{
  "id": "amenability-and-unitarizability",
  "title": "Amenability and unitarizability",
  "description": "OpenAI's answer (2026) to Dixmier's question for discrete groups: a discrete group is amenable exactly when every uniformly bounded representation on a Hilbert space is similar to a unitary one. The course proves the Day–Dixmier theorem and, for every nonamenable group, constructs representations with all norms at most 1 + ε that cannot be made unitary: nonamenability gives a contracting average of translations, Hall's theorem assigns the edges of a Cayley-type graph to their endpoints with small loads, random sign frames turn the assignment into operators with bounded conjugation differences, and a Hilbert–Schmidt energy bound shows that their cocycle has no bounded implementer. Corollaries for Thompson's group F and for group C*-algebras.",
  "units": [
    {
      "id": "uniformly-bounded-representations",
      "title": "Uniformly bounded representations",
      "reader": "uniformly-bounded-representations.html",
      "source": "src/uniformly-bounded-representations.md",
      "source_sha256": "7353989cf27faa3984729f5956e2743cd77ae2e8cfb36623bcd4c8887e179198",
      "status": "author_self_checked"
    },
    {
      "id": "contracting-averages-and-sparse-assignments",
      "title": "Contracting averages and sparse assignments",
      "reader": "contracting-averages-and-sparse-assignments.html",
      "source": "src/contracting-averages-and-sparse-assignments.md",
      "source_sha256": "1f8a4e7fc2986065a792c2cd310f0e8e4ce3197d3a98c38f85988184da576095",
      "status": "author_self_checked"
    },
    {
      "id": "random-sign-frames-and-masked-operators",
      "title": "Random sign frames and masked operators",
      "reader": "random-sign-frames-and-masked-operators.html",
      "source": "src/random-sign-frames-and-masked-operators.md",
      "source_sha256": "4f9b0e37b199e9baaba103c84182f1b7777d2be729210bed430085644e354bbb",
      "status": "author_self_checked"
    },
    {
      "id": "unitarizability-implies-amenability",
      "title": "Unitarizability implies amenability",
      "reader": "unitarizability-implies-amenability.html",
      "source": "src/unitarizability-implies-amenability.md",
      "source_sha256": "965681111a68c87fb4cb335318b5bdcb6e7a14920db2cdfc29ad70a28594b5cb",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "foundations-of-von-neumann-algebras lessons hilbert-spaces-and-compact-operators (Theorem 3.1, bounded sesquilinear forms), c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones (Proposition 8.5(10), positive square roots), hahn-banach-baire-and-the-basic-theorems-on-banach-spaces (Corollary 5.2, inverse mapping theorem), weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian (Theorem 3.1, Banach-Alaoglu)",
    "OA-ERGODIC lesson means-folner-sets-and-regular-representations (Section 1, means; Theorem 5.1, almost invariant vectors)",
    "core course Measure and Integration (D10): Fremlin, Measure Theory, Vol. 1, 134A (translation invariance) and Vol. 2, 263A (linear images of Lebesgue measure)",
    "kadisons-similarity-problem lesson kadisons-similarity-theorem (Theorem 2.1) and thompsons-group-f-and-amenability lessons 1 and 3 (Proposition 1.3, Theorem 5.1), both only for Section 5 of the fourth lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}