{
  "id": "kaplanskys-quasitrace-problem",
  "title": "Kaplansky's quasitrace problem",
  "description": "Kaplansky's quasitrace problem: is every normalized 2-quasitrace on a unital C*-algebra a tracial state? The answer is no: a separable unital C*-algebra has quasitraces, and every one of them fails additivity by at least 1/144 on one fixed pair of positive elements. The proof: a defect lemma from two relations between finitely many elements, bundles over products of projective spaces with a Chern class obstruction, generic expanding maps normalized by a determinant potential, and an algebra of weighted shifts. Proved in full from two named hypotheses: the existence theorem for quasitraces of Cuntz, Blackadar and Handelman, and standard facts on smooth manifolds. The course is an exposition of OpenAI's proof (2026), with a sharper constant.",
  "units": [
    {
      "id": "quasitraces-and-a-uniform-additivity-defect",
      "title": "Quasitraces and a uniform additivity defect",
      "reader": "quasitraces-and-a-uniform-additivity-defect.html",
      "source": "src/quasitraces-and-a-uniform-additivity-defect.md",
      "source_sha256": "19e539e29785f037c9e1e956d466a90edc630ce338e61c37b35c0bc212b75764",
      "status": "author_self_checked"
    },
    {
      "id": "expanding-maps-and-a-continuous-normalization",
      "title": "Expanding maps and a continuous normalization",
      "reader": "expanding-maps-and-a-continuous-normalization.html",
      "source": "src/expanding-maps-and-a-continuous-normalization.md",
      "source_sha256": "c33ee074c9de3f13f649d716a42471b11f3a10fb539abbc9d1136085edf3f3e8",
      "status": "author_self_checked"
    },
    {
      "id": "bundle-tests-with-a-chern-class-obstruction",
      "title": "Bundle tests with a Chern class obstruction",
      "reader": "bundle-tests-with-a-chern-class-obstruction.html",
      "source": "src/bundle-tests-with-a-chern-class-obstruction.md",
      "source_sha256": "fb4082ef31827ab4a02853a96a7fdfd2399fe9b8d001e8d2bcef36c3ee01b49c",
      "status": "author_self_checked"
    },
    {
      "id": "weighted-shifts-and-the-construction",
      "title": "Weighted shifts and the construction",
      "reader": "weighted-shifts-and-the-construction.html",
      "source": "src/weighted-shifts-and-the-construction.md",
      "source_sha256": "49c484d42663815f6222a4568a80a90c380876ab4a3a4743a4813001ae35e028",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Foundations of von Neumann algebras, lesson Order, local units and quotients of C*-algebras: C*-algebras, positivity, the continuous functional calculus, spectral permanence, generated C*-subalgebras",
    "course Characteristic classes, lessons Chern classes and the integral universal ring (definition (1.2), Proposition 1.1, Theorem 2.2) and The Gysin sequence and projective splitting (Theorems 3.2 and 4.1)",
    "linear algebra (core course Linear Algebra), Lebesgue measure on R^n, compactness",
    "the lessons prove every other fact they use, except the named hypotheses (CBH) of the first lesson and (SM) of the second lesson and the quoted consequences in Section 6 of the first lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}