Prerequisite proofs and reading order
The lesson arguments and all 504 worked solutions are supplied in this course. The general arguments use the exact preceding results listed below. A source comparison still in progress is a verification task; it does not by itself mean a mathematical proof is missing.
Links to current public provider editions are given where a suitable online route has been checked. They are not claims that the new edition has the same bytes as the earlier checked revision. Some exact provider routes are still pending, as identified below. The historical repository has been retired.
Read the introductory examples first, then follow the prerequisite order stated on the course page and in each lesson. Editable detailed prerequisite register.
| Preceding lesson | Programme course | Required proof content |
|---|---|---|
| Completely positive maps | 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 |
| The positive cone of a standard representation | Modular theory and weights | Section 6 support and orthogonality; Section 7 lower norm bound; Section 10 existence for every normal positive functional; Section 11 supports and norm correspondence; Section 5 full standard-form axiom span |
| Contractive retractions and the algebraic structure of 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 | 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 | Foundations of von Neumann algebras and its foundation remainder | Fact 2.7; Remark 3.2 |
| Spatial tensor products of von Neumann algebras | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | Modular theory and weights | Section 5 full JMJ commutation proof; Section 6 full covariance, automorphism and weight invariance proof |
| Trace densities and noncommutative integration | Modular theory and weights | Section 6 full isometric onto trace pairing and positive-cone proof |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 |
Online proof routes still pending
The following named proof providers have a checked local revision in the programme records, but their exact public edition has not yet been established for this release. Their precise required results are listed in the table. This is an online access or edition-verification issue; it is not evidence of reliance on paywalled material.
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
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Pointwise inner flows and continuous implementers
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
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
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Contractive retractions and the algebraic structure of expectations
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Crossed products and the flow of weights
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Every bounded functional becomes normal in one representation
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Injective von Neumann algebras
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Conditional expectations from modular invariance
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
The modular group and its analytic algebra
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
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.
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
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Trace densities and noncommutative integration
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Trace prerequisites: current available lessons and remaining public routes
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Trace Hilbert spaces, commutation, comparison and expectations
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
General weights: finite domains, GNS spaces, and normal representations
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.
Weights and the Hilbert spaces of multiplication
Exact public proof route pending. See the required locators and hypotheses in the register above and the invoking lesson.