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 original text; CC-BY-4.0 for the credited Westerbaan checkpoint.
• 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 original text; CC-BY-4.0 for the credited Westerbaan checkpoint · 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

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

• Lesson 4: the descent and free-product results used for K-amenability, for second countable locally compact groups and separable coefficient algebras.
• Lesson 12, equation (12.32): the Fack–Skandalis dual Thom product, with the stated suspension sign, Wiener–Hopf normalization and Takai identification. The arbitrary-coefficient extension is proved in the lesson conditional on that product assertion.
• Lesson 17: separable E-theory composition, extension exactness, stability and KK-to-E comparison; the compact-base Spin-c Thom convention; and the manifold wrong-way factorization and deformation-comparison theorems. The geometric groupoids, parameter extensions and noncompact Thom cycle are constructed in the lesson.

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, including CC BY 4.0 for the credited Westerbaan GNS checkpoint. Linked Fremlin-based readings retain the Design Science License.

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.