{
  "id": "characters-of-central-sequences-and-jiang-su-stability",
  "title": "Characters of central sequences and Jiang–Su stability",
  "description": "Kirchberg and Rørdam asked whether a unital separable C*-algebra A absorbs the Jiang–Su algebra Z whenever its central sequence algebra F_ω(A) has no characters. The answer is yes. The core is a finite theorem: every nonzero unital C*-algebra without characters has a finite maximal tensor power that receives a unital homomorphism from the dimension-drop algebra I(2,3). Its proof: the Rørdam–Winter criterion in an arbitrary unital C*-algebra; labels that stay linearly independent in every irreducible representation of a tensor power; a full square-zero element and a homomorphism from the matrix cone; a cube model of tensor powers of the cone. Proved in full from two named hypotheses: the theorem of Kirchberg and Rørdam relating Z-stability to F_ω(A), and (for infinite tensor powers) the theorem of Dadarlat and Toms. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "central-sequences-and-the-character-criterion",
      "title": "Central sequences and the character criterion",
      "reader": "central-sequences-and-the-character-criterion.html",
      "source": "src/central-sequences-and-the-character-criterion.md",
      "source_sha256": "fa0f5768745744dcba1985f0e605665189d093ec4f34de37df6df68582f82817",
      "status": "author_self_checked"
    },
    {
      "id": "cuntz-comparison-and-order-zero-maps",
      "title": "Cuntz comparison and order-zero maps",
      "reader": "cuntz-comparison-and-order-zero-maps.html",
      "source": "src/cuntz-comparison-and-order-zero-maps.md",
      "source_sha256": "1fa5c79584cec35626036bdfbccf01aba6c797f50b95f3686b92ae0d250a66dc",
      "status": "author_self_checked"
    },
    {
      "id": "the-rordam-winter-criterion",
      "title": "The Rørdam–Winter criterion",
      "reader": "the-rordam-winter-criterion.html",
      "source": "src/the-rordam-winter-criterion.md",
      "source_sha256": "276824290e1b0e9d71400bfe65869facdc1ffe2058d325a459d7863cd6609633",
      "status": "author_self_checked"
    },
    {
      "id": "labels-that-survive-representations",
      "title": "Labels that survive representations",
      "reader": "labels-that-survive-representations.html",
      "source": "src/labels-that-survive-representations.md",
      "source_sha256": "3f5f4c4690e0c0a594f65b4cc0bc8c9ca69650ac45b714c8859cb94c527daaf2",
      "status": "author_self_checked"
    },
    {
      "id": "a-full-cone-in-a-tensor-power",
      "title": "A full cone in a tensor power",
      "reader": "a-full-cone-in-a-tensor-power.html",
      "source": "src/a-full-cone-in-a-tensor-power.md",
      "source_sha256": "9bc775c858010ec644db3e310e4b8ca46a152eb300f6883a2bfb7d1d48521f43",
      "status": "author_self_checked"
    },
    {
      "id": "from-the-cone-to-the-dimension-drop-algebra",
      "title": "From the cone to the dimension-drop algebra",
      "reader": "from-the-cone-to-the-dimension-drop-algebra.html",
      "source": "src/from-the-cone-to-the-dimension-drop-algebra.md",
      "source_sha256": "d3f8956fd028b31f841ebf785ff75ab7ca98a32bf5aac81c54a2194ab096c9c3",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Foundations of von Neumann algebras: C*-algebras, the continuous functional calculus, positivity, quotients, automatic contractivity (lesson Order, local units and quotients of C*-algebras); representations, Schur's lemma, the Gelfand-Naimark theorem, norms in irreducible representations (lesson Building representations from positive functionals); the double commutant theorem, polar decompositions, weakly closed one-sided ideals (lesson The double commutant theorem); the universal enveloping von Neumann algebra (lesson The universal enveloping von Neumann algebra of a C*-algebra and W*-algebras)",
    "course Tensor products of operator algebras: algebraic tensor products (Lemma 1.1 of the first lesson), the minimal norm and the theorem of Murray and von Neumann on factors (Theorems 2.2 and 5.1 of the lesson Tensor products of C*-algebras and the minimal norm)",
    "lesson Regular values and degree modulo two (course Borsuk's conjecture in dimension nine): Sard's theorem and regular level sets (Theorem 2.1, Corollary 2.2, Proposition 3.1)",
    "lesson Norm ultrapowers and a group with property (T) (course Kirchberg's embedding problem): the norm of an ultrapower (Lemma 2.2)",
    "the lessons prove every other fact they use, except the named hypotheses (KR) and (DT) of the first lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}