Preparation and proof dependencies
The course assumes functional analysis, elementary topology, Banach and C*-algebras. Each lesson specifies the exact results used from other courses.
A missing public reading route and a missing mathematical proof are different. The entries below identify unavailable public prerequisite readings; they do not treat a continuing literature survey as a missing proof.
Remaining Bott–Dirac proof gap
The recursive Kasparov route still requires the scalar Cayley normalization and both Bott–Dirac product calculations in the planned KK lesson 12. They support KK lesson 18, Lemma 5.1, used in the Connes–Thom and descent route. This edition does not claim that recursive proof chain is complete. The K-theory lessons’ own Bott arguments retain their stated scope.
Unavailable separate readings
Hilbert modules and fields on the leaf space, Theorem 6.6
Optional comparison or further reading beside proofs supplied in this course; the separate public reading route is unavailable. Comparison of the disc/sphere difference-bundle definition with Foliations Theorem6.6; Lesson 12 proves its clutching/homeomorphism identification without importing an analytic-index theorem.
Used in Lesson 12.
Cited subsection: 6 sobolev modules and the analytic symbol class.
*K-theory of the leaf space*, “Suspension, Bott periodicity and extension boundaries”
Optional comparison or further reading beside proofs supplied in this course; the separate public reading route is unavailable. Convention comparison for the positive suspension/exponential boundary; Lesson 10 and Lesson 11 already prove their signs and boundary constructions.
Cited subsection: suspension bott periodicity and extension boundaries.
*Connections and the existence of the Kasparov product*, Proposition 7.1
Written programme proof; a current public reading route is unavailable. Proposition7.1 identifies imprimitivity zero-operator cycles with inverse conjugate modules for sigma-unital coefficient algebras, with both tensor evaluation products. Lesson 21 has separable algebras, satisfying sigma-unitality.
Used in Lesson 21.
*Descent and the K-theory of crossed products*, Lemma 9.1 and Theorem 9.2
Written programme proof; a current public reading route is unavailable. Lesson 21 Proposition6.3 imports the real Connes-Thom pair, both inverse products and its positive Fourier/right-Clifford normalization from KK18 §§4-6/Theorem6.1, and Proposition6.2 for the ordinary boundary comparison. KK18 is written, but its recursive KK12 Bott scalar/reverse-product input remains a concrete known gap.
Complete written programme proof
Written programme proof; a current public reading route is unavailable. HM Lesson15 Theorem6.1 proves irrational strong Morita classification by the full GL2(Z) fractional-linear orbit, including determinant -1 signs and tensoring Schwartz equivalences. Lesson 20 asserts this separate Morita statement as a comparison to its unital isomorphism classification.
Used in Lesson 20.
Hilbert C*-modules Lesson 15, Theorem 6.1
Written programme proof; a current public reading route is unavailable. HM Lesson15 Theorem6.1 proves irrational strong Morita classification by the full GL2(Z) fractional-linear orbit, including determinant -1 signs and tensoring Schwartz equivalences. Lesson 20 asserts this separate Morita statement as a comparison to its unital isomorphism classification.
Used in Lesson 20.
Cited subsection: 6 sufficiency signs and the groupoid picture.
Cyclic forms that survive norm completion, §4, Theorem 4.3, degree-zero paragraph
Optional comparison or further reading beside proofs supplied in this course; the separate public reading route is unavailable. Degree-zero agreement with Theorem4.3; Lesson 13 directly proves its bounded-trace K0 pairing.
Used in Lesson 13.
Cited subsection: 4 the form reaches k theory.
Cyclic forms that survive norm completion, §5, Theorem 5.3
Written programme proof; a current public reading route is unavailable. Theorem5.3 proves closability, matrix holomorphic calculus in the closed domain and the unique ambient K1 homomorphism for a dense antisymmetric derivation D -> B*. Lesson 14 explicitly invokes this more general dual-valued formulation beyond its separately proved smooth-action pairing.
Used in Lesson 14.
Cited subsection: 5 a dual valued derivation is the degree one case.
Cyclic forms that survive norm completion, §7, Lemma 7.3a
Written programme proof; a current public reading route is unavailable. Lemma7.3a supplies trace-class finite-rank density, Banach ideal completeness and bounded multiplier estimates, including rectangular corners used in Lesson 13 Theorem4.1. Full p=1 proof is written, not a paywalled-only assertion.
Used in Lesson 13.
Cited subsection: 7 geometric and operator examples.
Finite domains and the GNS space of a C*-weight, “The algebra of finite elements”, (CS.1)–(CS.3)
Written programme proof; a current public reading route is unavailable. CS.1-CS.3 construct the square-finite left ideal, positive linear extension and finite-product algebra; CS.5-CS.6 construct the GNS quotient/representation for any C*-weight. Lesson 13 checks the extra tracial ideal and density properties.
Used in Lesson 13.
Cited subsection: OA MOD CS 01.
(CS.5)–(CS.6)
Written programme proof; a current public reading route is unavailable. CS.1-CS.3 construct the square-finite left ideal, positive linear extension and finite-product algebra; CS.5-CS.6 construct the GNS quotient/representation for any C*-weight. Lesson 13 checks the extra tracial ideal and density properties.
Used in Lesson 13.
Cited subsection: OA MOD CS 02.
four multiplication axioms
Written programme proof; a current public reading route is unavailable. The opening HA subsection states all four Hilbert-algebra axioms: bounded left multiplication, adjoint identity, closable involution and Hilbert-norm density of products, without countability. Lesson 13 verifies them for its trace quotient.
Used in Lesson 13.
Cited subsection: OA MOD HA 01.
Trace densities and noncommutative integration, “A complete space of actual integrable operators”, “Complete norms on actual measurable operators”, and “Products, norming tests and exact factorization”, (TI.12)–(TI.18), (TI.27)–(TI.38)
Written programme proof; a current public reading route is unavailable. TI.12-TI.18 and TI.27-TI.38 prove actual integrable-space completeness, bounded multiplication, L2 L2 subset L1 and cyclicity for faithful normal semifinite traces. Lesson 13 permits nonseparable and non-sigma-finite cases.
Used in Lesson 13.
Cited subsection: OA MOD TI 05.
Weights and the Hilbert spaces of multiplication, “Completing the two multiplication domains” through “Recovering the representation and the full algebra”, (WH.6)–(WH.23)
Written programme proof; a current public reading route is unavailable. WH.6-WH.23 construct a faithful normal semifinite weight on the generated von Neumann algebra of a left Hilbert algebra, retaining exact foundational inputs. Lesson 13 needs arbitrary Hilbert spaces, possibly non-sigma-finite algebras and infinite identity trace.
Used in Lesson 13.
Cited subsection: OA MOD WH 03.
Spectral triples and dimension spectrum, §1
Optional comparison or further reading beside proofs supplied in this course; the separate public reading route is unavailable. Definition/context for a regular spectral triple. Lesson 15 proves the commutator-domain closedness, inverse closure and holomorphic calculus it uses, and only applies the regularity hypothesis to Lambda=|D|.
Used in Lesson 15.
Cited subsection: 1 differentiation represented on a hilbert space.
*Stable isomorphism and the Brown–Green–Rieffel theorem*, Theorem 2.1
Written programme proof; a current public reading route is unavailable. Theorem2.1 proves strong Morita equivalence iff stabilization isomorphism for sigma-unital A,B. Lesson 23 applies it to its separable AF/hereditary full-corner setup, satisfying the countability hypotheses.
Used in Lesson 23.
Cited subsection: 2 brown green rieffel.
The local index formula, §10, Lemma 10.1
Optional comparison or further reading beside proofs supplied in this course; the separate public reading route is unavailable. Lesson 13 line279 states the separately cited projection-pair index identity for orthogonal P,Q with compact difference and trace-class odd power. Lesson 13 Theorem4.1 proves its parametrix formula independently. This aside does import that identity if retained as an asserted comparison; it is not a premise of the independent parametrix theorem.
Used in Lesson 13.
Cited subsection: 10 computing the pairing by finite defects.
Open Mathematics Courses
Written programme proof; a current public reading route is unavailable. Exercise7.5 imports finite-factor faithful normal trace existence (Theorem5.2), trace equality iff projection equivalence (Corollary5.4), and unnormalized matrix amplification (Proposition6.5). The II1-factor application is not supplied merely by a normalized trace on a general algebra.
Used in Lesson 3.
Further development
The mapped source comparisons and pertinent proof additions include AF determinant criteria, automorphism paths, bundle classification and graded products, general trace pairings, coefficient Toeplitz sequences, interval trace constructions, ordered Cantor invariants and hereditary pure infiniteness. Broader subjects such as general Künneth theorems, general AT classification and converse Cantor realization remain separate developments. The exact recursive Bott–Dirac prerequisite gap above remains a proof gap.