Subfactors, index and finite-depth classification

A course on finite-index inclusions of II₁ factors, their Jones towers and tunnels, and classification through higher relative commutants and standard invariants. For proper extremal inclusions with separable predual, original strong amenability is also equivalent to a normal anti-isomorphism onto the whole physical opposite model, at arbitrary depth. The whole ordinary standard model characterizes original strong amenability at proper finite index and separable predual; its internal supported Jones tower gives the full converse. Global finite approximation, amenability relative to an ergodic actual core and original strong amenability are equivalent without ambient separability; exact finite-prefix matching retains every prescribed ordinary cup. Original strong amenability is equivalent to an ordinary generating tunnel at proper finite index and separable predual, with a direct compatible-hypertrace construction for every smooth representation. Full all-smooth representation amenability and original strong amenability are invariant under every nonzero physical corner and positive real amplification, including singular compatible hypertraces. Every nonzero physical corner and positive real amplification preserves both whole model pairs, with their separate traces and marked Jones data. The actual opposite model and its entire upper tower admit compatible tensor splittings, retaining the original and canonical Jones cups and their traces. Explicit commuting-square examples distinguish an abstract standard model from its realization as a core inside fixed ambient factors. The marked invariant reconstructs every ordinary standard model with its actual downward trace. Proper strongly amenable separable inclusions are classified at arbitrary depth, including nonextremal cases. The edition contains 99 readings: 96 teaching chapters and three supporting proof readings, with mathematical sources, solved exercises and editable figure sources. Several advanced mathematical statements remain unproved here; each argument declares its precise prerequisites.

Index and the basic construction

Traces and integrable densities; Jones projections, partial bases, module dimensions, finite group inclusions and index rigidity. Finite-dimensional and II₁ examples explain the normalizations.

Towers, graphs and finite-depth classification

Ordinary towers and tunnels, reflection and higher relative commutants; path algebras, principal graphs, commuting squares, connections, flatness, realization and the exceptional below-four cases. Exact nondegenerate finite-square presentations force an algebraic-integer index; the strongly amenable weighted-spin example shows why that presentation is an additional hypothesis.

Amenability and finite approximation

Smooth expected representations, compatible hypertraces, central dimensions and supported local approximation. The full finite-family partition retains each prescribed prefix and the entire residual. Canonical core representations allow nonfactor cores. Finite prefix approximation uses the stated ergodic-core hypothesis; the generating ordinary tunnel also requires separable predual.

Physical traces, analytic limits and explicit models

Physical and represented traces, modified cups, the actual tracial opposite-model tower, finite Haar averages, support obstructions and reconstruction maps. The index-ten original and opposite pairs have incompatible first relative-commutant trace weights; extremal strong amenability gives original-pair reconstruction at arbitrary depth with separable predual. Both actual opposite inclusions have finite bounded bases and trace-preserving center transport two steps apart. Adjacent centers have a common atomic-part projection; the opposite-model joint center is scalar, so both centers are atomic or both are diffuse. Extremal ordinary cores inherit this conclusion without finite depth. Weighted-spin, affine and lamp models distinguish proved estimates and counterexamples from the general assertions they do not settle. The original lamp model has positive finite cylinder bounds and certified normal count errors for each fixed finite input.

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.

Unfinished mathematical scope

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.

Offline reading and source

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.

Third-party notices