Sources and proof guide

Human mathematical sources

Jones provides the index and basic-construction framework; Pimsner–Popa provides finite-index inequalities, entropy and canonical modified projections. Popa’s Classification of amenable subfactors of type II, Acta Mathematica 172 (1994), 163–255, supplies the amenability, approximation and classification context, particularly §§1.3, 3.1–3.2 and 4.1–4.5. Article and DOI. Exact theorem, page and normalization references appear beside the corresponding arguments.

Finite-trace and functional-analysis references

• Takesaki, Theory of Operator Algebras I, 1979 edition: III.2.14, III.3.6–3.11, III.5.4, IV.5.4, V.1.37–1.38 and V.2.2–2.4.
• Blackadar, Operator Algebras, revised author edition, 8 February 2017: III.1.1 and III.2.1.5. Author edition.
• Van Neerven, Functional Analysis, version 7, 17 July 2025: 4.7–4.9, 4.32–4.33, 4.50 and 5.1. Source edition.
• Namioka–Asplund, A geometric proof of Ryll-Nardzewski’s fixed point theorem, 1967, pp.443–445.
• Anantharaman–Popa, An introduction to II1 factors, author draft: 4.1.3, 6.1.3, 8.6.1 and 11.2. Author draft.

Linked proof interfaces

• Finite-dimensional coefficient algebras: Theorem 2.4 and Sections 1(c),(f): matrix blocks, matrix norms and functional calculus in the fixed Gram corner of 53.2–53.4.
• Injectivity as a norm-one projection: Theorem 2.1 applies to the actual ucp projections in 75.6; the projections need not be normal.
• Individual hyperfiniteness of separable injective factors: Theorems 1.5–1.6 and their Section 9 assembly apply to each factor in 75.6. Separable predual is retained; the inclusion itself is not classified by this consequence.
• Traces on von Neumann algebras: Theorems 5.2/5.5 and Corollary 5.4: center-valued trace, scalar trace factorization and projection comparison on finite von Neumann algebras.
• Trace integration and the predual: TI-05–06: actual measurable L¹ operators and all normal functionals for a faithful normal semifinite trace. Normality is a hypothesis; singular states are not identified with densities.
• Semifinite trace inequalities: Theorem 3.1, Lemma 4.1(c) and Proposition 4.3: square-root and spectral-cut estimates at their stated semifinite trace hypotheses.
• Normal functional extension: Theorem 10.1: positive normal functionals as summable vector functionals on arbitrary Hilbert spaces.
• Finite central traces and actual density consequences: TE16–18: trace factorization, normal central densities, domination and trace-preserving expectation on integrable operators.
• Canonical expectations and full corners: Proposition 52.2a: normal faithful physical reduction, full-corner traces and centers, including arbitrary centers and cardinality.

General conditional expectations, weights and fusion are supplied by the modular-theory course; approximation inputs retain the hypotheses of their linked providers. Source access, permission to reuse expression and mathematical proof completeness are distinct. The course does not assert full prerequisite closure.

Mathematical scope and limits

The actual commutant expected tower has the original supported bases and exact physical trace correction (PH.9–PH.11). The physical returns have a bounded central limit h (PH.12–PH.15), with exact atomic index coefficient pᵢ/dᵢ, where pᵢ=τ(zᵢ) and dᵢ=[Uzᵢ: Nzᵢ] (PH.16–PH.18). The structural zero-one law gives h=0 or h=1 and limiting state distance zero or two, without amenability or a lower spectral bound (PH.22–PH.24). A nonzero limit, or one finite-index atomic corner, forces the whole physical bicommutant. The original strong-amenability implication still requires exclusion of h=0.

The physical forward shift has compatible full supported-matrix formulas, including a fractional last support (PH.25–PH.28). It defines a mixing normal trace-preserving automorphism of the entire relative-commutant envelope without extremality. The commutant of all bilateral cups there is scalar. For an original generating tunnel, the envelope is the entire physical completion, and the shift transports every stage, actual commutant and return density forward by two. The zero-return branch remains part of the original bicommutant question.

Weighted group models with a factorial balanced half have full physical bicommutant equality (PH.29–PH.31). Both unequal phase laws are retained; entropy transfer proves the two directions of mixing. Entire physical commutants and the original weighted cups give an actual ordinary Jones tower of index d=AB and return limit one. Subexponential-growth groups, including every finitely generated Abelian group, satisfy the factoriality condition automatically. The unrestricted remaining bicommutant and fixed-WM invariant endpoint remain separate questions.

Scalar central multiplicity has a quantitative obstruction in the index-ten affine model (IRG.1–IRG.3). Across every matrix deletion of one fixed actual core,a physical Folner net stays at normalized distance at least 3/110 from every one-level profile. A finite physical test set obstructs exact constant multiplicity. The freely selected-tunnel BF implication has the affine counterexample BFQ.3. The unrestricted physical bicommutant and the separate fixed-WM endpoint remain open.

Common-support frames and scalar multiplicity: Unrestricted common-support BF and constant-multiplicity Følner rounding have an actual affine counterexample (BFQ.1–BFQ.4), even when the ordinary tunnel may be chosen after the physical targets. The explicit BF construction and compatible finite tunnel maps would return nearly scalar profiles to one fixed core, contrary to its exact 3/110 affine profile bound. The conditional BF conversion theorems, factorial specializations, unequal-support first local form and full finite partition retain their stated hypotheses. The bare second central local form remains separate.

Tail projection profiles: Arbitrary tail projections have the exact weighted central-profile range TP.1. Every finite central partition is realized at the same stage,including its full residual. The support replacement estimate has the sharp concentration factor χ(q) = ‖T_C(q)‖∞/τ(q);a scalar profile in S alone does not remove it. Simultaneous physical estimates for the bare second local form remain a separate existence question.

Cup transfer and finite-depth profiles: Original cups give the exact weighted center transfer. Moving neighboring centers two steps preserves their joint trace. Tail projection profiles at every finite depth are ranges of the corresponding Markov iterates, with an explicit bound for almost stationary integrable profiles. Every finite partition includes its full residual. Simultaneous physical commutator and compression control for the bare second local form remains a separate requirement.

Physical moments and projection selection: Finite normal moments and the full central trace of a positive contraction can be realized by one projection at the same tail stage. The physical coefficients give an exact combined commutator and compression identity. The full complementary residual is retained; three explicit positive effects show why arbitrary partitions cannot retain every individual moment. The unrestricted second local form still requires positive inputs with small physical coefficient defects.

The full extremal opposite-model characterization: Lemma 17.17 transports whole ordinary tunnels and their complete supported right rows through normal anti-isomorphisms. Theorem 17.18 proves both directions of the physical opposite-model characterization in Popa, Theorem 5.1.1, at proper finite index, extremality and separable predual, without finite depth. Its reverse uses the independent normal extremal core/opposite comparison CTS.7 followed by the full ordinary-model converse, with two multiplication reversals and unchanged traces. The actual index-ten obstruction still excludes omission of extremality. The general physical bicommutant remains separate.

The internal core tower and the standard-model converse: Theorems 17.14–17.16 construct every internal ordinary basic construction with its full partial Gram corner, original cup, normal dual expectation and canonical trace on the whole positive cone. Factor endpoints give an actual generating tunnel of the entire core. A whole-pair model isomorphism returns this tunnel to the original inclusion, proving the full standard-model converse and Popa, Theorem 4.1.2(i)–(v), at proper finite index and separable predual. No extremality or finite depth is required. This does not identify an arbitrary represented model with an embedded core of fixed larger factors or settle the separate general physical bicommutant.

Global approximation and ergodic ordinary cores: Theorems 61.17–61.20 prove the full global-finite-approximation/strong-amenability equivalence for proper finite-index II₁ inclusions without ambient separability, as in Popa, Theorem 4.1.2(i)–(iii). Finite weighted cup matching retains every prescribed ordinary prefix exactly. A countable diagonal constructs an actual factorial core, while a separate directed finite-set net supplies a compatible possibly singular hypertrace for every smooth representation of the entire original factor. The proof does not identify a nonseparable ambient factor with its separable core or establish the separate general nonextremal physical bicommutant.

Generating tunnels and original representation amenability: Theorems 61.13–61.15 extend every smooth expected representation normally down the actual ordinary tunnel. Normal compatible states centralize its whole finite relative commutants, and a uniform estimate in the original physical trace returns a possibly singular hypertrace central over the entire original factor. Theorem 61.16 gives the full original strong-amenability/generating-tunnel equivalence for proper finite-index inclusions with separable predual, including every prescribed finite ordinary prefix. No extremality or finite depth is assumed. The separate general nonextremal physical bicommutant identity remains open.

Smooth representations, physical corners and amenability: Theorems 61.9–61.11 construct normal compression and finite induction for every smooth expected representation, retain the entire represented Jones tower, and return compatible hypertraces by finite UCP formulas. Full representation amenability is equivalent under every nonzero corner and every positive real amplification, without assuming that the hypertrace is normal. Corollary 61.12 separately preserves factoriality of the whole ordinary core and hence original strong amenability. No extremality, finite depth or ambient separability is assumed. The general nonextremal physical bicommutant remains open; the generating-to-amenability direction is proved in Theorem 61.15.

Corners, amplifications and the two model traces: Theorems 17.11–17.12 give exact physical corner Jones towers and normal trace-preserving comparisons of the entire ordinary and actual opposite models for every positive real amplification. Full finite marked rows, both original and canonical opposite cups, densities, expectations and canonical positive-cone traces are retained. The ordinary comparison uses its recovered downward trace, distinct from the upward physical opposite trace. Corollary 17.13 proves that the physical bicommutant equality is equivalent across amplitudes; the general nonextremal equality is not established by that equivalence. The index-one models remain scalar.

Opposite-model tensor stability: Proposition 50.6 and Theorems 50.7–50.8 prove the full proper-index opposite-model stability statement with arbitrary centers. One common hyperfinite factor splits the entire upper opposite tower. Compatible normal trace-preserving maps fix the original cups, their canonical rescalings and densities, both expectations and every canonical positive-cone trace. Any prescribed finite unital matrix factor in the smaller opposite algebra has the same complement property. Index one is excluded, and arbitrary prescribed infinite-factor removal is not asserted.

Represented standard models and embedded cores: Theorems 51.7–51.8 give an index-four controlled-unitary family of nondegenerate trace-preserving commuting squares. Each represented pair is normally trace-isomorphic to an actual standard model, but it is an embedded ordinary core exactly when the parameter unitary is scalar. A fixed first-relative-commutant unitary has squared distance 1−|τ(v)|² from the represented larger algebra. The example refutes the commuting-square embedded-core assertion in Popa, Section 1.4.4, Corollary, at its displayed hypotheses, including a hyperfinite finite-depth instance. The general physical bicommutant is not inferred.

Standard-model reconstruction and classification: Theorem 17.8 recovers every downward physical trace from a finite skipped-cup equation in the marked upward invariant. Theorem 17.9 normally reconstructs and compares whole ordinary core pairs at arbitrary depth, retaining all centers and the distinct physical opposite trace. Theorem 17.10 classifies proper strongly amenable separable II₁ inclusions without finite depth or extremality, using the original all-smooth amenability and ergodic-core hypotheses. The normal whole-pair converse is Theorem 17.16, whose internal tunnel retains the original cups, supported bases and traces. The different represented embedded-core assertion has the counterexample 51.7–51.8.

Finite bases and center types: Theorem 63.11 gives at most ⌈d⌉ bounded basis columns, both actual tracial positive bounds and trace-preserving two-step center transport for the opposite model. Theorem 63.12 identifies the actual atomic-part projections of adjacent Markov centers. Corollary 63.13 proves scalar joint center and the atomic/diffuse dichotomy for every actual opposite pair, and for every extremal ordinary core without finite depth or separability. These results do not prove the general physical bicommutant or common-support BF assertion.

Canonical tracial opposite-model tower: Lemmas63.6–63.7 and Theorem63.8 identify the full basic construction, actual rescaled Jones projections and canonical traces at every depth, allowing arbitrary centers. Original versus opposite pair: the actual strongly amenable index-ten model has incompatible first relative-commutant trace weights, excluding every unmarked pair isomorphism or anti-isomorphism. Corollary63.10 gives arbitrary-depth reconstruction under the stated extremal strong-amenability and separable-predual hypotheses. The general nonextremal physical bicommutant remains a separate question.

Exact finite squares and their index restriction: Proposition31.6 transports the existing matrix frame into the physical inclusion and proves the integer-matrix eigenvalue equation. Corollary31.7 excludes that presentation for the strongly amenable weighted-spin inclusion at every blocked depth, including common amplifications and corners in the smaller factor. Its bicommutant is proved by a different argument; the general nonextremal theorem remains open in this course.

The finite-family partition theorem retains the entire residual and each prescribed Jones prefix. The forward finite approximation and generating conclusions retain their ergodic-core hypotheses, and generation retains separable predual. These conclusions coexist with the separately stated counterexamples to stronger joint-balance, canonical-packing and specified-reflection assertions.

• The general nonextremal physical bicommutant theorem at its original hypotheses: exclusion of the actual zero-return branch h=0.
• The bare second central local form and remaining localization questions beyond the false unrestricted common-support BF/constant-multiplicity implication.
• The invariant zero-cost endpoint, or a positive floor for every invariant state, in the same fixed weighted lamp model, including arbitrary singular states.
• The statements and exercises of Takesaki III XIX.2–4 outside the proofs cited above.

Positive gaps at each fixed tower stage do not by themselves prove the general physical bicommutant. Its return martingale still requires preservation of unit mass under the original hypotheses. The specified reflection extends normally exactly in the extremal case; Theorems 17.16–17.18 give the whole ordinary-model and extremal physical opposite-model characterizations. The universal unmarked nonextremal pair comparison and the represented embedded-core assertion have the distinct counterexamples 63.9 and 51.7–51.8. Positive finite lamp-cylinder mass and certified normal count accuracy do not establish full derivative-tuple support, a sign decision or a uniform rate for moving word lists.

Files and attribution

The ZIP contains the full course reader, editable Markdown and figure sources, local rendering assets, license and notices. Extract it and open courses/OA-SUBFACTORS/public/html/index.html. Linked proofs in other courses and referenced books are outside this download; it is not a self-contained prerequisite collection.

Original course exposition and original figures by GPT-6.1 Sol (OpenAI), CC0 1.0. Human mathematical sources are credited in the readings. Component terms are recorded in the license and third-party notices.

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