# Commutative and noncommutative tori

*Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).*

*Independently authored CC0 lesson; self-checked by the writing AI.*

Adding a generator to a noncommutative torus doubles the total number of K-theory generators. Adding an implementing unitary for a linear torus automorphism can instead create torsion. The same exact sequence explains both phenomena: the first action is homotopic to the identity, whereas the second retains an integral matrix on the coordinate classes.

We use [Lesson 18's PV sequence](KT-OPK-18.md) and [Lesson 12's circle construction](KT-OPK-12.md#5-adding-circles-with-arbitrary-coefficients). The trace calculation for a two-generator rotation algebra is [Lesson 20](KT-OPK-20.md). All matrix trace extensions are unnormalized.

## 1. Building the universal torus one generator at a time

Let \(n\geq1\) and let \(\Theta=(\theta_{jk})\) be a real skew-symmetric \(n\)-by-\(n\) matrix. The noncommutative torus \(A_\Theta\) is the universal unital C*-algebra generated by unitaries \(u_1,\ldots,u_n\) satisfying

\[
u_j u_k=e^{2\pi i\theta_{jk}}u_k u_j.
\tag{1.1}
\]

For \(\Theta=0\), the universal commuting tuple identifies this algebra with \(C(\mathbb T^n)\). Indeed, continuous functions of the coordinate tuple give the homomorphism for every commuting unitary tuple, while the coordinate functions generate the continuous functions by Stone–Weierstrass.

**Proposition 1.1 (iterated crossed product).** If \(\Theta'\) is the leading \((n-1)\)-by-\((n-1)\) submatrix, then

\[
\begin{gathered}
A_\Theta\cong A_{\Theta'}\rtimes_{\alpha_n}\mathbb Z,\\
\alpha_n(u_j)=e^{2\pi i\theta_{nj}}u_j
\quad(1\leq j<n).
\end{gathered}
\tag{1.2}
\]

Here \(A_\Theta=C(\mathbb T)\) when \(n=1\).

*Proof.* Multiplying each preceding generator by a scalar of modulus one preserves every relation. Universality gives a homomorphism \(\alpha_n\); multiplying by the inverse scalars gives its inverse. The crossed product has the preceding generators and its implementing unitary \(w\), with \(wu_jw^*=\alpha_n(u_j)\), exactly the relations for \(u_n=w\). Conversely any tuple satisfying (1.1) supplies a representation of \(A_{\Theta'}\) covariant with its last unitary. The integrated map and the map from universal generators are inverse because both compositions fix the generators. This proves existence as well as the claimed universal identification. The coefficient algebra embeds faithfully in the crossed product. \(\square\)

There is an action of \(\mathbb T^n\) given by
\(\beta_t(u_j)=e^{2\pi it_j}u_j\). The inverse action multiplies by the inverse phases. On Laurent polynomials its dependence on \(t\) is norm continuous. Such polynomials are dense, and all these automorphisms are isometric, so polynomial approximation proves point-norm continuity on the whole algebra. This also justifies the continuous path

\[
\begin{gathered}
\alpha_{n,s}(u_j)=e^{2\pi i s\theta_{nj}}u_j,\\
0\leq s\leq1.
\end{gathered}
\tag{1.3}
\]

Every member is an automorphism. Thus \(\alpha_n\) is homotopic to the identity through automorphisms, with the required topology.

## 2. PV, free groups and the exterior labels

**Lemma 2.1.** Suppose \(\alpha\in\operatorname{Aut}(A)\) is point-norm homotopic to the identity and both K-groups of \(A\) are free abelian. For \(i=0,1\),

\[
K_i(A\rtimes_\alpha\mathbb Z)
\cong K_i(A)\oplus K_{1-i}(A).
\tag{2.1}
\]

*Proof.* Homotopy invariance gives \(\alpha_{*,i}=1\). The PV sequence reduces to

\[
\begin{gathered}
0\longrightarrow K_i(A)\xrightarrow{\iota_*}K_i(B),\\
K_i(B)\xrightarrow{d_i}K_{1-i}(A)\longrightarrow0,\\
B=A\rtimes_\alpha\mathbb Z.
\end{gathered}
\tag{2.2}
\]

Choose an antecedent for each basis element of the rightmost free group. Integer linear extension defines a right inverse of \(d_i\), and hence a splitting. This works for an infinite free basis as well, since every element has finite support. The splitting is not determined by exactness. \(\square\)

**Theorem 2.2 (all parameters).** For every real skew matrix \(\Theta\) of size \(n\geq1\),

\[
K_0(A_\Theta)\cong K_1(A_\Theta)
\cong\mathbb Z^{\,2^{n-1}}.
\tag{2.3}
\]

*Proof.* For \(n=1\), the circle has one generator in each parity. Apply Lemma 2.1 at each step of (1.2), using (1.3). Both groups remain free and their ranks each become the sum of the preceding two ranks. Induction gives \(2^{n-1}\) in each parity. No irrationality or simplicity assumption enters. For \(n=0\), the algebra is \(\mathbb C\), with groups \(\mathbb Z,0\); the displayed formula is not intended for that case. \(\square\)

One can index a basis by subsets of \(\{1,\ldots,n\}\): a subset not containing \(n\) labels a coefficient image, while a subset containing \(n\) labels a chosen lift of the opposite-parity preceding class. Even subsets label \(K_0\), odd subsets label \(K_1\). This proves the graded-group identification

\[
K_*(A_\Theta)\cong
\bigwedge\nolimits^*\mathbb Z^n.
\tag{2.4}
\]

This chosen-basis construction by itself does not produce a canonical splitting. Theorem 3.6 constructs the natural refinement from the integral Chern-character lattice, with its full deformation proof. Its odd part uses the specified central-circle suspension. These labels are determined by that deformation and the Chern convention, rather than an arbitrary choice of PV lifts. For a noncommutative algebra (2.4) is a statement about graded groups; it does not supply an internal multiplication of K-classes.

Freeness guarantees the splitting in this PV argument. Equality \(\alpha_*=1\) alone need not split an extension with torsion quotient. [Lesson 17, Corollary 4.3](KT-OPK-17.md#from-mapping-tori-to-crossed-products) proves the stronger assertion that crossed-product K-groups depend, up to isomorphism, only on the point-norm homotopy class in the automorphism group, with conjugacy allowed and without a freeness assumption. Its proof uses the actual mapping-torus isomorphism, Green's module, the linking-corner K-maps and the Wiener–Hopf boundaries. We use this exact programme proof for the general comparison formerly cited as Blackadar's Corollary 10.5.2; Lemma 2.1 also gives the specified coefficient inclusion and boundary splitting used in our computations.

**Corollary 2.3 (the AF obstruction).** For every \(n\geq1\) and every real skew \(n\)-by-\(n\) matrix \(\Theta\), the torus \(A_\Theta\) is not an AF algebra.

*Proof.* We first check the obstruction directly. Write a unital AF algebra \(B\) as the closure of an increasing union of finite-dimensional C*-subalgebras. The stages can be taken to have the unit of \(B\): choose a stage element \(a\) with \(\|1_B-a\|<1\), and let \(p\) be the unit of that stage. If \(p\ne1_B\), then
\((1_B-p)(1_B-a)=1_B-p\) implies \(\|1_B-a\|\geq1\), a contradiction. This stage and every later stage therefore have unit \(1_B\).

Let \(x\in GL_\ell(B)\). At a sufficiently late stage choose
\(y\in M_\ell(B_k)\) with
\(\|x^{-1}(y-x)\|<1\). The path \(x+t(y-x)\), \(0\leq t\leq1\), is invertible in \(M_\ell(B)\) by the Neumann series. The endpoint \(y\) is invertible in the finite-dimensional stage as well. Indeed, if its finite-dimensional block decomposition were singular in a block, there would be a nonzero stage projection \(q\) with \(yq=0\), contradicting ambient invertibility.

In each finite matrix block, polar decomposition joins \(y\) to its unitary polar part through invertibles: if \(y=v|y|\), use
\(v((1-t)|y|+t1)\).
A unitary block diagonalizes as
\(w\operatorname{diag}(e^{ia_j})w^*\), and
\(w\operatorname{diag}(e^{i(1-t)a_j})w^*\) joins it to the identity. Combining the finitely many block paths joins \(y\) to \(1_\ell\). Thus every \(GL_\ell(B)\) is path connected. The ordinary unital convention in [Lesson6, Proposition1.2](KT-OPK-06.md#1-a-definition-that-keeps-the-scalar-part-fixed) therefore gives \(K_1(B)=0\), also for the external scalar-identity definition.

By Theorem 2.2,
\(K_1(A_\Theta)\cong\mathbb Z^{2^{n-1}}\ne0\).
Since \(A_\Theta\) is unital, it cannot be AF. This argument applies to rational, irrational, degenerate, and zero parameters. The dimension-zero algebra \(\mathbb C\) is excluded by \(n\geq1\). \(\square\)

Rieffel's survey [1990, §3, author-numbered pp. 6–7] contrasts the parameter-independent K-groups with their limited ability to distinguish algebras and explains the two-dimensional AF obstruction. The corollary supplies the complete obstruction argument in every dimension. An embedding into an AF algebra can induce an isomorphism on \(K_0\) while killing all of \(K_1\); such an embedding does not make the domain AF.

## 3. Canonical traces and Elliott's formula

Write \(u^m=u_1^{m_1}\cdots u_n^{m_n}\) for \(m\in\mathbb Z^n\), with that fixed order.

**Proposition 3.1.** There is a faithful tracial state \(\tau_\Theta\) with

\[
\tau_\Theta(u^m)=
\begin{cases}1,&m=0,\\0,&m\ne0.\end{cases}
\tag{3.1}
\]

*Proof.* Average the gauge action \(\beta\) over the compact torus with probability Haar measure. On every monomial the average is zero except at \(m=0\), where it is one. Polynomial density shows that the average has scalar range; it is positive and unital, so it is \(a\mapsto\tau_\Theta(a)1\) for a state. It is faithful: if \(a\geq0\) is nonzero, choose a state \(\rho\) with \(\rho(a)>0\). The function \(t\mapsto\rho(\beta_t(a))\) is continuous, nonnegative and positive near the identity, so its integral is positive.

Monomials multiply by a scalar phase times \(u^{m+\ell}\). Both product orders have zero trace unless \(\ell=-m\). In that case the phases agree: moving factors into order contributes

\[
\exp\!\left(2\pi i\sum_{j>k}m_j\ell_k\theta_{jk}\right),
\tag{3.2}
\]

which has the same value for \((m,-m)\) and \((-m,m)\). Linearity and norm approximation prove the trace identity. \(\square\)

**Proposition 3.2 (two-coordinate copies).** For \(i<j\), the subalgebra generated by \(u_i,u_j\) is a faithful copy of the rotation algebra with parameter \(-\theta_{ij}\) in Lesson 20's order. Its trace range maps into that of \(A_\Theta\).

*Proof.* With \(u=u_i\), \(v=u_j\), relation (1.1) gives \(vu=e^{-2\pi i\theta_{ij}}uv\). Universality gives the unital map from that rotation algebra. Pulling back \(\tau_\Theta\) gives the canonical Haar trace, by the monomial rule (3.1). That trace is faithful for every parameter, including rational ones. If an element maps to zero, its squared norm element therefore has zero canonical trace and must vanish. Thus the map is injective. Matrix trace pairings commute with the inclusion, proving the range assertion. \(\square\)

Here is the trace theorem. Lemmas 3.4–3.5 and Theorem 3.6 below supply its full higher-dimensional proof.

**Theorem 3.3 (Elliott's trace formula).** For the canonical normalized trace,

\[
\begin{gathered}
\tau_{\Theta,*}(K_0(A_\Theta))\\
=\sum_{\substack{I\subseteq\{1,\ldots,n\}\\|I|\ \mathrm{even}}}
 \operatorname{Pf}(\Theta_I)\mathbb Z.
\end{gathered}
\tag{3.3}
\]

The empty Pfaffian is \(1\); \(\Theta_I\) is the principal submatrix with its indices in increasing order. [Rieffel 1988, §6, pp. 307–308] attributes the integral Chern-character lattice and exponential pairing to George A. Elliott and records their formulas. Those pages provide the historical and formula comparison. Lemmas 3.4–3.5 and Theorem 3.6 give the complete proof used here, for every real skew matrix, with the exact normalization in (3.9) and (3.20). The trace is the degree-zero component, which supplies the natural refinement mentioned after (2.4).

For a skew \(2r\)-by-\(2r\) matrix \(Q\), put

\[
P_\sigma(Q)=\prod_{k=1}^r Q_{\sigma(2k-1),\sigma(2k)}.
\]

The normalization is

\[
\operatorname{Pf}(Q)=\frac1{2^r r!}
\sum_{\sigma\in S_{2r}}\operatorname{sgn}(\sigma)P_\sigma(Q).
\tag{3.4}
\]

Thus in dimension three the stated range is
\(\mathbb Z+\theta_{12}\mathbb Z+\theta_{13}\mathbb Z+\theta_{23}\mathbb Z\). In dimension four one adds the term

\[
\theta_{12}\theta_{34}
-\theta_{13}\theta_{24}
+\theta_{14}\theta_{23}.
\tag{3.5}
\]

Rieffel's discussion at the end of §4, p. 297, corrects a sign in the earlier Chern-character conventions. The subgroup (3.3) is unchanged if \(\Theta\) is replaced globally by \(-\Theta\), since each even-size Pfaffian only changes by a sign. Equation (3.9) fixes the Chern orientation by comparison with Lesson 12, and (3.20) determines each labeled trace value. Nor does (3.3) assert trace injectivity: at \(\Theta=0\), the trace has rank-one image whereas \(K_0(C(\mathbb T^n))\) has rank \(2^{n-1}\).

### Smooth coefficients and deformation

The higher-dimensional trace formula needs more than the ranks in (2.3). We now construct the integral labels and compute their Chern forms. All parameters, including rational and degenerate skew matrices, are included.

Use the Weyl generators

\[
\begin{gathered}
b_\Theta(m)=\sum_{j<k}\theta_{jk}m_jm_k,\\
W_m=e^{-\pi i b_\Theta(m)}u^m,\\
W_mW_l=e^{\pi i m^T\Theta l}W_{m+l},\\
W_m^*=W_{-m}.
\end{gathered}
\tag{3.6}
\]

These identities follow by moving each factor in the second ordered monomial past the later factors in the first. Write \(\mathcal A_\Theta\) for the series \(\sum_m a_mW_m\) with rapidly decreasing coefficients: for every integer \(r\geq0\),
\(\sum_m(1+|m|)^r|a_m|<\infty\).
They form a dense algebra, with the seminorms just displayed. The product is twisted convolution by the phase in (3.6). The inequality
\(1+|m+l|\leq(1+|m|)(1+|l|)\)
proves its continuity for every seminorm. Matrices over this algebra use matrix coefficients and the ordinary matrix trace.

The gauge derivations and the differential are

\[
\begin{gathered}
\delta_j(W_m)=2\pi i m_jW_m,\\
d a=\sum_j\delta_j(a)e_j,\qquad e_j=dx_j.
\end{gathered}
\tag{3.7}
\]

The symbols \(e_j\) anticommute and commute with coefficients. Products of forms combine algebra multiplication with exterior multiplication. The canonical trace, extended to matrices and then coefficientwise to forms, is denoted \(T\). It is a graded trace and \(T(d\omega)=0\): the first assertion uses Proposition 3.1 and the exterior sign, and the second uses \(T\delta_j=0\).

An element belongs to \(\mathcal A_\Theta\) exactly when its gauge orbit is smooth in norm. Indeed smoothness bounds every Fourier coefficient by the norm of any corresponding derivative; using derivatives of order greater than \(r+n\) makes the weighted coefficient sum converge. Conversely rapid decrease permits termwise differentiation in norm, to every order. Consequently this algebra is closed under inversion of its invertible elements in \(A_\Theta\). The inverse gauge orbit is smooth because inversion in a Banach algebra is smooth, with derivative \(-a^{-1}\delta_j(a)a^{-1}\). The same reasoning applies to a resolvent, so contour functional calculus stays in \(\mathcal A_\Theta\).

**Lemma 3.4 (evaluation and smooth representatives).** Let \(\Theta(s)\) be a smooth path of skew matrices on \([0,1]\). There is a unital C*-algebra \(\mathfrak A\) with central coefficient algebra \(C([0,1])\) and unitaries satisfying (1.1) with \(\Theta\) replaced by \(\Theta(s)\). Every fiber evaluation
\(\operatorname{ev}_s:\mathfrak A\to A_{\Theta(s)}\)
induces isomorphisms on both K-groups. Every element of \(K_0(\mathfrak A)\) is a difference of projections whose Fourier coefficients are smooth in \(s\) and rapidly decreasing, uniformly with every derivative on \([0,1]\).

*Proof.* Construct \(\mathfrak A\) by the iterated crossed products of Proposition 1.1, beginning with \(C([0,1])\). At each step the multiplying phases are central continuous unitary functions. Universality identifies the quotient at \(s\) with the required torus: the quotient and its reverse map both fix every generator. The coefficient algebra is faithful at each step, so no fiber relation is lost.

For the first coefficient algebra evaluation is a K-isomorphism, by contraction of the interval. Suppose it is one for the algebra with \(n-1\) generators. Evaluation intertwines its automorphism with the fiber automorphism. Naturality of the six-term PV sequence gives a diagram of exact cyclic sequences. Four surrounding coefficient maps are isomorphisms; the exact-row diagram chase gives that the middle crossed-product maps are isomorphisms as well. Explicitly, lift the boundary of a proposed target class through the preceding coefficient isomorphism; exactness lifts it to a source crossed-product class, and its remaining difference comes from the coefficient inclusion. For injectivity, a class mapping to zero has zero boundary, hence comes from a coefficient class; the image of that coefficient class lies in the image of the preceding endomorphism, and the coefficient isomorphisms pull this correction back. This proves the induction in both degrees without choosing PV splittings.

Here are the regularity details. Let \(P\) be a projection in a finite matrix algebra over \(\mathfrak A\). Finite Fourier polynomials with continuous scalar coefficient functions are dense, by the iterated construction. Approximate those finitely many functions by smooth functions, and symmetrize in the Weyl coordinates; the involution there is independent of \(s\). We obtain a self-adjoint finite Fourier polynomial \(q(s)\) arbitrarily close to \(P\) in the total C*-norm. Choose the approximation so that
\(w=1-(2q-1)^2\) has norm at most \(\rho<1\). Then

\[
\begin{gathered}
p=\tfrac12+(q-\tfrac12)\sum_{k=0}^{\infty}c_k w^k,\\
c_k=4^{-k}\binom{2k}{k}.
\end{gathered}
\tag{3.8}
\]

The scalar binomial identity proves \(p^2=p=p^*\). This is the spectral projection of \(q\) near 1; as the approximation tends to \(P\), it tends in norm to \(P\). A sufficiently close projection has the same K-class, by Lesson 3's projection homotopy.

The series also has the asserted smoothness. Derivatives of the phase in a product of \(k\) Weyl factors apply pairs of gauge derivations to these factors; a derivative of order \(h\) distributes at most \(2h\) such derivations. A further \(v\) gauge derivatives distribute at most \(v\) more. All derivatives of the finite polynomial \(w\) have uniformly bounded fiber norms. Each resulting product has at most \(2h+v\) differentiated factors; its undifferentiated factors, in the consecutive blocks between them, have total norm bounded by \(\rho^{k-2h-v}\). The number of terms is bounded by a constant times a fixed power of \(k\), for fixed \(h,v\). Thus the differentiated series is dominated by a polynomial in \(k\) times this geometric bound. It converges uniformly in every gauge derivative norm. The Fourier estimate following (3.7), applied with sufficiently many further gauge derivatives, makes it converge in every weighted coefficient seminorm too. This justifies every parameter derivative and its uniform continuity. Endpoints can be treated by extending the finitely many smooth coefficients and the path a little beyond them; the same norm bounds hold on small endpoint neighborhoods. Finally \(K_0\) is defined by differences of finite projections, so the construction applies to every class. \(\square\)

For clarity, the norm bound in this proof does not require a pre-existing continuous-field theorem. It follows in every fiber from the norm in the single total algebra. Norms of a section of a central \(C([0,1])\)-algebra are upper semicontinuous: a representative within a prescribed quotient norm can be chosen modulo the fiber ideal; its error is a finite sum of central functions vanishing at that parameter times bounded elements, up to an arbitrarily small remainder. This proves the needed bound on a neighborhood. It also justifies the endpoint extension in the last paragraph.

### Chern forms and their derivative

For a smooth projection \(p\) define the even form

\[
\begin{gathered}
\operatorname{Ch}_\Theta(p)=\sum_{r\geq0}C_r(p),\\
C_r(p)=\frac{(-1)^rT\bigl(p(dp)^{2r}\bigr)}{r!(2\pi i)^r}.
\end{gathered}
\tag{3.9}
\]

Terms of degree greater than \(n\) vanish. This defines an additive map on \(K_0(A_\Theta)\). To check homotopy invariance directly, put \(x=dp\). Differentiating \(p^2=p\) gives \(px+xp=x\), so \(x\) is off diagonal for \(p\oplus(1-p)\). For a projection homotopy at fixed multiplication, \(p'\) is off diagonal as well. Graded cyclicity and \(T d=0\), applied to each differentiated factor, give
\(\frac{d}{ds}T(px^{2r})=(2r+1)T(p'x^{2r})=0\).
Here \(x^{2r}\) is diagonal; for \(r=0\), the trace of an off-diagonal element is also zero. A continuous C*-projection homotopy can be approximated and corrected by (3.8), uniformly along the homotopy. Its corrected endpoints are smoothly joined to the original smooth projections by the same construction applied to their straight interpolation. Thus the check covers the K-theory relations, not only already smooth homotopies. Block sums prove additivity.

Let \(\iota_j\) be contraction by the \(j\)-th coordinate vector. For a skew matrix \(Q=(q_{ij})\), set

\[
\iota_Q=\sum_{i<j}q_{ij}\,\iota_j\iota_i.
\tag{3.10}
\]

In particular \(\iota_Q(e_i\wedge e_j)=q_{ij}\). These conventions specify the sign below.

**Lemma 3.5 (the contraction identity).** For any smooth family of projections over \(A_{\Theta(s)}\),

\[
\frac{d}{ds}C_r(p)=\iota_{\Theta'}C_{r+1}(p).
\tag{3.11}
\]

*Proof.* The derivative of Weyl multiplication is

\[
\begin{aligned}
\partial_s(ab)&=a'b+ab'+D(a,b),\\
D(a,b)&=\frac1{4\pi i}\sum_{i<j}\theta'_{ij}B_{ij},\\
B_{ij}&=\delta_i(a)\delta_j(b)\\
&\quad-\delta_j(a)\delta_i(b).
\end{aligned}
\tag{3.12}
\]

Indeed differentiating \(e^{\pi i m^T\Theta(s)l}\) gives this formula on two Weyl monomials. Absolute convergence of the differentiated convolution extends it to all smooth coefficients and forms. A product of several factors has one such term for each ordered pair of factor positions, in addition to the individual factor derivatives.

We give the full finite calculation. It suffices to consider one pair \(i<j\), omit its scalar coefficient \(\theta'_{ij}\), and put
\(y=\delta_i p\), \(z=\delta_j p\), \(X=dy\), \(Y=dz\), \(S=2p-1\), and \(x=dp\). The elements \(x,y,z\) are off diagonal for \(p\), and \(S\) anticommutes with them. The diagonal parts obtained by differentiating \(p^2=p\), and then \(py+yp=y\), are

\[
\begin{gathered}
(p')_{\mathrm{diag}}=-\frac{S(yz-zy)}{4\pi i},\\
X_{\mathrm{diag}}=-S(xy+yx),\\
Y_{\mathrm{diag}}=-S(xz+zx).
\end{gathered}
\tag{3.13}
\]

Only the indicated summand of \(p'\) is used in the first line; summing pairs gives its whole diagonal part. All off-diagonal contributions have trace zero in the computations that follow.

First take \(N=2r\geq2\), and define, for \(0\leq b\leq N\),

\[
U_b=T\bigl(S(yx^bz-zx^by)x^{N-b}\bigr).
\tag{3.14}
\]

The terms from differentiating the factors themselves give
\(-(N+1)U_0/(4\pi i)\). This uses the homotopy calculation above, now with the diagonal part of \(p'\) in (3.13).

For completeness consider all the multiplication terms. After removing their common coefficient \(1/(4\pi i)\), pairs involving the initial \(p\) contribute
\(\sum_{b=0}^{N-1}(-1)^{b+1}V_b\), where

\[
\begin{aligned}
V_b&=T\bigl(Xx^bz x^{N-1-b}\bigr)\\
&\quad-T\bigl(Yx^by x^{N-1-b}\bigr).
\end{aligned}
\tag{3.15}
\]

This follows by integrating \(Y=dz\) or \(X=dy\) by parts, using \(T d=0\). A pair involving two of the \(N\) factors \(x\), with \(b\) intervening factors, gives \((-1)^{b+1}V_b\) by the same integration and graded cyclicity. There are \(N-1-b\) such pairs. Thus their total, including the initial-factor pairs, is
\(\sum_{b=0}^{N-1}(N-b)(-1)^{b+1}V_b\).
The factor following \(X\) or \(Y\) is diagonal. Substitution of (3.13), followed by moving its initial \(x\) cyclically past the other forms, gives
\(V_b=-U_b-U_{b+1}\).
Also \(U_{N-b}=U_b\): move the final \(N-b\) copies of \(x\) to the front, use graded cyclicity and \(Sx=-xS\), then move the last degree-zero factor cyclically. Its anticommutation with \(S\) interchanges the two terms. In particular \(U_N=U_0\).

Consequently the multiplication terms are
\((N-1)U_0+\sum_{b=1}^{N-1}(-1)^{b+1}U_b\), divided by \(4\pi i\). Adding the individual derivatives gives

\[
\begin{gathered}
\frac{d}{ds}T(px^N)\\
=-\frac{2U_0+\sum_{b=1}^{N-1}(-1)^bU_b}{4\pi i}.
\end{gathered}
\tag{3.16}
\]

Here is the corresponding contraction count. Contracting two of the \(N+2\) factors \(x\) in \(T(px^{N+2})\), with \(b\) intervening factors, contributes
\((-1)^b T(px^{a-1}(yx^bz-zx^by)x^{N+1-a-b})\),
where \(1\leq a\leq N+1-b\). Move the initial \(a-1\) factors cyclically. The projection switches to \(p\) or \(1-p\) according to their parity; writing those projections as \((1\pm S)/2\) separates a scalar part and an \(S\) part. The scalar part has the alternating sum over \(a\), which is zero for odd \(b\) and one for even \(b\). For even \(b\), graded cyclicity gives
\(T((yx^{N-b}z-zx^{N-b}y)x^b)=-T((yx^bz-zx^by)x^{N-b})\).
Thus the scalar parts for \(b\) and \(N-b\) cancel, including the zero middle term when they agree. The \(S\) parts give

\[
\begin{gathered}
\iota_j\iota_iT(px^{N+2})\\
=\tfrac12\sum_{b=0}^N(N+1-b)(-1)^bU_b\\
=\tfrac{N+2}{4}
 \left(2U_0+\sum_{b=1}^{N-1}(-1)^bU_b\right).
\end{gathered}
\tag{3.17}
\]

The second equality pairs \(b\) with \(N-b\). Equations (3.16)–(3.17) therefore give
\(\partial_sT(px^{2r})=-(2\pi i(r+1))^{-1}\iota_j\iota_iT(px^{2r+2})\).
Multiplying by \((-1)^r/(r!(2\pi i)^r)\) proves (3.11) for \(r\geq1\).

For \(r=0\), differentiation of the projection relation and traciality give directly

\[
\begin{gathered}
T(p')=-2T(pD(p,p)),\\
T(p')=-\frac{T(p(yz-zy))}{2\pi i}\\
=\iota_j\iota_iC_1(p).
\end{gathered}
\tag{3.18}
\]

Here \(T(D(p,p))=0\), since it is a sum of traced commutators. This separate calculation also checks the normalization of the contraction. Restore the factors \(\theta'_{ij}\) and sum over pairs. \(\square\)

### The integral lattice and the Pfaffian range

At \(\Theta=0\), (3.9) is the ordinary even Chern character of a bundle, represented by constant forms on the torus. We justify that comparison and its integrality before using it. The range bundle of a smooth projection has the Grassmann connection \(p d\), whose curvature is \(p(dp)^2p\); the identity \(pdp p=0\) gives its \(r\)-th traced power as \(\operatorname{tr}(p(dp)^{2r})\). The trace-exponential form is independent of the connection: for \(\nabla_t\), differentiating curvature gives \(\dot F_t=[\nabla_t,\dot\nabla_t]\); the Bianchi identity and graded trace give
\(\partial_t\operatorname{tr}(F_t^r)=r d\operatorname{tr}(\dot\nabla_t F_t^{r-1})\).
The Bianchi identity itself follows by expanding \([\nabla_t,\nabla_t^2]=0\). The sign in (3.9) agrees with Lesson 12's tautological-line convention. On the line \(\mathbb C(1,\zeta)\), its unit frame has connection form \((\overline\zeta\,d\zeta-\zeta\,d\overline\zeta)/(2(1+|\zeta|^2))\) and curvature \(d\overline\zeta\wedge d\zeta/(1+|\zeta|^2)^2\). The integral of \(-F/(2\pi i)\) over the complex-oriented projective line is \(-1\). Since restriction to this line is an isomorphism on the degree-two cohomology of every projective space, this curvature class is the \(-h\) of Lesson 12, (6.3). Pullback of the tautological connection along the classifying map, and connection independence, give the same comparison for every line bundle. Thus the trace exponential with the signs in (3.9) represents \(\exp(c_1)\).

Pull a bundle to the successive projective flag spaces of Lesson 12, Lemma 6.3. It splits into lines there, and pullback is injective on rational cohomology. Direct sums add both the trace exponential and the Newton-polynomial character. Their equality on lines, and this injectivity, prove their equality on every bundle. Finally averaging a closed form over translations does not change its class: the homotopy from identity to any fixed translation gives the difference as the integral of the Cartan formula \(L_v=d\iota_v+\iota_vd\). Integrating this identity over translations proves the averaging assertion. Coefficientwise averaging is exactly \(T\) in (3.9), so the comparison holds with its stated sign.

The full integral character lattice on a commutative torus is

\[
\operatorname{ch}\bigl(K^*(\mathbb T^n)\bigr)
=\bigwedge\nolimits^*\mathbb Z^n.
\tag{3.19}
\]

This stronger assertion does not follow just from the rational character theorem. Induct on the number of circles using Lesson 12, Theorem 5.1. Evaluation at the distinguished point of the new circle splits off the preceding torus; its kernel is the inclusion of the suspended coefficient algebra. The two character identities in Lesson 12, (6.15), send these suspension summands to the exterior product with the new integral degree-one generator, with signs \(+1\) for \(\beta\) and \(-1\) for \(\theta\). Thus each step has, in both theories, the preceding lattice and precisely its copy wedged with this new generator. Starting with rank on a point proves both inclusion and surjectivity in (3.19). The same induction proves injectivity of the character here, since its two summands are injective at the preceding step. This avoids inferring an integral result from a rational isomorphism.

**Theorem 3.6 (Elliott's integral labels and trace).** There is an isomorphism
\(\mu_\Theta:K_0(A_\Theta)\to\bigwedge^{\mathrm{even}}\mathbb Z^n\)
such that

\[
\operatorname{Ch}_\Theta(x)=e^{\iota_\Theta}\mu_\Theta(x).
\tag{3.20}
\]

Its labels are natural for changes of the integral generating coordinates and for coordinate inclusions. The odd labels are obtained by the specified central-circle suspension. In particular Theorem 3.3 holds in every dimension and for every real skew matrix.

*Proof.* Apply Lemma 3.4 to \(\Theta(s)=s\Theta\). Define
\(\mu_\Theta=\operatorname{ch}\,(\operatorname{ev}_0)_*(\operatorname{ev}_1)_*^{-1}\),
using the even isomorphism of (3.19). A total K-class has smooth projection-difference representatives by Lemma 3.4. Lemma 3.5 gives the finite-dimensional differential equation
\(\partial_s\operatorname{Ch}_{s\Theta}=\iota_\Theta\operatorname{Ch}_{s\Theta}\).
Its solution is the finite polynomial exponential in (3.20): differentiating the polynomial verifies the equation, and multiplication by \(e^{-s\iota_\Theta}\) proves uniqueness. It is invertible with inverse \(e^{-\iota_\Theta}\). Thus the complete Chern-character lattice is precisely the exponential contraction of the integral exterior lattice; no arbitrary PV lift has selected these labels.

Naturality can be checked before taking K-theory. An integral matrix \(M\) sends the generators for \(M^T\Theta M\) to the Weyl generators \(W_{Me_j}\) for \(\Theta\), giving the corresponding homomorphism of the whole path algebras. For \(M\in GL_n(\mathbb Z)\) the inverse generating change gives an isomorphism; a coordinate inclusion is the same construction with selected columns. At parameter zero these are the usual torus pullbacks, whose characters act by the appropriate exterior powers of \(M\). Evaluation and its K-theory inverse commute with these maps, proving the asserted naturality. Gauge phases act homotopically trivially and give the same labels. The numerical coordinate parameter \(\Theta\) remains part of this specification; identifying algebras after an integral shift of its entries need not preserve the chosen deformation labels.

For odd classes adjoin a central circle, with its coordinate first, so the enlarged skew matrix is \(0\oplus\Theta\). Lesson 12's circle splitting identifies \(K_1(A_\Theta)\) with the kernel of circle evaluation on \(K_0(C(\mathbb T,A_\Theta))\), using \(j_*\theta\). The preceding natural even labels take this kernel exactly to \(e_0\wedge\bigwedge^{\mathrm{odd}}\mathbb Z^n\). Define the odd label to be minus its \(e_0\)-coefficient, because \(\operatorname{ch}_0\theta=-\sigma\operatorname{ch}_1\). This supplies the natural odd exterior isomorphism with the ordinary winding normalization, and applies the same contraction to it, since \(\iota_{0\oplus\Theta}\) leaves \(e_0\) untouched.

The degree-zero term of (3.20) is the matrix trace pairing. For an ordered even subset \(I\) of size \(2r\), the degree-zero term of \(\iota_\Theta^r e_I/r!\) is \(\operatorname{Pf}(\Theta_I)\): expand the contractions; an ordering of the \(r\) pairs contributes the permutation sign, and the \(r!\) orders of those pairs cancel the factorial. Reversing each pair changes both its skew entry and its exterior sign, agreeing with normalization (3.4). Hence

\[
\begin{gathered}
\tau_{\Theta,*}(x)\\
=\sum_{\substack{I\subseteq\{1,\ldots,n\}\\|I|\ \mathrm{even}}}
 \operatorname{Pf}(\Theta_I)\,\mu_\Theta(x)_I.
\end{gathered}
\tag{3.21}
\]

Every integral exterior vector occurs, so (3.21) proves exactly (3.3). The empty subset gives the normalized unit. For two coordinates the trace is rank plus \(\theta_{12}\) times the specified topological Chern coordinate. With the rotation parameter \(-\theta_{12}\) of Proposition 3.2 this gives the same two-generator trace subgroup, now with a fully specified Chern orientation. \(\square\)

The construction gives the integral labels and the trace range, not a multiplication on the K-theory of a noncommutative algebra. Faithfulness of the trace does not imply its injectivity on \(K_0\), as the commutative example after (3.5) already shows.

### The two-dimensional Chern label and spectral gaps

For the rotation algebra in Lesson 20's convention, write its generators as \(U,V\), with

\[
\begin{gathered}
VU=e^{2\pi i a}UV,
\\ a\in\mathbb R.
\end{gathered}
\tag{3.22}
\]

Thus its skew matrix in this lesson is \(\Theta_a=\left(\begin{smallmatrix}0&-a\\a&0\end{smallmatrix}\right)\), in the coordinate order \(U,V\). On its smooth algebra use \(\delta_1(U)=2\pi iU\), \(\delta_2(V)=2\pi iV\), with the other two generator derivatives zero. For a smooth matrix projection \(p\), put

\[
\kappa(p)=\frac{1}{2\pi i}\,T\bigl(p[\delta_1p,\delta_2p]\bigr),
\tag{3.23}
\]

where \(T\) is the canonical trace with the unnormalized matrix trace. This is the normalized two-derivation pairing used in Connes's discussion of the two-torus. Equation (3.9) specifies its relation to this lesson's Chern form: the degree-two term is \(-\kappa(p)e_1\wedge e_2\).

**Corollary 3.7 (trace and Chern number together).** For every real \(a\), (3.23) descends additively to an integer-valued map \(\kappa:K_0(A_a)\to\mathbb Z\). The map

\[
\begin{gathered}
\Phi_a:K_0(A_a)\longrightarrow\mathbb Z^2,
\\ \Phi_a(x)
\\ =\bigl(\tau_{a,*}(x)-a\kappa(x),\kappa(x)\bigr).
\end{gathered}
\tag{3.24}
\]

is an isomorphism, taking the unit to \((1,0)\). Equivalently, the joint range of trace and Chern number is exactly

\[
\bigl\{(m+ak,k):m,k\in\mathbb Z\bigr\}.
\tag{3.25}
\]

If \(a\) is irrational, the trace value alone determines both integers. If \(a\) is rational, the Chern number remains an independent invariant which the trace does not determine.

*Proof.* Smooth projection differences represent every K-class by Lemma 3.4, or by Lesson 15's matrix holomorphic-calculus theorem. The homotopy calculation following (3.9) shows that the coefficient in (3.23) is invariant under all stable projection homotopies, including norm-continuous ambient homotopies corrected to smooth projections. Block sums give additivity, so it defines a map on K-theory.

Write the integral label of Theorem 3.6 as \(\mu_{\Theta_a}(x)=m+c\,e_1\wedge e_2\). In two dimensions the exponential in (3.20) has only its constant and linear terms. Since \(\iota_{\Theta_a}(e_1\wedge e_2)=-a\), it gives

\[
\begin{gathered}
\operatorname{Ch}_{\Theta_a}(x)
\\ =(m-ac)+c\,e_1\wedge e_2.
\end{gathered}
\tag{3.26}
\]

Comparison with (3.23) gives \(c=-\kappa(x)\), and hence \(\tau_{a,*}(x)=m+a\kappa(x)\). Both \(m\) and \(c\) are integers, and every such pair occurs because \(\mu_{\Theta_a}\) is an isomorphism. Replacing \(c\) by \(-k\) proves (3.24) and (3.25), including injectivity and surjectivity. The unit has label \(1\).

For irrational \(a\), equality \(m+ak=m'+ak'\) forces \(k=k'\) and then \(m=m'\). For \(a=r/s\) in lowest terms, with \(s>0\), the nonzero pair \((-r,s)\) has trace zero and Chern number \(s\). Its antecedent under (3.24) is a nonzero virtual K-class. It need not be a positive projection class: faithfulness of the trace makes a projection of trace zero vanish. Thus the failure of trace injectivity is a failure for the group of virtual classes, fully consistent with trace faithfulness. \(\square\)

**Corollary 3.8 (labels are constant along a spectral gap).** Fix any real \(a\). Let \(t\mapsto h_t\) be a norm-continuous path of selfadjoint elements of \(M_N(A_a)\), each smooth for the gauge action. Let \(t\mapsto b_t\) be a continuous real function with \(b_t\notin\operatorname{Spec}(h_t)\) for every \(t\in[0,1]\). Then

\[
p_t=1_{(-\infty,b_t)}(h_t)
\tag{3.27}
\]

is a norm-continuous path of smooth projections. Its K-class, trace, and Chern number are independent of \(t\). For a single \(h\), these quantities are constant as the threshold varies in one connected component of \(\mathbb R\setminus\operatorname{Spec}(h)\).

*Proof.* Put \(k_t=h_t-b_t1_N\). At a fixed \(t_0\), zero is outside the compact real spectrum of \(k_{t_0}\). Choose a positively oriented rectangular contour enclosing exactly its negative spectrum, with right vertical side on the imaginary axis and all sides in the resolvent set. If the negative spectrum is empty, the same rectangle may enclose none of the spectrum. The remaining sides can be chosen beyond the finite spectral bounds. The inverses \((z-k_{t_0})^{-1}\) have a common norm bound on this compact contour. Norm continuity and the Neumann series then keep every \(z-k_t\) invertible on the same contour for \(t\) near \(t_0\). Their inverse norms remain uniformly bounded there.

The resolvent identity shows that the contour formula

\[
p_t=\frac{1}{2\pi i}\int_\Gamma(z-k_t)^{-1}\,dz
\tag{3.28}
\]

depends continuously on \(t\) in norm. For nearby \(t\), the contour still encloses precisely the negative spectrum: it stays outside the spectrum, its other sides lie beyond the spectral bounds, and its right side separates the negative and positive real axes. The formula is therefore (3.27). This local argument at every \(t_0\) proves norm continuity on the entire interval. Matrix holomorphic functional calculus for the gauge-smooth algebra makes each \(p_t\) smooth. No continuity of the gauge derivatives in the parameter \(t\) is required.

A norm-continuous projection path has one K-class. Corollary 3.7 consequently gives one Chern number and one trace value for the whole path. For a fixed \(h\), two thresholds in the same spectral-gap interval select exactly the same part of its spectrum, and therefore exactly the same projection; this also proves the last assertion directly. \(\square\)

### An integer pairing for invertible symbols

Let \(B_a=C^\infty(S^1,\mathcal A_a)\), with circle coordinate \(t\in[0,2\pi]\) oriented increasingly. Extend the spatial derivations to its coefficients and put \(\partial_1=\delta_1\), \(\partial_2=\delta_2\), \(\partial_3=\partial_t\). For smooth matrix coefficients define

\[
\begin{gathered}
\rho_1(\sigma^0,\sigma^1)
\\ =\int_0^{2\pi}T(\sigma^0\partial_t\sigma^1)\,dt,
\\ \rho_3(\sigma^0,\sigma^1,\sigma^2,\sigma^3)
\\ =\int_0^{2\pi}\sum_{\pi\in S_3}\operatorname{sgn}(\pi)
\\ T(\sigma^0\partial_{\pi(1)}\sigma^1
\partial_{\pi(2)}\sigma^2\partial_{\pi(3)}\sigma^3)\,dt.
\end{gathered}
\tag{3.29}
\]

These are the ordered one- and three-derivation cochains displayed in Connes's IV section 6. We prove their needed integer combination directly.

**Corollary 3.9 (integral symbol pairing).** For every real \(a\) and every smooth invertible matrix symbol \(\sigma\), the number

\[
\begin{aligned}
J_a(\sigma)&=\frac{a}{6(2\pi i)^2}
\\ &\quad\cdot\rho_3(\sigma^{-1},\sigma,\sigma^{-1},\sigma)
\\ &\quad-\frac{1}{2\pi i}\rho_1(\sigma^{-1},\sigma).
\end{aligned}
\tag{3.30}
\]

is an integer. It is additive on stable block sums and invariant under ambient invertible homotopy. Under Lesson 12's circle splitting
\(K_1(C(S^1,A_a))\cong K_1(A_a)\oplus K_0(A_a)\), it vanishes on the constant summand and is \(-m\) on a suspension class whose second summand has the pair \((m,k)\) of (3.24).

*Proof.* Use the algebra of forms with coefficient algebra \(B_a\), differential
\(d=\delta_1e_1+\delta_2e_2+\partial_tdt\), and orientation \(e_1\wedge e_2\wedge dt\). Integrating the trace of the top coefficient is a closed graded trace: spatial derivatives have canonical trace zero, and a time derivative integrates to zero by periodicity. For \(\Omega=\sigma^{-1}d\sigma\), differentiation of the inverse gives

\[
\begin{gathered}
\rho_3(\sigma^{-1},\sigma,\sigma^{-1},\sigma)
\\ =-\int T(\Omega^3).
\end{gathered}
\tag{3.31}
\]

Here and below an integral of a form means its top coefficient in the displayed orientation. For a smooth homotopy, put \(\eta=\sigma^{-1}\partial_s\sigma\). The product rule gives
\(\partial_s\Omega=d\eta+[\Omega,\eta]\) and \(d\Omega=-\Omega^2\). Thus \(d(\Omega^2)=0\). Graded cyclicity cancels the commutator contribution and makes the three differentiated factors in the cubic trace equal. Consequently

\[
\begin{gathered}
\partial_s\int T(\Omega^3)
\\ =3\int T(d\eta\,\Omega^2)
\\ =3\int T(d(\eta\Omega^2))=0.
\end{gathered}
\tag{3.32}
\]

For \(\rho_1\), the time component of the same identity is
\(\partial_s\Omega_t=\partial_t\eta+[\Omega_t,\eta]\). Taking trace and integrating again gives zero. Block sums add both cochains, and an identity block contributes zero.

These checks also give homotopy invariance in the ambient C*-algebra. Finite circle Fourier polynomials with gauge-smooth coefficients are dense in \(C(S^1,A_a)\): circle Fejer sums converge uniformly, and their finitely many coefficients can each be approximated by gauge-smooth elements. On any continuous path of invertibles the inverse norms have a common finite bound \(M\). Choose division steps and smooth node errors below \(1/(4M)\); each interpolating segment then stays within \(1/(2M)\) of the original invertible at its initial node. Divide the path into finitely many sufficiently small steps. Approximate its values at the division points by elements smooth for the circle and gauge actions, retaining already smooth endpoints. The approximations and step sizes can be chosen so that consecutive values are within a Neumann neighborhood of an invertible. Their straight segments are therefore invertible and have smooth coefficients. Reparametrizing these finitely many segments by a smooth function flat at its endpoints gives a smooth path with the same endpoints. Density of the smooth algebra and this path argument show that the two cochain expressions descend to the stable ambient K-group; matrix holomorphic calculus gives the same conclusion for inverses. Thus we may evaluate (3.30) using the circle splitting.

On a constant loop both expressions vanish, since there is no time derivative. The suspension of a smooth projection \(p\) is represented by the positive loop

\[
u_p(t)=e^{it}p+(1-p).
\tag{3.33}
\]

This is the projection-corner version of the positive suspension map \(\beta\) in Lesson 12, Theorem 5.1. We calculate the cochains, including their signs. Put \(z=e^{it}\), \(b=z^{-1}-1\), \(c=z-1\), and \(d_0=\delta_1e_1+\delta_2e_2\). The derivative of a projection is off diagonal, so direct multiplication gives

\[
\begin{aligned}
u_p^{-1}du_p&=\Omega_0+ip\,dt,\\
\Omega_0&=-b\,p\,d_0p
\\ &\quad+c(1-p)d_0p,\\
\Omega_0^2&=-bc\,(d_0p)^2.
\end{aligned}
\tag{3.34}
\]

The three terms containing one \(dt\) in the traced cubic are equal by graded cyclicity. Hence
\(T((u_p^{-1}du_p)^3)=-3i\,bc\,T(p(d_0p)^2)\,dt\).
The elementary scalar identity is \(bc=2-z-z^{-1}\), whose integral is \(4\pi\). Equations (3.31) and (3.34) therefore give

\[
\begin{gathered}
\rho_1(u_p^{-1},u_p)=2\pi i\,T(p),
\\ \rho_3(u_p^{-1},u_p,u_p^{-1},u_p)
\\ =6(2\pi i)T(p[\delta_1p,\delta_2p]).
\end{gathered}
\tag{3.35}
\]

It follows that \(J_a(u_p)=a\kappa(p)-\tau_{a,*}[p]=-m\), by Corollary 3.7. Projection differences give every second summand in the circle splitting, so additivity gives the same formula for each of them. Constants give the first summand and contribute zero. This proves integrality for every class, and hence every smooth invertible matrix symbol. \(\square\)

For the book's real-line symbol algebra, \(0<\theta\leq1\) and \(a=1/\theta\), so (3.30) is exactly the numerical expression in its Corollary 4(b). The proof here establishes that expression's integrality, including matrix symbols and rational parameters. Identifying it with the analytic index of an arbitrary elliptic difference-differential operator requires the further elliptic theorem and its analytic proof.

The mathematical input to a spectral-gap label is explicit: a smooth spectral projection and the integer pairing (3.23). Connes's Chapter IV, section 6 relates this pairing to Hall conductivity by the Kubo formula and extends the discussion to localization. Those assertions involve further analytic and physical hypotheses. Corollary 3.8 treats bounded smooth Hamiltonians and genuine spectral gaps; it does not identify a conductivity or replace a spectral gap by a gap of extended states.

## 4. The actual action of a linear two-torus map

Let \(M\in SL_2(\mathbb Z)\). Regard points \(x\in\mathbb T^2\) as columns, and set

\[
\begin{gathered}
T_M(x)=Mx,\\
\alpha_M(f)=f\circ T_M^{-1},\\
B_M=C(\mathbb T^2)\rtimes_{\alpha_M}\mathbb Z.
\end{gathered}
\tag{4.1}
\]

Write \(w\) for its implementing unitary and \(z_j(x)=e^{2\pi ix_j}\). Lesson 12 gives \(K_1(C(\mathbb T^2))=\mathbb Z[z_1]\oplus\mathbb Z[z_2]\).

**Lemma 4.1.** With exponent columns in this basis, the action on \(K_1\) is

\[
M'=M^{-T}.
\tag{4.2}
\]

Its action on \(K_0(C(\mathbb T^2))\) is the identity.

*Proof.* For \(a\in\mathbb Z^2\), the monomial \(z^a\) has class \(a_1[z_1]+a_2[z_2]\). This follows from the product relation in stable \(K_1\), not from a pointwise homotopy between the monomials. Pullback gives
\(z^a(M^{-1}x)=z^{M^{-T}a}(x)\), proving (4.2).

For the integral two-dimensional step, use the CW presentation of the torus with two one-cells and one two-cell attached by their commutator. A rank-\(r>0\) complex bundle is trivial over the one-skeleton: choose vertex frames and extend them along each edge, correcting the terminal frame by a path in \(U(r)\). Such paths exist because a finite-dimensional unitary diagonalizes as \(W=V\operatorname{diag}(e^{ia_j})V^*\), and \(V\operatorname{diag}(e^{ita_j})V^*\) joins identity to it. It is trivial over the characteristic disk by the cylinder transport of Lesson 2. Its comparison along the disk boundary is therefore a loop \(v:S^1\to U(r)\), taking coefficients in the disk frame to coefficients in the one-skeleton frame. The complete determinant calculation in [Lesson 6, §5](KT-OPK-06.md#5-components-detected-by-spectra-winding-and-index) proves that determinant winding classifies this loop up to homotopy. A change of frame on the disk gives a null-homotopic boundary loop. A change on the one-skeleton has winding zero on the attaching commutator, since winding is additive and inverses negate it. Conversely, a homotopy of comparison loops gives a bundle on the cylinder and hence an isomorphism between the endpoint bundles. Every loop produces a bundle by gluing; local triviality follows by extending its unitary comparison to a collar of the disk boundary. Thus the integer \(k=\operatorname{wind}(\det v)\) classifies bundles of each fixed positive rank.

The sign of its first Chern number is explicit with this comparison direction. The bundle difference, trivialized on the one-skeleton, pulls back to the disk triple \((\mathbf1^r,\mathbf1^r,v)\). [Lesson 12, Lemma 2.2 and §6](KT-OPK-12.md#odd-classes-and-both-connecting-maps) identify that triple with \(k\mathfrak b\), and \(\operatorname{ch}_0(\mathfrak b)=-\eta_2\). The oriented disk cell generates \(H^2(\mathbb T^2;\mathbb Z)\): its cellular boundary is the commutator, whose incidence numbers on both one-cells are zero. Consequently the Chern number of the glued bundle is \(-k\). Rank and first Chern class are therefore complete integral invariants. Group completion gives \(K^0(\mathbb T^2)=\mathbb Z^2\): every integer Chern number is realized already by a line comparison loop, and equal rank/Chern invariants for virtual bundles become equal fixed-rank bundles after adding trivial summands. The latter addition can always make the rank positive, so the fixed-rank classification proves injectivity as well as surjectivity.

Choose a line bundle \(L\) of Chern number one by this gluing, and put \(\eta=[L]-[1]\). Thus \([1],\eta\) form a basis of \(K_0\). Pullback preserves rank and multiplies the Chern number by the degree of \(T_M^{-1}\), namely \(\det(M^{-1})=1\). The degree assertion follows directly from its pullback on the oriented area form \(dx_1\wedge dx_2\). Both invariants are unchanged, so the induced K-map is the identity. \(\square\)

## 5. Kernels, cokernels and torsion

Put \(D=1-M'\), acting on integral columns.

**Theorem 5.1.** There are split short exact sequences giving

\[
\begin{gathered}
K_0(B_M)\cong\mathbb Z^2\oplus\ker D,\\
K_1(B_M)\cong\mathbb Z^2\oplus\operatorname{coker}D.
\end{gathered}
\tag{5.1}
\]

The splittings need not be canonical. If \(\operatorname{tr}M\ne2\), then \(\ker D=0\) and the cokernel is finite of order \(|2-\operatorname{tr}M|\).

*Proof.* Insert Lemma 4.1 into the two halves of the PV sequence. They become

\[
\begin{gathered}
0\longrightarrow\mathbb Z^2\xrightarrow{\iota_*}K_0(B_M),\\
K_0(B_M)\xrightarrow{d_0}\ker D\longrightarrow0;\\
0\longrightarrow\operatorname{coker}D\xrightarrow{\bar\iota_*}K_1(B_M),\\
K_1(B_M)\xrightarrow{d_1}\mathbb Z^2\longrightarrow0.
\end{gathered}
\tag{5.2}
\]

The kernel of an integer matrix is free, and the second quotient is free, so choosing lifts splits each sequence. For a two-by-two determinant-one matrix, \(\det(1-M')=2-\operatorname{tr}(M')\). Since \(\operatorname{tr}(M')=\operatorname{tr}M\), this gives \(\det D=2-\operatorname{tr}M\). If it is nonzero, \(D\) is injective over \(\mathbb Q\), hence over \(\mathbb Z\). Its integral image has index \(|\det D|\), as follows from integer row and column reduction to Smith form. This proves the last assertion. \(\square\)

To determine the finite group, and not only its cardinality, let \(s_1\) be the gcd of all entries of a nonsingular \(D\). Then its Smith factors are \(s_1\) and \(s_2=|\det D|/s_1\), with \(s_1\mid s_2\), and

\[
\operatorname{coker}D\cong
\mathbb Z/s_1\mathbb Z\oplus\mathbb Z/s_2\mathbb Z.
\tag{5.3}
\]

The Euclidean algorithm implemented by integral row and column operations gives a diagonal matrix with divisibility of the first diagonal entry into the second. Those operations preserve both the gcd of entries and determinant magnitude. This proves the stated factors and index. In particular, the torsion group need not be cyclic.

If \(\operatorname{tr}M=2\) but \(M\ne1\), the matrix \(D\) has rank one. Its Smith form is \(\operatorname{diag}(s_1,0)\), so \(\ker D\cong\mathbb Z\) and \(\operatorname{coker}D\cong\mathbb Z\oplus\mathbb Z/s_1\mathbb Z\). Thus \(K_0\cong\mathbb Z^3\), whereas \(K_1\cong\mathbb Z^3\oplus\mathbb Z/s_1\mathbb Z\). When \(M=1\), both groups are \(\mathbb Z^4\), in agreement with \(B_1=C(\mathbb T^3)\).

## 6. Examples and their generators

**Example 6.1 (the basic shear).** For

\[
M=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad
D=\begin{pmatrix}0&0\\1&0\end{pmatrix},
\tag{6.1}
\]

the kernel is generated by \([z_2]\), and the cokernel by the coset of \([z_1]\). Hence both groups of \(B_M\) are \(\mathbb Z^3\). The coefficient classes \(\iota_*[1],\iota_*\eta\) and any class \(b\) with \(d_0b=[z_2]\) are a K-zero basis. Since \(z_2\) and \(w\) commute, the image of the two-torus Bott difference class in their subalgebra supplies such \(b\), after choosing its sign. To check this, apply PV naturality to the inclusion of the invariant coefficient circle generated by \(z_2\), with its trivial action. For that product circle the kernel of \(d_0\) is the unit summand, so the other primitive Bott summand maps to \(\pm[z_2]\).

A K-one basis consists of \(\iota_*[z_1]\), \([w]\), and a lift \(\ell\) with \(d_1\ell=\eta\). Scalar-system naturality gives \(d_1[w]=\pm[1]\). The lift \(\ell\) exists by (5.2); changing it by a multiple of \(\iota_*[z_1]\) changes the chosen splitting. We have specified its boundary, without pretending that exactness selects a unique unitary representative.

The mapping torus of the shear on \(\mathbb T^2\) is the usual Heisenberg nilmanifold, a circle bundle over a two-torus. Its fundamental group is \(\mathbb Z^2\rtimes_M\mathbb Z\): the implementing generator commutes with one coordinate generator and takes the other to its product with that central coordinate. Proposition 6.3 below gives the degree-one KK comparison with \(B_M\), including the gluing convention and the two inverse products.

**Example 6.2 (two hyperbolic matrices).** For
\(M=\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)\), one has \(\operatorname{tr}M=3\), hence \(\det D=-1\); both groups are \(\mathbb Z^2\). For
\(M=\left(\begin{smallmatrix}3&1\\2&1\end{smallmatrix}\right)\), one has

\[
\begin{gathered}
M'=\begin{pmatrix}1&-2\\-1&3\end{pmatrix},\\
D=\begin{pmatrix}0&2\\1&-2\end{pmatrix}.
\end{gathered}
\tag{6.2}
\]

Its determinant is \(-2\) and the gcd of its entries is one. Thus \(K_0(B_M)=\mathbb Z^2\) and \(K_1(B_M)\cong\mathbb Z^2\oplus\mathbb Z/2\mathbb Z\). The torsion is the coefficient class \(\iota_*[z_1]\): the two column relations kill \([z_2]\) and twice \([z_1]\).

**Proposition 6.3 (the mapping-torus KK comparison).** Let \(A\) be a separable C*-algebra, possibly nonunital, and let \(\alpha\in\operatorname{Aut}(A)\). Put

\[
\begin{gathered}
M_\alpha=\{F\in C([0,1],A):\\
F(1)=\alpha(F(0))\},\\
B=A\rtimes_\alpha\mathbb Z,\\
C=M_\alpha\rtimes_\tau\mathbb R,\\
(\tau_rF)(t)=F(t+r),\\
F(t+n)=\alpha^n(F(t)).
\end{gathered}
\tag{6.3}
\]

There are inverse classes \(x_\alpha\in KK^1(M_\alpha,B)\) and \(y_\alpha\in KK^1(B,M_\alpha)\).

*Proof.* The extension and translation in (6.3) are the ones checked in [Lesson 17, Proposition 4.2](KT-OPK-17.md#from-mapping-tori-to-crossed-products). In particular \(\tau\) is point-norm continuous. Green's induced algebra for \((\mathbb R,\mathbb Z,\alpha)\) consists of functions \(f(t+n)=\alpha^{-n}(f(t))\). Reflection \(F(t)=f(-t)\) identifies it with \(M_\alpha\), and takes left translation to \(F(t+r)\). Thus [*Green's imprimitivity theorem*, Theorem 7.4 and equations (7.45)–(7.47)](https://kokunoyumeto.github.io/open-math-courses-public/courses/KT-CP/KT-CP-07.html) supplies a \(C\)–\(B\) imprimitivity module \(X_\alpha\), obtained from \(C_c(\mathbb R,A)\) with its proved inner products and full crossed-product norms.

Both coefficient algebras and both crossed products are separable, since the groups are second countable. Hence the Green module and its conjugate \(\overline X_\alpha\) are countably generated: their compact-operator algebras are the separable left corners. Their zero-operator cycles define even classes. The inner-product evaluation unitaries give

\[
\begin{gathered}
{}[X_\alpha]\widehat\otimes_B[\overline X_\alpha]=1_C,\\
{}[\overline X_\alpha]\widehat\otimes_C[X_\alpha]=1_B.
\end{gathered}
\tag{6.4}
\]

These are the actual Morita inverse products of [*Connections and the existence of the Kasparov product*, Proposition 7.1](https://kokunoyumeto.github.io/open-math-courses-public/courses/KT-OPK/preparation.html#dependency-c9822bcc4659). Their evaluation maps are proved in [*Imprimitivity bimodules and Morita equivalence*, Theorem 3.2](https://kokunoyumeto.github.io/open-math-courses-public/courses/KT-CP/prerequisites/hilbert-c-star-modules-and-morita-equivalence/imprimitivity-bimodules-and-morita-equivalence.html).

For the real action \(\tau\), use the Thom class \(t_\tau\in KK^1(M_\alpha,C)\) and its inverse \(d_\tau\in KK^1(C,M_\alpha)\) from [*Descent and the K-theory of crossed products*, §§4–6, Theorem 6.1](https://kokunoyumeto.github.io/open-math-courses-public/courses/KT-OPK/preparation.html#dependency-e674f564ac7f). In that lesson's positive Fourier and right-Clifford conventions, \(d_\tau=-t_{\widehat\tau}\widehat\otimes m_\tau\), where \(m_\tau\) is the Takai Morita class. Both inverse identities are part of that theorem. Define

\[
\begin{aligned}
x_\alpha&=t_\tau\widehat\otimes_C[X_\alpha],\\
y_\alpha&=[\overline X_\alpha]\widehat\otimes_C d_\tau.
\end{aligned}
\tag{6.5}
\]

Associativity and (6.4) now give

\[
\begin{aligned}
x_\alpha\widehat\otimes_B y_\alpha
 &=t_\tau\widehat\otimes_C d_\tau=1_{M_\alpha},\\
y_\alpha\widehat\otimes_{M_\alpha}x_\alpha
 &=[\overline X_\alpha]\widehat\otimes_C[X_\alpha]=1_B.
\end{aligned}
\tag{6.6}
\]

All factors remain in their stated order; the Green classes have even degree. This proves the claimed KK-equivalence.

Its K-theory normalization can also be compared precisely with Lesson 17. That lesson defines \(\Xi_{\alpha,i}=\partial_i^\tau\,\mu_{\overline X_\alpha,i}\), using the ordinary Wiener–Hopf boundary. Proposition 6.2 of the KK descent lesson says that multiplication by \(t_\tau\) on \(K_j(M_\alpha)\) is \((-1)^{j+1}(\partial_{j+1}^\tau)^{-1}\). Inverting this map and then composing with the conjugate Green map therefore gives

\[
\begin{gathered}
(y_\alpha)_*:K_i(B)\longrightarrow K_{i-1}(M_\alpha),
\\
(y_\alpha)_*=(-1)^i\Xi_{\alpha,i}.
\end{gathered}
\tag{6.7}
\]

Thus the degree-zero map agrees with \(\Xi_{\alpha,0}\), and the degree-one map is its negative. The reflection and the specified Thom inverse fix these signs. \(\square\)

For the geometric application, write
\[
N_M=(\mathbb T^2\times[0,1])/
\bigl((x,1)\sim(T_Mx,0)\bigr).
\]
Because \(\alpha_M f=f\circ T_M^{-1}\), the endpoint condition identifies \(M_{\alpha_M}\) with \(C(N_{M^{-1}})\). Interval reversal \([x,t]\mapsto[x,1-t]\) is a homeomorphism \(N_{M^{-1}}\to N_M\): the images of \((x,1)\) and \((T_M^{-1}x,0)\) are equivalent under the \(T_M\)-gluing. Transporting the inverse pair (6.5) through its pullback gives a degree-one KK-equivalence between \(C(N_M)\) and \(B_M\). In Example 6.1 this is the asserted Heisenberg-nilmanifold comparison.

## 7. Exercises with solutions

**Exercise 21.1 (basic: three commuting circles).** Compute both K-groups of \(C(\mathbb T^3)\), specifying four generators in each parity.

*Solution.* Use the recursive classes \(g_I\) of Lesson 12, Corollary 5.2. In degree zero the subsets are \(\varnothing,\{1,2\},\{1,3\},\{2,3\}\); in degree one they are \(\{1\},\{2\},\{3\},\{1,2,3\}\). The empty subset is the unit, the singletons are coordinate unitaries, and adjoining a new coordinate to a preceding class means the ideal inclusion followed by the typed suspension isomorphism. At every stage the circle extension is split, so these classes form a basis, rather than merely a generating list. Both groups are \(\mathbb Z^4\). The same ranks hold for every noncommutative three-torus by Theorem 2.2, with PV lifts replacing the specified commutative suspensions. \(\square\)

**Exercise 21.2 (intermediate: shear generators).** Recompute Example 6.1 from the chosen action and verify its bases.

*Solution.* The inverse pullback satisfies \(\alpha_M(z_1)=z_1z_2^{-1}\) and \(\alpha_M(z_2)=z_2\). Thus \(M'\) has columns \((1,-1)^T,(0,1)^T\), so \(D\) sends \((a,b)^T\) to \((0,a)^T\). Its kernel is \(\mathbb Z(0,1)^T\), and its image is the second coordinate subgroup. The cokernel is the first coordinate copy of \(\mathbb Z\). The sequences (5.2) give exactly the bases described in Example 6.1. Applying the boundary to a relation among the K-zero basis first kills the coefficient terms and forces the coefficient of \(b\) to vanish; injectivity then proves independence of the two remaining terms. For K-one, applying \(d_1\) first proves independence of \([w],\ell\) modulo the coefficient image; the injective cokernel map handles \(\iota_*[z_1]\). Subtracting the corresponding lifts proves spanning. \(\square\)

**Exercise 21.3 (intermediate: hyperbolic torsion).** Compute the torsion for the two matrices of Example 6.2, including its generator.

*Solution.* For \(\left(\begin{smallmatrix}3&1\\2&1\end{smallmatrix}\right)\), formula (6.2) makes the image relations \(e_2=0\) and \(2e_1-2e_2=0\). The quotient is \(\mathbb Z/2\) on \(e_1\), and \(e_1\) is nonzero because reduction of its first coordinate modulo two kills both image columns. The determinant is nonzero, so the kernel vanishes. The PV injection of this quotient identifies its generator with \(\iota_*[z_1]\) in \(K_1(B_M)\), while the free quotient contributes two additional lifts. For \(\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)\), one has \(D=\left(\begin{smallmatrix}0&1\\1&-1\end{smallmatrix}\right)\), an integral invertible matrix. Both kernel and cokernel are zero, so there is no torsion and both groups are \(\mathbb Z^2\). \(\square\)

**Exercise 21.4 (advanced: the three rotation traces).** For a skew three-by-three \(\Theta\), determine the trace ranges contributed by its three two-coordinate rotation algebras. Prove their inclusion without using Elliott's formula.

*Solution.* Proposition 3.2 identifies the copies for \((1,2),(1,3),(2,3)\) with parameters \(-\theta_{12},-\theta_{13},-\theta_{23}\) in Lesson 20's convention. Their canonical trace ranges are respectively \(\mathbb Z+\theta_{12}\mathbb Z\), \(\mathbb Z+\theta_{13}\mathbb Z\), and \(\mathbb Z+\theta_{23}\mathbb Z\), valid for rational as well as irrational entries. Trace compatibility on matrix projection differences shows that each number in each of these groups is a trace value in the larger algebra. Since its range is a subgroup, it contains their sum
\[
\begin{gathered}
\mathbb Z+\theta_{12}\mathbb Z\\
{}+\theta_{13}\mathbb Z+\theta_{23}\mathbb Z.
\end{gathered}
\]
Theorem 3.3 states that there are no additional values in dimension three, but that reverse inclusion is not supplied by the two-coordinate argument. Negative values here are trace values of virtual classes, not of positive projections. \(\square\)

**Exercise 21.5 (advanced: order is not the group).** Take \(M=-1_2\). Compute both K-groups and explain why the torsion is not cyclic of order four.

*Solution.* Here \(M'=M=-1_2\), so \(D=2\cdot1_2\). Its kernel is zero, and its image is \(2\mathbb Z^2\). Consequently the cokernel is \((\mathbb Z/2)^2\), with independent classes of the two coordinate vectors. Formula (5.1) gives \(K_0(B_M)\cong\mathbb Z^2\) and \(K_1(B_M)\cong\mathbb Z^2\oplus(\mathbb Z/2)^2\). Its torsion subgroup has four elements, as \(|2-\operatorname{tr}M|=4\) predicts, but every nonzero element has order two. Thus its cardinality does not determine its isomorphism type; the two Smith factors do. \(\square\)

## Proof inputs and source comparison

The group computations use exact PV maps and the coefficient-circle results of Lessons 18 and 12. The universal identifications, point-norm homotopies, canonical trace, two-coordinate embeddings, actual inverse-transpose action, bundle-invariant argument and integer matrix calculations are proved here. Lemmas 3.4–3.5 and Theorem 3.6 prove Elliott's natural integral Chern lattice and the full Pfaffian trace range: evaluation K-isomorphisms, smooth representatives, the finite differentiated-trace identity, the integral commutative lattice and both parities are all included. Proposition 6.3 proves the mapping-torus composition and both inverse products, using the real Thom pair of KT-KK Lesson 18, Theorem 6.1, and the Green module of KT-CP Lesson 7, Theorem 7.4. Equation (6.7) records its normalization relative to the K-group comparison of Lesson 17.

Blackadar's Corollary 10.5.2 concerns homotopy in the automorphism group, not just equality of induced K-maps. Its remarks after Exercise 10.11.6 mention higher tori without giving the Pfaffian computation. Rosenberg's assigned §4 treats two-dimensional rotation algebras. It supplies comparison and context, rather than a higher-dimensional trace formula. Rieffel's §6, pp. 307–308, recalls Elliott's lattice and trace formulas before proving its separate positive-cone theorem; the recalled formulas are not proved on those pages. Its orientation correction on pp. 297–298 supplies the sign comparison. The proof used here is the evaluation construction, smooth-representative argument, finite contraction calculation and integral-lattice argument of Lemmas 3.4–3.5 and Theorem 3.6. Yashinski's deformation formulas motivate the contraction calculation; Lemma 3.5 gives the complete finite proof directly, without importing the general Gauss–Manin connection theorem. The sign is independently calibrated against Lesson 12's tautological line. Finally, the shear's relation to a Heisenberg nilmanifold is a relation through its mapping torus and a degree shift; the crossed product remains noncommutative.

Connes's Chapter IV, section 6 uses the ordered two-derivation cocycle \(T(a^0(\delta_1a^1\delta_2a^2-\delta_2a^1\delta_1a^2))\). Corollaries 3.7-3.9 give its integral two-torus label, the complete spectral-gap constancy argument, and the integer symbol pairing of that section's Corollary 4(b), calibrated against (3.9), (3.20), and the positive circle suspension. In that section the real-line elliptic regularity and index results are Theorems 2 and 3, attributed to [98]; their full analytic proofs are not supplied there. The cited 1980 note, also freely available from the author, states its pseudodifferential and index theorems without full proofs. The Kubo and localization assertions are further external inputs. These statements are therefore not being used as completed proofs of the present corollaries.

## References

- **[Blackadar 1998]** Bruce Blackadar, *K-Theory for Operator Algebras*, second edition, Cambridge University Press, 1998, Proposition 10.5.1, Corollary 10.5.2, and the remarks after Exercise 10.11.6. [Author's book](https://www.bruceblackadar.com/Mathematics/book6.pdf).
- **[Rosenberg 2012]** Jonathan Rosenberg, *Examples and applications of noncommutative geometry and K-theory*, in *Topics in Noncommutative Geometry*, Clay Mathematics Proceedings 16, 2012, pp. 93–129, §4. [Publisher's volume](https://www.claymath.org/wp-content/uploads/2022/03/cmip016c.pdf).
- **[Emerson 2024]** Heath Emerson, *An Introduction to C*-Algebras and Noncommutative Geometry*, Birkhäuser, 2024, §11.5, especially Corollaries 11.5.12–11.5.16.
- **[Rieffel 1988]** Marc A. Rieffel, *Projective modules over higher-dimensional non-commutative tori*, Canadian Journal of Mathematics 40 (1988), 257–338, the lattice and trace comparison at the start of §6, pp. 307–308, and the orientation correction at the end of §4, pp. 297–298. [Author's free paper](https://math.berkeley.edu/~rieffel/papers/projective88.pdf). These passages attribute the lattice and trace formulas to George A. Elliott. The proof used in this lesson is given in Lemmas 3.4–3.5 and Theorem 3.6; Rieffel's recalled statement does not replace that proof, and the original 1984 paper is not a proof prerequisite.

- **[Yashinski 2014]** Allan Yashinski, *The Gauss-Manin connection and noncommutative tori*, arXiv:1210.4531, version 2 (27 December 2014), Theorem 6.8, Corollaries 6.9–6.10 and Theorem 6.11, pp. 23–27. [Author's preprint](https://arxiv.org/abs/1210.4531). Deformation-formula comparison; Lemma 3.5 proves the needed identity directly with the normalization of (3.9).

[Rieffel1990, rank and AF comparison] M. A. Rieffel, *Non-Commutative Tori — A Case Study of Non-Commutative Differentiable Manifolds*, Contemporary Mathematics105(1990),191–211, complete §3, author-numberedpp6–7. [Freely readable author survey](https://math.berkeley.edu/~rieffel/papers/non_com_tori.pdf). Theorem2.2 and Corollary2.3 supply the full rank and AF-obstruction arguments for every real skew parameter and n≥1. The survey refers to separate Pimsner–Voiculescu and Kumjian papers for its AF embeddings and inductive presentation; those constructions are not imported from statements. Its generator order corresponds to the negative of (1.1)'s parameter matrix, which does not change the ranks or this obstruction.
- **[Connes 1994]** Alain Connes, *Noncommutative Geometry*, Chapter IV, section 6, pp. 356-375, with the two-torus cocycle at pp. 359 and 369 and the symbol expression at pp. 360-361. [Author's free book](https://alainconnes.org/wp-content/uploads/book94bigpdf.pdf). Corollaries 3.7-3.9 use the independently proved integral labels, coefficient-circle splitting and holomorphic calculus of this programme; the chapter is the result-level comparison source for the elliptic and Hall-effect statements.
- **[Connes 1980/2001]** Alain Connes, *C*-algèbres et géométrie différentielle*, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A599-A604. [Original note](https://alainconnes.org/wp-content/uploads/note80.pdf); [author's English translation](https://alainconnes.org/wp-content/uploads/notetranslated.pdf), 2001. The original's Proposition 9 and Theorems 10-11 correspond to Proposition 8 and Theorems 9-10 of the translation; these are statements without full analytic proofs in the note.