Prerequisite proofs and reading order

Building an intrinsic flow from modular coordinates

Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.

Extending the dual weight beyond the common involution domain

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Pointwise inner flows and continuous implementers

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Continuous decompositions, modular invariants, and canonical extensions

Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.

How the dual weight moves crossed-product generators

Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.

Changing the Hilbert space of a regular crossed product

Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.

Fixed elements and changes of density

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Contractive retractions and the algebraic structure of expectations

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Crossed products and the flow of weights

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Every bounded functional becomes normal in one representation

The complete four-positive-part and vector-coefficient proof precedes the all-state bidual construction. Restriction identifies its concrete predual isometrically with the original dual; positive functionals extend normally, positively and with the same norm. PCY02 proves the nonunital approximate-identity norm limit, PCY03 constructs the essential cyclic vector, and PCY04 proves state norming and order detection. The scalar, concrete-predual and bounded-density arguments have exact linked proof bodies. The representation-extension and ideal proof route follows.

Lemma 8.1 and Theorem 8.3 give the increasing one-sided ideal net, its strong projection limit, all six closures and centrality for two-sided ideals. The inverse-order proof is Proposition 8.5(7); Theorem 6.1 gives the full bounded monotone-net proof. Lemma 2.1 constructs the normal representation extension, central kernel and unit-ball surjection, treating the essential space explicitly. Its onto isometric preadjoint proves normality of the inverse. The following second-adjoint proof identifies inclusion ranges by annihilators. Proposition 4.1 supplies the ideal/quotient bidual splitting with a direct normal-inverse proof. The full quotient-predual argument supplies the required isometric duality and weak* topology.

Injective von Neumann algebras

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Conditional expectations from modular invariance

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The modular group and its analytic algebra

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Density from finite vector tests

ODF03 proves simultaneous finite-vector bicommutant approximation. ODF05 turns self-adjoint approximants into contractions by two resolvent identities. ODF06 gives positive matrix approximants with the target’s norm bound; ODF01 proves their ultraweak convergence. ODF02 includes nonunital, degenerate and zero representations, and ODF07 proves the full contraction-ball version used in the enveloping-algebra construction. The classical density theorem and resolvent method are due to Irving Kaplansky.

Predual compactness and the Bernoulli example

PC00–PC08 prove the exact quadratic compactness criterion and faithful-state specialization used by ordinary multipliers. The proof constructs positive vector envelopes, proves its needed sequential extraction, and makes the weak* closure normal by a real graph extension. PC09 constructs the Bernoulli probability, multiplication algebra, exact L1 predual and weak-null coordinate functionals.

Scalar measures, compact rectangles and completion

RM01–RM05 construct the unique finite regular Borel measure representing a positive real-linear functional on a compact metric space. The proof includes metric separation, Borel measurability, full Borel regularity, the integral identity and uniqueness. SC01 proves compactness of a closed rectangle; SC02 constructs the completion of any measure, including an infinite or non-sigma-finite one. Read the linked scalar integration and compact-metric proofs before these constructions. These are the scalar inputs to the bounded spectral and modular arguments used in NC00.

The positive cone of a standard representation

Natural-cone construction constructs the chosen faithful normal semifinite weight, its normal GNS representation and full original left Hilbert algebra, then proves positive form extension, endpoint duality, both quarter-power descriptions, self-duality and algebra preservation. It includes the universal cyclic adjoint core and central conjugation. Cyclic realization constructs every normal positive functional by a positive Fourier correction with residual norm reduced by a factor of one quarter, then uses a direct real-separation density proof. Cone geometry gives the support formula, orthogonal decomposition, squared-distance bound and arbitrary-algebra corner passage. The construction permits nonfaithful functionals and arbitrary Hilbert-space cardinality. The classical cone and correction methods are described in the free accounts of Hiai, Araki and Haagerup.

Trace inequalities for finite von Neumann algebras

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Trace densities and noncommutative integration

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Trace prerequisites: current available lessons and remaining public routes

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Trace Hilbert spaces, commutation, comparison and expectations

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General weights: finite domains, GNS spaces, and normal representations

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Weights and the Hilbert spaces of multiplication

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Periodic fixed-corner spectra and the modular intersection

PP00–PP04 prove the complete periodic action-spectrum formula and give an actual nonzero eigenoperator of every active degree in every nonzero fixed corner when the fixed algebra is a factor. PP05 applies this to the constant CAR product. The subtype uses the separate all-weight comparison in MG0–4, including balanced modular weights and exact type III corner transport, at its separable-predual factor scope.

General modular foundations and fixed corners

BG1–6 construct the modular conjugation, group, fullification and complete closed domains from bounded graph projections for every left Hilbert algebra. FC1–5 identify the full GNS and modular graphs of every nonzero modular-fixed corner of a faithful normal semifinite weight, including infinite corner weight. They supply the pure normalization and fixed-corner steps used by MG0–4; that every-weight comparison retains its finite-ideal/filter, balanced-weight and type III corner-transport hypotheses and inputs.

Normal GNS representations and whole weight domains

The full normal-functional GNS coefficient proof proves ultraweak continuity on the entire algebra, using the concrete norm-closed predual and summable vector tests. NF5 gives the corresponding theorem for every normal weight. EW1–5 prove the whole-cone normality/minorant theorem and bounded ultraweak-times-weak GNS graph closure, including infinite weight values. The human convexity component CV4 retains its CC BY4.0 terms.

Balanced weights and exact GNS graphs

BC1–5 give all finite matrix-weight domains, canonical unitary cocycles, their chain law and scalar normalization. BC3 identifies every matrix subcorner through its complete GNS graph and all spectral-power domains; BC4 treats identical weights by the full orthogonal-sum graph; BC5 proves isomorphism covariance by the full graph unitary. These arguments permit infinite corner weight and do not require the KMS uniqueness branch.

Whole bounded vectors and canonical weights

BW1–5 proves the entire multiplication ideals, adjoint intersections and mixed multiplication from BG1–6. Its BW4 proves modular covariance on all bounded vectors; BW5 supplies the bounded polar column. These are the explicit foundations of WF1–6, WR1–7 and MW1–4, preserving arbitrary faithful semifinite weights, infinite values and full closed GNS domains.

Finite interpolation and exact lifting

The direct compact-level interpolation proof, dual-ball construction, and scalar dilation and faithful representation are complete written bodies. The exact lifting diagram records the bounded cutoff, net selection, unchanged quotient and proved geometric tail. All original clauses and full solutions remain. The actual free source methods and their referred-out steps are distinguished in the lessons and version register; historical paid-expression/organization and earlier prerequisite qualification remain individual.

Tensor norms and nonunital nuclearity

The tensor construction proves the faithful spatial bound, commuting-action norm and split unitization sequence. Its proof diagram displays the exact finite-coordinate Gram mass. The eleven original statements and seven complete solutions are preserved. Actual source reading, existing written prerequisite bodies and their remaining source and historical-structure qualification are recorded separately.

Scalar GNS and matrix norms

The scalar positive-extension proof leads to GNS and a faithful representation before matrix norms are constructed. The boundedness and complete-positivity proof avoids assuming continuity of a positive functional. All six arguments include their stated arbitrary-cardinality, nonunital, degenerate and zero cases. The diagram shows the exact order and estimates. The version register qualifies the actual free scalar-source clauses and records the steps supplied locally; inherited lesson and solution expression/organization remains separately unfinished.

Normal positive parts with exact norm

The norm-attainment proof and rational range cutoffs give the two normal positive parts of a hermitian normal functional, with exactly additive norms. Norm equality proves uniqueness. The diagram shows the actual unitary and corner maps. This supplies the selected marginal-correction input without importing a general polar or bidual theorem; other course prerequisites and inherited expression/organization remain separately qualified.

Dilation and finite central covers

The nonunital tensor-factor construction uses the full operator-system extension theorem and unital dilation. The finite-cover argument uses normal representation extensions, central corner retractions, finite products, bounded monotone convergence and pointwise ultraweak compactness. Commutant permanence and the injective/semidiscrete implication retain their full exact natural-cone and weight obligations.

Inputs of normal representation comparison

The complete comparison proof uses CP-04 and CP-06, together with continuous square roots and bounded Hilbert forms from CF Sections 6–8. It requires no trace-class spectral decomposition or positive-functional vector theorem in the standard cone. Commutant permanence additionally requires a conjugate-linear isometric involution with JMJ equal to the commutant. Its general existence, and the arbitrary-weight/core inputs of Theorem 4.1, retain their full prerequisite obligations.

Inputs of the coefficient-compression transfer

Normal compression uses CP-04–06 summable predual coefficients and the concrete bicommutant property. Identity-neighborhood maps use H0.1 compact cutoffs, positivity and compact finiteness of the fixed left Haar measure, and the regular coefficient action. Finite-matrix transfer uses the normal recording-map definition of semidiscreteness. This transfer proof has no standard-cone, commutant tensor characterization, trace or modular-theory input. General Haar/integration and normal-representation topology retain their own full prerequisite source obligations.

General Haar existence and the integration boundary

HR-06 constructs Haar measure on every locally compact Hausdorff group from compact-content HR-04 and open-valuation HR-01, whose Borel measure step uses SC-01. Compact cutoffs and finite products are proved in H0; the ultrafilter input is the earlier CF proof. This does not require second countability or an invariant mean. The full HR-03 compact-continuous density argument uses SC-03–07; complete scalar-integration source qualification and the vector-valued integration inputs remain separate.

Scalar convergence and the vector-valued boundary

CF-01 proves the real mean-value and continuous Riemann/FTC tools before SC-00 real powers. SC-01–07 then prove measure construction, simple integration, monotone and complex dominated convergence, every Holder/Minkowski endpoint and full Lp completeness without a sigma-finiteness assumption. HR-03 combines these with H0 cutoffs and the outer-regular Radon convention for finite-exponent compact-continuous density. The regular prelude explicitly matches the scalar/tensor inner-product convention. Vector integration and carrier-qualified Radon products are proved by the complete arguments below. The group Haar normalization and translation arguments are supplied below. The standard J-pair is constructed below; operator-algebra modular theory and core inputs retain their exact separate obligations.

Vector integration and arbitrary Hilbert spaces

L24 Section2 and HR-08–09 give the exact finite-exponent completed/local Haar comparison. L24 Propositions4.1–4.2, SC-03–07, HR-03 and H0 prove Banach-valued integration and the onto Hilbert-valued L2 identification. The owned argument makes strong measurability, finite simple approximation and bounded coefficient-field multiplication explicit without assuming a separable Hilbert space or a sigma compact group. The full scalar Radon-product and L24 Proposition 4.3 vector Fubini arguments are supplied below at their explicit carrier conditions. The group Haar/modular/translation arguments are supplied below; the standard J-pair is constructed below, and the other original operator-algebra modular, core and hypertrace inputs retain separate obligations.

Radon products and vector Fubini

The complete HR-05 proof constructs the full Borel Radon product, proves its open/compact section formulae, carrier-qualified Tonelli/Fubini, qualified rectangle formula and scalar Hilbert tensor identification. H0 common-support product approximation uses the complete CF Stone–Weierstrass proof. The owned vector expansion proves the complete L24 Proposition 4.3 limit step, including outer strong measurability and carrier-qualified null sections, without separability of the ambient Hilbert space or group. Operator-algebra modular theory, weights/core/dual-weight, hypertrace and the remaining original course source/structure obligations remain separate.

Group modular identities and regular continuity

HR-07, full Radon products, compact cutoffs and positive-open-set Haar measure give uniqueness at arbitrary LCH-group generality. L24 Section 3 gives Delta, inversion and finite-p translation norms. The complete owned argument supplies the weighted-Radon regularity and Borel inversion extension, then full left-L2 strong continuity for arbitrary Hilbert spaces. Group modular normalization is thus separate from the remaining operator-algebra modular, weight/core/dual-weight and hypertrace arguments; the original course expression/organization audit also remains open.

The standard J-pair and an arbitrary-algebra weight

The full owned construction uses an arbitrary orthogonal family of normal vector-state supports, finite-subsum weight, direct normal Hilbert sum and a whole graph-closability argument. The complete bounded modular construction, with its exact free scalar/spectral/density inputs, gives JMJ=M′. This is the complete existence input of Theorem 1.3; only faithful normal semifinite weight existence is used here by Theorem 4.1. No faithful state, countable support family or separable predual is imposed. Natural-cone consumers, general modular theory for a given weight, dual weights/core/trace, hypertrace and the remaining original course expression/organization duties remain separate.

The full given-weight GNS inputs

The complete given-weight argument covers nonfaithful and nonsemifinite normal weights, including null Hilbert spaces. It proves finite ideals and the quotient first; then the support-state closed-form argument proves the finite-functional order/ultraweak equivalence and every GNS coefficient is normal. Semifinite cutoffs and finite-star density have complete proofs. The exact GW1–5, NF1–5 and FF4 representation inputs are bound to their free spectral/Hilbert/predual roots. Theorem4.1 still needs the separate given-weight involution, modular, dual-weight and core/trace arguments; this GNS proof does not clear those uses or the whole course.

The given-weight involution, recovery and modular action

The complete owned proof uses the whole-cone normal-minorant theorem to construct bounded intertwiners and polar vectors, then proves closability on the entire finite-star domain. The exact full bounded-vector ideals and weight recovery identify every finite and graph domain. The complete modular/opposite consequences give normal point-ultraweak continuity and invariance including infinite values, without a faithful-state or countability restriction. Theorem4.1 still requires its separate dual-weight/core/trace and hypertrace arguments; this qualification does not clear KMS uniqueness or the whole original course.

The full KMS and trace-criterion prerequisites

The whole owned proof establishes the full spectral KMS strip and analytic finite-star core, general-weight modular-group uniqueness and the whole-cone trace criterion. The continuous-vector integrals are proved for arbitrary Hilbert spaces. General-weight uniqueness retains weight invariance as a hypothesis; finite-functional invariance follows from KMS. Theorem 4.1 still requires its full dual-weight/core/trace and hypertrace arguments. No natural-cone, normalized cocycle or operator-valued weight theorem is inferred.

Full centralizer perturbation and trace conversion

The complete owned proof supplies whole bounded-density additivity, arbitrary unbounded-density finite/GNS domains, all infinite centralizer corners, the modular formula and exact normalized cocycle. The balanced graphs, whole analytic right-ideal argument and density-domain construction are at current complete free roots. Theorem4.1 uses the separate full dual-weight and affiliated-density construction now supplied below. Conditional trace conversion by itself does not establish those inputs, arbitrary cocycle realization or the full Radon–Nikodym classification converse.

Continuous-core weight, full domains and trace

The complete construction now supplies the dual-weight and trace-existence prerequisite used by Theorem4.1 at arbitrary algebra and Hilbert cardinality. Whole closed graphs and bounded-vector ideals lead to the full faithful normal semifinite weight; the resolvent supplies the affiliated implementing density, and the complete centralizer perturbation supplies the trace and scaling. Only GDW4 and RF5 are newly selected. General noninvariant-action comparison and cocycle realization are not premises. The scalar-trace corner passage is supplied below; the wider projection-semifinite route and full-course proof and inherited-expression/organization audits remain separate.

Finite-trace corners of the continuous core

The complete construction uses the faithful normal semifinite core trace. Nonzero finite-trace cuts produce a maximal orthogonal family; its finite sums increase strongly to one. Every corner has its specified faithful normal tracial state. Theorem5.1 and corner injectivity supply the matrix models, and the complete normal-functional corner passage controls the two errors. Theorem4.1 retains its arbitrary-algebra conclusion. The wider projection-semifinite theorem and whole-course prerequisite and inherited-structure audits remain separate.

Tensor lifts and common states

The completed-residual construction builds projective lifts directly, while the full Gaussian kernel proof establishes their uniqueness. A nearest-orthant calculation supplies the separating weights for one common pair of states. The repeated constant improvement uses finite third-root averages and continuous polar cutoffs. The scalar C*-calculus, universal-bidual and scalar complex-analysis inputs retain their specified proof scopes.

Finite modes, product overlaps and spectral bands

The finite compressed occupation densities construct positive states before the doubled Fock identification. The expectation tower gives the exact product-innerness criterion. The dyadic spectral-band construction supplies a small Hilbert–Schmidt diagonal perturbation. The constant-state parameter calculation uses the entire product modular domain and comparison with every faithful normal semifinite weight. The finite-ideal, balanced-weight, fixed-corner and scalar calculus prerequisites retain their stated proof scopes.

Complete solved problems and finite traces

The seven linked problem sets give complete arguments for product vectors and finite modes, Hilbert–Schmidt perturbations, the periodic core, tensor clipping, state domination and the sharp bilinear inequality. Every original mathematical condition, example and requested conclusion is retained. The convex-separation proof includes the full gauge construction. The finite-factor trace construction uses its supplied coefficient, compactness and fixed-point proofs. The all-weight modular comparison and broader source-organization qualifications retain their separately stated scopes.

Tensor lifting and the sharp bilinear bound

The completed lift retains approximation errors and every residual term. The kernel proof includes norm-preserving extension, arbitrary dual-ball compactness and the full Gaussian argument. The sharp bound derives its scalar strip estimate, original-algebra power curves and norm-preserving unitization extension. Its canonical bidual and polar inputs are the selected all-state, coefficient, operator-density and concrete polar constructions. These mathematical scopes are specified separately from the wider inherited-source and course-organization audit.

Weights, corners and the periodic CAR calculation

The modular intersection retains every faithful normal semifinite weight. Its consumed proofs include complete finite-ideal filters, closed involution graphs and balanced matrix corners. Coefficient closure and scalar normalization establish the predual duality, ultraweak density, corner-factor inference and Fourier constants used by this calculation. The selected weight foundations are specified separately from the broader CAR and inherited-source organization audit.

Spatial tensors and product-state supports

The complete proof component proves spatial commutation in arbitrary represented Hilbert spaces, whole tensor involution graphs, normal representation corners and slices, and arbitrary-index nonfaithful product supports. It distinguishes the state support from the central kernel of its normal GNS representation. The ITP application supplies the general modular graph and the CAR application retains all endpoint states. The retained foundation versions preserve their original licence notices. General inner-flow, density and broader inherited-source qualifications remain separately recorded.

General semifiniteness

The complete supporting proofs construct the support corners, traces, densities and continuous internal unitary group needed by the general infinite tensor product criterion. They include the full Borel-selection and measurable-operator prerequisites and retain arbitrary normal states, NSF traces and the original separable-predual hypothesis. Exact versions and source rights are in the source ledger. The course-wide wording, selection and organization audit remains unfinished.

Tensor norms and universal biduals

The complete supporting proofs construct the minimal tensor norm by finite Hilbert compressions and the universal bidual by state coefficients. They prove injective tensor maps, unit padding, arbitrary disjoint-block regrouping, normal extension, its whole unit-ball image and the globally normal supported inverse, including nonunital algebras and degenerate representations. The infinite tensor product universal-limit proof uses these constructions and the complete coefficient predual. Exact source versions and separate reuse terms are in the source ledger. The inherited wording, selection and organization audit remains unfinished.

From matrix norms to operators

The complete realization proof obtains one pair of finite-matrix states by an explicit distance-to-orthant argument, then constructs conjugate coefficient operators detecting every matrix norm. The full Gram quotient and arbitrary Hilbert direct sum are included. All three results and six solved exercises retain their mathematical scope. Exact scalar and Hilbert inputs, source versions and distinct human-source terms are in the ledger. The inherited course-wide wording, selection and organization audit remains unfinished.

Complete matrix dilation and bidual arguments

Completely bounded extension and factorization includes the full matrix-unit decomposition and finite-matrix bidual identification, with the entire contraction-ball and predual norm arguments. The exact scalar extension, dilation, matrix-norm and universal-bidual prerequisites are bound in the ledger. All eight original results and nine solved exercises retain their scope, including nonunital algebras and arbitrary Hilbert spaces. Historical wording, selection and organization remain a separate unfinished course audit.

Trace adjoints and rational matrix channels

The complete tracial-adjoint lesson supplies finite Rayleigh diagonalization, trace-coordinate positivity and normality, the weighted Hilbert adjoint and exact norm bound, and both rational-channel Kraus maps with the full commutator estimate. The scalar and coefficient-predual inputs are bound to exact retained proofs. The free publisher comparison is Zhou–Fang, Lemmas 3.5–3.6; its general Radon–Nikodym reference is replaced by finite coordinates. All three original results and ten full solved exercise duties retain their scope. Historical wording, selection and organization remain a separate unfinished course audit.

Trace-preserving finite models

The complete finite-model lesson proves the normal unit-defect correction and trace-2-norm convergence needed by its direct faithful-trace hypertrace input. The weighted Hilbert adjoint and rational Kraus channel supply both exact trace identities. Complete selected scalar/coefficient, trace-realization, density and bounded-polar arguments are bound to their actual versions; no continuous core, factor or separability hypothesis is imported. All five original results and ten full solved exercises are retained. The free publisher method comparison is Zhou–Fang, Lemmas 3.5–3.6 and the scalar Claim 1 of Theorem 3.2. Historical wording, topic/exercise selection and organization remain a separate unfinished audit.

Natural cones and exact marginals

Analytic multiplication, support corners and positive-functional realization supply the full arbitrary-algebra cone route. The proofs include complex-power domains, the positive form extension, actual ultraweak corner transport and the normalized Fourier correction. The independent normal Jordan argument and finite-matrix support normalization complete exact marginal correction and the generic semidiscrete commutant converse. All seven semidiscrete results have exact free mathematical input routes; all eleven tensor result routes are supplied in the subsequent tensor section. All18 original statements and13 full solved exercises are retained. Inherited wording, exercise/topic selection and organization keep their separate unfinished reviews. This private working repair does not certify a public snapshot.

Tensor nuclearity and type I algebras

The complete quotient construction, universal predual and bidual, positive-matrix density and arbitrary-dimensional type I spatial proof supply the remaining nuclearity inputs. All eleven original tensor results now have exact full free mathematical input routes, with all seven complete solved exercises preserved. The dual middle matrix space retains its trace-class norm. Inherited wording, topic/exercise selection and organization keep their separate unfinished audit. This private working edition does not certify a public snapshot.

Nuclear biduals and extensions

The full lesson proves the ideal projection, nonunital dilation retraction, finite central covers and predual route back to nuclearity. Balanced-weight graph restrictions, the continuous core and finite-trace corners supply the exact injectivity and semidiscreteness inputs. Coordinate duality, cyclic central support and explicit interval models cover the exercise prerequisites. All six original statements and eight full solutions are preserved. The dual matrix norm remains trace class. Inherited wording, selection and organization keep their separate unfinished audit; this private working repair does not certify a public snapshot.

Complete averaging and injectivity inputs

The twelve original arguments and ten full solutions have complete free mathematical inputs, including arbitrary-group regular models, finite matrix blocks and the type I corner construction. Inherited wording, selection and organization remain under review. Private working repairs precede Qual’s public snapshots.

Read the type I corner construction · Exact proof and source versions

Free-group norm averaging

Seven original results and ten whole solutions have complete freely readable proof inputs, including word normal forms, arbitrary-index Hilbert spaces and the faithful trace. The proof prose and organization have been rebuilt. Preserved result and exercise ancestry remains under review.

Read the complete averaging mechanism · Exact source and proof records

Projections in the reduced free group algebra

The seven original results and ten complete solutions have full freely readable mathematical inputs: compact spectral decomposition, the usual complete trace ideal and generic holomorphic calculus. The proof prose and organization have been rebuilt. Preserved result and exercise ancestry remains under review.

Read the integer-trace mechanism · Read the complete analytic proofs · Exact source and proof records

Topological inputs for central sequences

Complete TS1–6 proofs supply Polish metric constructions, Borel refinement, analytic separation, injective Borel images and the category/open-mapping argument. The two Polish-group results retain their full original scope. The normal-predual, general center-valued-trace and finite AFD uniqueness inputs of the rest of the lesson remain under review, as does inherited expression and organization.

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Preduals and faithful-state topology

NT1–2 proves normal-functional decomposition, predual module bounds and density, the Schwarz step, faithful-state bounded strong-star topology and separability of the GNS space. Eighteen exact complete earlier provider spans supply the twelve-node prerequisite chain. At the earlier normal-topology checkpoint, five original results had complete routes. The current finite-trace record below supersedes that frontier: the current complete record below closes all ten original results and all eleven whole exercise routes, while original-selection review remains separate.

Read the full functional and state arguments · Exact mathematical records

Finite traces with arbitrary centers

The complete retained PC0–7 and FCT0–7 arguments prove projection comparison, finite cancellation, central tilings and the normal faithful center-valued trace on every finite von Neumann algebra. No countability reduction or assumed scalar trace is used. This completes Corollary 1.2 at its full original scope. The preceding finite-trace checkpoint closed nine original routes. The complete current record below also supplies finite AFD uniqueness; all ten original routes and all eleven whole exercise routes are now complete. Literary selection review remains separate.

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Complete central sequences and factor examples

All ten original results and all eleven whole exercises now have complete free mathematical routes. The proofs retain arbitrary finite algebras, every separable-predual algebra with its center, all Polish groups, all countable free-group ranks at least two, and the full permutation-factor identification with R. AF1–5 supplies finite trace projections, scalar cuts, projection repair, exact dyadic containment and the normal tracial-completion isomorphism.

Read finite AFD uniqueness · Read the complete lesson · Exact proof versions and dependencies

Proof, problem and solution narratives have been reconstructed. Original selection ancestry and the full-course literary audit remain separate.

Central-sequence quotients and exact lifts

All six original results and ten whole solved exercises have complete free mathematical routes. CQ1–7 proves the C* quotient norm, faithful trace representation, metric completion and normal automorphisms, finite-subpiece cancellation with possibly infinite endpoints, small infinite pieces in both state seminorms, minimal-projection scalarity, and quantitative balancing of finite partitions. The lesson retains exact prescribed endpoints and every nonunital matrix support.

Read the whole lesson · Read the full prerequisite proofs · Exact source versions and dependencies

The exposition and organization have been rebuilt. The documented original selection ancestry and whole-course literary audit remain separate.

Fullness, hypercentrality and every corner

All six original results and ten whole solved exercises have complete free mathematical routes. FH0–5 proves normality of finite-factor tracial states, division with the center retained, compact-orbit displacement, the exact normal matrix tail map, the whole nonfactor example, and both-square estimates for varying centralizing multipliers. Ordinary and every-ultrafilter assertions are written separately, and type II1 remains distinct from a scalar center.

Read the whole lesson · Read the full prerequisite proofs · Exact proof versions and dependencies

The current proof route uses the complete programme inputs and the accessible Connes comparison. An inaccessible historical MathNet endpoint is not a required proof source. Original selection ancestry and the whole-course literary audit remain separate.

Matrix averages and tensor absorption

All five original results and ten whole solved exercises have complete free mathematical routes. SS0–4 proves the finite sign/permutation average and p^(3/2) bound, whole-domain normality, the complete variable matrix product, normal product-GNS splitting and the full stable type I example. Both summability theorems retain nonfactor algebras and infinitely many nontrivial matrix factors. All six absorption alternatives and every free-ultrafilter assertion retain their full scope.

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Original result/problem selection and whole-course organization remain a separate audit.

Centrally trivial automorphisms and decreasing AFD factors

All eight original results and twelve whole solved exercises have complete free mathematical routes. CT0–5 supplies normal transport, approximate innerness on R, the tensor relative commutant, complete finite and infinite Bernoulli actions, direct irreducibility and diffuse AFD finite fixed algebras, and compact twisted-orbit minimization. Both fixed states, all original constants, arbitrary II1 factors, every free ultrafilter and the scalar intersection retain their full scope.

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Finite outer actions and whole trace commutants

All six original results and eight whole solved exercises have complete free mathematical routes. FO0–4 supplies the whole trace representation and commutant, normal faithful ranges and entire-domain inverses, arbitrary closed central kernels, matrix-base Bernoulli products and the full finite Hilbert projection. General finite fixed-factor irreducibility and typeII1 retain nonseparable preduals; countable invariant stages give R, and directed stages retain the local-AFD boundary.

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Continuous outer actions and regular/CAR proofs

The three original results and eight whole solved exercises have complete free mathematical routes. CO0–5 supplies compact cutoffs, the exact complete Haar construction, whole L2 and regular representation, direct tracial CAR completion, full Hilbert–Schmidt innerness and point-predual continuity, and normal target transport. Second-countable locally compact Hausdorff groups include noncompact and nonunimodular cases; finite and trivial groups are treated explicitly.

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Bogoliubov generators and complete proof inputs

Six original results and twelve whole solved exercises have exact complete free mathematical routes. The generator proof keeps arbitrary unbounded self-adjoint h on its whole common domain and every real time. The scalar proofs cover the sine chord, logarithmic product bound, full diagonal singular sums and both series examples. The whole tracial CAR, diagonal/general innerness and full automorphism-group density are supplied by complete local components.

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Hyperfinite finite factors: complete algebra and vacuum inputs

Nine original results and ten whole solved exercises have complete free mathematical routes. The local proofs cover arbitrary-subalgebra trace expectations, all-state cutoffs, every finite center, full dyadic containment and corner converse, arbitrary Hilbert normal commutants, any-algebra finite-level implementation and the whole vacuum/tracial direct-sum example. Separability is retained exactly where the original assertion requires it.

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Completely positive finite models: complete inputs

Sixteen original results and eight whole solved exercises have complete free mathematical routes. The inputs include every nonunital and arbitrary-Hilbert tensor/dilation case, intrinsic finite-algebra units, the full commutative representation theorem and all matrix/compactness/separation constructions.

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Tracial adjoints and rational records: complete inputs

Three original results and ten whole scattered solved exercises have complete free mathematical routes. The lesson contains all finite-coordinate, normality, positivity, Hilbert-space and exact multiplicity inputs, for any given faithfully tracial von Neumann algebra and every faithful rational-spectrum matrix density.

Read the lesson · Read P00–P06 · Exact proof versions and dependencies

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Trace-preserving finite models: complete inputs

Five original results and ten whole solved exercises have complete free mathematical routes, including the incoming faithful-trace construction, exact unit correction, both rational perturbations and every trace and error estimate. The full stated scope includes arbitrary faithfully tracial injective algebras and unitary replacement for any unital completely positive finite-matrix map.

Read the lesson · Read the incoming normalization · Exact proof versions and dependencies

Inherited expression, original result/problem selection and course organization remain separately assessed.

Hypertraces: complete center and normality inputs

Thirteen original results and eight entire solved exercises have complete free mathematical routes, including arbitrary finite centers, possibly singular tracial states, uniform averaging and both semifinite corner passages. The bounded-functional normality theorem supplies the topology of the finite-trace corner state.

Read the lesson · Read the supporting constructions · Exact proof versions and dependencies

Inherited expression, original result and problem selection, and course organization remain separately assessed.

Completely bounded extension: complete inputs

Eight original results and nine entire solved exercises have complete free mathematical routes, including arbitrary nonclosed subspaces and Hilbert spaces, nonunital matrix biduals, exact balanced factorization and the multiplier conclusion for surjective two-isometries.

Read the lesson · Read the matrix bidual construction · Exact proof versions and dependencies

Inherited expression, original result and problem selection, and course organization remain separately assessed.

Complete abstract operator space realization inputs

The whole lesson proves all three original results and preserves all six complete solved exercises, without completeness or separability assumptions. The finite-state compactness and orthant-distance argument, conjugate coefficient operators, arbitrary Hilbert direct sum and minimum matrix-norm example are complete.

The exact retained scalar and Hilbert foundations and positive-functional and norming-state proof supply the prerequisites. Gupta–Mandayam–Sunder v3, Theorem 1.3.1, is the freely readable coefficient-method comparison; its omitted inequality (1.3) is proved locally. Effros–Ruan Theorem 2.2(1) states the representation result by reference and is not treated as its proof.

Exact complete proof bindings retain versions, locations, dependencies and existing licence notices. This mathematical closure does not clear inherited expression, original problem selection or course organization.

Complete completely positive lifting inputs

The whole lifting lesson contains all14 original results and9 complete solved exercises. It proves the quasicentral cutoff construction, quotient norm, exact gluing, summable selection and full nonunital norm control. The quotient algebra and ideal need no separability assumption.

The complete finite-model proofs, retained scalar and matrix foundations, bounded square-root construction and full nuclearity equivalence supply the exact prerequisites. Gabe v4 and Olesen Section2.4 provide freely readable comparisons; delegated or scope-mismatched steps are proved locally.

Complete versioned proof bindings retain dependencies and licence notices. Historical restricted-source reading and inherited expression, selection and organization remain separately unresolved.

Whole tensor positivity and nuclearity inputs

The whole lesson preserves all11 results and7 entire solved exercises. Its complete tensor constructions, normal marginal correction, nuclearity equivalence, dual Banach norm arguments and arbitrary-cardinality type I spatial construction are version-bound.

Every solution has its own complete proof binding. In particular the ideal-permanence solution uses the full finite-model ideal-compression argument and its explicit approximate-identity prerequisites. Author-hosted Blackadar, university-hosted Olesen and the publisher papers provide free mathematical comparisons; omitted source proofs are supplied locally.

Complete versioned proof records retain source and licence boundaries. Historical target selection and inherited wording and organization remain separately unresolved.

Whole semidiscrete finite-model inputs

The whole lesson preserves all7 results and6 entire solved exercises. Its commutant criterion, independent normal-representation construction, predual matrix norms, extension argument and norm-one projection criterion have complete versioned prerequisite proofs.

The tensor positivity arguments supply the full normal exact-marginal route. The representation comparison keeps arbitrary Hilbert cardinalities. Tomiyama provides the credited norm-one projection method; the author-hosted Anantharaman–Popa draft supplies the dominated-form comparison, with the current and historical PDF versions recorded separately.

Complete versioned proof records preserve source and licence boundaries. Inherited wording, historical target selection and course organization remain separately unresolved.

Whole nuclear-bidual and extension inputs

The whole lesson preserves all six results, eight complete solved exercises, both examples and the interval diagram. Its ideal projection, normal quotient inverse, nonunital dilation, finite central covers, core and predual arguments have complete versioned proof bindings.

Each solution has its own prerequisite route. The retraction criterion uses the full ucp retraction equivalence; the passage through preduals uses the full predual finite-model equivalence. The selected balanced-weight and continuous-core proofs retain their exact qualified versions. The complete retained strip, graph and finite-domain arguments preserve the reviewed prerequisites through later programme revisions.

Complete versioned proof records preserve source rights and earlier substantive reconstruction evidence. Inherited expression, original target and problem selection, and course organization remain separately unresolved.

Whole averaging and crossed-product inputs

The complete lesson preserves twelve results and ten entire solved exercises. Its independent normal-representation comparison, arbitrary-algebra standard pair, whole weight and KMS domains, core trace, Haar and regular model, normal coefficient compression and type I corner construction have complete versioned proof routes.

Each solution has its own prerequisite route. The four bounded-strip, pointwise-graph, entire finite-domain and modular-uniqueness links lead to their exact reviewed arguments. The human-source reference distinguishes posed exercises from supplied proofs.

The proof record gives all 352 acyclic nodes and 325 complete provider bindings. A bounded source comparison identifies specific construction differences while retaining six historical target correspondences. Inherited expression, original problem/result selection and whole-course organization remain separately unclosed.

Free-group norm averaging: proof dependencies

The lesson constructs reduced words, the regular operator algebra and its faithful trace before deriving finite conjugation averages. The cancellation argument works with two fixed generators for an arbitrary generator set. The orthogonal-range estimate applies to nonselfadjoint operators, and polynomial approximation gives the full norm-limit consequences.

The CF1–10 foundations supply Hilbert-space adjoints and positive C*-algebra calculus. The rank-one counterexample uses explicit approximate eigenvectors; it requires neither a Fourier transform nor character classification.

De la Harpe’s freely readable account gives the Powers partition criterion and range-averaging method. The local proof supplies the finite word partition directly and uses the convergent near-unit inverse series in the difference from the identity. The historical exercise-part correspondences and specific cancellation recipe remain documented in the source record; no conclusion of independent original selection or organization is asserted.

Projections in the reduced free-group algebra: analytic dependencies

The projection argument realizes a second representation with one missing basis vector. Finite generator differences extend to a dense trace-class graph algebra, whose actual representation units are retained in the resolvent formula.

The analytic module proves compact spectral decomposition, the ordinary Banach trace ideal and its absolute diagonal trace, and holomorphic calculus on arbitrary given neighborhoods. Spectral pairing gives integer trace; a nearby projection and an explicit unitary transfer the conclusion to the full reduced algebra.

Schick’s freely readable account supplies a comparison for the diagonal trace and different-unit method. The local argument requires no Hilbert-module relative theorem or K-theory calculation. The historical six-part exercise correspondence and modified-generator construction are documented in the source record; independent original selection or organization is not asserted.

Invariant states and trace Hilbert spaces

The invariant-state lesson passes from a possibly singular invariant state to a normal density and one finite-trace spectral level. The square-root and level estimates cover positive elements of the full semifinite trace Hilbert space.

Trace Hilbert spaces and finite trace expectations constructs the complete operator space, sharp product bound, cyclic pairing, finite trace expectations, decreasing-space intersection, exact factor trace cuts and ordinary Hilbert-space rank. The finite injective algebra corollary uses the hypertrace and an explicit bounded Gram correction.

The near-covering family uses the complete global scalar-pinching theorem, including compatible matrices and the decreasing AFD construction. The human one-corner Rohlin lemmas have a smaller conclusion. Exact proof versions and the historical invariant-state, finite-rank and pinching correspondence are given in the source record; independent original expression, selection or organization is not asserted.

Semiregular MASAs and scalar pinching

The semiregular-MASA lesson matches equal-trace projections by partial normalizers, proves corner regularity, and constructs a dyadic AFD factor containing the entire given MASA. Each diagonal is chosen in the current first corner and transported through compatible matrix rows.

Exact diffuse trace cuts and controlled dyadic transfers supply equal atoms, including the support reduction for a nonfaithful finite abelian trace. Finite-stage trace projections and the actual trace-GNS unitary identify the factor. The finite trace expectation proof supplies its expectations and decreasing-space limit.

The scalar theorem uses decreasing irreducible AFD factors and two distinct refinements. All five statements and twelve full solutions have exact routes in the source record. The historical normalizer, AFD-inclusion and scalar-pinching progression is retained as provenance; independent original expression, selection or organization is not asserted.

Small matrix corners

Repairing small matrix corners cuts away a spectral defect and removes the ranges already used. Exact partial isometries then give a finite matrix system with one common initial projection.

The full Gram induction keeps the trace-loss recurrence and the square-root trace ratio in coefficient estimates. Its first three results apply to arbitrary finite algebras with faithful normal tracial state. The near-covering corollary uses the global scalar-pinching selection, retaining its factor and separability hypotheses. Its initial projections are orthogonal; its range supports may overlap.

The source record binds every result and all ten complete solutions to full trace, spectral and scalar arguments. The author draft supplies the complete deformation-lemma comparison. Historical polar, range, Gram and coefficient-transfer selection remains documented without asserting independent original expression or organization.

Small corners and maximality

Small corners and the second injective proof selects one almost invariant matrix corner from an external finite-rank model, then fills the identity by a quantitative maximality argument.

The local-to-global theorem applies to every finite von Neumann algebra with faithful normal tracial state. Its proof treats increasing support projections, bounded density on the support corner, the Hilbert projection limit and orthogonal squared errors. The injective-to-small-corner construction retains its separable factor hypothesis. The final identification with the tracial matrix product uses the separate exact-containment theorem.

The source record binds four full results, the small-corner definition and all ten solved exercises to complete finite-trace, corner and dyadic arguments. Popa’s publisher article and the author draft provide the human comparisons. The historical result selection and order are documented without asserting independent organization.

Properly infinite algebras and dyadic approximation

Properly infinite injective algebras and dyadic approximation constructs unitary approximants to an isometry, handles unbounded approximants through resolvents, and extends finite matrix blocks to dyadic factors.

The local injective conclusion holds without separability or a faithful normal state. A generating dyadic sequence uses separable predual, an explicit countable dense set and exact containment through the relative commutant. Normal GNS representations supply the vector estimates; the complete internal Choi calculation gives one compression by an isometry.

The source record binds seven complete results and ten solved exercises to full projection, normal-state, Choi and semidiscrete proofs. Elliott–Woods supplies the complete human dyadic comparison. The historical result order and reconstruction method are documented without asserting independent organization.

Central traces and finite AFD structure

Central traces and AFD finite algebras cuts a projection to a prescribed central dimension, rounds varying matrix ranks on finite central partitions and enlarges a chosen dyadic factor exactly.

The cuts and refinement require neither factoriality nor separability. Separable predual supplies the increasing sequence for the center-times-tracial-product theorem. The directed AFD injectivity proof uses closed commutants, finite matrix-unit averages and a normal standard representation. Finite type I, finite type II and properly infinite central parts have explicit sequence constructions.

The source record binds five whole results and ten solved exercises to complete center-trace, projection, expectation and dyadic arguments. The human author draft supplies factor comparisons; its unrestricted center-trace reference is not a substitute for the local PC/FCT construction. Historical selection and method order are documented without asserting independent organization.

Balanced reconstruction and unitary couplings

Balancing Kraus families and unitary couplings fills matched positive defects, averages the outgoing Kraus sum in a matrix relative commutant, and constructs normal trace-preserving ucp extensions with both Kraus sums equal to one.

The two-defect lemma retains center-valued trace equality in arbitrary faithfully tracial finite algebras. The simultaneous norm averages have both finite projection-cut and spectral-contraction proofs. Balanced reconstruction and its injective application require a type II factor, with no separability assumption.

The source record binds all five results and ten whole solutions to complete PC/FCT, finite trace expectationTL3, internal Choi Kraus and trace-preserving finite-model arguments. Historical method and result selection remain documented without claiming independent organization.

Multiplier ultraproducts and normal embeddings

Multiplier ultraproducts and normal embeddings constructs the multiplier quotient of bounded strongstar-null sequences, its faithful normal state and the normal constant copy. Two-vector metric completeness gives closure for arbitrary strong Cauchy nets.

A faithful normal state suffices for the full quotient; separable predual is retained for the finite centralizing subalgebra. Chosen convergent automorphism families act normally, with an explicit inverse predual estimate. Possibly nonunital dense subalgebras supply bounded representatives, and type I factors have constant quotients.

The source record binds all nine results and fourteen complete solutions to full C*quotient, predual, faithful-state topology, spectral and bounded-density proofs. Ocneanu supplies the direct free antecedent; the Ando-Haagerup support-corner method is compared with its by-reference prerequisite kept distinct. Historical selection and organization are documented without claiming independence.

Unitary couplings and finite injective factors

Unitary couplings and finite injective factors samples balanced intertwiners by finite fourth-root averages, clips their norms by trace-Hilbert functional calculus and obtains a single unitary through finite center-valued projection comparison in a normal ultrapower.

Sections 1–4 allow every faithfully tracial finite algebra. The injective II₁ application is local without separability; separable predual gives a generating dyadic sequence and its tracial product identification. The II∞ corollary constructs countably many equivalent finite corners, absorbs the residual projection and proves the full spatial matrix decomposition.

The source record gives exact whole proof and prerequisite locators for all seven results and ten complete solutions. Haagerup supplies the sampling, clipping and ultrapower method. Anantharaman–Popa supplies finite matrix coordinates and the statement of the countable decomposition; the latter has an explicit proof in Corollary 5.2.

Central disintegration and measurable matrix assembly

Central disintegration and measurable matrix assembly constructs the central tracial field of a finite algebra with separable predual. Stabilized dyadic matrix models yield finite-factor approximants, which are selected by countably many rational tests and assembled by Gram repair on central pieces.

Countably generated measurable fields and their algebras proves the Gram-field construction, the diagonal commutant, compact continuous-equation selections, generated algebras, centers and fixed-base uniqueness, and tensor stabilization. The measure representation uses a full compact-metric positive-cover construction.

The source record gives exact full proof and prerequisite locators for four results and twelve complete solutions, including scalar integration, compact measure, faithful trace topology, finite-factor approximation, retractions and Gram repair. Effros supplies the general measurable-choice context; Elliott and Haagerup supply the nonfactor antecedents.

Injective algebras and separable tracial envelopes

Injective algebras and separable tracial envelopes constructs a separable injective algebra around any countable set in a faithfully tracial algebra. Exact trace-preserving matrix maps, positive integrable densities and central support cuts give finite approximation at every normal neighborhood without countability assumptions.

The full local equivalence combines finite and properly infinite summands and proves the converse by explicit finite invariant averages. A complete nonfactor matrix reduction gives the center-times-R structures, and trace-preserving expectations give the finite central decomposition of every von Neumann subalgebra of R. The source record gives full current proof and prerequisite locators for six results and twelve complete solutions.

Ultraproduct expectations and fast reindexing

Ultraproduct expectations and semi-lift ambiguity constructs the faithful normal expectation, its original-center-valued trace and covariance for a specified convergent automorphism family. Exact tensor-tail matrices distinguish different semi-lifts with the same limit and show the obstruction to unrestricted equivariance.

Fast reindexing with liftable actions proves the constant-lift theorem with an ordered M-valued expectation identity. It includes countable data, all finite selections, fixed two-sided multiplier moduli and the full normal injective extension. The source records give complete current proof and prerequisite locators for nine results and eighteen complete solutions, with general and separable hypotheses distinguished.

Slow reindexing and semi-lift compatibility

Slow reindexing first selects fixed inner operators and then nested outer neighborhoods. Its full image commutes with the centralizing part of a prescribed algebra. The construction preserves the ordered M-valued expectation identity and makes each actual semi-lift agree on the image with its associated constant lift. The lesson gives the complete multiplier and normal-extension arguments, four results, twelve complete solutions and a tensor example distinguishing this compatibility from stronger domain equivariance.

Ordinary multipliers and predual compactness

Ordinary multipliers proves the two-sided multiplier criterion with both adjoint directions, its all-free-ultrafilters formulation and the quadratic-orbit weak-compactness characterization. The full positive-controller argument includes its Baire and spectral-cut proof, normality by graph extension and the exact Bernoulli probability and L1 predual. Four results and twelve complete solutions include rank-one, Bernoulli and scalar examples explaining the adjoint, weak-limit and closure conditions.

Index selection and uniform multiplier control

Index selection proves the diagonal construction under common row multiplier moduli, with expectation, constant, centralizing and actual semi-lift action clauses. The exact state kernel describes its injective quotient. The full finite-trace case includes nonfactor algebras, and a two-dimensional swap-unitary example proves that unrestricted row membership cannot replace common control. Five results and twelve complete solutions include the complete countable data, finite selections, multiplier and norm-extension proofs.

Abelian pinching and relative commutants

Abelian pinching develops finite state-preserving expectations, arbitrary corner commutants and simultaneous binary reduction of the full tuple energy. Directed scalar-gap limits characterize maximal abelianness; scalar approximation and exact dyadic trace transfers supply equivalent atoms. Seven results and twelve complete solutions use the full faithful-state expectation and projection/trace foundations, with no factor or separability restriction beyond the stated specializations.

MASAs in expected semifinite subfactors

Expected semifinite subfactors constructs an ambient MASA semiregular in the subfactor and regular there in the AFD case. Four results and twelve complete solutions include the full normal-support faithfulness proof, coherent finite dyadic matrices and trace-GNS factor identification, countable semifinite coordinates, both type I endpoints, independent finite-corner injectivity and the irreducible state-centralizer corollary.

Group MASAs and singular affine examples

Group MASAs develops the subgroup-orbit criterion, affine ICC and malnormality, finite-support coefficient inequalities and singularity for every countably infinite field and infinite dilation subgroup. Six results and ten complete solutions use bounded subgroup compressions and direct regular-group trace and commutant proofs. The exact double-coset bijection covers finite and infinite multiplicative indices.

Double cosets and intrinsic MASA multiplicity

Double cosets identifies the canonical central subgroup projection and every nontrivial regular product orbit, calculates the full matrix commutant and proves intrinsic finite or countably infinite multiplicity under abstract pair isomorphisms. Four results and ten complete solutions include the finite-subgroup counterexample, all inter-copy operators, exact trace transport, finite character-rank proofs and the proper infinite shift.

Finite-field towers and prescribed MASA multiplicities

Finite-field towers develops finite-field existence and cyclicity, an elementary cyclotomic prime argument, compatible primary decompositions and an exact cyclic quotient. Five results and ten complete solutions give singular MASAs of every positive finite multiplicity and a rational example of countably infinite multiplicity, with full group, commutant and pair-isomorphism prerequisites.

Symmetric tensor norms and Rademacher truncation

Symmetric tensor norms constructs state detectors, calculates the complete noncommuting fourth moment, and balances clipping cost against a contracting residual. Five results and ten complete solutions give projective lifts for arbitrary completed self-adjoint tensors, with every approximation error and an explicit choice for any positive final tolerance. The finite affine example includes its exact nearest-pair proof.

Tensor completion, injectivity and state domination

Tensor completion and state domination proves that projective tensors remain distinct in the symmetric completion. A Gaussian norm estimate tests the kernel; unique lifts give the real norm comparison, and a compact affine argument chooses states for every input. Six results and ten complete solutions include arbitrary dual-ball compactness and explicit subnet constructions.

The sharp noncommutative bilinear inequality

The sharp bilinear inequality combines norm-holomorphic polar powers, a scalar strip estimate and exact finite third-root moments to improve a finite coefficient to the form norm. Four results and ten complete solutions include common-state selection, norm-preserving nonunital extensions, canonical bidual and polar foundations, and an orthogonal-isometry proof of optimality.

Infinite tensor products and reference states

Infinite tensor products develops directed limits, pointed Hilbert products and their normal operator algebras. Thirteen results and twelve complete solutions cover normal expectations, commutants and factors, faithful traces, the full product modular operator, inner automorphisms and the semifiniteness criterion at its separable predual hypothesis, including nonfaithful reference states.

Fermions, Fock space and quasi-free factors

Fermions and quasi-free factors develops arbitrary exterior Fock spaces, finite CAR matrix algebras and positive states for every covariance. Seven results and ten complete solutions cover even-leg tensor norms, factorality and finite-dimensional approximation in every representation, the unique tracial half-state, and faithful constant-state modular periods and finite-corner obstructions.

Constant CAR states and the periodic core

Constant CAR states and the periodic core gives eight results and twelve complete solutions. The faithful countable product leads to a normal occupation diagonal and a complex Bernoulli invariance argument, a type II1 centralizer, the Fourier corners of the period-circle crossed product and the center of the continuous core. Explicit coefficient resolvents and full arbitrary-weight GNS filters, balanced weights and fixed-corner arguments determine the intersection over every faithful normal semifinite weight. The hypotheses distinguish the infinite nontracial product from finite matrices, the half-state and the endpoint states.

Finite measured groupoids

The finite-stage labels identify each stage with a matrix field algebra. Discretizing the measurable center gives finite-dimensional approximation, including for nonergodic relations. The ergodic type II₁ conclusion uses the invariant-diagonal center, separability and finite-factor uniqueness. The lesson contains six results and ten solved exercises; its counting, diagonal-closure and equivalent-measure arguments are given in Section 8. Exact proof inputs are listed in the source records.