Preparation and proof dependencies

Hilbert modules and fields on the leaf space, Theorem 6.6

This is an optional foliation comparison, not a missing proof prerequisite. Its separate reading route is unavailable here. Lesson 12, Theorem 3.1, proves the disc/sphere difference-bundle and trivial-line Thom identification using its specified operator K-theory proofs; it does not import an analytic-index theorem from the comparison. The singular-cohomology and characteristic-class proofs used later have exact programme routes in Section 6.

Used in Lesson 12.

K-theory of the leaf space: suspension and boundary conventions

This optional comparison chapter uses the positive projection loop and positive exponential boundary. Its separate public edition is not yet available. The complete proofs used here are in this course: Lesson 8, Theorem 2.1, Lesson 10, Theorem 4.1, and Lesson 11, Theorems 1.1–2.1. The comparison chapter refers back to those proofs; it is not a prerequisite or a circular justification of them.

Used in Lesson 10, Lesson 11.

Imprimitivity inverse products

Connections and the existence of the Kasparov product, Proposition 7.1 proves the zero-operator cycles and the two inner-product evaluation unitaries for an imprimitivity bimodule over sigma-unital algebras. The separable coefficient and crossed-product algebras in Lesson 21 satisfy these hypotheses.

Used in Lesson 21, Proposition 6.3.

Thom equivalence and the K-amenable quotient

For the mapping-torus comparison, Descent and the K-theory of crossed products, Sections 4–6, Theorem 6.1 and Proposition 6.2 supply the real Thom pair, both inverse products, and the positive Fourier/right-Clifford boundary normalization. Its scalar foundation is the Bott–Dirac calculation linked above.

For the full-to-reduced quotient, Theorem 8.2 constructs both inverse coefficient KK-classes. Lemma 9.1 and Theorem 9.2 give the tree homotopy and the weakly regular unit cycle for a free group; Theorem 9.5 also proves countable free-product permanence. Lesson 23, Corollary 6.5, uses a finite-rank free group and a separable, possibly nonunital coefficient algebra.

Used in Lesson 21, Proposition 6.3, and Lesson 23, Corollary 6.5.

Small equivalence modules and cancellation

Lesson 20, Lemma 5.5 uses Hilbert C*-modules Lesson 15, Theorem 3.2, the trace identity (4.4) and Theorem 6.1. These construct full finite projective right modules over an irrational rotation algebra with arbitrarily small positive trace dimension and with a rotation algebra as their entire adjointable endomorphism algebra. They are an essential input to the full cancellation proof, not just a classification comparison.

For positive irrational t, the constructed left A_1/t, right A_t module induces a K-group map from A_1/t to A_t with trace multiplier t; the inverse has multiplier 1/t. Tensor products multiply these factors in the stated direction by Lesson 14, Proposition 7.5. The compact-identity criterion is Finite projective modules, Theorem 2.2.

The coefficient positivity and frame are proved in Connections and curvature, Lemma 7.1 and Theorem 7.2. Inverse modules and tensor transitivity are Imprimitivity bimodules, Theorem 3.2 and Corollary 4.2. The torus provider uses only Lesson 20, Theorems 2.1, 3.1 and 5.1, not its cancellation theorem; this dependency is acyclic.

Used in Lesson 20.

Strong Morita classification of irrational rotation algebras

Hilbert C*-modules Lesson 15, Theorem 6.1 proves the full GL₂(ℤ) fractional-linear orbit criterion for irrational real parameters. The proof includes determinant −1 signs, nonzero denominators, the Schwartz equivalence and tensor transitivity. It is separate from the unital isomorphism classification in Lesson 20.

Used in Lesson 20.

Degree-zero trace pairing: comparison

Cyclic forms that survive norm completion, Theorem 4.3, degree-zero paragraph proves extension of a bounded trace, its unnormalized matrix pairing and the scalar-zero convention for relative K₀. This agrees with Lesson 13, Theorem 1.1, whose proof is given directly in that lesson. It is a comparison, not an additional missing proof.

The neighbouring Theorem 3.1 and Theorem 4.1 explain the inverse-closure and approximation mechanism; the general higher-degree pairing is separate from this degree-zero comparison.

Used in Lesson 13.

Dense dual-valued derivations and the odd pairing

Cyclic forms that survive norm completion, Theorem 5.3 proves closability for a densely defined antisymmetric derivation from a Banach algebra to its dual. Lemma 5.1 gives matrix holomorphic functional calculus in the closed graph domain. The inverse and polygonal approximation arguments of Theorem 4.1 supply representatives and homotopies for every ambient K₁ class; an arbitrary initial dense domain is not assumed to do so.

Lesson 14, Section 5 constructs d(a)(b) = τ(bδa). Boundedness of τ makes this a bounded functional in b; τδ = 0 and cyclicity give antisymmetry. The matrix pairing uses the unnormalized trace, with the factor 1/(2πi) for winding number. Extension by zero on the new scalar unit handles relative classes in a nonunital algebra.

The smooth-action case has its own proof in Connections and curvature, Theorem 5.2, using Lemma 1.1 for density and matrix functional calculus. These arguments do not require the separate weak-amenability result.

Used in Lesson 14.

Trace-class approximation and rectangular cyclicity

Cyclic forms that survive norm completion, Lemma 7.3a proves the Schatten Banach-ideal theorem, finite-rank density and bounded multiplier estimates. Its p = 1 case supplies the trace-norm completeness and continuity used in Lesson 13, Theorem 4.1.

A compact operator from H₁ to H₀ embeds in an off-diagonal corner of H₀ ⊕ H₁, preserving its nonzero singular values. Finite-rank approximation and the multiplier estimate therefore give the rectangular trace identity proved in Lesson 13. In the Calderón–Fedosov formula only the stated powers of the two parametrix errors are assumed trace class; the errors themselves need not be.

Used in Lesson 13.

Finite domains of an arbitrary C*-weight

Finite domains and the GNS space of a C*-weight, CS.1–CS.3 proves the square-finite left ideal, its finite-product algebra and the unique positive linear extension on the finite cone. No faithfulness, continuity or density hypothesis is used for these constructions; the algebra may be nonunital or zero.

Lesson 13, Lemma 5.0 proves the additional two-sided ideal and cyclicity properties from the trace assumption. These do not hold for an arbitrary weight. Norm density is a separate hypothesis in Theorem 5.1 and Lemma 6.1.

Used in Lesson 13.

The weight GNS quotient and representation

Finite domains and the GNS space of a C*-weight, CS.5–CS.6 proves Cauchy–Schwarz, the null-space quotient, Hilbert completion, bounded left representation and uniqueness. Inner products are linear in their first variable. The representation need not be faithful, and no single cyclic vector is asserted.

Lesson 13, Lemma 6.1 uses this quotient for a possibly nonfaithful trace, then constructs a faithful trace on the generated von Neumann algebra. Equality with the original weight is proved for finite values and infinity; faithfulness is not silently added to the original trace.

Used in Lesson 13.

The four Hilbert-algebra axioms

Building the two multiplication actions of a Hilbert algebra, starting data states bounded left multiplication, the adjoint identity, closability of the involution and Hilbert-norm density of products. This is the definition, not an omitted theorem.

Lesson 13, Sections 6.1–6.2 checks all four properties for its coefficient and convolution quotients. Neither a unit nor separability is assumed. Density of products in the Hilbert norm is not confused with being a core for the graph norm.

Used in Lesson 13.

Integration on actual measurable operators

Trace densities and noncommutative integration, TI.12–TI.18 constructs the complete L¹ space of closed measurable operators, with bounded multiplier estimates, adjoint isometry and the continuous trace. TI.27–TI.30 proves the complete finite-exponent norms, and TI.31–TI.38 proves product integrability, Hölder estimates and cyclicity, including L²L² ⊆ L¹.

The hypothesis is a faithful normal semifinite trace on a von Neumann algebra, on an arbitrary Hilbert space. Finite projections form a net, not an assumed countable exhaustion. Lesson 13 applies these results to its constructed faithful trace and uses uniqueness of measure limits to prove the graph-norm domains complete. The original C*-trace may be nonfaithful, and the identity may have infinite trace.

Used in Lesson 13.

Constructing the weight from a Hilbert algebra

Weights and the Hilbert spaces of multiplication, WH.6–WH.9 constructs the bounded multiplication domains and their full completion. WH.11–WH.18 defines the weight by square-root vectors and proves its exact finite ideals and GNS norm. WH.19–WH.20 proves faithfulness, arbitrary-net normality and semifiniteness. WH.21–WH.23 recovers the representation and the full algebra.

The original algebra need not already be full; Hilbert spaces and index sets are arbitrary. Lesson 13 proves that its closed involution is an everywhere-defined antiunitary, and uses WH.8 and WH.16 to make the constructed weight a trace. This extra tracial conclusion is checked in the consumer, not assumed for every Hilbert algebra. The provider retains its stated Hilbert-algebra, graph, polar and weak-compactness foundations.

Used in Lesson 13.

Spectral triples and dimension spectrum: regularity context

The definition and regularity condition in Section 1 provide optional context. The density and complete commutator-domain arguments are proved in Lesson 15, Section 4. They do not use the separate chapter’s invertibility or summability assumptions, or its local index formula. That chapter is not a missing proof prerequisite.

Used in Lesson 15.

Stable isomorphism and the Brown–Green–Rieffel theorem

Hilbert C*-modules Lesson 13, Theorem 2.1 proves that strong Morita equivalence is equivalent to isomorphism after stabilization when both C*-algebras are σ-unital. Lesson 23, Section 2 proves its AF corner is full; the corner and crossed product are separable, hence both σ-unital. The conclusion retains a tensor factor of the compact operators on each side.

The absorption argument uses Kasparov stabilization, Theorem 3.1, the compact-algebra countable-generation criterion, inverse-module evaluation, interior tensor associativity and the exterior compact-operator identification. It uses two explicitly constructed absorption isomorphisms, not cancellation of infinite direct sums.

Used in Lesson 23.

The projection-pair index identity

The local index formula, Section 10, Lemma 10.1 proves Index(QP : PH → QH) = Tr((P − Q)^2j+1) for orthogonal projections with compact difference and trace-class odd power. The parametrix gives Fredholmness; anticommuting operators pair the interior positive and negative eigenvalues, leaving the two finite defect spaces.

This supplies the asserted comparison in Lesson 13, Section 4 with its stated sign and hypotheses. The Calderón–Fedosov proof in Theorem 4.1 remains independent of this comparison; no claim about the other local-index theorems follows.

Used in Lesson 13.

Finite-factor traces and matrix amplification

Projection comparison and normal center-valued traces, CT constructs the faithful normal center-valued trace on every finite von Neumann algebra. In a factor its scalar values give the normalized faithful normal tracial state. The argument uses normal central states and monic projection decompositions, not an assumed pre-existing scalar trace.

C02 proves arbitrary projection suprema and strong convergence; C05 proves general comparison. Traces on von Neumann algebras, Corollary 5.4 gives equivalence exactly when the faithful trace values agree. Proposition 6.5, using Theorem 6.2, proves faithful normal matrix amplification by the sum of diagonal traces, without division by the matrix size.

No separability or countable-decomposition hypothesis is imposed. Exercise 7.5 itself proves the full interval of projection traces, then the real-valued K-theory isomorphism, using faithfulness in the amplified factor. Its proof is unchanged.

Used in Lesson 3.