{
  "id": "kirchbergs-embedding-problem",
  "title": "Kirchberg's embedding problem",
  "description": "Kirchberg's embedding problem: does every separable C*-algebra embed into the norm ultrapower of the Cuntz algebra O₂? The answer is no: the full C*-algebra of an explicit group has no unital embedding into the norm ultrapower of any nuclear C*-algebra. The proof: a spectral test for representations of a group with property (T), a finiteness statement in nuclear algebras, and an endomorphism of finite index that forces chains of representations. Proved in full from two named hypotheses: property (T) of the group ℤ³ ⋊ SL₃(ℤ), and the nuclearity of O₂. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "norm-ultrapowers-and-a-group-with-property-t",
      "title": "Norm ultrapowers and a group with property (T)",
      "reader": "norm-ultrapowers-and-a-group-with-property-t.html",
      "source": "src/norm-ultrapowers-and-a-group-with-property-t.md",
      "source_sha256": "93ad22e107dd06062cc4c519121fc0557be323fc04ceee498e13e6bb071cff96",
      "status": "author_self_checked"
    },
    {
      "id": "a-spectral-test-and-nuclear-finiteness",
      "title": "A spectral test and nuclear finiteness",
      "reader": "a-spectral-test-and-nuclear-finiteness.html",
      "source": "src/a-spectral-test-and-nuclear-finiteness.md",
      "source_sha256": "c804d9863e777ade04dc657f507c5a9468a6b318c22d1345c3ffb8e97080ed50",
      "status": "author_self_checked"
    },
    {
      "id": "predecessors-and-the-contradiction",
      "title": "Predecessors and the contradiction",
      "reader": "predecessors-and-the-contradiction.html",
      "source": "src/predecessors-and-the-contradiction.md",
      "source_sha256": "4cd32a11844d2a5051794b1ace6ab59f3cd4eb5600b450ded95e90c8c07cd4d8",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Foundations of von Neumann algebras: C*-algebras, the continuous functional calculus, positivity, quotients, injective *-homomorphisms, spectral permanence (lesson Order, local units and quotients of C*-algebras); faithful representations (lesson Building representations from positive functionals); Hilbert tensor products and compact operators; trace-class and Hilbert-Schmidt operators",
    "course Positive maps and finite-dimensional approximation (OA-APPROX), lesson Tensor positivity and nuclearity: the minimal and the maximal tensor product and nuclearity as their equality",
    "unitary representations of groups of finite dimension (defined in the lessons)",
    "the lessons prove every other fact they use, except the named hypotheses (T) and (O) of the first lesson and the quoted consequences in the last section of the third lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}