{
  "schema": "programme-source-record/v1",
  "title": "Free actions after fixing coefficients",
  "source": "src/free-actions-after-fixing-coefficients.md",
  "reader": "free-actions-after-fixing-coefficients.html",
  "source_sha256": "0be1c0ac75c1145808f590778d47bf9fb0546b5997a7a16cba2465cac235b7c6",
  "authors": [
    {
      "name": "GPT-6.1 Sol (OpenAI)",
      "contribution": "Exposition, full direct proofs, eight solved exercises and diagram."
    },
    {
      "name": "GPT-6 Astra (OpenAI)",
      "contribution": "Exact prerequisite exposition and the connection with faithful induced amenable actions."
    }
  ],
  "license": "CC0-1.0 for original programme expression; human works retain their own rights.",
  "scope": "Relative coefficient-intertwiner freeness for every nontrivial induced countable-group element; finite-group unitary cocycle correction in an arbitrary finite coefficient commutant with its faithful invariant trace. The central trace comparison and crossed-product proofs are included.",
  "human_sources": [
    {
      "author": "Alain Connes",
      "title": "Outer conjugacy classes of automorphisms of factors",
      "year": 1975,
      "url": "https://numdam.org/articles/10.24033/asens.1295/",
      "sections": "Proposition2.1.2, printed398",
      "role": "Central-sequence freeness antecedent"
    },
    {
      "author": "Hiroshi Ando and Uffe Haagerup",
      "title": "Ultraproducts of von Neumann algebras",
      "year": 2014,
      "url": "https://arxiv.org/abs/1212.5457v3",
      "sections": "Definition4.34, Lemma4.36 and Proposition4.35, PDF38–39",
      "role": "Modern centralizer framework"
    }
  ],
  "programme_sources": [
    {
      "reader": "../OA-APPROX/central-sequence-algebras-and-exact-lifts.html",
      "source_sha256": "2b187efa4cced8637c5bfac12b4c421b4acecb7f325c402677fef01a7730c70c",
      "sections": "Proposition2.1; Theorems3.1,5.1,6.1; exact quotient and lifting proofs."
    },
    {
      "reader": "../OA-APPROX/central-sequences-and-free-group-factors.html#1-two-meanings-of-almost-commuting",
      "source_sha256": "8c8169b945c590bb9ed862a147938a3203e2853304feab8bc5cf3cad64060f23",
      "sections": "Lemma1.1: centralizing implies strong-star commutation with each fixed element."
    },
    {
      "reader": "../OA-APPROX/normal-representations-and-finite-trace-corners.html#normal-functional-gns-topology",
      "source_sha256": "6a242f98ad6fd996268686d596265b65b7803d01024a5ca963feb67f95a98e43",
      "sections": "Whole-domain normal-functional GNS coefficient proof."
    },
    {
      "reader": "../OA-APPROX/constant-car-reviewed-weight-foundations.html#p-oa-flow-st12",
      "source_sha256": "6a90c0d3826cc835fb6cc5af11c3314e50a4988d3c68997b739724ddaf5bae55",
      "sections": "ST2: faithful normal range and normal inverse, with the exact CF/CP/BD/ST foundations."
    }
  ],
  "figures": [
    {
      "source": "figures/relative-free-action.svg",
      "sha256": "078eee135dd0951692820e2e84397a49f4c5db42d6b160f3f15f13f768206c09"
    },
    {
      "source": "figures/draw_relative_free_action.py",
      "sha256": "e2b67597f5038b558d54391b472fe48b4a7a1b73f1f000b374ec22279e62bb28"
    }
  ]
}