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  "title": "Common gauge invariants and outer conjugacy",
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  "scope": "Common-gauge invariants for automorphisms of approximately finite type III-zero factors, the exact kernel of the separate-class projection, a realized sign obstruction and conditional outer-conjugacy classification with normal crossed-product descent; eight solved exercises.",
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      "author": "Colin E. Sutherland and Masamichi Takesaki",
      "title": "Actions of discrete amenable groups on injective factors of type III_lambda, lambda not equal to one",
      "year": 1989,
      "url": "https://msp.org/pjm/1989/137-2/pjm-v137-n2-p08-s.pdf",
      "scope": "Proposition1.1p409; Theorem2.2(2.4)p416; Assumption4.1/Theorem4.2pp427–428.",
      "role": "Extended characteristic framework, cocycle transport and semifinite classification."
    },
    {
      "author": "Yasuyuki Kawahigashi, Colin E. Sutherland and Masamichi Takesaki",
      "title": "The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions",
      "year": 1992,
      "url": "https://projecteuclid.org/journals/acta-mathematica/volume-169/issue-none/The-structure-of-the-automorphism-group-of-an-injective-factor/10.1007/BF02392758.pdf",
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      "role": "Centrally trivial automorphisms and the normal-form/tower prerequisites."
    },
    {
      "author": "Alain Connes",
      "title": "Outer conjugacy classes of automorphisms of factors",
      "year": 1975,
      "url": "https://www.numdam.org/item/ASENS_1975_4_8_3_383_0/",
      "scope": "Introductionpp383–385 and the stated classical outer-period framework.",
      "role": "Ordinary scalar obstruction; its elementary invariance is proved in the linked programme lesson."
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      "scope": "Setting and Sections 1–5: canonical extensions, functoriality and continuity."
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      "source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
      "scope": "Section 1, CDEC.1.f–o: the full normal amplification isomorphism and inverse; Sections 2 and 4: type reflection and recovery of the specified trace-scaling system."
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