926a49c8b19a261702e6f0a807013f77bf7dc05c00e4392925126fa02cc7f",
      "status": "author_self_checked"
    },
    {
      "id": "graphical-presentations-and-coned-diagrams",
      "title": "Graphical presentations and coned diagrams",
      "reader": "graphical-presentations-and-coned-diagrams.html",
      "source": "src/graphical-presentations-and-coned-diagrams.md",
      "source_sha256": "bfe5e8266a5637bbd7c6bcff21d473813b2ae1bbd3b07738056bae1831b61881",
      "status": "author_self_checked"
    },
    {
      "id": "thin-triangles-and-simultaneous-overlaps",
      "title": "Thin triangles and simultaneous overlaps",
      "reader": "thin-triangles-and-simultaneous-overlaps.html",
      "source": "src/thin-triangles-and-simultaneous-overlaps.md",
      "source_sha256": "0c351e98236785fe54539679fc52ae338af76deae862609bac982321bb2515f5",
      "status": "author_self_checked"
    },
    {
      "id": "labelled-expanders-and-cycle-spaces",
      "title": "Labelled expanders and cycle spaces",
      "reader": "labelled-expanders-and-cycle-spaces.html",
      "source": "src/labelled-expanders-and-cycle-spaces.md",
      "source_sha256": "3414b81e4b4f5dffb61baf88a952d3fdd9f4391406c351d12aa742533d14883c",
      "status": "author_self_checked"
    },
    {
      "id": "codes-and-the-lamp-module",
      "title": "Codes and the lamp module",
      "reader": "codes-and-the-lamp-module.html",
      "source": "src/codes-and-the-lamp-module.md",
      "source_sha256": "9ce3be0bde59f690dff35f779cd922e596e551ab267a46f0beb084f44e422120",
      "status": "author_self_checked"
    },
    {
      "id": "fourier-decomposition-and-enabled-graphs",
      "title": "Fourier decomposition and enabled graphs",
      "reader": "fourier-decomposition-and-enabled-graphs.html",
      "source": "src/fourier-decomposition-and-enabled-graphs.md",
      "source_sha256": "0a660128b40fd44d2adeab81ee8272122c084319377fb684829413d84f5cd77d",
      "status": "author_self_checked"
    },
    {
      "id": "the-spectral-gap-and-an-irrational-trace",
      "title": "The spectral gap and an irrational trace",
      "reader": "the-spectral-gap-and-an-irrational-trace.html",
      "source": "src/the-spectral-gap-and-an-irrational-trace.md",
      "source_sha256": "571f7ea1406dfa852149bf297883904a4e4ad3e382c30c28ecd9fa0482d3e01f",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "programme course Foundations of von Neumann algebras: lesson C*-algebras, continuous functional calculus, automatic continuity and positive cones (Theorems 3.2 and 5.1, Corollaries 4.6 and 5.4)",
    "programme course The Kervaire conjecture (the-kervaire-conjecture): lesson The regular trace and spectral phase (Lemmas 1.1 and 1.3)",
    "core course Measure and Integration (D10): infinite products of probability spaces (Fremlin, Measure Theory, Volume 2, Section 254)",
    "programme course Van Kampen diagrams (van-kampen-diagrams): lessons Plane maps (Propositions 1.2, 2.1, 2.2 and 2.3) and Van Kampen's lemma (Definition 1.1, Theorems 3.1, 4.1 and 5.1), which prove the statements (VK) and (MAP)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}