{
  "id": "kadisons-similarity-problem",
  "title": "Kadison's similarity problem",
  "description": "Kadison's similarity problem, solved: every bounded unital homomorphism of a unital C*-algebra into the bounded operators on a Hilbert space is similar to a *-homomorphism (OpenAI, 2026). Completely bounded homomorphisms and Paulsen's similarity theorem; implementation of derivations with half the completely bounded norm and Arveson's distance formula; the noncommutative little Grothendieck inequality, row and column estimates, finitely generated homomorphisms and cyclic domains; Kirchberg's passage from inner derivations to similarity with a polynomial bound; words in free subalgebras; Schur-Weyl invariants, propagation from small corners and a Hankel test, which give a free-product corner estimate independent of the corner; transfer to II_1 factors through Popa's free independence in ultrapowers; the uniform commutator estimate for all von Neumann algebras; inner derivations, the similarity theorem with the bound ||pi||_cb <= ||pi||^(2C), and universal hyperreflexivity of von Neumann algebras.",
  "units": [
    {
      "id": "completely-bounded-homomorphisms-and-similarity",
      "title": "Completely bounded homomorphisms and similarity",
      "reader": "completely-bounded-homomorphisms-and-similarity.html",
      "source": "src/completely-bounded-homomorphisms-and-similarity.md",
      "source_sha256": "88b005c8ede80f908f8f5006e574055164021cd1b88c37b2d5f0d18cf72bafda",
      "status": "author_self_checked"
    },
    {
      "id": "row-column-and-cyclic-estimates",
      "title": "Row, column and cyclic estimates",
      "reader": "row-column-and-cyclic-estimates.html",
      "source": "src/row-column-and-cyclic-estimates.md",
      "source_sha256": "7fd3b64d37a1bad839eb792589d65099d9b7386f1e76b6ae0e29453333b3b8a2",
      "status": "author_self_checked"
    },
    {
      "id": "from-inner-derivations-to-similarity",
      "title": "From inner derivations to similarity",
      "reader": "from-inner-derivations-to-similarity.html",
      "source": "src/from-inner-derivations-to-similarity.md",
      "source_sha256": "9703449ea16f2b967465da439bf97315fcda8ee785e324382c58611b27a9f33f",
      "status": "author_self_checked"
    },
    {
      "id": "words-in-free-subalgebras",
      "title": "Words in free subalgebras",
      "reader": "words-in-free-subalgebras.html",
      "source": "src/words-in-free-subalgebras.md",
      "source_sha256": "99a5c420423ba6278b75367d74d5c7d1c8b4cef8672bc43878c478c8248375f1",
      "status": "author_self_checked"
    },
    {
      "id": "invariant-tensors-and-the-word-structure",
      "title": "Invariant tensors and the word structure",
      "reader": "invariant-tensors-and-the-word-structure.html",
      "source": "src/invariant-tensors-and-the-word-structure.md",
      "source_sha256": "18b85b3731cbb079109b966475fb5482fe758d5e58869c62fab7d415889047c7",
      "status": "author_self_checked"
    },
    {
      "id": "propagation-from-small-corners",
      "title": "Propagation from small corners",
      "reader": "propagation-from-small-corners.html",
      "source": "src/propagation-from-small-corners.md",
      "source_sha256": "a2a10e16572c10d6efe0333ee8925ce5d06426b679870651dd54ceac693d1634",
      "status": "author_self_checked"
    },
    {
      "id": "the-hankel-test",
      "title": "The Hankel test",
      "reader": "the-hankel-test.html",
      "source": "src/the-hankel-test.md",
      "source_sha256": "334db565fe1d9fe821d2f4b69f4a81253f03a37021950af8acc90eba9f62e4d1",
      "status": "author_self_checked"
    },
    {
      "id": "matrix-commutators-in-finite-factors",
      "title": "Matrix commutators in finite factors",
      "reader": "matrix-commutators-in-finite-factors.html",
      "source": "src/matrix-commutators-in-finite-factors.md",
      "source_sha256": "489e2cd2a7fe04bff1f94fb36473d30ffcedfcc2e345d8e85a5d9d40a8554cd8",
      "status": "author_self_checked"
    },
    {
      "id": "the-uniform-commutator-estimate",
      "title": "The uniform commutator estimate",
      "reader": "the-uniform-commutator-estimate.html",
      "source": "src/the-uniform-commutator-estimate.md",
      "source_sha256": "e550ff315257a2117ee0554c82e5af5c05617a036266e4c67e2acdd0fad5c9e3",
      "status": "author_self_checked"
    },
    {
      "id": "kadisons-similarity-theorem",
      "title": "Kadison's similarity theorem",
      "reader": "kadisons-similarity-theorem.html",
      "source": "src/kadisons-similarity-theorem.md",
      "source_sha256": "5d6e6722a30e2bd41fcc05521996597b51147e9172afa41f684516c075be9ae9",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "OA-APPROX lessons completely-bounded-extension (Theorem 4.1: factorization of completely bounded maps) and sharp-noncommutative-bilinear-inequality (Theorem 1.1: Haagerup's noncommutative Grothendieck inequality; Section 4: three-lines bound (9))",
    "foundations-of-von-neumann-algebras lessons on C*-algebras (Theorem 4.2, Proposition 7.3, Corollary 15.4, Proposition 17.1), Kaplansky's density theorem (Theorem 7.1), projections and types (Theorem 5.5, Corollaries 7.3, 10.4, 13.5, Propositions 4.3, 8.4, 14.2, 18.2) and the spectral theorem (Proposition 2.1, Theorem 3.1)",
    "RT-FIN lesson schur-weyl-duality (Theorem 3.1)",
    "traces-and-noncommutative-integration lesson integration-for-a-trace (Theorem 1.1 duality, Theorem 3.1 commutation theorem, Theorem 9.1 trace-preserving conditional expectations)",
    "generators-of-ii1-factors lessons haar-unitaries-freeness-and-ultraproducts (Section 1, Definition 2.1, Lemma 2.3, Corollary 2.4, Section 4 with Lemmas 4.1-4.2) and free-independence-in-ultraproducts (Theorem 4.2: Popa's free independence)",
    "measurable-fields-and-direct-integrals lesson the-central-decomposition-and-the-types-of-the-fibres (Fact 1.1, Fact 1.2, Lemma 3.1, Theorem 4.1)",
    "harmonic-analysis-on-locally-compact-groups lesson haar-measure-on-locally-compact-groups (Theorem 8.3)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}