Fredholm modules and analytic K-homology

Written by GPT-6.1 Sol (OpenAI). Public domain (CC0).

The shift obstruction in extension theory can be recorded before taking a quotient. A representation supplies the algebra action; an operator that commutes with this action modulo compacts supplies the obstruction. A Fredholm module is this pair of data, together with a parity. The parity determines whether we will test it against projections or unitaries.

Throughout, \(A\) is a separable complex C*-algebra and Hilbert spaces are separable. Representations may be degenerate. Compact operators between two Hilbert spaces mean norm limits of finite-rank operators. We use the Fredholm criterion, index invariance under compact perturbations and norm-continuous Fredholm paths, and index additivity proved in Lemmas 1.1–1.2 of the extension lesson. The preceding two lessons supply the Busby correspondence and the complementary-corner construction for a completely positive lift.

1. The cycle and its three compact defects

An odd Fredholm module over \(A\) is a triple \((H,\pi,F)\), where \(\pi:A\to\mathcal B(H)\) is a *-representation and \(F\) is bounded, such that, for every \(a\in A\),

An even Fredholm module has in addition an orthogonal grading \(H=H_+\oplus H_-\). Its grading operator \(\Gamma\) is \(1\) on \(H_+\) and \(-1\) on \(H_-\). We require \(\Gamma\pi(a)=\pi(a)\Gamma\) and \(\Gamma F=-F\Gamma\). Thus

The module is degenerate if all three expressions in (1.1) are zero, for every \(a\). Such a module contributes no obstruction. In particular, a self-adjoint unitary commuting with the representation is degenerate. The zero representation makes every bounded \(F\) degenerate.

For nonunital \(A\), the last two defects need not themselves be compact. Their compactness is required only where the algebra acts. This distinction is necessary, for example, when a cycle has an infinite-dimensional zero-representation summand.

Unitary equivalence means a unitary intertwining both \(\pi\) and \(F\); in the even case it must also preserve the grading. An operator homotopy has fixed \(H,\pi\), and grading, and a norm-continuous path \(F_t\) of operators satisfying (1.1). Direct sums use block representations and operators, and the sum grading in the even case.

We quotient cycles by the equivalence relation generated by unitary equivalence, operator homotopy, and the relation identifying a cycle with its direct sum with a degenerate cycle. We also identify every degenerate cycle with the zero cycle. Write the resulting sets as \(K^0(A)\) and \(K^1(A)\). Superscripts refer to analytic K-homology; subscripts \(K_0(A),K_1(A)\) refer to the K-theory that will test it.

The comparison with the general Hilbert-module homotopy definition of \(KK^i(A,\mathbb C)\) is a theorem, rather than part of the meaning of operator homotopy. Its full proof must compare Hilbert-module homotopy with the operator-homotopy and degenerate-summand relations used here. That comparison remains unproved here.

2. Normalization without a unitality assumption

Before writing \(P=(1+F)/2\), we must make \(F\) a self-adjoint involution. We develop the quotient-ideal and doubled-operator construction using Blackadar, Sections 17.4–17.5. Its operators must be checked against the locally compact defects in (1.1), since neither a unit nor a nondegenerate representation is assumed here.

For a fixed representation put

Lemma 2.1. The algebra of compact defects

\(\mathcal D_\pi\) is a unital C*-algebra and \(\mathcal J_\pi\) is a closed two-sided *-ideal in it. For \(T\in\mathcal D_\pi\), membership in \(\mathcal J_\pi\) is equivalent to compactness of \(\pi(a)T\) for all \(a\). The cycle conditions say exactly that the image of \(F\) in \(\mathcal D_\pi/\mathcal J_\pi\) is a self-adjoint involution.

Proof. For each \(a\), the commutator map \(T\mapsto[T,\pi(a)]\) is norm-continuous and the compact ideal is norm closed. The intersection of its inverse images is therefore closed. The identity belongs to it, and the identities \([ST,\pi(a)]=S[T,\pi(a)]+[S,\pi(a)]T\) and \([T^*,\pi(a)]=-[T,\pi(a^*)]^*\) prove multiplication and adjoint closure. It is consequently a unital C*-subalgebra of \(\mathcal B(H)\).

If \(T\in\mathcal D_\pi\), the equality \(\pi(a)T=T\pi(a)-[T,\pi(a)]\) proves the left-right equivalence. The maps \(T\mapsto T\pi(a)\) are also continuous, so their compact inverse images form a closed subspace. If \(T\) lies in that subspace, \(T^*\pi(a)=(\pi(a^*)T)^*\) is compact. For \(S\in\mathcal D_\pi\), \(ST\pi(a)\) is compact by the compact ideal property, and \(TS\pi(a)=T\pi(a)S+T[S,\pi(a)]\) is compact as well. These calculations prove exactly the closed two-sided *-ideal assertion. The adjoint and square defects of a cycle belong to this ideal by the left-right equivalence, so its quotient image is a self-adjoint element whose square is the identity. Conversely those quotient equalities are precisely the required locally compact defects. \(\square\)

Proposition 2.2. Locally compact perturbations

If \(F,F'\) are cycle operators for the same representation and parity, and \((F'-F)\pi(a)\) is compact for every \(a\), then the segment \(F_t=(1-t)F+tF'\) is an operator homotopy. In particular an ordinary compact perturbation gives an operator homotopy.

Proof. Put \(K=F'-F\). Both endpoints lie in \(\mathcal D_\pi\), and the assumed compact products put \(K\) in \(\mathcal J_\pi\). For \(F_t=F+tK\), \[ F_t^2-1=(F^2-1)+t(FK+KF)+t^2K^2, \qquad F_t-F_t^*=(F-F^*)+t(K-K^*). \] Every term on the right belongs to \(\mathcal J_\pi\), using the ideal property for the products; their multiplication on the represented algebra is therefore compact. The commutator is \([F,\pi(a)]+t[K,\pi(a)]\), a sum of compacts. This proves every cycle condition without imposing global compactness on \(K\). The path is norm-continuous and is odd for the grading whenever its endpoints are. An ordinary compact \(K\) satisfies the assumed products. \(\square\)

Proposition 2.3. An exact involution representative

Every even or odd module is equivalent to one with \(F=F^*\) and \(F^2=1\). The construction can be performed continuously on a norm-continuous family of cycle operators.

Proof. Lemma 2.1 permits functional calculus in the quotient \(\mathcal D_\pi/\mathcal J_\pi\). Set \(F_s=(F+F^*)/2\). It has the same quotient image as \(F\), so the segment joining them is allowed by Proposition 2.2. That quotient image is an involution. For the odd continuous clipping function \(g(t)=\max(-1,\min(t,1))\), put \(f=g(F_s)\). Its quotient image is still the same involution, since \(g(1)=1\) and \(g(-1)=-1\); hence \(f-F_s\in\mathcal J_\pi\) and a second allowed segment joins them. The operator \(f\) is a self-adjoint contraction. If the starting operator is odd, conjugating by the grading replaces \(F_s\) by \(-F_s\), and the oddness of \(g\) makes \(f\) odd too.

The positive operator \(s=(1-f^2)^{1/2}\) commutes with \(f\). Its quotient image is zero, so \(s\in\mathcal J_\pi\). On \(H\oplus H\), with representation \(\pi\oplus0\), form

For an even cycle use the grading \(\operatorname{diag}(\Gamma,-\Gamma)\). The square root \(s\) is even, because \(f^2\) is even, and the opposite second grading therefore makes its off-diagonal positions odd. All \(G_t\) are self-adjoint. Direct multiplication, using \(fs=sf\), gives \[ G_t^2-1=(t^2-1) \begin{pmatrix}s^2&0\\0&s^2\end{pmatrix}. \] Its product with the represented algebra is compact because \(s^2\in\mathcal J_\pi\). The only other commutator blocks are \([f,\pi(a)]\), \(-t\pi(a)s\) and \(ts\pi(a)\); these are compact by \(f\in\mathcal D_\pi\) and \(s\in\mathcal J_\pi\). Thus (2.2) is an operator homotopy. At \(t=0\) the first block is the clipped cycle and the second is a zero-representation degenerate cycle. At \(t=1\) the displayed square defect is exactly zero. These successive segments and the degenerate-summand relation produce the required representative.

For a norm-continuous original family the self-adjoint parts are uniformly bounded on compact parameter intervals. Polynomial approximation to \(g\) on one common spectral interval proves norm continuity of the clipping; polynomial approximation to the square root on \([0,1]\) proves norm continuity of \(s\). Matrix assembly then proves continuity of \(G_1\). This argument includes operators whose spectra cross zero; no gap is used. \(\square\)

This construction deliberately allows a zero-representation summand. For an even cycle of nonzero index, one cannot in general obtain an exact odd involution by a compact perturbation on the original graded space: an exact odd involution is an isomorphism between its two graded summands.

3. Addition, inverses and pullback

Theorem 3.1. Analytic K-homology is an abelian group

Direct sum makes \(K^0(A)\) and \(K^1(A)\) abelian groups. An inverse of an odd cycle is \((H,\pi,-F)\). An inverse of an even cycle is \((H^{\mathrm{op}},\pi,-F)\), where \(H^{\mathrm{op}}\) has the opposite grading.

Proof. Direct sum preserves the defects and all generating relations. Rebracketing and flipping summands are grading-preserving unitaries where appropriate, giving associativity and commutativity. The zero cycle is an identity by the degenerate-summand relation.

We may first replace \(F\) by its self-adjoint part, by Proposition 2.2. Add the proposed inverse and use, on the two copies,

for \(0\leq\theta\leq\pi/2\), with representation \(\pi\oplus\pi\). In the even case the opposite second grading makes the off-diagonal identity odd. Squaring gives \[ F_\theta^2-1 =\cos^2\theta \begin{pmatrix}F^2-1&0\\0&F^2-1\end{pmatrix}. \] Its product with the representation is compact. The commutator has diagonal blocks \(\cos\theta[F,\pi(a)]\) and their negatives and zero off-diagonal blocks; self-adjointness is exact. At \(\theta=0\) we have the cycle plus its proposed inverse. At \(\pi/2\) we have the self-adjoint exchange unitary, commuting exactly with the representation. It is degenerate, so the sum represents zero. Finally \(-F\) and the grading reversal respect each generating equivalence relation, so the inverse is well defined on classes. \(\square\)

Proposition 3.2. Contravariance

A *-homomorphism \(\alpha:A\to B\) induces \[ \begin{aligned} \alpha^*&:K^i(B)\longrightarrow K^i(A),\\ [(H,\pi,F)]&\longmapsto[(H,\pi\alpha,F)]. \end{aligned} \] It is a group homomorphism, and \((\beta\alpha)^*=\alpha^*\beta^*\), \(1_A^*=1\).

Proof. Each compact condition over \(A\) is the corresponding condition over \(B\), evaluated at \(\alpha(a)\). Degeneracy, unitary equivalence and operator homotopy are preserved, as is direct sum. Composition and the identity follow by composing the representations. \(\square\)

When \(A\) is unital, one may also remove a degenerate part of its representation. Put \(p=\pi(1)\). The off-diagonal blocks of \(F\) relative to \(pH\oplus(1-p)H\) are compact, because \([F,p]\) is compact. Delete them by Proposition 2.2. The remaining zero-representation part is degenerate, and the part on \(pH\) has a unital representation. The grading respects this decomposition for even cycles. This reduction does not require a nondegeneracy convention at the start.

4. From a differential operator to a bounded cycle

We state a spectral argument with all its operator hypotheses. It will apply to Dirac operators and to compact-resolvent spectral triples.

Lemma 4.0. Spectral calculus at compact resolvent

Let \(D\) be a densely defined self-adjoint operator with compact \((1+D^2)^{-1}\). There is an orthonormal basis \(e_j\) of eigenvectors, with real eigenvalues \(\lambda_j\), and \[ \operatorname{Dom}D=\left\{\sum_j u_je_j:\sum_j(1+\lambda_j^2)|u_j|^2<\infty\right\}, \qquad D\left(\sum_j u_je_j\right)=\sum_j\lambda_ju_je_j . \] Only finitely many eigenvalues, counted with multiplicity, lie in any bounded interval. Bounded scalar functions define bounded diagonal operators \(f(D)\), preserving products and adjoints. Uniformly bounded pointwise convergence of such functions on the eigenvalues gives strong operator convergence. If \(f(t)\to0\) as \(|t|\to\infty\), then \(f(D)\) is compact.

Proof. For \(\alpha>0\), \(\|(D\pm i\alpha)u\|^2=\|Du\|^2+\alpha^2\|u\|^2\). Closedness of \(D\) makes these ranges closed. Their orthogonal complements are the kernels of \(D\mp i\alpha\), which are zero by the same equality and self-adjointness. Thus both ranges are all \(H\), and their inverse norms are at most \(1/\alpha\). Set \(R=(D-i)^{-1}\). Testing the adjoint identity on the two domains gives \(R^*=(D+i)^{-1}\), and the resolvent identity gives \[ R-R^*=2iRR^*=2iR^*R. \] Hence \(T=RR^*\) is positive, injective, and has dense range. The product maps into \(\operatorname{Dom}D^2\): \(R^*v\in\operatorname{Dom}D\), \(DRw=w+iRw\), and \(R\) maps \(\operatorname{Dom}D\) into \(\operatorname{Dom}D^2\). Multiplying out on this domain proves \(T=(1+D^2)^{-1}\), so \(T\) is compact.

Here is the compact positive spectral argument used, including completeness. If \(T\ne0\), put \(r=\|T\|\). The self-adjoint norm identity in Hilbert spaces and compact operators, Corollary 3.2, and positive functional calculus give \(\sup_{\|u\|=1}\langle u,Tu\rangle=r\) and \(0\leq T^2\leq rT\). Choose unit \(u_n\) with \(\langle u_n,Tu_n\rangle\to r\). Then \[ \|(T-r)u_n\|^2 \leq r^2-r\langle u_n,Tu_n\rangle\longrightarrow0. \] Compactness of \(T\) gives a convergent subsequence of \(Tu_n\), hence of \(u_n\), whose unit limit is an eigenvector of eigenvalue \(r\). Each nonzero eigenspace is finite dimensional: infinitely many orthonormal vectors in one such space would have images under \(T\) with no convergent subsequence. The orthogonal complement of any collected eigenspaces reduces \(T\), and, if its restriction is nonzero, the same argument supplies another positive eigenvalue. There can be only finitely many eigenvectors with eigenvalues at least any fixed \(\delta>0\), by compactness. Taking the closed span of all positive eigenspaces leaves a reducing complement on which \(T=0\), because a nonzero restriction would again supply a positive eigenvector. Injectivity makes that complement zero.

Every positive eigenspace of \(T\) is contained in \(\operatorname{Dom}D^2\), since \(v=\theta^{-1}Tv\). It is invariant under \(D\), because the resolvents, hence \(T\), commute with \(D\) on its domain. The restriction of \(D\) to this finite-dimensional space is self-adjoint and diagonalizable. Its eigenvectors satisfy \(Te_j=(1+\lambda_j^2)^{-1}e_j\). The preceding compactness property proves the finiteness assertion for bounded \(\lambda_j\).

For \(u\in\operatorname{Dom}D\), self-adjointness gives \(\langle e_j,Du\rangle=\lambda_j\langle e_j,u\rangle\), so the indicated weighted sum is finite. Conversely its finiteness makes finite eigenvector truncations converge together with their \(D\)-images; closedness puts the limit in the domain and proves the formula. Bounded diagonal multipliers have the stated products and adjoints by their coordinates. Strong convergence follows by choosing a finite coordinate truncation of a vector first, using the uniform bound for its tail, and then passing to the pointwise limit on the finite part. Vanishing at infinity makes the norm of the remaining diagonal tail tend to zero; the finite truncations are finite rank. This proves compactness. Finite-dimensional \(H\) satisfies the same conclusions with a finite basis. \(\square\)

Theorem 4.1. The bounded transform

Let \(D\) be self-adjoint on \(H\), with \((1+D^2)^{-1}\) compact. Suppose a dense *-subalgebra \(\mathcal A\subset A\) satisfies \[ \pi(a)\operatorname{Dom}D\subset\operatorname{Dom}D \tag{4.1} \] for every \(a\in\mathcal A\), and every commutator \([D,\pi(a)]\) extends to a bounded operator. Then \[ F=D(1+D^2)^{-1/2} \tag{4.2} \] defines an odd Fredholm module. If \(H\) is graded, \(\pi\) is even, the domain of \(D\) is grading invariant, and \(D\) is odd on that domain, it defines an even module.

Proof. Lemma 4.0 defines (4.2) as the bounded diagonal scalar function \(t/\sqrt{1+t^2}\) of \(D\). It gives \[ F=F^*,\quad \|F\|\leq1,\quad F^2-1=-(1+D^2)^{-1}\in\mathcal K(H). \tag{4.3} \] We prove the commutator assertion rather than assuming that bounded commutators with \(D\) automatically pass through functional calculus.

Set \(\alpha(s)=\sqrt{1+s^2}\) and \(R_\pm(s)=(D\pm i\alpha(s))^{-1}\). These operators are compact. Indeed they are \((1+D^2)^{-1/2}\) times a bounded continuous function of \(D\); the square root of a positive compact operator is compact. The domain invariance in (4.1), applied also to \(a^*\), justifies the resolvent identity \[ [R_\pm(s),\pi(a)]=-R_\pm(s)[D,\pi(a)]R_\pm(s). \tag{4.4} \] For example apply both sides to a vector, use that each resolvent takes it into \(\operatorname{Dom}D\), and multiply by \(D\pm i\alpha\); the identity follows there and extends boundedly.

The scalar identity \[ \frac{t}{\sqrt{1+t^2}} =\frac1\pi\int_0^\infty \left(\frac1{t-i\alpha(s)}+\frac1{t+i\alpha(s)}\right)\,ds \tag{4.5} \] holds because its truncation at \(R\) is \(\frac{2t}{\pi\sqrt{1+t^2}}\arctan(R/\sqrt{1+t^2})\). These scalar truncations have absolute value at most one and converge pointwise. On every finite interval the resolvents are norm-continuous, so their operator integrals exist; applying the integral to each eigenvector gives exactly the displayed scalar truncation. Lemma 4.0 now makes these integrals converge strongly to \(F\). No unbounded spectral theorem is assumed in this passage.

Taking their commutators is better behaved: (4.4) gives a compact integrand of norm at most \[ \frac{2\|[D,\pi(a)]\|}{\pi(1+s^2)}. \] It is norm-continuous in \(s\), and this bound is integrable. Thus its integral converges in norm to a compact operator. The strong limit of the same commutators is \([F,\pi(a)]\), so that operator is compact for \(a\in\mathcal A\). Since \(\|[F,\pi(a)]\|\leq2\|F\|\|a\|\), density proves compactness for all \(a\in A\).

In the graded case \(\Gamma D\Gamma=-D\) with its domain retained. Functional calculus therefore gives \(\Gamma F\Gamma=-F\), since the scalar function in (4.2) is odd. This proves the required parity and completes the proof. \(\square\)

Any continuous real normalizing function \(\chi\), with \(\chi(t)\to\pm1\) as \(t\to\pm\infty\), gives the same class, provided it is odd when a grading is present. The difference from \(t/\sqrt{1+t^2}\) vanishes at infinity. A continuous function of \(D\) vanishing at infinity is compact: truncate it to a bounded spectral interval, whose spectral projection is finite rank because \((1+D^2)^{-1}\) is compact, and estimate the remaining norm by its supremum. Thus the two transforms differ by a compact operator, and Proposition 2.2 applies.

Let \(M\) be a closed smooth manifold and \(E\) a smooth Hermitian vector bundle of finite rank. Fix any smooth positive density on \(M\), which defines \(L^2(M,E)\). The following proofs apply to this density and require no orientation.

The elementary measure inputs are the actual earlier sigma-finite product and real-line measure lemma, Lemma 0.1(a)–(c): Fubini, interval measure and one-dimensional \(C_c^\infty\)-density in \(L^2\). Orthogonal projections, Hilbert-space forms, orthonormal expansions and Parseval are proved in Hilbert spaces and compact operators, Theorems 2.2, 3.1 and 4.1. The finite-dimensional chain rule and change-of-variables formula are used in smooth coordinates. No elliptic regularity, parametrix or pseudodifferential boundedness result is an input.

Lemma 4.1a. Finite-chart Sobolev density and compactness

The space \(H^1(M,E)\) is equivalently the completion of smooth sections in a finite-chart first-derivative norm or the space of \(L^2\) sections having locally square-integrable weak first derivatives in every smooth frame. These definitions and their equivalent norms are independent of the finite charts and cutoffs. Smooth sections are dense in both \(H^1(M,E)\) and \(L^2(M,E)\). The inclusion \(H^1(M,E)\hookrightarrow L^2(M,E)\) is a norm limit of finite-rank operators. Multiplication by any smooth scalar function or smooth bundle endomorphism is bounded on \(H^1\), and smooth functions are uniformly dense in \(C(M)\).

Proof. We first give the finite Fourier argument, including completeness. On an interval of length \(\ell\), rescale the circle exponentials to the normalized functions \(\ell^{-1/2}e^{2\pi ikx/\ell}\). Integration proves orthonormality. The nonnegative circle polynomial \[ K_N(t)=\frac1N\left|\sum_{j=0}^{N-1}e^{ijt}\right|^2 \] has normalized integral one and satisfies \(K_N(t)\leq [N\sin^2(\delta/2)]^{-1}\) for \(\delta\leq |t|\leq\pi\). The integral is one because only the equal-index terms survive; the inequality follows by summing the geometric progression. For a continuous periodic function, convolution with \(K_N\) is a trigonometric polynomial. Splitting its difference from that function into \(|t|<\delta\) and its complement, uniform continuity controls the first part and the displayed bound controls the second. Thus trigonometric polynomials are uniformly dense in the continuous periodic functions. The earlier interval-density lemma gives \(C_c^\infty(0,\ell)\)-density in \(L^2(0,\ell)\); these functions extend periodically and continuously by zero. Uniform approximation therefore proves completeness of the one-dimensional Fourier family.

Products give a complete orthonormal family on any periodic cube \(Q=(-\ell/2,\ell/2)^n\). Here is the induction proving completeness. If \(g\in L^2(Q)\) is orthogonal to every product exponential, for each \(k'\in\mathbb Z^{n-1}\) form its partial Fourier coefficient in the first \(n-1\) coordinates. Fubini and Cauchy–Schwarz put that coefficient in the \(L^2\) space of the last coordinate. All its one-dimensional coefficients vanish, so it is zero almost everywhere. Intersect the countably many full-measure sets for \(k'\), and also the full-measure set on which the sections of \(g\) are in \(L^2\). Each remaining section has all its \((n-1)\)-dimensional coefficients zero, hence vanishes by induction. Fubini gives \(g=0\). This proves completeness for every finite \(n\), also for vector-valued functions component by component.

Write \[ e_k(x)=\ell^{-n/2}e^{i\kappa_k\cdot x}, \qquad \kappa_k=\frac{2\pi}{\ell}k, \qquad \widehat v(k)=\int_Q \overline{e_k(x)}\,v(x)\,dx . \] Parseval identifies the periodic first-derivative space with \[ \|v\|_{H^1(Q)}^2 =\sum_{k\in\mathbb Z^n}(1+|\kappa_k|^2)\|\widehat v(k)\|^2 =\|v\|_2^2+\sum_{j=1}^n\|\partial_jv\|_2^2 . \] To verify the weak-derivative assertion rather than assume it, testing a weak derivative against \(e_k\) gives coefficients \(i\kappa_{k,j}\widehat v(k)\). Conversely, if the weighted sum is finite, the finite Fourier sums converge in \(L^2\) together with their derivatives. Integration by parts against a smooth periodic test function passes to these limits and proves that the derivative limits are the weak derivatives. This also proves completeness of this space, since the weighted sequence space is complete. A function supported in the interior of \(Q\), extended by zero and periodically, has the same weak-derivative interpretation: in the testing identity insert a smooth cutoff equal to one near its support, so no boundary term is present.

For the global construction choose finitely many bundle charts with coordinate cubes \(Q_j\) having compact closures inside their charts, and smooth real functions \(\chi_j\) supported in their interiors, with \(\sum_j\chi_j^2=1\). These cutoffs can be constructed without an analytic existence theorem: in each chart a translated and rescaled version of \(\exp(-1/(1-|x|^2))\) on \(|x|<1\), extended by zero, is positive on a smaller ball. Finitely many such smaller balls cover \(M\); divide the cutoffs by the positive smooth square root of the sum of their squares. Smoothness at the ball boundary follows because every derivative is an exponential times a rational expression, and \(t^{-a}e^{-1/t}\to0\) for every \(a\) as \(t\downarrow0\), as follows from the exponential series. Smooth local frames may be made orthonormal by the finite-dimensional Gram–Schmidt formulas. On the compact coordinate cubes the density is \(\rho_j(x)\,dx\), with \(0<c_j\leq\rho_j\leq C_j\).

Let \(S_ju\) be the coordinate vector of \(\chi_ju\), extended periodically on \(Q_j\), and for smooth \(u\) define \[ \|u\|_{H^1}^2=\sum_j\|S_ju\|_{H^1(Q_j)}^2 . \] This norm controls the global \(L^2\) norm, since the local \(L^2\) norms are equivalent to their density-weighted norms and \(\sum_j\chi_j^2=1\). Its completion embeds injectively in \(L^2\). Indeed, if a Cauchy sequence converges to zero in \(L^2\), each coordinate vector converges to zero there; its derivative limit is zero by integration by parts against compactly supported tests. Thus its \(H^1\) limit is zero.

We check all changes of coordinates and reconstructions used next. On a compact overlap, a localized frame change and coordinate pullback has the form \(v\mapsto C(x)v(y(x))\), with a smooth compactly supported matrix factor \(C\). The ordinary chain rule gives [ \partial_\ell(C,v\circ y) =(\partial_\ell C)v\circ y

• C\sum_m(\partial_\ell y_m)(\partial_m v)\circ y . ] The matrix coefficients and first derivatives are bounded there, as are the Jacobian of the coordinate change and its reciprocal. The change-of-variables formula consequently bounds the output \(H^1\) norm by a constant times the input norm. For a periodic \(H^1\) vector the same formula follows by approximating it by its finite Fourier sums. Products with the cutoffs have support away from the chart boundary, so the zero extensions are legitimate weak-derivative extensions. Define \(R_jv\) to be the section with local vector \(\chi_jv\), extended by zero. The preceding estimate proves that \(R=\sum_jR_j\) is bounded from the finite product of the coordinate \(H^1\) spaces into the global completion. It is also bounded on the corresponding \(L^2\) spaces. On smooth sections \(RSu=u\), because \(\sum_j\chi_j^2=1\), and this identity extends by \(L^2\) limits.

These facts prove the claimed weak-derivative description. A completion element has its local weak derivatives by the preceding limit argument. Conversely, a section whose localized coordinate vectors have weak derivatives in \(L^2\) has \(Su\) in the product \(H^1\) space, so \(u=RSu\) belongs to the completion. The same chain-rule estimate, using \(RSu=u\), compares any two finite-chart norms on smooth sections and hence on the completions. A covariant-derivative definition gives the same space: in a frame \(\nabla_j=\partial_j+C_j(x)\), with smooth bounded matrices on the compact supports, and the two first-derivative norms therefore bound one another.

Let \(\Pi_{j,R}\) retain the finitely many modes with \(|\kappa_k|\leq R\) in \(Q_j\). Then \(T_R=\sum_jR_j\Pi_{j,R}S_j\) is a finite-rank map with smooth output. The Fourier formula gives \[ \|(1-\Pi_{j,R})v\|_2 \leq(1+R^2)^{-1/2}\|v\|_{H^1(Q_j)} . \] Boundedness of the reconstruction proves \(\|u-T_Ru\|_{L^2}\leq C(1+R^2)^{-1/2}\|u\|_{H^1}\). This is the asserted norm approximation of the inclusion by finite-rank operators. For a fixed \(v\in H^1(Q_j)\), the weighted Fourier tail tends to zero as well; the \(H^1\)-bounded reconstruction therefore proves \(T_Ru\to u\) in \(H^1\). For \(u\in L^2(M,E)\), ordinary Fourier-tail convergence and \(L^2\)-bounded reconstruction prove \(T_Ru\to u\) in \(L^2\). This proves both smooth-density assertions.

Multiplication by a smooth matrix is \(H^1\)-bounded by the displayed product rule and the compact-support coefficient bounds. To check the final uniform-density assertion explicitly, take a nonnegative smooth integral-one function \(\eta\) supported in the unit ball, obtained by normalizing the exponential cutoff above, and put \(\eta_\varepsilon(x)=\varepsilon^{-n}\eta(x/\varepsilon)\). For \(f\in C(M)\), the local function \(\chi_jf\), extended by zero, is continuous with compact support. Its convolutions with \(\eta_\varepsilon\) are smooth and converge uniformly to it: the error is at most the supremum of \(|(\chi_jf)(x-z)-(\chi_jf)(x)|\) for \(|z|\leq\varepsilon\). Reconstructing with \(\chi_j\) gives smooth global functions tending uniformly to \(\sum_j\chi_j^2f=f\). On a zero-dimensional component all these spaces are finite dimensional and the assertions are immediate; compactness permits only finitely many such chart components. \(\square\)

Lemma 4.1b. The first-order elliptic estimate

Let \(D:C^\infty(M,E)\to C^\infty(M,E)\) be any smooth first-order elliptic differential operator. There are constants \(C,C'>0\) such that \[ \|u\|_{H^1}\leq C(\|Du\|_2+\|u\|_2), \qquad \|Du\|_2\leq C'\|u\|_{H^1}, \qquad u\in C^\infty(M,E). \] Neither estimate assumes a Dirac-type symbol or invertibility of \(D\).

Proof. In a local frame write \(P=\sum_{j=1}^n A_j(x)\partial_j+B(x)\), with smooth matrices. Our principal-symbol convention is \(\sigma_P(x,\xi)=i\sum_jA_j(x)\xi_j\). At a fixed point \(x_0\), ellipticity and compactness of the unit spheres in the finite-dimensional covector and vector spaces give a constant \(c>0\) such that \[ \left\|i\sum_jA_j(x_0)\xi_j v\right\| \geq c|\xi|\|v\| . \] Indeed the continuous left side on the product of the two unit spheres is positive and attains a positive minimum. Let \(P_0=\sum_jA_j(x_0)\partial_j\). For a smooth vector \(v\) compactly supported in a coordinate cube, extend it periodically and use Lemma 4.1a. The Fourier coefficient of \(P_0v\) is \(i\sum_jA_j(x_0)\kappa_{k,j}\widehat v(k)\). Squaring the preceding inequality and summing by Parseval gives \(\|\nabla v\|_2\leq c^{-1}\|P_0v\|_2\).

Shrink the coordinate neighborhood \(W\) of \(x_0\) until \((\sum_j\|A_j(x)-A_j(x_0)\|^2)^{1/2}\leq c/2\) on it. For \(v\in C_c^\infty(W)\), Cauchy–Schwarz in the finite sum gives \[ \begin{aligned} \|\nabla v\|_2 &\leq c^{-1}\big(\|Pv\|_2 +(c/2)\|\nabla v\|_2+\|B\|_{\infty,W}\|v\|_2\big),\\ \|\nabla v\|_2 &\leq 2c^{-1}\big(\|Pv\|_2+\|B\|_{\infty,W}\|v\|_2\big). \end{aligned} \] This absorbs the coefficient variation and proves the local estimate. Density weights and smooth-frame norms change its constant only by their bounded upper and lower factors.

Choose finitely many such neighborhoods covering \(M\), with cutoffs supported compactly inside them as in Lemma 4.1a. The norm defined from this refined cover is equivalent to the earlier \(H^1\) norm. Apply the local estimate to each \(\chi_ju\), using \[ D(\chi_ju)=\chi_jDu+[D,\chi_j]u, \qquad [D,\chi_j]=\sum_\ell A_\ell\,\partial_\ell\chi_j . \] The commutator is a smooth bounded multiplication operator. Summing the finitely many estimates gives the first global inequality. In the same charts the bounded coefficients and the product rule give the second inequality, by reconstructing \(u=\sum_j\chi_j(\chi_ju)\). A zero-dimensional component contributes only a finite-dimensional norm bound and the harmless \(\|u\|_2\) term. This proves both estimates in the full stated generality. \(\square\)

Lemma 4.1c. Friedrichs regularization and the weak domain

For the operator \(D\) in Lemma 4.1b, \[ u\in L^2(M,E),\quad Du\in L^2(M,E) \text{ in the sense of distributions} \quad\Longleftrightarrow\quad u\in H^1(M,E). \] The forward implication has the same bound as Lemma 4.1b. Locally the mollifier commutator extends to uniformly bounded operators on \(L^2\) and tends strongly to zero. In particular smooth approximations can be chosen to converge in the graph norm.

Proof. We supply the commutator bound and convergence before using them. On a fixed coordinate neighborhood extend the local coefficients smoothly to \(\mathbb R^n\), agreeing on a larger neighborhood of the compact support under consideration and having bounded coefficients and first derivatives; multiply by a smooth cutoff inside the original chart to do this. These extensions need not be elliptic outside that neighborhood. Let \[ P=\sum_jA_j(x)\partial_j+B(x),\qquad J_\varepsilon u=\eta_\varepsilon*u , \] where \(\eta\) is the nonnegative smooth integral-one ball kernel constructed in Lemma 4.1a. For a nonnegative integrable kernel \(k\) of mass \(m\), Cauchy–Schwarz in its measure gives \[ \left|\int k(z)v(x-z)\,dz\right|^2 \leq m\int k(z)|v(x-z)|^2\,dz . \] Integration and Fubini give \(\|k*v\|_2\leq m\|v\|_2\). The same estimate applied to norms controls a matrix kernel bounded in norm by \(k\). Thus \(\|J_\varepsilon\|\leq1\). For a compactly supported vector in the interior of a fixed cube and small \(\varepsilon\), convolution equals periodic convolution there; its Fourier multiplier is \(\int\eta(z)e^{-i\varepsilon\kappa_k\cdot z}\,dz\), bounded by one and tending to one for each \(k\). Finite Fourier truncation followed by a tail estimate proves \(J_\varepsilon u\to u\) in \(L^2\), and also in \(H^1\) when \(u\in H^1\). It gives a smooth function: each derivative falls on the smooth kernel, and Cauchy–Schwarz on its compact support and dominated differentiation justify the integral derivatives. Compactly supported smooth vectors are dense in \(L^2(\mathbb R^n)\): first truncate a vector to an inner cube, use its Fourier approximation in a larger cube, and multiply the finite Fourier sums by a smooth cutoff equal to one on the inner cube. The cutoff does not increase the \(L^2\) error. Consequently the contraction bound extends the \(L^2\) convergence to every vector.

For a compactly supported smooth \(u\), integration by parts in the convolution gives the exact formula \[ \begin{aligned} R_\varepsilon u(x) &:=(PJ_\varepsilon-J_\varepsilon P)u(x)\\ &=\sum_j\int \big(A_j(x)-A_j(y)\big) (\partial_j\eta_\varepsilon)(x-y)u(y)\,dy\\ &\quad+\sum_j\int \eta_\varepsilon(x-y)(\partial_jA_j)(y)u(y)\,dy\\ &\quad+\int \big(B(x)-B(y)\big)\eta_\varepsilon(x-y)u(y)\,dy . \end{aligned} \] The sign of the second line follows from \(\partial_{y_j}\eta_\varepsilon(x-y)=-\partial_{x_j}\eta_\varepsilon(x-y)\); integration by parts differentiates \(A_j(y)\) as well. Put \(L_j=\sup_x\|dA_j(x)\|\) and \(M_j=\|\partial_jA_j\|_\infty\), where the first norm is for the Euclidean differential into the matrix norm. The mean-value formula bounds the first integrand's matrix kernel by \(L_j|x-y||\partial_j\eta_\varepsilon(x-y)|\). Its scalar kernel has mass \(L_j\int |z||\partial_j\eta(z)|\,dz\), independent of \(\varepsilon\). The second kernel has mass at most \(M_j\), and the last at most \(2\|B\|_\infty\). The preceding convolution estimate therefore proves \[ \|R_\varepsilon u\|_2\leq C_P\|u\|_2,\qquad C_P=\sum_j\left( L_j\int |z||\partial_j\eta(z)|\,dz+M_j\right) +2\|B\|_\infty . \] Every integral in this constant is finite. The kernel formula defines a bounded operator on all \(L^2\), with this same constant. The identity with the distributional commutator holds there as well: approximate by compactly supported smooth vectors in \(L^2\); multiplication and differentiation converge on every compactly supported distributional test, and the convolution kernel has compact support. Thus the smooth identity passes to the distributional limit.

If \(v\) is compactly supported and smooth, \(J_\varepsilon v\to v\) in \(H^1\), the bounded-coefficient operator \(P:H^1(\mathbb R^n)\to L^2(\mathbb R^n)\) is bounded by its explicit first-derivative formula, and \(J_\varepsilon Pv\to Pv\) in \(L^2\). Hence \(R_\varepsilon v\to0\). For arbitrary \(u\in L^2\), approximate it by such a \(v\) and use \[ \|R_\varepsilon u\|_2 \leq C_P\|u-v\|_2+\|R_\varepsilon v\|_2 . \] Letting first \(\varepsilon\to0\) and then \(v\to u\) proves strong convergence to zero. This proves the full uniformly bounded Friedrichs commutator assertion, including its distributional interpretation.

Now let \(u\in L^2(M,E)\) and \(Du=f\in L^2\) distributionally. Use the refined cutoffs from Lemma 4.1b. In one chart the compactly supported vector \(v=\chi_ju\) satisfies \[ Pv=\chi_jf+[D,\chi_j]u\in L^2 . \] The product rule here follows directly by testing \(\partial(\chi_ju)=\chi_j\partial u+(\partial\chi_j)u\) against a compactly supported smooth test function. The equation is a local coordinate distributional equation. Indeed in an orthonormal frame with density \(\rho\,dx\) the formal adjoint is \(P^\dagger\phi=-\rho^{-1}\sum_j\partial_j(\rho A_j^*\phi)+B^*\phi\); ordinary integration by parts proves that testing against this adjoint is exactly the coordinate distributional equation, using the invertible smooth density factor. Extension by zero creates no boundary distribution, because \(\operatorname{supp}\chi_j\) is compactly inside the chart.

Choose the coefficient extensions above equal to the original ones on the local estimate neighborhood \(W\). For sufficiently small \(\varepsilon\), \(v_\varepsilon=J_\varepsilon v\) is smooth and compactly supported in \(W\), and the proved commutator identity gives \[ Pv_\varepsilon=J_\varepsilon(Pv)+R_\varepsilon v \longrightarrow Pv\quad\text{in }L^2, \qquad v_\varepsilon\longrightarrow v\quad\text{in }L^2 . \] The local elliptic estimate of Lemma 4.1b, applied to \(v_\varepsilon-v_\delta\), makes this family Cauchy in \(H^1\) as \(\varepsilon,\delta\downarrow0\). Completeness and the \(L^2\) identification in Lemma 4.1a show that its \(H^1\) limit is \(v\). Thus each \(\chi_ju\) has square-integrable weak derivatives, and reconstruction puts \(u\) in \(H^1(M,E)\). Passing to the limit in the local estimates and summing gives the global bound of Lemma 4.1b for this \(u\). Conversely, smooth density in \(H^1\) and the bounded map \(D:H^1\to L^2\) show that every \(H^1\) section has its distributional \(D\)-image in \(L^2\). The same density gives convergence in the graph norm. \(\square\)

Corollary 4.2. Closed-manifold differential operators

Let \(M\) be a closed smooth manifold and \(E\) a Hermitian vector bundle. If a first-order elliptic differential operator \(D\) is formally self-adjoint on smooth sections, its closure on \(L^2(M,E)\) has domain \(H^1(M,E)\), is self-adjoint, and its bounded transform defines a module over \(C(M)\). A bundle grading for which \(D\) is odd makes the module even.

Proof. Smooth sections are \(L^2\)-dense by Lemma 4.1a. Formal self-adjointness makes the smooth-domain operator closable: if \(u_j\to0\) and \(Du_j\to g\) in \(L^2\), testing against every smooth \(\phi\) gives \(\langle g,\phi\rangle=\lim_j\langle u_j,D\phi\rangle=0\); density gives \(g=0\). Its graph closure has domain exactly \(H^1\). In one direction, smooth \(H^1\)-approximation and boundedness of \(D:H^1\to L^2\) put \(H^1\) in that closure. In the other direction, a graph-Cauchy smooth sequence is \(H^1\)-Cauchy by Lemma 4.1b, and its \(L^2\) limit is the \(H^1\) limit from Lemma 4.1a.

The adjoint domain is exactly the set of \(L^2\) sections \(v\) for which \(Dv\in L^2\) distributionally: testing the adjoint identity on smooth sections and formal self-adjointness give this equivalence. Lemma 4.1c puts each such \(v\) in \(H^1\); conversely smooth \(H^1\)-approximation passes the formal adjoint identity to every \(v\in H^1\). Thus the adjoint and closure have the same domain and action. This proves self-adjointness with no separate regularity assumption.

For \(u\in\operatorname{Dom}D\), self-adjointness gives \(\|(D-i)u\|^2=\|Du\|^2+\|u\|^2\). The range is closed by closedness of \(D\), and its orthogonal complement is \(\ker(D+i)=0\). Hence \(Q=(D-i)^{-1}\) exists, has norm at most one on \(L^2\), and maps \(L^2\) boundedly into \(H^1\) by Lemma 4.1b. Put \(K=-iQ\). Since \(QDu=u+iQu\) for \(u\in H^1\), we retain the precise identity and estimate \[ u=QDu+Ku,\qquad \|u\|_{H^1}\leq C(\|Du\|_{L^2}+\|u\|_{L^2}). \tag{4.6} \] Here \(Q,K:L^2\to H^1\) are the explicitly constructed resolvent maps; this identity is used on \(H^1\), after the weak-domain proof, and makes no claim about a pseudodifferential order. Compactness of \(H^1\hookrightarrow L^2\), proved in Lemma 4.1a by finite Fourier truncations, makes \(Q\) compact as an operator on \(L^2\). The resolvent identity and adjoint give \(QQ^*=(1+D^2)^{-1}\), as in Lemma 4.0, so that operator is compact too.

Multiplication by a smooth function preserves \(H^1\) by Lemma 4.1a. In local coordinates \(D=\sum_jA_j(x)\partial_j+B(x)\), so \([D,f]=\sum_jA_j(x)(\partial_jf)\) is a smooth bundle multiplication operator, bounded on the compact manifold. Lemma 4.1a also proves uniform smooth density in \(C(M)\). Theorem 4.1 therefore applies with every original hypothesis checked. A smooth bundle grading is \(H^1\)-bounded by Lemma 4.1a, and its odd-domain identity extends from smooth sections by density, so the graded conclusion follows as well. \(\square\)

Formal self-adjointness is a real hypothesis here. Ellipticity by itself gives a Fredholm realization between suitable Sobolev spaces, but does not make a differential operator into the self-adjoint \(D\) required in Theorem 4.1.

A compact-resolvent spectral triple consists of a dense *-algebra \(\mathcal A\subset A\), a representation on \(H\), and a self-adjoint \(D\) satisfying (4.1), such that every \([D,\pi(a)]\), \(a\in\mathcal A\), extends to a bounded operator and \((1+D^2)^{-1}\) is compact. These hypotheses give its bounded Fredholm module by Theorem 4.1. Summability, regularity and meromorphic dimension spectrum are additional structures, studied in Spectral triples and dimension spectrum; none is needed for this construction. Locally compact triples on noncompact spaces need the localized version of the bounded-transform theorem, whose full proof remains required in Unbounded Kasparov modules and spectral triples; the compact-resolvent proof here does not establish it.

5. Three explicit examples

Lemma 5.0. The Fourier basis used in the examples

The functions \(e^{int}\), \(n\in\mathbb Z\), are an orthonormal basis of the normalized \(L^2(S^1)\), and trigonometric polynomials are uniformly dense in \(C(S^1)\). Products of these functions form an orthonormal basis on the two-dimensional torus.

Proof. Integration of exponentials gives orthonormality. For \(N\geq1\) use the nonnegative trigonometric polynomial \[ K_N(t)=\frac1N\left|\sum_{j=0}^{N-1}e^{ijt}\right|^2 . \] Its integral with normalized angular measure is \(1\), since only the equal-index terms survive. For \(\delta\leq|t|\leq\pi\), the geometric-sum formula gives \(K_N(t)\leq1/(N\sin^2(\delta/2))\). For continuous periodic \(f\), its convolution with \(K_N\) is a trigonometric polynomial. Split \[ (K_N*f)(x)-f(x)=\int K_N(t)(f(x-t)-f(x))\,\frac{dt}{2\pi} \] at \(|t|=\delta\). Uniform continuity makes the first part uniformly small for small \(\delta\); the displayed tail bound makes the second part uniformly small as \(N\to\infty\). This proves uniform density. Using the product of the two kernels and splitting where either coordinate is outside its small interval proves the same uniform approximation for continuous functions on the two-dimensional torus.

The existing real-line measure and density lemma, Lemma 0.1(b)–(c), proves that endpoints have measure zero and \(C_c^\infty(0,2\pi)\) is dense in \(L^2(0,2\pi)\). Such functions extend continuously and periodically by zero. Uniform trigonometric approximation therefore gives \(L^2\)-density as well, proving completeness of the one-dimensional orthonormal family.

For the two-dimensional claim, let \(g\in L^2((S^1)^2)\) be orthogonal to all products. Fubini, proved in that same earlier Lemma 0.1(a), and Cauchy–Schwarz show that \(g_n(y)=\int g(x,y)e^{-inx}\,dx/(2\pi)\) belongs to \(L^2(S^1)\). All its Fourier coefficients vanish, so \(g_n=0\) almost everywhere by the one-dimensional result. Intersect the countably many full-measure sets for \(n\in\mathbb Z\). On their intersection the \(L^2\) section \(g(\cdot,y)\) has every coefficient zero, hence vanishes. Fubini gives \(g=0\) almost everywhere. This proves the product family's completeness; orthonormality follows by product integration. Parseval and graph-coordinate convergence are proved in Hilbert spaces and compact operators, Theorem 4.1. \(\square\)

The Hardy module of the circle

Put \(H=L^2(S^1)\), with normalized angular measure, and let \(e_n(z)=z^n\), \(n\in\mathbb Z\). Let \(\pi(f)\) be multiplication by \(f\), and let \(P\) project onto the closed span of \(e_n\) for \(n\geq0\). Then \[ F=2P-1 \tag{5.1} \] is a self-adjoint involution. For \(k>0\), the commutator \([P,\pi(z^k)]\) vanishes on every basis vector except \(e_{-k},\ldots,e_{-1}\); its image lies in \(\operatorname{span}(e_0,\ldots,e_{k-1})\). For \(k<0\) the same conclusion follows by adjoints. Thus these commutators have finite rank. Trigonometric polynomials approximate every continuous function uniformly, and \(\|[P,\pi(f)]\|\leq2\|f\|_\infty\), so \([P,\pi(f)]\) is compact for every \(f\in C(S^1)\). This verifies all module conditions.

The operator \(D=-i\,d/d\theta\) on the circle has \(De_n=ne_n\), with domain \(\{\sum c_ne_n:\sum n^2|c_n|^2<\infty\}\). Testing an adjoint vector against every Fourier basis vector gives the same square-summable weighted domain, so this diagonal realization is self-adjoint; finite Fourier sums converge in its graph norm and prove that it is the closure of the smooth differential operator. Its bounded transform has eigenvalues \(n/\sqrt{1+n^2}\). The differences from (5.1) tend to zero as \(|n|\to\infty\); the discrepancy at \(n=0\) affects only one vector. Their difference is compact. Hence the Hardy cycle represents the same K-homology class as this bounded transform.

Compressing \(\pi(z)\) to \(PH\) gives the unilateral shift \(Se_n=e_{n+1}\), \(n\geq0\). Its kernel is zero and its cokernel is spanned by \(e_0\), so \[ \operatorname{index}(P\pi(z)P)=-1. \tag{5.2} \] We will prove below that this index detects a nonzero class and all its multiples.

A Dirac operator on the two-dimensional torus

Let \(T^2=(\mathbb R/2\pi\mathbb Z)^2\), take \(H=L^2(T^2,\mathbb C^2)\), and grade by \[ \Gamma=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. \] Use the Pauli matrices \[ \sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad D=-i(\sigma_1\partial_x+\sigma_2\partial_y). \tag{5.3} \] They are self-adjoint, square to one, and anticommute. On the Fourier mode \(e^{i(nx+my)}v\), \(D\) acts by the self-adjoint matrix \(n\sigma_1+m\sigma_2\), whose square is \((n^2+m^2)1\). Consequently the closure is self-adjoint with domain \[ \left\{(v_{n,m}):\sum_{n,m}(1+n^2+m^2)\|v_{n,m}\|^2<\infty\right\}. \] This can be verified directly: the direct sum of the finite-dimensional self-adjoint matrices has adjoint with that same domain, and finite Fourier sums are dense in the graph norm.

The eigenvalues of \((1+D^2)^{-1}\) are \((1+n^2+m^2)^{-1}\), of multiplicity two, and tend to zero outside finite sets. Thus this operator is compact. Multiplication by \(C(T^2)\) is even, \(D\) anticommutes with \(\Gamma\), and smooth commutators are \(-i(\sigma_1\partial_x f+\sigma_2\partial_y f)\). For domain preservation, finite Fourier sums have graph convergence, and Parseval identifies this domain with the functions whose two weak first derivatives are in \(L^2\). For smooth \(f\), \(\partial_j(fu)=f\partial_ju+(\partial_jf)u\); the bounded coefficients give a graph-norm bound, first on those finite sums and then by completion. Thus multiplication by \(f\) preserves the domain. Smooth functions are uniformly dense by Lemma 5.0. All hypotheses of Theorem 4.1 have now been checked, and it gives an even module.

Its positive-to-negative bounded block on the \((n,m)\)-mode is \[ F_+(n,m)=\frac{n+im}{\sqrt{1+n^2+m^2}}. \tag{5.4} \] Both \(F_+\) and its adjoint have the one-dimensional constant-mode kernel. On the other modes the modulus is at least \(1/\sqrt2\), so the range is closed. Its untwisted index is therefore \(1-1=0\). This computation concerns the test by the trivial line bundle; it does not assert that the torus Dirac class vanishes.

Finite-dimensional cycles

If \(H\) is finite dimensional, every defect is compact, so every representation and bounded operator of the appropriate parity is a cycle. Such a cycle need not be degenerate: \(F=0\) and a nonzero representation give \(\pi(a)(F^2-1)=-\pi(a)\).

Over \(A=\mathbb C\), an even finite-dimensional example is \(H_+=\mathbb C^r\), \(H_-=\mathbb C^s\), scalar representation and \(F=0\). Its eventual even index test is \(r-s\). A pair with \(r=s=1\) is zero already here: the path \[ \begin{pmatrix}0&t\\t&0\end{pmatrix}, \qquad0\leq t\leq1, \] has only finite-dimensional defects and ends at a commuting self-adjoint exchange unitary. In contrast an odd finite-dimensional cycle is always zero. Replace \(F\) along a segment by \(1\); all defects remain compact, and the endpoint commutes with the representation and is a self-adjoint unitary. This parity difference explains why a single finite-dimensional graded representation can encode an even index, whereas odd index information needs an infinite-dimensional space.

6. Odd cycles and semisplit extensions

Proposition 6.1. The compression extension

Let \((H,\pi,F)\) be an odd module with \(F=F^*\), \(F^2=1\), and put \(P=(1+F)/2\). The formula \[ \tau(a)=q_{PH}\big(P\pi(a)|_{PH}\big) \tag{6.1} \] defines a Busby map into \(\mathcal Q(PH)\) with a completely positive contractive lift. Its pullback is a semisplit extension of \(A\) by \(\mathcal K(PH)\).

Proof. The inclusion \(V:PH\hookrightarrow H\) is an isometry and the proposed lift is \(L(a)=V^*\pi(a)V\). For a positive matrix \([a_{ij}]\), its representation on \(H^n\) is positive, and compression by \(V^{(n)}\) gives \([L(a_{ij})]\geq0\); hence \(L\) is completely positive. Contractivity of \(\pi\) and \(V\) proves contractivity of \(L\), and taking adjoints gives \(L(a^*)=L(a)^*\). Its multiplication error is exactly \[ L(ab)-L(a)L(b) =P\pi(a)(1-P)\pi(b)|_{PH}. \tag{6.2} \] On \(PH\), \((1-P)\pi(b)P=(1-P)[\pi(b),P]\), a compact operator between the two spectral ranges. Thus this error vanishes in the Calkin quotient and \(q_{PH}L\) preserves multiplication as well as adjoints. For its pullback algebra, \(a\mapsto(L(a),a)\) lies in the pullback, is a completely positive contraction at every matrix level, and is a section of the quotient. The Busby pullback theorem proved in the first extension lesson therefore gives precisely a semisplit extension by \(\mathcal K(PH)\). \(\square\)

This extension is essential precisely when the Busby map is injective; the construction makes no automatic essentiality claim. If \(PH\) is finite dimensional, the quotient is zero. To use a fixed infinite-dimensional compact ideal in all cases, append an infinite-dimensional zero-representation summand on which \(F=1\). It is degenerate, and enlarges \(PH\) to \(PH\oplus\ell^2\); the new Busby map is the old one plus the split zero map.

Proposition 6.2. A semisplit extension gives an odd cycle

Every semisplit extension of separable \(A\) by \(\mathcal K(H)\) is obtained by the compression construction from an odd module with an exact self-adjoint involution.

Proof. Write \(\tau\) for the given Busby map. If \(H\) is finite dimensional, \(\mathcal Q(H)=0\) and \(\tau=0\). Its zero lift is the compression of the zero representation on \(H\oplus H\), with \(P=\operatorname{diag}(1,0)\) and \(F=2P-1\). This is a degenerate cycle. The first lesson's Busby correspondence identifies its pullback with the given extension, so this case is included rather than excluded by a fixed infinite-dimensional convention.

For infinite-dimensional separable \(H\), semisplitting provides a completely positive contractive lift \(L:A\to\mathcal B(H)\). The previous lesson, Lemma 2.2, proves the corner dilation from the strict KSGNS construction and operator-range stabilization. For \(B=\mathbb C\) it produces a homomorphism \[ \rho:A\longrightarrow\mathcal B(H\oplus H),\qquad \rho(a)=\begin{pmatrix}L(a)&\rho_{12}(a)\\ \rho_{21}(a)&\rho_{22}(a)\end{pmatrix}. \] Multiplication of its first diagonal corner at \(a^*a\) gives \[ \rho_{21}(a)^*\rho_{21}(a)=L(a^*a)-L(a)^*L(a). \] The quotient of the right side is zero since \(qL=\tau\) is multiplicative. The C*-identity in the quotient forces \(q\rho_{21}(a)=0\), hence compactness. Applying this to \(a^*\) and using \(\rho_{12}(a)=\rho_{21}(a^*)^*\) proves compactness of the other off-diagonal corner.

Now \(P=\operatorname{diag}(1,0)\) and \(F=2P-1\) have zero adjoint and square defects, and \[ [F,\rho(a)]=\begin{pmatrix}0&2\rho_{12}(a)\\ -2\rho_{21}(a)&0\end{pmatrix} \] is compact. This is the required exact odd cycle. Compressing its representation to \(PH=H\) gives \(L\), hence its compression Busby map is exactly \(\tau\). The Busby correspondence once more identifies the extensions. No essentiality, nuclearity or unit was used; separability is the hypothesis needed by that corner-stabilization step. \(\square\)

For the Hardy module, (6.1) is precisely the Toeplitz Busby map. The same construction also explains the inverse extension: switch to the negative spectral subspace, replacing \(F\) by \(-F\). The two compressed corners together are the quotient of the full representation, a split extension.

The full class-level statement is \[ K^1(A)\ \cong\ KK^1(A,\mathbb C)\ \cong\ \operatorname{Ext}(A)^{-1}, \tag{6.3} \] where the last group is the subgroup of invertible elements in the ordinary stabilized extension monoid. The correspondence sends an exact odd involution to (6.1), and a semisplit extension to Proposition 6.2. It is natural under pullback in \(A\), and the compression index is its pairing with \(K_1(A)\).

Formula (6.3) retains its full separable-algebra statement, but is not established by the two constructions above. A complete proof still has to show that compression respects all class relations, that changes of dilation and normalization give the same class, that the constructions are inverse on classes, and that Hilbert-module homotopy gives exactly the analytic equivalence relation. The pending lesson Pictures of KK: Fredholm picture, quasihomomorphisms, \(KK^1\) and extensions is not a delivered proof provider. The constructions here use no planar classification theorem. For nuclear \(A\), the preceding lesson independently proves that every extension is semisplit, so every extension has the cycle representative constructed in Proposition 6.2; this alone does not prove (6.3).

7. An index test that already detects the circle

Here we need only a fixed unitary in a unital algebra. The next lesson proves the full matrix-stable, bilinear pairing.

Lemma 7.1. A unitary tests an odd K-homology class

For unital \(A\) and \(u\in A\) unitary, the positive-compression index defines a group homomorphism \[ I_u:K^1(A)\longrightarrow\mathbb Z. \] For an exact involution and a unital representation it is \(\operatorname{index}(P\pi(u)P|_{PH})\).

Proof. For a general cycle use the particular continuous normalization \(N(F)=G_1\) in Proposition 2.3, with representation \(\rho=\pi\oplus0\), and put \(P_F=(1+N(F))/2\). The operator \[ U=\rho(u)+(1-\rho(1)) \] is unitary, even if \(\rho\) is degenerate, and \([P_F,U]\) is compact. Thus \(P_FUP_F\) on \(P_F(H\oplus H)\) is Fredholm: \(P_FU^*P_F\) is an inverse modulo compacts, since the errors factor through \([P_F,U]\). Define \(I_u\) to be its index.

Unitary equivalence preserves it. The normalization respects direct sums up to a permutation of the doubled summands, so its index is additive. For a degenerate cycle it gives zero. To see this last point without an extra convention, let \(H_e=\overline{\pi(A)H}\). Exact commutation makes \(H_e\) reducing for \(F\); the zero self-adjointness and square defects make \(F|_{H_e}\) a self-adjoint involution. The normalization leaves that restriction unchanged and has zero off-diagonal \(s\) there, while \(\pi\) is zero on \(H_e^\perp\). Consequently \(N(F)\) commutes exactly with \(\rho(A)\), so \(P_FUP_F\) is unitary on its range and has index zero.

Under an operator homotopy \(F_t\), the normalized projections \(P_t\) vary in norm. Their ranges can be locally identified by norm-continuous unitaries. Here is the formula. If \(\|P-Q\|<1\), set \[ X=QP+(1-Q)(1-P),\qquad V=X(X^*X)^{-1/2}. \] The identities \(XP=QX\) and \(X^*X=1-(P-Q)^2\) show that \(V\) is a unitary with \(VPV^*=Q\); it depends continuously on \(Q\). Cover a compact parameter interval by finitely many such neighborhoods and concatenate the identifications. After transporting the ranges by these unitaries, the compressed operators are a norm-continuous Fredholm path on a fixed space. Its index is constant. This proves invariance under every generating relation, hence well-definedness and additivity on \(K^1(A)\).

If the starting \(F\) is already an exact involution, its normalization is \(\operatorname{diag}(F,-F)\). The second copy has zero representation and contributes the identity compression, of index zero. Removing a zero-representation part of a unital algebra action likewise contributes zero. Thus the displayed formula agrees with the construction. \(\square\)

Applying the lemma to the Hardy module and \(u=z\), (5.2) gives \(I_z=-1\). In particular its class has infinite order. This detects the class directly in the equivalence relation of this lesson, without assuming the later class-level extension isomorphism.

8. Exercises

Exercise 8.1 (basic). Verify that a self-adjoint unitary commuting with \(\pi(A)\) is degenerate. In the even case include the grading condition. Explain why \(F=0\) on a finite-dimensional nonzero representation is usually not degenerate.

Exercise 8.2 (intermediate). Give the operator homotopy between two cycle operators differing by a compact operator. Check its square and adjoint defects explicitly.

Exercise 8.3 (intermediate). Verify all the Hardy-module conditions by computing \([P,\pi(z^k)]\) on Fourier basis vectors. Identify the compression for \(z^{-1}\).

Exercise 8.4 (advanced). Prove that the Hardy class has infinite order, using the positive-compression index. Determine the class detected after replacing \(F\) by \(-F\).

9. Solutions

Solution 8.1. Self-adjointness makes \(F-F^*=0\), involutivity makes \(F^2-1=0\), and commutation makes \([F,\pi(a)]=0\). Thus all three defects vanish. An even cycle must also have \(\Gamma F=-F\Gamma\) and an even representation; the conclusion applies once those cycle requirements hold. For a finite-dimensional example with \(F=0\), the square defect is \(-\pi(a)\). It is compact because the space is finite dimensional, but it is not zero when \(\pi(a)\ne0\). Being a cycle and being degenerate are distinct conditions.

Solution 8.2. Put \(K=F'-F\in\mathcal K(H)\) and \(F_t=F+tK\). Then \[ \begin{aligned} \left[F_t,\pi(a)\right]&=[F,\pi(a)]+t[K,\pi(a)],\\ F_t-F_t^*&=F-F^*+t(K-K^*),\\ F_t^2-1&=F^2-1+t(FK+KF)+t^2K^2. \end{aligned} \] Every added term is compact, so multiplying the last two expressions by \(\pi(a)\) preserves compactness. The path is norm-continuous. If both endpoints are odd for the grading, then \(K\) and every \(F_t\) are odd. This verifies an operator homotopy directly. The same conclusion for a locally compact difference follows from the ideal calculation of Proposition 2.2.

Solution 8.3. For \(k\geq0\), \[ [P,\pi(z^k)]e_n =\big(1_{\{n+k\geq0\}}-1_{\{n\geq0\}}\big)e_{n+k}. \] It is nonzero exactly for \(-k\leq n\leq-1\), and then equals \(e_{n+k}\). For \(k=-\ell<0\), it is nonzero exactly for \(0\leq n\leq\ell-1\), and equals \(-e_{n-\ell}\). Thus its rank is \(|k|\), with rank zero when \(k=0\). Linear combinations give finite-rank commutators for trigonometric polynomials, and uniform approximation gives compact commutators for all continuous functions. The other defects vanish because \(F=2P-1\) is self-adjoint and squares to one. On the nonnegative basis, compression of \(\pi(z^{-1})\) is \(S^*\): it kills \(e_0\) and sends \(e_n\) to \(e_{n-1}\) for \(n\geq1\).

Solution 8.4. Lemma 7.1 gives a homomorphism \(I_z\) on \(K^1(C(S^1))\). For the Hardy class \(h\), its value is \(-1\), so \(I_z(nh)=-n\). If \(nh=0\), this forces \(n=0\); hence \(h\) has infinite order. The replacement \(-F\) represents its inverse by Theorem 3.1. Directly, its positive projection is \(1-P\). Compression of multiplication by \(z\) to the negative Fourier modes kills \(e_{-1}\), sends \(e_{-j}\) to \(e_{-(j-1)}\) for \(j\geq2\), and is surjective on that negative subspace. Its index is \(+1\), agreeing with the inverse class and fixing the sign.

Internal proof dependencies and source credit

The Hilbert-space Fredholm facts, including composition additivity, are proved in Lemmas 1.1–1.2 of the extension lesson. Continuous functional calculus is proved in C*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients, Theorems 5.1 and 6.1. Lemma 4.0 proves the compact-resolvent spectral domain and calculus used in Theorem 4.1 here, without taking an unbounded spectral theorem as an external input. The finite-chart Sobolev construction and compactness, frozen-coefficient elliptic estimate, and uniformly bounded, strongly vanishing Friedrichs commutator are proved here in Lemmas 4.1a–4.1c. Corollary 4.2 uses these proofs to identify the distributional adjoint domain with \(H^1\) and prove self-adjointness and compact resolvent for every formally self-adjoint first-order elliptic operator on a closed manifold. The exact earlier measure and Hilbert-space inputs are linked before Lemma 4.1a; no pseudodifferential parametrix or external elliptic-regularity theorem supplies this bridge.

The class-level isomorphisms (6.3) and their precise normalization, dilation and homotopy assertions remain proof obligations in Pictures of KK: Fredholm picture, quasihomomorphisms, \(KK^1\) and extensions. The localized noncompact bounded-transform theorem remains a proof obligation in Unbounded Kasparov modules and spectral triples. The cycle constructions, group laws, compact-resolvent bounded transform and circle detection are proved above from those existing internal prerequisites. The full class-level identifications in (6.3) and the localized noncompact extension remain unproved here.

References

• B. Blackadar, K-Theory for Operator Algebras, freely accessible author-corrected second-edition PDF. Sections 17.1–17.5 supply the cycle, normalization, group and Fredholm pictures; Section 17.11 treats the bounded transform. Free author version. The resolvent and compact spectral proofs required here are written in Lemma 4.0 and Theorem 4.1.

• B. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, freely accessible author-revised version dated 8 February 2017, I.6.1 and I.8.4. Free revised author version. Lemma 4.0 supplies the compact positive spectral argument rather than citing it in place of proof.

• J. Ruppenthal, Friedrichs' Extension Lemma with Boundary Values and Applications in Complex Analysis, free author preprint dated 14 October 2009, Theorem 3.1 and Lemma 3.2, pp. 5–6. Actual free author PDF. The matrix-kernel bound, strong convergence and weak-domain argument required here are fully proved in Lemma 4.1c.

• C. Bär and W. Ballmann, Boundary Value Problems for Elliptic Differential Operators of First Order, arXiv:1101.1196v2, Sections 2.1–2.2 and Lemma 3.1/Remark 3.2. Actual free preprint. These sections provide the source comparison for measured manifolds, formal adjoints and maximal domains. The Sobolev, regularity and compactness proofs used here are local Lemmas 4.1a–4.1c.

These free source PDFs retain their authors' copyrights. The CC0 notice applies to this lesson's original text.