Scope and prerequisites

This selection contains 59 lessons, 410 solved exercises and seven proof companions. Each lesson states the hypotheses and prerequisite theories of its results.

The lessons develop projection correction, central sequences, relative freeness, covariance-preserving reindexing, Rokhlin towers from finite orbit classes or descending Følner packing, scalar and unitary cocycle correction, quantitative original-factor correction, two-sided convergence, generator tests, alternating and joint quotient comparison, scalar associators, measurable selection and characteristic classification of constant semifinite fields, characteristic-compatible approximations, isotropy-action gluing, characteristic models and phase/trace alignment, tensor absorption, outer conjugacy, core normalizers and Powers factors, including state-preserving modular-invariant Powers subfactors in arbitrary separable type III-lambda factors. The selection includes 58 of the original 67 course drafts and a separate projection-correction lesson; the remaining 9 original drafts are outside this selection.

The flow applications compare trace-scaling actions over their centers, reconstruct continuous comparisons from return arrows, and realize prescribed properly ergodic flows by skew odometers. The realization lesson includes the invariant-coordinate calculation, compatible matrix inclusions and the regular abelian crossed-product model, with seven solved exercises.

Proof companions

Bounded topology and faithful states: Hilbert geometry, concrete preduals, bounded calculus and density.
Normality and tracial representations: normal functionals and weights, support projections and threshold cuts.
Normal tensor tests and tracial products: product-functional density, specified tracial identifications, normal homomorphism and completely positive tensor maps.
Trace existence and central averaging: arbitrary finite center-valued traces, normal corner extension, semifinite scalar traces and simultaneous norm averaging.
Discrete trace-scaling cocycles: the counting and Fourier argument, with T1 supplied by the trace companion and W1–W3 stated as additional inputs.
Dual fixed points, core commutants and central cocycles: specified-cocycle realization and faithful-state core charts.
Crossed-product recognition and injective cores: algebra-valued expectations, finite-dimensional averaging, corners, the same-parameter classification reduction and the full factorial-centralizer spectral theorem, norm-one retractions and invariant-state finite classification, including tracial quotients, normal embeddings and bounded trace expectations with their stated foundations. The tensor companion proves normal homomorphism and completely positive tensor maps in arbitrary faithful normal representations, including exact kernels, ranges and normal dilation.

The general spectral, extended-positive, modular-transfer and classification foundations have separate scope.

Sources

The lessons cite their mathematical antecedents with exact theorem and page locators. Bibliographic source details and programme proof locators are collected in the sources and authorship record.

Access to a secondary exposition does not resolve its inherited methods. The tower-packing and action-comparison constructions have classical ancestry that requires its own comparison. A different link, notation or exercise does not establish independent expression or teaching organization. The full source ancestry, expression, organization and transitive programme proofs remain outside any claim of completed review.

Each lesson states the assumptions of its results. General modular theory, standard forms, spectral theory and measurable-field constructions remain prerequisites where specified.

The sources and authorship record includes reading links for Connes’s article on almost-periodic states and Connes–Størmer’s article on state-space homogeneity. Those articles retain their copyright.

Sources and mathematical scope · Sources and authorship

Full-group criterion

Proposition 2.A detects full-group membership using bounded asymptotically invariant functions for any nonsingular aperiodic transformation. Its finite-marker construction and two-colour separation include both directions of the criterion. Corollary 2.B implements the base correction by an explicit strong-star sum of partial isometries. The variable-field route gives coefficient innerness at its stated foundations. The constant-field argument is a conditional alternative.

The implementing-unitary cocycle

Proposition 2.C and Corollary 2.D use the full varying-roof suspension and its exponential trace to remove a specified integer cocycle. The resulting inner correction fixes the crossed-product implementer. The variable-field route gives coefficient innerness at its stated foundations. The constant-field argument is a conditional alternative.

Lacunary centralizer corners

Proposition 2.E compares two centralizer corners, including all mixed matrix entries and normal inverse maps. Proposition 2.F transports a matched cut to the whole factor with its lacunary weight. Under central triviality, the partial orbit criterion and Theorem 2.H match the two cuts. The variable-field route gives coefficient innerness at its stated foundations. The constant-field argument is a conditional alternative.

Matching under central triviality

Theorem 2.H uses the partial center map, asymptotically invariant tests and its orbit implementer to identify the full matched coefficient corner. The resulting inner perturbation preserves a lacunary centralizer. This statement uses central triviality; it asserts no corresponding normalization for every automorphism.

Variable coefficient automorphisms

Lemma 6.2, Proposition 6.3 and Corollary 6.4 use countable measurable phase tests, a coefficient-algebra central sequence, the nonsingular return tower and a measurable implementer. This route proves the core-normalizer equality at the stated standard-form, field, semifinite and phase-witness foundations.

Periodic cocycles and return phases

Periodic cocycles and the characteristic square gives the circuit invariant, two transfer constructions, explicit core realizers, six exact sequences and the cyclic characteristic phases. It includes seven solved exercises and an editable diagram of the Fourier construction. The tensor-action comparison retains its separate classification hypothesis.

Modular cyclic returns

Classifying modular cyclic returns computes the inner-return lattice and its characteristic phases, specializes the one-factor comparison and constructs the normal crossed-product isomorphism. Seven solved exercises distinguish trace kernels, inner kernels, scalar defects and return representatives. The classification retains its stated foundational hypotheses.

Crossed-product types

Crossed-product types from cyclic return data determines the centre and factor subtypes through the inner-return lattice, the trace kernel and finite central returns. It gives the full finite-matrix maps and exact integer formulas, with eight solved exercises. The type conclusions retain their stated foundational hypotheses.

Weighted orbit matching

Lattice derivatives and weighted orbit matching develops homogeneous transports, weighted matrix arrays and quantitative partition approximation. Its compatible refinements preserve all old array maps. The type III application retains the stated modular foundations; six solved exercises cover derivative orientation, centralizers and finite models.

Weighted finite models

Weighted finite models and biased tail classification proves simultaneous binomial capacity bounds, finite orbit retention and exact weighted refinement. Its address construction includes the Borel inverse and both directions of orbit equivalence, with six solved exercises and a reproducible capacity figure.