Prerequisite proofs

The course uses the complete supporting readings below. Each link is bound to its actual mathematical source and component terms in the source records. Included chapters retain their mathematical hypotheses, formulas, proof labels, attributions and component terms.

AN 03

• Banach estimates, quotient spaces and compact parameter arguments — CC0-1.0.
• Singularities along a submanifold and smooth boundary passage — CC0-1.0.
• Dirichlet realizations, spectral projectors, and local extensions — CC0-1.0.
• From local energy to global divergence equations — CC0-1.0.
• Symbols, operators and Sobolev scales — CC0-1.0.
• Finite defects under perturbation — CC0-1.0.
• Quadratic Fourier multipliers at a moving scale — CC0-1.0.
• Detecting regularity without choosing coordinates — CC0-1.0.
• Symbols, finite defects, and the index on a closed manifold — CC0-1.0.
• Local inverses and distance-weighted elliptic estimates — CC0-1.0.
• Spectral measures with the original operator domain retained — CC0-1.0.
• Metric and topological foundations — CC0-1.0.
• Localizing symbols with moving metrics — CC0-1.0.
• When a moving symbol scale controls an operator — CC0-1.0.
• Positivity through a moving family of scalar probes — CC0-1.0.
• Fourier transforms, finite spectra and convex separation — CC0-1.0.
• Boundary energy, local inverses, and harmonic data — CC0-1.0.
• Stable modes and the algebra of boundary data — CC0-1.0.
• Polynomial and contour interfaces for stable boundary models — CC0-1.0.
• Curved weights and the directions in which support can end — CC0-1.0.
• Totally characteristic operators on the half space — CC0-1.0.
• Traces that survive passage to cohomology — CC0-1.0.
• From Weyl symbols to operators and changes of coordinates — CC0-1.0.
• Two measuring scales, one Weyl product — CC0-1.0.

HA LCA

• Characters and the dual group — CC0-1.0 AND LicenseRef-Design-Science-License.
• The dual group as the Gelfand spectrum of \(L^1(G)\) — CC0-1.0 AND LicenseRef-Design-Science-License.
• The Fourier inversion theorem and the dual Haar measure — CC0-1.0 AND LicenseRef-Design-Science-License.
• The Plancherel theorem — CC0-1.0 AND LicenseRef-Design-Science-License.
• The Pontryagin duality theorem — CC0-1.0 AND LicenseRef-Design-Science-License.
• Unitary representations of abelian groups: the spectral theorem — CC0-1.0 AND LicenseRef-Design-Science-License.

KT OPK

• Idempotents, projections and their equivalences — CC0-1.0.
• Vector bundles and finitely generated projective modules — CC0-1.0.
• The Grothendieck group and \(K_0\) of a unital algebra — CC0-1.0.
• Nonunital algebras: unitization, relative classes and half-exactness — CC0-1.0.
• Matrix stability, stability and continuity of \(K_0\) — CC0-1.0.
• Invertibles, unitaries and \(K_1\) — CC0-1.0.
• The index map and the exact sequence at \(K_0\) — CC0-1.0.
• Suspension, higher K-groups and the long exact sequence — CC0-1.0.
• Toeplitz operators and the index theorem on the circle — CC0-1.0.
• Bott periodicity — CC0-1.0.
• The six-term exact sequence and the exponential map — CC0-1.0.
• Topological K-theory of spaces, pairs and vector bundles — CC0-1.0.
• The mapping torus — CC0-1.0.
• The Pimsner–Voiculescu exact sequence — CC0-1.0.

NCG CYCLIC

• Connections and curvature from symmetries of an algebra — CC0-1.0.
• Frequency calculus for an action of Euclidean space — CC0-1.0.
• Cyclic cohomology: traces, differentials and symmetry — CC0-1.0.

NCG FOLIATIONS

• The C*-algebra of a foliation — CC0-1.0.

Representation and integration foundations

The repaired group and covariance readings are included with their five complete earlier foundation notes and the explicit bounded-form and separation corollaries. They supply the group representation correspondence, regular L1 faithfulness, faithful-coefficient norm independence, and the exact free-group norm gap used in Lessons 1, 2 and 4.

• Continuous calculus, positivity and Hilbert spaces — CC0 1.0.
• Scalar measure, convergence and calculus — CC0 1.0.
• Compact topology and the Hilbert tensor construction — CC0 1.0.
• Radon representation, qualified products and Haar measure — CC0 1.0.
• Positive functionals and nonunital representations — CC0 1.0.
• Unitary representations and the two group C* completions — CC0 1.0.
• Covariance and crossed products with nonunital coefficients — CC0 1.0.

Bounded forms and finite-dimensional separation supplies the two standard corollaries used by the imported chapter.

OA FLOW

• Recovering a group action from its integrated operators — CC0-1.0.
• Recovering covariance with nonunital coefficients — CC0-1.0.

OA MOD

• Analytic kernels for unbounded modular operators — GFDL-1.2-or-later.
• Spectral calculus with its domains retained — GFDL-1.2-or-later.

analytic elements strips and kms

• Holomorphy in Banach spaces, Stone's theorem and resolvent convergence — CC0-1.0.

foundations of von neumann algebras

• Finite-dimensional approximations and AF-algebra classification — CC0-1.0.
• Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory — CC0-1.0.
• Order, local units and quotients of C*-algebras — CC0-1.0.
• Cauchy's theorem for cycles and its consequences — CC0-1.0.
• Compact and trace-class operators, the predual of B(H), and the operator topologies — CC0-1.0.
• Hahn–Banach, Baire and the basic theorems on Banach spaces — CC0-1.0.
• Hilbert spaces and compact operators — CC0-1.0.
• Kaplansky's density theorem and its consequences — CC0-1.0.
• Building representations from positive functionals — CC0-1.0.
• The double commutant theorem — CC0-1.0.
• Order, local units and quotients of C*-algebras — CC0-1.0 · proof edition used by the cited result.
• Building representations from positive functionals — CC0-1.0 · proof edition used by the cited result.

function algebras and approximation

• The Stone–Weierstrass theorem for functions vanishing at infinity — CC0-1.0 · proof edition used by the cited result.
• The Stone–Weierstrass theorem for functions vanishing at infinity — CC0-1.0.

harmonic analysis on locally compact groups

• Haar measure on locally compact groups — CC0-1.0.
• Measure and Hilbert space tools for Haar integration — CC0-1.0.
• Measure and Hilbert space tools for Haar integration — CC0-1.0 · proof edition used by the cited result.

hilbert c star modules and morita equivalence

• The Rieffel correspondence and induced representations — CC0-1.0 · proof edition used by the cited result.
• Adjointable operators — CC0-1.0.
• Compact operators, multipliers and the strict topology — CC0-1.0.
• Continuous fields over locally compact spaces — CC0-1.0.
• Finite projective modules, frames and \(K_0\) — CC0-1.0.
• Hilbert C*-modules — CC0-1.0.
• Imprimitivity bimodules and Morita equivalence — CC0-1.0.
• Kasparov's stabilization theorem — CC0-1.0.
• Morita invariance of K-theory and maps induced by correspondences — CC0-1.0.
• Tensor products and C*-correspondences — CC0-1.0.
• The Rieffel correspondence and induced representations — CC0-1.0.

index theory of elliptic operators

• Covering dimension and finite trivializing covers — CC0-1.0.
• Hermite functions, tempered distributions and the Schwartz kernel theorem — CC0-1.0.
• The Bott operator, suspension, and reduction of the index to Euclidean space — CC0-1.0.

Bivariant prerequisites

Lesson 4 has supplied programme proofs: KT-KK-18, Theorem 2.2 constructs full and reduced descent; Theorem 3.2 proves product compatibility; and Theorem 8.2 constructs both inverse KK-classes for the full-to-reduced quotient. The quotient theorem assumes a second-countable locally compact K-amenable group and separable graded coefficients. Theorem 9.2 covers finite- and countable-rank free groups; Theorem 9.5 covers free products of finitely or countably many countable discrete K-amenable groups. These proofs are online programme dependencies, not embedded in the KT-CP download. The descent proof uses the earlier universal, regular and amenable crossed-product constructions, not Lesson 4’s subsequent K-amenability application.

Lesson 12, Theorem 12.6 and equation (12.32), has a supplied proof: KT-KK-18, Lemma 5.1 and equations (5.2)–(5.3) fix the scalar sign and Takai Morita class; Theorem 6.1 and Proposition 6.2 prove both inverse identities and the Wiener–Hopf comparison. Lesson 12 explicitly converts the raw negative product to the positive double-product convention, compares regular-module models and proves the extension from separable to arbitrary coefficients. This KK proof is linked online, not embedded in the download.

Lesson 17 now uses supplied online proofs: KT-KK-21, Theorem 3.1, Lemma 4.1, Lemma 6.1 and Theorem 8.2 for separable E-theory composition, stability, the KK-to-E functor and extension exactness; KT-KK-22, Lemma 22.1, Theorem 22.2 and Proposition 22.4 for actual deformation families and their K-action; and KT-KK-13, Theorem 5.3, Proposition 7.2 and Corollary 7.3 for the compact-base Thom theorem and the outward-versus-ordered sign. Lesson 17, Proposition 17.9 proves the equality of its evaluation-inverse class and that actual family without a semisplitting assumption. These KK-course readers are online dependencies, not embedded in the download. The compact-base theorem does not replace the separate noncompact cycle proof in Lesson 17.

The following remaining assertions are conditional on proofs not included in this edition:

• Lesson 17: the general identification of its Thom/deformation composite with the manifold wrong-way class, including full orientation and integer sign normalization. Factorization independence, homotopy invariance, composition and projection formulas for that deformation construction remain tied to this comparison. The groupoids, extensions, noncompact Thom cycle and E-classes themselves are supplied.

Additional readings records later examples and further readings outside the proof sections used here.

Component terms

The independently written AN-03 elliptic-analysis readings and original course figures carry CC0 1.0. Their rights notice, title page, history and CC0 dedication accompany them. Other components retain their individually stated terms. Linked Fremlin-based readings are CC0 and cite Fremlin.

The rendered selection adds course navigation, mathematical anchors and links. Reader presentation is by GPT-6.1 Sol (OpenAI), Ultra, October 2026; each component retains its specifically recorded terms. The original sources and notices accompany the selection.