{
  "id": "a-counterexample-to-naimarks-problem",
  "title": "A counterexample to Naimark's problem",
  "description": "A unital, infinite-dimensional, simple C*-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent, constructed in ZFC: a negative answer to Naimark's problem without additional set-theoretic axioms. Excision of pure states, expectations onto algebras of peaking projections, reduced amalgamated free products and reduced HNN adjunctions give a bridge that joins two chosen classes of pure states while every pure state keeps a unique extension; transfinite iteration with closed subiterations and countable determination treats every pure state. The course is an exposition of OpenAI's construction (2026); it uses the homogeneity theorem of Kishimoto, Ozawa and Sakai without proof.",
  "units": [
    {
      "id": "pure-states-excision-and-peaking-projections",
      "title": "Pure states, excision and peaking projections",
      "reader": "pure-states-excision-and-peaking-projections.html",
      "source": "src/pure-states-excision-and-peaking-projections.md",
      "source_sha256": "c05c19d7165615328f73ab4ad042098fce524114a9a8d1874d9c1e31693e2cc5",
      "status": "author_self_checked"
    },
    {
      "id": "expectations-from-peaking-projections",
      "title": "Expectations from peaking projections",
      "reader": "expectations-from-peaking-projections.html",
      "source": "src/expectations-from-peaking-projections.md",
      "source_sha256": "eba6d69eea9e38ef4516a6ce9de54d3bffb8318c912844e18f6f88563808e203",
      "status": "author_self_checked"
    },
    {
      "id": "reduced-amalgamated-free-products-of-tracial-algebras",
      "title": "Reduced amalgamated free products of tracial algebras",
      "reader": "reduced-amalgamated-free-products-of-tracial-algebras.html",
      "source": "src/reduced-amalgamated-free-products-of-tracial-algebras.md",
      "source_sha256": "bd7575a044f7fc117760266f8158ebd08bbd02ec4ae4d7e2f21b30007aadbf7a",
      "status": "author_self_checked"
    },
    {
      "id": "reduced-hnn-adjunctions",
      "title": "Reduced HNN adjunctions",
      "reader": "reduced-hnn-adjunctions.html",
      "source": "src/reduced-hnn-adjunctions.md",
      "source_sha256": "2dac0551d1cbdb1ed2ebf3dd897cecc43a823d8d67eaab4d4ab22952a33a47a6",
      "status": "author_self_checked"
    },
    {
      "id": "the-bridge",
      "title": "The bridge",
      "reader": "the-bridge.html",
      "source": "src/the-bridge.md",
      "source_sha256": "46bb1f6b8c22d16141d287a124e3b33686115fd54a5729ba52b43377190ad162",
      "status": "author_self_checked"
    },
    {
      "id": "closed-subiterations-and-countable-determination",
      "title": "Closed subiterations and countable determination",
      "reader": "closed-subiterations-and-countable-determination.html",
      "source": "src/closed-subiterations-and-countable-determination.md",
      "source_sha256": "b9c5d57614916f45accbe2ba56eb15e2f82789e6991edd40d8eae68dc4e6f563",
      "status": "author_self_checked"
    },
    {
      "id": "the-construction-in-zfc",
      "title": "The construction in ZFC",
      "reader": "the-construction-in-zfc.html",
      "source": "src/the-construction-in-zfc.md",
      "source_sha256": "9bbe75de18921b35dd734e836e52dd6a6ddcf0b9adf86891f994744d0f40ffa4",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course foundations-of-von-neumann-algebras, lessons Representations and positive functionals: the GNS construction and the Gelfand-Naimark theorem (Proposition 3.2, Lemma 3.3, Theorems 5.4 and 5.5, Proposition 7.1a, Corollary 2.3, Theorem 8.3, Proposition 8.4), The universal enveloping von Neumann algebra of a C*-algebra and W*-algebras (Theorem 3.3, Lemma 2.1(2), Lemma 7.2, Proposition 7.3, Theorem 8.5, Lemma 11.1), Kaplansky's density theorem and its consequences (Theorem 1.3, Corollary 1.4, Theorem 11.1), C*-algebras: continuous functional calculus, automatic continuity, positive cones (Theorems 1.3 and 5.3, Corollary 4.6, Proposition 8.5), Completely positive maps (Proposition 3.2(1), Theorem 5.1(1)), Hahn-Banach, Baire and the basic theorems on Banach spaces (Theorems 8.4 and 8.5), Numerical ranges, positive matrices and co-Souslin sets (Theorem 0.1)",
    "course hilbert-c-star-modules-and-morita-equivalence, lessons Hilbert C*-modules (Theorem 2.3), Adjointable operators (Theorem 2.2), Tensor products and C*-correspondences (Theorems 1.2, 2.1, 3.1 and Proposition 3.2)",
    "the named hypothesis (KOS), the homogeneity theorem of Kishimoto, Ozawa and Sakai, stated in the seventh lesson and used without proof"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}