# Prerequisite proofs and reading order

The lessons state their hypotheses at each use. The following register identifies the actual programme lessons supplying the selected proofs. These are written proof interfaces checked by the author; this course does not certify completion of every provider course or all of its upstream prerequisites. Public links identify available editions; the checked mathematical revisions and any public-edition differences are retained in the source records.

Start with completely positive finite models, completely bounded extension and matrix norms. Continue through tensor positivity, lifting, semidiscreteness, hypertraces, averaging and the bidual theorem. These supply the approximation tools used later. The tensor-product and CAR lessons precede finite AFD uniqueness, central sequences, strong stability and MASAs. The three bilinear-inequality lessons form their own consecutive chain.

For the first finite injective proof, read weighted tracial adjoints, trace-preserving finite models, balanced Kraus families and unitary couplings in that order. Central disintegration and tracial envelopes then give the general nonfactor result. The second finite proof runs from invariant states through repaired small matrix corners to maximality.

The finite expected-subfactor theorem and scalar MASA pinching precede that second proof. The semifinite MASA regularity conclusion additionally uses the finite injective theorem: read its finite branch first and return to its type II-infinity regularity step after unitary couplings. The finite scalar-pinching argument does not use that later branch.

The basic entry requirements are Hilbert-space and Banach-space functional analysis, Hahn–Banach separation, weak-star compactness, measure integration, C*-functional calculus, states and GNS representations. Arbitrary-cardinality or nonunital inputs are retained where stated. Separable-predual, faithful-state, trace, normal-expectation and irreducibility hypotheses are imposed only at their actual uses.

| Preceding programme lesson | Responsible programme course | Selected proof content and locators |
| --- | --- | --- |
| [Completely positive maps](https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/completely-positive-maps.html) | Foundations of von Neumann algebras and its foundation remainder | Theorem 6.1 and normal/commutant dilation span; Proposition 7.1 dual order; Theorem 7.3 dominated-functional complete order isomorphism; Theorem 7.3(4) full dominated composition and norm proof |
| [Operator density from finite vector tests](operator-density-foundations.html) | Positive maps and finite-dimensional approximation | ODF01 bounded product and ultraweak series-tail estimate; ODF02 supported nonunital representation; ODF03 finite-vector bicommutant; ODF04 real self-adjoint density; ODF05 direct rational resolvents; ODF06 positive matrix density with the target norm bound; ODF07 strong* and ultraweak contraction-ball density. Human method: [Kaplansky, Theorem1 and Lemmas1–5](https://msp.org/pjm/1951/1-2/pjm-v1-n2-p06-p.pdf). |
| [A finite regular measure from positive function covers](compact-rmk-from-function-covers.html) and [compact rectangles and measure completion](scalar-topology-completion-bridges.html) | Positive maps and finite-dimensional approximation; scalar integration and compact-metric topology | RM01–RM05 construct the finite regular Borel measure of a positive real-linear functional on a compact metric space, prove its mass, Borel measurability, outer and compact-inner regularity, representation and uniqueness. SC01 proves compactness of every closed rectangle; SC02 constructs completion of any measure without finiteness or sigma-finiteness. SS1–SS2 and SS4 supply scalar Hilbert completeness, compact-metric facts and measurable representatives. Classical positive-functional methods are credited to [Fremlin, Chapter43](https://www1.essex.ac.uk/maths/people/fremlin/chap43.tex). |
| [Natural-cone construction](natural-cone-construction.html), [cyclic realization](cyclic-cone-realization.html), and [cone geometry](standard-cone-geometry.html) | Positive maps and finite-dimensional approximation; the earlier modular operator proofs | NC01–NC01b construct the chosen faithful normal semifinite weight, its normal GNS representation, original involution graph and full left Hilbert algebra. NC00–NC08 construct the cone from bounded multiplication, including the original graph cores, positive form extension, endpoint duality, both quarter-power descriptions, Gaussian self-duality and algebra preservation. CR01–CR07 prove the mass-one Fourier correction, support-corner transport, quarter-error iteration, direct real-separation density and cyclic normal-functional realization. CG01–CG10 prove supports, orthogonal decomposition, squared-distance estimates, corner commutants and the arbitrary-algebra passage. Classical methods: [Hiai, Lemmas3.3 and3.17–3.19](https://arxiv.org/pdf/2004.02383v1), [Araki](https://msp.org/pjm/1974/50-2/pjm-v50-n2-p02-p.pdf), and [Haagerup](https://journals.msp.org/mscand/article/download/2067/2066/2098). |
| [Contractive retractions and the algebraic structure of expectations](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#contractive-expectations) | Modular theory and weights | Section 3 orthogonal left support estimate; Section 4 nonunital retraction theorem; Section 5 matrix positivity test; Section 6 CP and complete contractivity |
| [Injective von Neumann algebras](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#injective-von-neumann-algebras) | Injective factors | Definition 1.2; Theorem 2.1; Corollary 2.2; B5 normal representation comparison; Proposition 1.4 opposite systems; Proposition 2.3(a)–(d) full corner/product/arbitrary-amplification proofs; Lemma 4.1 and Theorems 4.2–4.3 full commutant and directed permanence proofs; Theorem 5.1 full amenable-normalizer averaging proof |
| [Multiplicity of a von Neumann algebra on a Hilbert space](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FOUND-REMAINDER/reader/supplements/normal-representation-comparison.html) | Foundations of von Neumann algebras and its foundation remainder | Fact 2.7; Remark 3.2 |
| [Spatial tensor products of von Neumann algebras](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FOUND-REMAINDER/reader/supplements/spatial-tensor-products.html) | Foundations of von Neumann algebras and its foundation remainder | Proposition 3.1(4), full positive normal vector realization proof; Theorem 8.2, full arbitrary-cardinality cyclic decomposition proof; Theorem 9.2 full normal slice/module/Fubini proof; Theorem 10.1 full normal product/predual/support proof; Section 11 Lemmas 11.1–11.3 and Theorem 11.4 full commutation proof |
| [Trace inequalities for finite von Neumann algebras](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#trace-inequalities-for-finite-von-neumann-algebras) | Injective factors | Theorem 2.1 full semifinite joint-distribution proof, reread with Background B1–B9; Theorem 3.1 full Powers–Stormer proof; Lemma 4.1 layer-cake proof; Proposition 4.3 full spectral-truncation proof; Example 4.4 stricter bound obstruction |
| [Traces on von Neumann algebras](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FOUND-REMAINDER/reader/supplements/traces-on-von-neumann-algebras-part-a-def-v-2-1-to-def-v-2-17.html) | Foundations of von Neumann algebras and its foundation remainder | Proposition 1.6(c) full semifinite projection proof; Theorem 5.2 full center-valued trace proof; Lemmas 5.6 and 5.7 and full Theorem 5.5 proof for normal or singular finite traces; Corollary 5.12 full central decomposition proof; Corollary 5.4 full center-valued trace projection comparison proof |
| [Integration for a trace, the commutation theorem, and applications](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#traces-on-von-neumann-algebras-part-b-thm-v-2-18-to-prop-v-2-36-exerci) | Foundations of von Neumann algebras and its foundation remainder | Proposition 2.4 full bounded-vector proof; Theorem 3.1 full commutation theorem proof; Theorem4.1 and Corollary4.2 full semifinite commutant/type comparison proofs; Proposition8.10 full finite-projection extended-trace criterion proof |
| [Projections and types of von Neumann algebras](https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras.html) | Foundations of von Neumann algebras and its foundation remainder | Theorem 14.1 full finite joins/sums and modularity proof; Proposition 11.2 and Corollary 11.3 full faithful normal abelian-commutant proof; Proposition 3.5 full central-support and induced-kernel proof; Proposition13.3 full type II halving proof; Proposition13.4 and Corollary13.5 full properly infinite halving/division proofs; Proposition14.2 full finite complementary-projection equivalence proof; Proposition15.2 complete countable absorption and sigma-finite infinite-projection equivalence proof; Lemma7.4 full finite full-central-support projection proof; Lemma8.2 and Proposition8.4 and Lemma8.5 full matrix splitting proof; Proposition9.1, Lemmas9.3/9.5/10.2 and Theorem10.3 full finite type I structure proof |
| [Weights and the Hilbert spaces of multiplication](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#weight-hilbert-algebra) | Modular theory and weights | Section 13 complete existence proof, orthogonal normal-state supports and finite-subset sum |
| [General weights: finite domains, GNS spaces, and normal representations](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#weight-gns) | Modular theory and weights | Section 8 full finite positive cutoff equivalence proof; Section 7 full normal GNS representation proof for arbitrary normal weights, including states |
| [Fixed elements and changes of density](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#centralizers-and-perturbations) | Modular theory and weights | Section 8 full construction, normality, semifiniteness and exact support proof; Section 11 full supported modular-group proof; Section 5 full finite-domain cyclicity and unitary weight invariance criterion |
| [Changing the Hilbert space of a regular crossed product](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#15-regular-model-independence) | Crossed products and the flow of weights | Haar and operator-algebra prerequisite contract; Coefficient operators and left translations, full construction/covariance proof (R2–R3); Normality from compact monotonicity, full faithful normal construction proof (R4–R6) |
| [Building an intrinsic flow from modular coordinates](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#04-core-and-flow) | Crossed products and the flow of weights | Core in a weight chart, Stone generator and affiliation; Why the dual weight has an inner modular action, full proof; Removing the modular density, full proof; only existence/NSF trace is used |
| [Extending the dual weight beyond the common involution domain](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#13-dual-weight-construction) | Crossed products and the flow of weights | Two domains and representations; General left Hilbert algebra-to-weight prerequisite contract; Constructing the weight on the crossed product (W1), full existence argument |
| [How the dual weight moves crossed-product generators](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#14-dual-weight-modular-action) | Crossed products and the flow of weights | General closed Tomita graph and modular implementation prerequisites (A6–A7); Modular operator of the dual weight (A8–A10), full identification argument; Two generator formulas (A11–A16), full coefficient/group computation |
| [Every bounded functional becomes normal in one representation](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#cstar-universal-bidual) | Modular theory and weights | Prerequisite contract; Section 3–05 full all-state universal representation, predual isometry and bidual structure proofs; Section 6–08 full normal extension, central kernel, compact-ball surjectivity and normal-inverse proofs; Section 9 full second-adjoint functoriality, isometric inclusion and annihilator-range proof |
| [AF-algebras](https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/af-algebras.html) | Foundations of von Neumann algebras and its foundation remainder | Definition 4.1 and Proposition 4.2 full norm/universal construction; Example 4.3; Lemma 4.4; Theorem 4.5 and Remark 4.6 full compatibility proofs; Definition 4.7 and Proposition 4.8 full sequential AF/separability proof; Example 5.3 full UHF/CAR finite-string and inductive-limit comparison |
| [The modular group and its analytic algebra](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#modular-fundamental-theorem) | Modular theory and weights | Section 5 full JMJ commutation proof; Section 6 full covariance, automorphism and weight invariance proof |
| [Trace densities and noncommutative integration](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#trace-integration) | Modular theory and weights | Section 6 full isometric onto trace pairing and positive-cone proof |
| [Pointwise inner flows and continuous implementers](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#136-pointwise-inner-flows-and-continuous-implementers) | Crossed products and the flow of weights | Complete theorem and proof, equations (1)–(46), including Borel section, alternating bicharacter, measurable twist, integrability, common-conull disintegration and character cancellation |
| [Conditional expectations from modular invariance](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#modular-expectations) | Modular theory and weights | Section 1–08 full forward existence, compression normality, weight preservation and uniqueness/GNS projection proofs |
| [Polish spaces and standard Borel spaces](https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces.html) | Foundations of von Neumann algebras and its foundation remainder | Background F1–F4 prerequisite declaration; Theorem3.6 complete refinement, tree-separation, Borel image and inverse proof |
| [Compact and trace-class operators, the predual of B(H), and the operator topologies](https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html) | Foundations of von Neumann algebras and its foundation remainder | Theorem2.3 full Schmidt/self-adjoint compact spectral decomposition proof; Definition4.1, Lemma4.2, Theorem4.3, Corollary4.4 and Theorem4.6 full trace-class completeness, summability, ideal and absolute trace proofs |
| [Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory](https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.html) | Foundations of von Neumann algebras and its foundation remainder | Definition6.4 and Proposition6.5 complete surrounding-cycle/germ proof; Theorems6.7 and6.10 full homomorphism, spectral mapping and composition proofs; Theorems8.1 and8.2 full spectral semicontinuity and resolvent Lipschitz proof |
| [Compact and trace-class operators, the predual of B(H), and the operator topologies](https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html) | Foundations of von Neumann algebras and its foundation remainder | Lemma3.1 and Theorem3.3 full basis-independent Hilbert-Schmidt, ideal, compactness and Hilbert completeness proofs; Lemma11.1 and Theorem11.2 full finite-rank spectral-block approximation and summable off-diagonal removal proof |
| [Polar decomposition of functionals and weak compactness in preduals](https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html) | Foundations of von Neumann algebras and its foundation remainder | Lemma10.1 full bidual closure compactness proof; Theorem10.2 full seven-condition equivalence, including all proof directions; Lemmas10.3 and10.4 full reduction, spectral cutoff, Baire and finite-functional proofs |
| [Measurable fields of Hilbert spaces and their direct integrals](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FOUND-REMAINDER/reader/supplements/measurable-fields-direct-integrals.html) | Foundations of von Neumann algebras and its foundation remainder | Definitions2.1/8.1; Lemma2.2; full Theorems3.1/5.1/6.2/8.2/9.1/10.1 and their construction proofs |
| [Direct integrals of von Neumann algebras](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FOUND-REMAINDER/reader/supplements/direct-integrals-of-von-neumann-algebras-commutants-and-centres-takesa.html) | Foundations of von Neumann algebras and its foundation remainder | BackgroundB1-B12; full Proposition1.3; Propositions2.2/2.3; Lemma3.1; Theorems3.2/4.1/4.3; Lemma5.1 full countable generator proof |
| [Continuous decompositions, modular invariants, and canonical extensions](https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-APPROX/prerequisites.html#139-continuous-decomposition-invariants-and-canonical-extensions) | Crossed products and the flow of weights | Sections The three type III subtypes and Modular spectrum from the kernel of the center flow, including equations14-18 and the full supplied proof; use only the periodic case for a typeIII factor |

The finite dyadic uniqueness and general injectivity/AFD proofs also supply the corresponding approximation bridges for the injective-factor course. The theorem about every representation of a general connected locally compact group requires further representation theory and is not asserted here. General weight, expectation and fusion constructions retain their modular-course ownership; factor classification and the subfactor chapter are outside this course.