{
  "id": "strong-kadison-kastler-stability",
  "title": "Strong Kadison–Kastler stability",
  "description": "Close von Neumann algebras are conjugate by a unitary near the identity, with a tolerance that depends only on the required closeness, after OpenAI's 2026 papers. Finite factors through boundary means of countable groups, measurable cocycles, double ergodicity and common tracial position; infinite factors through a fixed III_1 tensor factor, rescaled Tomita graphs, infinitesimal normalizers, analytic multipliers, a common modular action and common crossed products; all algebras through Dixmier's theorem and the central decomposition. Two counterexamples: close separable C*-algebras without unitary conjugacy, and one-sided near inclusions without small spatial embeddings.",
  "units": [
    {
      "id": "close-operator-algebras",
      "title": "Close operator algebras",
      "reader": "close-operator-algebras.html",
      "source": "src/close-operator-algebras.md",
      "source_sha256": "4afda06a2945b9751caa2514464a447440974a687980212c3d163beaa75ff082",
      "status": "author_self_checked"
    },
    {
      "id": "averaging-and-the-completely-positive-criterion",
      "title": "Averaging and the completely positive criterion",
      "reader": "averaging-and-the-completely-positive-criterion.html",
      "source": "src/averaging-and-the-completely-positive-criterion.md",
      "source_sha256": "5e60568f27d1e8e6010780d8778a6347eab05f8a64e4b0921ba2d9fefe375361",
      "status": "author_self_checked"
    },
    {
      "id": "boundary-means-and-measurable-cocycles",
      "title": "Boundary means and measurable cocycles",
      "reader": "boundary-means-and-measurable-cocycles.html",
      "source": "src/boundary-means-and-measurable-cocycles.md",
      "source_sha256": "feae712e01f1a8f764787b5709d55344ec035f98f370af4f773e793c194d9d0c",
      "status": "author_self_checked"
    },
    {
      "id": "double-ergodicity-and-diagonal-comparison",
      "title": "Double ergodicity and diagonal comparison",
      "reader": "double-ergodicity-and-diagonal-comparison.html",
      "source": "src/double-ergodicity-and-diagonal-comparison.md",
      "source_sha256": "4cd526343da4d52bff777baa38c16430e3a292da66d4a4c361832be10e0f720b",
      "status": "author_self_checked"
    },
    {
      "id": "irreducible-hyperfinite-subfactors",
      "title": "Irreducible hyperfinite subfactors",
      "reader": "irreducible-hyperfinite-subfactors.html",
      "source": "src/irreducible-hyperfinite-subfactors.md",
      "source_sha256": "d9022c3ac4961c93e22f4c0dc36b32e171af0ab73794962322ded8ac57905ab5",
      "status": "author_self_checked"
    },
    {
      "id": "common-tracial-position-and-jordan-maps",
      "title": "Common tracial position and Jordan maps",
      "reader": "common-tracial-position-and-jordan-maps.html",
      "source": "src/common-tracial-position-and-jordan-maps.md",
      "source_sha256": "6b88ef5ac39043aed0907600d789460ca8d0fa5fab28d5e2578ee6ad5fe5c5e6",
      "status": "author_self_checked"
    },
    {
      "id": "stability-of-finite-factors",
      "title": "Stability of finite factors",
      "reader": "stability-of-finite-factors.html",
      "source": "src/stability-of-finite-factors.md",
      "source_sha256": "05003e0c32eae378106680cc31e1f7c56df57cfb76ad1ba210ac9c433ab55d81",
      "status": "author_self_checked"
    },
    {
      "id": "preparing-infinite-factors",
      "title": "Preparing infinite factors",
      "reader": "preparing-infinite-factors.html",
      "source": "src/preparing-infinite-factors.md",
      "source_sha256": "969cc7086925bc66779d250e52cce7090c203223fddd9b2d9672fdad8a500066",
      "status": "author_self_checked"
    },
    {
      "id": "rescaled-tomita-graphs",
      "title": "Rescaled Tomita graphs",
      "reader": "rescaled-tomita-graphs.html",
      "source": "src/rescaled-tomita-graphs.md",
      "source_sha256": "61c799445c6595a6b473cdc4487dd47ca3a2d234c8287ada97c64603e968abdb",
      "status": "author_self_checked"
    },
    {
      "id": "infinitesimal-normalizers-of-type-iii-factors",
      "title": "Infinitesimal normalizers of type III factors",
      "reader": "infinitesimal-normalizers-of-type-iii-factors.html",
      "source": "src/infinitesimal-normalizers-of-type-iii-factors.md",
      "source_sha256": "0724cebfa1878a6741d14acabee70e5e9d3ef8d805725df482531baaedc3c3d0",
      "status": "author_self_checked"
    },
    {
      "id": "frequency-nests-and-analytic-multipliers",
      "title": "Frequency nests and analytic multipliers",
      "reader": "frequency-nests-and-analytic-multipliers.html",
      "source": "src/frequency-nests-and-analytic-multipliers.md",
      "source_sha256": "5e1742f3bd239af12fdbf12700aaeff7c6f3d2eaede6a1b8ba082d7095e211b8",
      "status": "author_self_checked"
    },
    {
      "id": "near-normalizers-at-every-time",
      "title": "Near normalizers at every time",
      "reader": "near-normalizers-at-every-time.html",
      "source": "src/near-normalizers-at-every-time.md",
      "source_sha256": "af165daad0a9117fee3c3b754ee18486811c956cd56302a35966938fd8fe0d18",
      "status": "author_self_checked"
    },
    {
      "id": "a-common-modular-action",
      "title": "A common modular action",
      "reader": "a-common-modular-action.html",
      "source": "src/a-common-modular-action.md",
      "source_sha256": "b7cc2d28775e11aed3294c7400337a9031414ff5081057c6b72b27553a9a6c1a",
      "status": "author_self_checked"
    },
    {
      "id": "common-crossed-products",
      "title": "Common crossed products",
      "reader": "common-crossed-products.html",
      "source": "src/common-crossed-products.md",
      "source_sha256": "f330eee584f460f8d924bd567ace64fefb7b8f3c734790d69254a79aaf5624f5",
      "status": "author_self_checked"
    },
    {
      "id": "stability-of-separably-acting-factors",
      "title": "Stability of separably acting factors",
      "reader": "stability-of-separably-acting-factors.html",
      "source": "src/stability-of-separably-acting-factors.md",
      "source_sha256": "0d8224825fedbcd9f9bc6ddc1c226ffd49a6465963bf32005f2a413923d1b307",
      "status": "author_self_checked"
    },
    {
      "id": "centers-and-measurable-assembly",
      "title": "Centers and measurable assembly",
      "reader": "centers-and-measurable-assembly.html",
      "source": "src/centers-and-measurable-assembly.md",
      "source_sha256": "3f52b44b784f936b3762a60a0127ea5328d4ac9eefbd1b9372193f5406945202",
      "status": "author_self_checked"
    },
    {
      "id": "stability-on-arbitrary-hilbert-spaces",
      "title": "Stability on arbitrary Hilbert spaces",
      "reader": "stability-on-arbitrary-hilbert-spaces.html",
      "source": "src/stability-on-arbitrary-hilbert-spaces.md",
      "source_sha256": "c8ad73a0aa8727cc916fc5b7234fbed3436836f67d4063fb60fe72717bb3e918",
      "status": "author_self_checked"
    },
    {
      "id": "convergent-sequences-and-a-relative-tensor-norm",
      "title": "Convergent sequences and a relative tensor norm",
      "reader": "convergent-sequences-and-a-relative-tensor-norm.html",
      "source": "src/convergent-sequences-and-a-relative-tensor-norm.md",
      "source_sha256": "b0f1a1707baee0edff164b50a0a999414494e809bf0872c0c9116ddacd85c4fb",
      "status": "author_self_checked"
    },
    {
      "id": "a-free-group-deformation-and-close-separable-algebras",
      "title": "A free-group deformation and close separable algebras",
      "reader": "a-free-group-deformation-and-close-separable-algebras.html",
      "source": "src/a-free-group-deformation-and-close-separable-algebras.md",
      "source_sha256": "5841f0f677f702fbe419aec95b20efe8ca8d65f71d99f80fe995486643896f4a",
      "status": "author_self_checked"
    },
    {
      "id": "a-circle-family-and-a-near-inclusion",
      "title": "A circle family and a near inclusion",
      "reader": "a-circle-family-and-a-near-inclusion.html",
      "source": "src/a-circle-family-and-a-near-inclusion.md",
      "source_sha256": "7788ab2ef16b323f5f247725f5c2959beb6277aef0e72375b955c4229a172e6f",
      "status": "author_self_checked"
    },
    {
      "id": "almost-commuting-matrices-and-small-embeddings",
      "title": "Almost commuting matrices and small embeddings",
      "reader": "almost-commuting-matrices-and-small-embeddings.html",
      "source": "src/almost-commuting-matrices-and-small-embeddings.md",
      "source_sha256": "6484c29bf65da63a3320365171742c064daf59c2b439b74fb07d2897907eddae",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "Foundations of von Neumann algebras: operator topologies and the double commutant theorem, Kaplansky's density theorem, the continuous and Borel functional calculus, the weak compactness of the unit ball of B(H), projections and types, normality of *-isomorphisms, completely positive maps, Polish spaces and measurable selection, derivations of von Neumann algebras, traces on finite factors (course Foundations of von Neumann algebras, OA-FND-BI-06, OA-FND-KD-07, OA-FND-CF-07, OA-FND-CF-11, OA-FND-LT-10, OA-FND-TY-03, OA-FND-TY-17, OA-FND-WA-18, OA-FND-WA-19, OA-FND-PB-09, OA-FND-PB-11, OA-FND-INNER-THEOREM; supplement OA-FND-TA-11, OA-FND-TA-12)",
    "Arveson's distance formula 2 dist(x, A') = ||ad x|_A||_cb and the uniform commutator estimate (course Kadison's similarity problem: Completely bounded homomorphisms and similarity, Corollary 4.5; The uniform commutator estimate, Theorem 3.1)",
    "Traces and noncommutative integration: the commutation theorem, trace-preserving conditional expectations, commutants of semifinite algebras, minimal projections (course Traces and noncommutative integration)",
    "Spatial tensor products of von Neumann algebras: matrix units, slice maps and the commutation theorem (course Tensor products of operator algebras)",
    "Haar measure on compact groups (course Harmonic analysis on locally compact groups, OA-FND-HM-05)",
    "Injective von Neumann algebras and the uniqueness of the injective II_1 factor (course Injective factors); an injective subalgebra with the same core commutant (course Relative bicentralizers and core commutants)",
    "Tomita-Takesaki theory, the standard form, the modular expectation theorem and the continuous core (courses OA-MOD and OA-FLOW); ultraproducts of type III factors (course Type III factors); Stone's theorem (course Analytic elements, strips and KMS); the central decomposition (course Measurable fields and direct integrals)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}