39.06.03 · operator-algebras / spectral-triples-ncg

Fredholm Modules and the K-Theory/K-Homology Index Pairing

shipped3 tiersLean: none

Anchor (Master): Connes *Noncommutative Geometry* (1994) Ch. IV; Higson & Roe *Analytic K-Homology* Ch. 8–9; Blackadar *K-Theory for Operator Algebras* §17 (Kasparov KK)

Intuition Beginner

An operator can have an index: a single whole number that counts how many independent solutions an equation has, minus how many obstructions block solving it. The remarkable thing about this number is its toughness. You can wiggle the operator, perturb it, deform it by anything small, and the index does not budge. It is an integer carved out of a sea of continuous data, and that rigidity is what makes it worth computing.

Noncommutative geometry turns this into a game played between two sides. On one side you have shapes built from projections and reversible elements — the K-theory of an algebra you already met. On the other side you have a single abstract operator, a kind of "sign of the Dirac operator," that knows the geometry. When you bring the two together, the operator measures the shape and hands back an integer. Pair an algebra-shape with an operator and you get a number, always a whole number, never a fraction.

The payoff is that this pairing, in the familiar case of an ordinary curved space, is exactly the count produced by the celebrated index theorem for the Dirac operator twisted by a bundle. The abstract machine and the classical theorem give the same integer, so the noncommutative side is a genuine generalization, not a new invention.

Visual Beginner

Two kinds of data meet in the middle and produce one integer: the shapes (projections and loops) on the left, the abstract operator on the right, and the index in the centre.

The dictionary reads: the left pan holds K-theory classes (projections for the even case, unitaries for the odd case); the right pan holds the operator $F$ , a symmetry squaring to one whose commutators with the algebra are small (compact); and the pivot returns an integer. Squeeze $F$ between the projection on both sides and read off the index of what remains.

Worked example Beginner

Take the simplest abstract operator on a sequence space. Picture two stacks of basis vectors, a left stack and a right stack, and let $F$ be the operator that swaps them: it sends each left vector to the matching right vector and back. This $F$ is its own inverse applied twice gives the identity, so $F$ squared is one, exactly the rule the operator must satisfy.

Now pair it against the simplest loop, the shift that pushes the right stack one step forward and pulls the left stack one step back. Squeezing the shift between the two halves of $F$ produces a one-sided shift: an operator that is onto but has a one-dimensional space sent to zero on one side. Count solutions minus obstructions:

index = (dimension sent to zero) - (dimension missed) = 1 - 0 = 1.

The pairing of this loop with this operator is the integer $1$ .

What this tells us: bringing together a reversible element (the shift) and the abstract operator $F$ produced a one-sided shift whose index is a whole number. The same recipe, with a more interesting operator and a more interesting shape, is how the noncommutative index theorem assigns integers to spaces with no points.

Check your understanding Beginner

Formal definition Intermediate+

Let $A$ be a $*$ -algebra (a dense $*$ -subalgebra of a C*-algebra, as for the spectral triples of 39.06.01). A Fredholm module over $A$ is a triple $(H, F, π)$ where $π : A \to B (H)$ is a $*$ -representation on a Hilbert space $H$ and $F \in B (H)$ is a bounded operator with

F = F^{*}, F^{2} = 1, [F, π (a)] \in K (H) for all a \in A,

where $K (H)$ is the compact operators. (Allowing $F^{2} - 1$ and $F - F^{*}$ to be compact rather than zero gives the same theory up to homotopy; we take the normalized form.) The module is even if there is a $Z /2$ -grading $γ = γ^{*} \in B (H)$ , $γ^{2} = 1$ , with $γ π (a) = π (a) γ$ for all $a \in A$ and $γ F = - F γ$ ; it is odd if no grading is given. An even module decomposes $H = H^{+} \oplus H^{-}$ into the $\pm 1$ eigenspaces of $γ$ , and the odd operator $F$ has block form $$ F = \begin{pmatrix} 0 & F^- \ F^+ & 0 \end{pmatrix}, \qquad F^+ : \mathcal{H}^+ \to \mathcal{H}^-, \quad F^- = (F^+)^*. $$

A Fredholm module is $p$ -summable ( $1 \leq p < \infty$ ) if $[F, π (a)] \in L^{p} (H)$ for all $a \in A$ , where $L^{p}$ is the Schatten ideal of operators with $\sum_{n} μ_{n} (T)^{p} < \infty$ ( $μ_{n}$ the singular values); it is ** $θ$ -summable** if $e^{- t D^{2}}$ is trace-class for all $t > 0$ in the unbounded model below. The classes of Fredholm modules over $A$ , modulo unitary equivalence, addition of degenerate modules (those with $[F, π (a)] = 0$ for all $a$ ), and operator homotopy, form the K-homology groups: even modules give $K^{0} (A)$ and odd modules give $K^{1} (A)$ . K-homology is the functor dual to the K-theory $K_{*} (A)$ of 39.02.02.

A Fredholm module is the bounded avatar of a spectral triple $(A, H, D)$ 39.06.01. Given $D$ self-adjoint with compact resolvent and bounded commutators $[D, π (a)]$ , set $$ F = D|D|^{-1} = \operatorname{sign}(D) $$ on $(ker D)^{⊥}$ (and $F = 1$ on $ker D$ , a finite-dimensional space). Then $F = F^{*}$ , $F^{2} = 1$ , and each $[F, π (a)]$ is compact, so $(H, F, π)$ is a Fredholm module: the phase of $D$ . The even/odd parity and the grading transfer unchanged. Passing from $D$ to $F$ forgets the metric (the eigenvalue spacing) but keeps the homotopy-theoretic and index content; the unbounded picture is the geometry, the bounded picture is its K-homology shadow.

The index pairing is a bilinear map $$ \langle ,\cdot,, ,\cdot, \rangle : K^(A) \times K_(A) \longrightarrow \mathbb{Z}, $$ pairing K-homology against K-theory by even-with- $K_{0}$ and odd-with- $K_{1}$ . For an even module $[F] \in K^{0} (A)$ and a projection $e \in M_{n} (A)$ representing $[e] \in K_{0} (A)$ , set $P = e \oplus \dots \oplus e$ acting with $π$ amplified to $M_{n}$ and form the operator $e F^{+} e$ (more precisely $P (F^{+} \otimes 1_{n}) P$ ) as a map from $e H_{n}^{+}$ to $e H_{n}^{-}$ . This operator is Fredholm because $[F, e]$ is compact, and $$ \langle [F], [e] \rangle = \operatorname{index}\big( e F^+ e : e\mathcal{H}^+_n \to e\mathcal{H}^-_n \big) \in \mathbb{Z}. $$ For an odd module $[F] \in K^{1} (A)$ and a unitary $u \in M_{n} (A)$ representing $[u] \in K_{1} (A)$ , set $P = \frac{1}{2} (1 + F)$ (the spectral projection onto the $+ 1$ eigenspace of $F$ ) amplified to $M_{n}$ , and $$ \langle [F], [u] \rangle = \operatorname{index}\big( P u P : P\mathcal{H}_n \to P\mathcal{H}_n \big) = -\operatorname{sf}(D, u D u^) \in \mathbb{Z}, $$ the index of the compression $P u P$ , equal up to sign to the spectral flow of the path from $D$ to $u D u^$ (the net number of eigenvalues crossing zero). In the commutative case this is a winding number.

Counterexamples to common slips

A Fredholm module is not required to have $F$ invertible in any algebraic sense beyond $F^{2} = 1$ ; $F$ is a symmetry, not a Fredholm operator on the nose. What is Fredholm is the compression $P u P$ or $e F^{+} e$ , and only after squeezing by a K-theory class. A bare $F$ with $F^{2} = 1$ has index $0$ on each block — the geometry enters through the algebra element.
$p$ -summability is a condition on the commutators $[F, π (a)]$ , not on $F$ itself. The operator $F$ is bounded with $∥ F ∥ = 1$ and is never trace-class on an infinite-dimensional space; demanding $F \in L^{p}$ would force $dim H < \infty$ . Summability constrains how fast $[F, π (a)]$ decays, which is what makes the Chern character converge in a finite cyclic degree.
The bounded transform $F = D ∣ D ∣^{- 1}$ loses information that the unbounded $D$ retains: the metric (Connes distance 39.06.01) and the dimension spectrum live in $D$ , not in $F$ . Two spectral triples with the same $F$ but different $∣ D ∣$ are K-homologically equal yet metrically distinct; the index pairing sees only $F$ .

Key theorem with proof Intermediate+

Theorem (the index pairing is well-defined and integer-valued). Let $(H, F, π)$ be an even Fredholm module over $A$ and $e \in M_{n} (A)$ a projection. Then $e F^{+} e$ is a Fredholm operator from $e H_{n}^{+}$ to $e H_{n}^{-}$ , its index depends only on the class $[F] \in K^{0} (A)$ and the class $[e] \in K_{0} (A)$ , and the resulting pairing $K^{0} (A) \times K_{0} (A) \to Z$ is additive in each variable. ^{[Connes Ch. IV; Higson-Roe Ch. 8]}

Proof. Work first with $n = 1$ ; the general case amplifies $π$ and $F$ to $M_{n}$ without change. Write $P = π (e)$ , a projection on $H$ commuting with the grading $γ$ (since $e$ commutes with $γ$ through $π$ ), so $P = P^{+} \oplus P^{-}$ on $H^{\pm}$ . Consider $T = P^{-} F^{+} P^{+} : P^{+} H^{+} \to P^{-} H^{-}$ . Its candidate parametrix is $S = P^{+} F^{-} P^{-}$ . Compute $$ ST - P^+ = P^+ F^- P^- F^+ P^+ - P^+. $$ Since $F^{-} F^{+} = (F^{2})^{+} = 1$ on $H^{+}$ , inserting $P^{-} = 1 - (1 - P^{-})$ gives $$ ST - P^+ = P^+ F^- F^+ P^+ - P^+ F^-(1 - P^-)F^+ P^+ - P^+ = -P^+ F^-(1-P^-)F^+P^+. $$ Now $(1 - P^{-}) F^{+} P^{+} = (1 - P^{-}) (F^{+} P^{+} - P^{-} F^{+}) + (1 - P^{-}) P^{-} F^{+} = (1 - P^{-}) [F^{+}, P]_{+}$ , using $(1 - P^{-}) P^{-} = 0$ , so it is a product involving the commutator $[F, π (e)]$ , which is compact by hypothesis. Hence $S T - P^{+}$ is compact; symmetrically $T S - P^{-}$ is compact. Therefore $T$ is invertible modulo compacts, i.e. Fredholm, and $e F^{+} e = T$ has a well-defined integer index by Atkinson's theorem 39.02.02.

Homotopy invariance in $[F]$ : an operator homotopy $F_{t}$ through Fredholm modules gives a norm-continuous path of Fredholm operators $e F_{t}^{+} e$ , along which the integer index is locally constant, hence constant. Adding a degenerate module (with $[F, π (a)] = 0$ ) contributes a block on which $e F^{+} e$ commutes with $e$ and restricts to an invertible operator $F^{+}$ between equal-dimension spaces, index $0$ ; so degenerate modules do not change the pairing. Invariance in $[e]$ : if $e \sim e^{'}$ are Murray–von Neumann equivalent via a partial isometry $v$ with $e = v^{*} v$ , $e^{'} = v v^{*}$ 39.02.02, then conjugation by $π (v)$ identifies $e F^{+} e$ with $e^{'} F^{+} e^{'}$ modulo a compact perturbation (again because $[F, π (v)]$ is compact), preserving the index; homotopic projections give norm-close Fredholm operators, hence equal index. Additivity: $e \oplus e^{'}$ yields the direct sum $e F^{+} e \oplus e^{'} F^{+} e^{'}$ , whose index adds; orthogonal direct sums of modules add the operators $F$ , so the index adds in $[F]$ as well. $□$

Bridge. The integrality of the pairing builds toward the entire analytic index theory of noncommutative geometry, and it appears again in 39.07.01 where the Chern character converts both $[F]$ and $[e]$ into cyclic (co)homology classes whose natural pairing computes this same integer by a local formula. The foundational reason the proof works is exactly the compactness of $[F, π (a)]$ : it is what makes the compression $e F^{+} e$ Fredholm rather than merely bounded, so the abstract elliptic condition of a spectral triple becomes the Atkinson-Fredholm condition of 39.02.02. This is exactly the dual of the K-theory construction: where $K_{0} (A)$ packaged projections into a group, K-homology packages operators $F$ into the dual group $K^{0} (A)$ , and the index pairing is the perfect bilinear pairing between them — the central insight that K-homology is the homology theory pairing against K-theory. Putting these together, the bridge is that a spectral triple $(A, H, D)$ gives a K-homology class $[D] = [F]$ through its phase $F = D ∣ D ∣^{- 1}$ , and pairing $[D]$ against $K_{*}$ generalises the index of the Dirac operator from 03.09.10 to algebras with no underlying manifold.

Exercises Intermediate+

Exercise 5 (medium, short-answer).

Show that adding a degenerate Fredholm module (one with $[F_{0}, π_{0} (a)] = 0$ for all $a$ ) to a given module does not change any index pairing.

Hint

On the degenerate summand, $e F_{0}^{+} e$ commutes with $e$ ; compute its index between equal-dimension spaces.

Answer

On the degenerate summand $(H_{0}, F_{0}, π_{0})$ , the projection $π_{0} (e)$ commutes with $F_{0}$ because $[F_{0}, π_{0} (e)] = 0$ . Hence $e F_{0}^{+} e$ restricts to the operator $F_{0}^{+} : e H_{0}^{+} \to e H_{0}^{-}$ , which is invertible there (its inverse is $F_{0}^{-} = (F_{0}^{+})^{*}$ , since $F_{0}^{2} = 1$ and $F_{0}$ commutes with $e$ ), so its index is $0$ . The pairing with $[F \oplus F_{0}]$ is therefore $⟨[F], [e]⟩ + 0 = ⟨[F], [e]⟩$ . This is why K-homology is defined modulo degenerate modules.

Rubric: full credit for the commuting-projection step and the index-zero conclusion.

Exercise 6 (hard, short-answer).

Prove that for an even Fredholm module the index pairing can be written $⟨[F], [e]⟩ = \frac{1}{2} Tr (γ e (F [F, e]))$ -type Calderón formula: namely that when $[F, π (e)] \in L^{p}$ with $p$ even, $⟨[F], [e]⟩ = c_{p} Tr (γ π (e) [F, π (e)]^{p})$ for a universal constant $c_{p}$ . Sketch why a Schatten condition is needed.

Hint

Calderón's formula expresses the Fredholm index of $e F^{+} e$ as a trace of a sufficiently high power of the commutator; the power must be large enough that $[F, π (e)]^{p}$ is trace-class.

Answer

For a Fredholm operator $T = e F^{+} e$ with parametrix $S = e F^{-} e$ , the index equals $Tr (1 - S T)^{m} - Tr (1 - T S)^{m}$ for any $m$ large enough that $(1 - S T)^{m}, (1 - T S)^{m}$ are trace-class (Calderón / Fedosov formula). The computation in the Key theorem showed $1 - S T$ and $1 - T S$ are built from $[F, π (e)]$ ; if $[F, π (e)] \in L^{p}$ then $(1 - S T)^{m} \in L^{p / m}$ , which is trace-class once $m \geq p$ . Assembling the even-graded version with $γ$ folds the two traces into a single supertrace $Tr_{s} (\cdot) = Tr (γ \cdot)$ , giving $⟨[F], [e]⟩ = c_{p} Tr (γ π (e) [F, π (e)]^{p})$ with $c_{p}$ a normalization. The Schatten condition is essential: without $[F, π (e)] \in L^{p}$ the trace diverges and the formula is meaningless — summability is exactly the convergence requirement that makes the Chern character land in cyclic degree $p$ .

Rubric: full credit for the Calderón trace formula, the Schatten-power bookkeeping making the trace finite, and the supertrace assembly.

Exercise 7 (hard, short-answer).

For the canonical even spectral triple of a closed even-dimensional spin manifold $M$ with Dirac operator $\neq D$ , identify the index pairing $⟨[\neq D], [e]⟩$ with the Atiyah-Singer index of a twisted Dirac operator, and write the resulting integer as a characteristic-class integral.

Hint

A projection $e \in M_{n} (C^{\infty} (M))$ is a bundle $E = e \underline{C^{n}}$ ; coupling $\neq D$ to $E$ gives $\neq D_{E} = e \neq D^{+} e$ , and Atiyah-Singer 03.09.10 computes its index.

Answer

A projection $e \in M_{n} (C^{\infty} (M))$ defines a smooth vector bundle $E = e \underline{C^{n}}$ over $M$ (Serre-Swan 03.08.01), and the compression $e \neq D^{+} e$ is precisely the chiral Dirac operator $\neq D_{E}^{+}$ twisted by $E$ . Hence $$ \langle [\not{!D}], [e]\rangle = \operatorname{index}(\not{!D}^+E) = \int_M \widehat{A}(M),\operatorname{ch}(E), $$ the Atiyah-Singer index formula 03.09.10, with $A (M)$ the $A$ -genus and $ch (E)$ the Chern character of $E$ . So the noncommutative index pairing of the K-homology class $[\neq D]$ against the K-theory class $[E]$ reproduces the classical twisted-Dirac index. In Connes' language $\operatorname{ch}[\not{!D}] $i s t h ecy c l i c - co h o m o l o g y C h er n c ha r a c t er w h ose p ai r in g w i t h$ \operatorname{ch}_[E]$ is this integral.

Rubric: full credit for the bundle identification, the twisted-Dirac index, and the $A ch$ integral.

Lean formalization Intermediate+

lean_status: none — Mathlib has Fredholm operators with an integer index, compact and bounded operators, and C*-algebras, but it does not bundle a representation $π$ , a symmetry $F = F^{*}$ with $F^{2} = 1$ , and the compactness of $[F, π (a)]$ into a Fredholm module, nor does it carry the even/odd grading, the K-homology groups, the index pairing, the Schatten-summability hierarchy, or the bounded transform $F = D ∣ D ∣^{- 1}$ from a spectral triple.

The intended statement reads schematically:

import Mathlib.Analysis.CStarAlgebra.Basic
import Mathlib.Analysis.Fredholm

/-- A Fredholm module over a *-algebra A: a representation π on H together with
a self-adjoint symmetry F (F = F*, F² = 1) whose commutators with the algebra
are compact. (Schematic; the compact-commutator predicate and the even grading
need the not-yet-bundled K-homology infrastructure.) -/
structure FredholmModule (A : Type*) (H : Type*)
    [CStarAlgebra A] [NormedAddCommGroup H] [InnerProductSpace ℂ H] where
  rep            : A →⋆ₐ[ℂ] (H →L[ℂ] H)
  F              : H →L[ℂ] H
  selfAdjoint    : IsSelfAdjoint F
  symmetry       : F ∘L F = ContinuousLinearMap.id ℂ H
  compactCommutator : ∀ a, True   -- [F, rep a] ∈ CompactOperators

/-- The index pairing of an even module against a projection (Schematic):
index of the compression e F⁺ e is an integer, well-defined on K-classes. -/
theorem index_pairing_integer {A H : Type*} (m : FredholmModule A H)
    (e : A) (he : IsIdempotentElem e) :
    ∃ k : ℤ, True :=   -- k = index (compress m.F e)
  sorry

The Connes-Moscovici local index formula — that the residues of $Tr (\cdot ∣ D ∣^{- s})$ assemble into a cyclic cocycle computing this pairing — is a far-horizon target requiring the full Schatten/Dixmier and cyclic-cohomology stack alongside the K-homology bundling.

Advanced results Master

K-homology as the dual theory. The Fredholm-module groups $K^{0} (A)$ (even) and $K^{1} (A)$ (odd) are the K-homology of $A$ : a homotopy-invariant, half-exact, stable bifunctor, contravariant where K-theory is covariant. For $A = C (X)$ commutative, $K^{*} (C (X)) = K_{*} (X)$ is Atiyah's topological K-homology, the dual of the K-theory $K^{*} (X) = K_{*} (C (X))$ of 39.02.02. The index pairing $K^{*} (A) \times K_{*} (A) \to Z$ is the noncommutative incarnation of the cap-product duality between homology and cohomology, and for a closed spin manifold the fundamental K-homology class $[\neq D] \in K_{*} (C (X))$ implements Poincaré duality $K^{*} (X) ≅ K_{*} (X)$ — the spectral-triple Poincaré duality axiom of 39.06.01 is exactly the statement that capping with $[\neq D]$ is an isomorphism.

The bounded/unbounded dictionary. The map $(A, H, D) \mapsto (H, D ∣ D ∣^{- 1}, π)$ from spectral triples to Fredholm modules is surjective on K-homology: every class is the phase of some unbounded representative (Baaj-Julg), and the unbounded picture is the one in which the Kasparov product is computable by a connection formula rather than by the intractable bounded product. The bounded transform is continuous for operator homotopy, so it descends to $K^{*} (A)$ ; its fibres are the spectral triples sharing a K-homology class but differing in metric data. This is the precise sense in which a spectral triple is "an unbounded Fredholm module": it refines the bounded datum by a metric, recovering the Connes distance 39.06.01 that $F$ alone discards.

Summability and the Chern character. A $p$ -summable Fredholm module has a Chern character $ch^{*} [F]$ in the cyclic cohomology $H C^{2 k} (A)$ (even) or $H C^{2 k + 1} (A)$ (odd) for $2 k \geq p$ , given by the cyclic cocycle $$ \tau_{2k}(a^0, a^1, \ldots, a^{2k}) = c_k \operatorname{Tr}s\big(\pi(a^0)[F,\pi(a^1)]\cdots[F,\pi(a^{2k})]\big), $$ with $Tr_{s} = Tr (γ \cdot)$ the supertrace; the Schatten condition $p \leq 2 k$ makes the trace converge. The index pairing factors through the Chern characters of both sides: $$ \langle [F], [e] \rangle = \langle \operatorname{ch}^[F], \operatorname{ch}_[e] \rangle, $$ the right pairing being the canonical $HC^* \times HC* \to \mathbb{C} $o f [39.07.01], w hi c h l an d s in$ \mathbb{Z} $b ec a u se t h e l e f t s i d e d oes . T hi s i s t h e ana l y t i c - t o - cy c l i c t r an s l a t i o n; t h e * * C o nn es - M osco v i c i l oc a l in d e x f or m u l a * * co m p u t es$ \operatorname{ch}^*[D] $f r o m t h er es i d u eso f$ \zeta_b(s) = \operatorname{Tr}(b,|D|^{-s}) $o v er t h e d im e n s i o n s p ec t r u m [39.06.01], r e pl a c in g t h e d i v er g e n t co mm u t a t or t r a ces w i t h l oc a l r es i d u ecoc hain s an d r eco v er in g, in t h eco mm u t a t i v ec a se, t h e$ \widehat{A}\operatorname{ch}$ integrand of the heat-kernel proof of Atiyah-Singer.

The commutative case is Atiyah-Singer. For the canonical even module of a closed even-dimensional spin manifold $M$ , a projection $e \in M_{n} (C^{\infty} (M))$ is a bundle $E$ , the compression $e \neq D^{+} e$ is the twisted chiral Dirac operator $\neq D_{E}^{+}$ , and the index pairing is $$ \langle [\not{!D}], [E]\rangle = \operatorname{index}(\not{!D}^+_E) = \int_M \widehat{A}(M),\operatorname{ch}(E), $$ the twisted spin index theorem, a special case of Atiyah-Singer 03.09.10 ^{[Connes-Moscovici 1995]}. The odd module on an odd-dimensional manifold pairs with $[u] \in K_{1}$ to give the spectral flow of $\neq D$ along the path $u \neq D u^{*}$ , equal to a $\int_{M} A (M) ch (u)$ odd Chern-Simons-type integral. Thus the index theorem is the commutative shadow of a pairing defined for any algebra; the noncommutative torus 39.06.04, the leaf space of a foliation, and the algebra of a discrete group all carry the same machine, with no manifold underneath.

Synthesis. The Fredholm module is the foundational reason the index theorem survives the passage to noncommutative spaces, and the index pairing is exactly the duality that makes K-homology the homology dual of K-theory. The bounded picture is dual to the unbounded one: a spectral triple is an unbounded Fredholm module, and its phase $F = D ∣ D ∣^{- 1}$ is the central insight that lets the metric geometry of 39.06.01 reduce to the homotopy-theoretic datum that pairs integrally with 39.02.02. Putting these together, the Chern character is the bridge to cyclic cohomology 39.07.01: it converts both the operator $[F]$ and the projection $[e]$ into cyclic classes whose pairing computes the integer, and the Connes-Moscovici local formula generalises the heat-kernel proof of Atiyah-Singer 03.09.10 to the residue cocycle of the dimension spectrum. This is exactly the analytic index theory of noncommutative geometry: the same integer, computed three ways — as a Fredholm index of a compression, as a cyclic pairing of Chern characters, and (commutatively) as the $A ch$ characteristic integral — and the assertion that these three coincide is the noncommutative index theorem.

Full proof set Master

Proposition (the bounded transform of a spectral triple is a Fredholm module). Let $(A, H, D)$ be a spectral triple 39.06.01 and $F = D ∣ D ∣^{- 1}$ on $(ker D)^{⊥}$ , $F = 1$ on the finite-dimensional $ker D$ . Then $F = F^{*}$ , $F^{2} = 1$ , and $[F, π (a)]$ is compact for every $a \in A$ .

Proof. Self-adjointness and $F^{2} = 1$ are immediate from the spectral calculus of the self-adjoint $D$ : $sign$ is a real function with $sign^{2} = 1$ . For compactness of the commutator write, for $a \in A$ , $$ [F, a] = [D|D|^{-1}, a] = [D, a]|D|^{-1} + D[,|D|^{-1}, a,]. $$ The first term is bounded $\times$ compact: $[D, a]$ is bounded (spectral-triple axiom) and $∣ D ∣^{- 1}$ is compact since $D$ has compact resolvent and $∣ D ∣^{- 1} = (1 + D^{2})^{- 1/2} \cdot (bounded)$ up to the finite kernel, so $(1 + D^{2})^{- 1/2}$ is compact. The second term is handled by the integral formula $∣ D ∣^{- 1} = \frac{1}{π} \int_{0}^{\infty} (D^{2} + λ)^{- 1} λ^{- 1/2} d λ$ , giving $[∣ D ∣^{- 1}, a] = - \frac{1}{π} \int_{0}^{\infty} (D^{2} + λ)^{- 1} [D^{2}, a] (D^{2} + λ)^{- 1} λ^{- 1/2} d λ$ , where $[D^{2}, a] = [D, a] D + D [D, a]$ is $D$ -bounded; each resolvent factor $(D^{2} + λ)^{- 1}$ is compact, so $D [∣ D ∣^{- 1}, a]$ is a norm-convergent integral of compacts, hence compact. Therefore $[F, a]$ is compact. $□$

Proposition (compression of a symmetry by a projection is Fredholm iff the commutator is compact). Let $F = F^{*}$ , $F^{2} = 1$ on $H$ , $P = P^{*} = P^{2}$ a projection, and suppose $[F, P]$ is compact. Then $P F P : P H \to P H$ (odd case) is Fredholm, and in the even case $e F^{+} e$ is Fredholm.

Proof. In the odd case set $T = P F P$ on $P H$ . Then $T^{2} = P F P F P = P F (P F) P = P F (F P - [F, P]) P = P F^{2} P - P F [F, P] P = P - P F [F, P] P$ . Since $[F, P]$ is compact and $F$ bounded, $T^{2} = P - K$ with $K$ compact, so $T^{2}$ is invertible modulo compacts in $B (P H)$ , hence $T$ is Fredholm (a bounded operator whose square is invertible mod compacts is itself invertible mod compacts: $T (T P - \dots)$ supplies a parametrix). The even case is the off-diagonal block computation of the Key theorem with $e$ in place of $P$ . $□$

Proposition (the odd pairing equals minus a spectral flow / winding number). Let $(H, F, π)$ be an odd Fredholm module arising as the phase of $(A, H, D)$ , $u \in A$ unitary, $P = \frac{1}{2} (1 + F)$ . Then $index (P u P) = - sf (D, u D u^{*})$ , the negative spectral flow of the straight-line path from $D$ to $u D u^{*}$ .

Proof. The compression $P u P$ on $P H$ is a Toeplitz operator with symbol $u$ ; its index is computed by counting eigenvalue crossings of the path $D_{t} = (1 - t) D + t u D u^{*}$ , $t \in [0, 1]$ , which connects $D$ to its unitary conjugate. The phase $F_{t} = sign (D_{t})$ varies, and the net change in the rank of the spectral projection $\frac{1}{2} (1 + F_{t})$ across $t \in [0, 1]$ is the spectral flow $sf (D, u D u^{*})$ — the signed count of eigenvalues passing through $0$ . A standard winding computation (the suspension of the Toeplitz index theorem 39.01.04) identifies $index (P u P) = - sf (D, u D u^{*})$ . For $A = C^{\infty} (T)$ and $D = - i \frac{d}{d θ}$ , $u = e^{ik θ}$ , the spectral flow is $k$ , recovering $index (P u P) = - k$ , the negative winding number of Exercise 3. $□$

Proposition (the index pairing descends to $K$ -theory and $K$ -homology and is bi-additive). The map $([F], [e]) \mapsto index (e F^{+} e)$ is well-defined on $K^{0} (A) \times K_{0} (A)$ and additive in each variable.

Proof. Well-definedness in $[e]$ and $[F]$ and additivity were established in the Key theorem (homotopy invariance of the integer index, vanishing on degenerate modules, conjugation-invariance under Murray–von Neumann equivalence, and direct-sum additivity). It remains to extend from projections to formal differences: $K_{0} (A)$ consists of differences $[e] - [e^{'}]$ , and one sets $⟨[F], [e] - [e^{'}]⟩ = ⟨[F], [e]⟩ - ⟨[F], [e^{'}]⟩$ , consistent because the pairing is additive on projections and the index is a group homomorphism out of the projection monoid into $Z$ . Stability under $A \mapsto M_{n} (A)$ holds because amplifying $π$ and $F$ to $M_{n}$ commutes with the index of the compression 39.02.02. Hence the pairing is a well-defined bi-additive map $K^{0} (A) \times K_{0} (A) \to Z$ , and identically $K^{1} (A) \times K_{1} (A) \to Z$ in the odd case. $□$

Proposition (commutative reduction to the twisted-Dirac index). For the canonical even spectral triple of a closed even-dimensional spin manifold $M$ and a projection $e \in M_{n} (C^{\infty} (M))$ defining a bundle $E$ , $⟨[\neq D], [E]⟩ = \int_{M} A (M) ch (E)$ .

Proof. The compression $e \neq D^{+} e$ is the chiral Dirac operator twisted by $E = e \underline{C^{n}}$ (Serre-Swan identification of $e$ with sections of $E$ 03.08.01): coupling $\neq D$ to the connection $e d e$ on $E$ gives exactly $\neq D_{E} = e (\neq D \otimes 1) e$ , an elliptic first-order operator. By the Atiyah-Singer index theorem for twisted spin Dirac operators 03.09.10, $index (\neq D_{E}^{+}) = \int_{M} A (M) ch (E)$ , where $A (M)$ is the $A$ -genus of the Levi-Civita connection and $ch (E)$ the Chern character of $E$ . The left side equals $⟨[\neq D], [E]⟩$ by definition of the index pairing, completing the identification. $□$

Connections Master

Spectral Triples and the Reconstruction Theorem 39.06.01 — the unbounded operator $D$ of a spectral triple yields its bounded phase $F = D ∣ D ∣^{- 1}$ , which is the Fredholm module of this unit; the Poincaré-duality axiom there is precisely the nondegeneracy of the index pairing capping against the fundamental class $[D]$ , and the metric data $D$ discards in passing to $F$ is the Connes distance of that unit.
Operator K-Theory: $K_{0}$ and $K_{1}$ 39.02.02 — K-homology $K^{*} (A)$ built from Fredholm modules is the functor dual to the K-theory $K_{*} (A)$ there, and the index pairing $K^{*} (A) \times K_{*} (A) \to Z$ is the perfect bilinear pairing between them; the Atkinson-Fredholm index used throughout is the same integer invariant that the index map $\partial : K_{1} (A / J) \to K_{0} (J)$ computes for the Toeplitz extension.
Atiyah-Singer index theorem 03.09.10 — the commutative case of the index pairing is exactly the index of the twisted Dirac operator $\neq D_{E}$ , equal to $\int_{M} A (M) ch (E)$ ; the Connes-Moscovici local index formula generalises the heat-kernel proof to the residue cocycle of the dimension spectrum, so this unit is the noncommutative envelope of that theorem.
Cyclic cohomology and the Chern character 39.07.01 — the Chern character $ch^{*} [F]$ sends a $p$ -summable Fredholm module to a cyclic cocycle, and the index pairing factors as the pairing of $ch^{*} [F]$ with $ch_{*} [e]$ in $H C^{*} \times H C_{*}$ ; summability is the convergence condition that fixes the cyclic degree.
Toeplitz and Cuntz Algebras and Extensions 39.01.04 — the odd index pairing on $C (T)$ is the Toeplitz index $index (P u P) = - wind (u)$ , the Noether-Gohberg-Krein theorem; the Toeplitz extension realises the odd Fredholm module whose connecting map is the winding-number index computed here.
The Noncommutative Torus as a Noncommutative Geometry 39.06.04 — the flat spectral triple on $A_{θ}$ has a bounded phase whose index pairing with the K-theory $K_{0} (A_{θ}) ≅ Z^{2}$ computes the integer-valued Connes pairing detecting the Powers-Rieffel projection, a manifold-free instance of the machine of this unit.

Historical & philosophical context Master

The index of an elliptic operator entered topology with the Atiyah-Singer theorem of 1963, and Atiyah's 1969–1970 reformulation recast elliptic operators as cycles for a homology theory dual to K-theory, the genesis of analytic K-homology ^{[Atiyah 1969]}. Atiyah's abstract elliptic operators — bounded $F$ with $F^{2} - 1$ and $[F, a]$ compact — are the Fredholm modules of this unit, and Brown-Douglas-Fillmore theory (1977) gave the odd case through extensions of $C (X)$ by the compacts, identifying $K^{1} (C (X))$ with essential-unitary classes. Kasparov's $K K$ -theory (1980) unified K-theory and K-homology into a bivariant functor $K K (A, B)$ with $K_{*} (A) = K K (C, A)$ and $K^{*} (A) = K K (A, C)$ , the index pairing becoming the Kasparov product $K K (A, C) \times K K (C, A) \to K K (C, C) = Z$ .

Connes placed Fredholm modules at the centre of noncommutative geometry in his 1985 paper and the 1994 book Noncommutative Geometry ^{[Connes Ch. IV]}, introducing the $p$ -summability hierarchy, the Chern character into cyclic cohomology, and the bounded/unbounded dictionary that makes a spectral triple an unbounded Fredholm module. The Connes-Moscovici local index formula of 1995 ^{[Connes-Moscovici 1995]} computed the Chern character of $[D]$ from the residues of the zeta functions $Tr (b ∣ D ∣^{- s})$ , giving a noncommutative residue formula that specialises to the $A ch$ integrand of the heat-kernel proof of Atiyah-Singer. Higson and Roe's Analytic K-Homology (2000) ^{[Higson-Roe Ch. 8]} gave the definitive operator-theoretic account, deriving the index pairing and the Kasparov product from the Fredholm-module definition. The lineage runs from Noether and Gohberg-Krein's winding-number index for Toeplitz operators, through Atiyah-Singer, to Connes' realisation that the index is a pairing defined for any algebra, with the manifold an inessential commutative special case.

Bibliography Master

Connes, A., Noncommutative Geometry, Academic Press, 1994. Ch. IV.
Atiyah, M. F., "Global theory of elliptic operators", in Proc. Int. Symp. on Functional Analysis (Tokyo, 1969), University of Tokyo Press, 1970, 21–30.
Atiyah, M. F. & Singer, I. M., "The index of elliptic operators I", Annals of Mathematics 87 (1968), 484–530.
Brown, L. G., Douglas, R. G. & Fillmore, P. A., "Extensions of C*-algebras and K-homology", Annals of Mathematics 105 (1977), 265–324.
Kasparov, G. G., "The operator K-functor and extensions of C*-algebras", Math. USSR Izvestija 16 (1981), 513–572.
Connes, A. & Moscovici, H., "The local index formula in noncommutative geometry", Geometric and Functional Analysis 5 (1995), 174–243.
Higson, N. & Roe, J., Analytic K-Homology, Oxford University Press, 2000. Ch. 8–9.
Baaj, S. & Julg, P., "Théorie bivariante de Kasparov et opérateurs non bornés dans les C*-modules hilbertiens", C. R. Acad. Sci. Paris 296 (1983), 875–878.

Operator-algebras spine, noncommutative-geometry chapter. The K-homology unit: Fredholm modules as bounded/abstract elliptic operators, the bounded phase $F = D ∣ D ∣^{- 1}$ of a spectral triple, the integer-valued index pairing $K^(A)\times K_*(A)\to\mathbb{Z} $, t h eo dd p ai r in g a ss p ec t r a l f l o w / w in d in g n u mb er,$ p $- an d$ \theta$-summability, the Chern character into cyclic cohomology, and the commutative reduction to the Atiyah-Singer twisted-Dirac index. Builds on spectral triples (39.06.01) and operator K-theory (39.02.02).*

Prerequisites

39.06.01
39.02.02

Tier anchors

beginner: Connes-style 'an abstract elliptic operator paired with a vector bundle counts solutions minus obstructions'; the index of an operator as a robust integer, in the spirit of Atiyah's K-theory survey
intermediate: Connes & Marcolli *Noncommutative Geometry, Quantum Fields and Motives* §1; Gracia-Bondía, Várilly & Figueroa *Elements of Noncommutative Geometry* Ch. 8; Higson & Roe *Analytic K-Homology* Ch. 8
master: Connes *Noncommutative Geometry* (1994) Ch. IV; Higson & Roe *Analytic K-Homology* Ch. 8–9; Blackadar *K-Theory for Operator Algebras* §17 (Kasparov KK)

References

Connes, A. — Noncommutative Geometry · Ch. IV, Fredholm modules, K-homology, the Chern character and the index pairing
Higson, N. & Roe, J. — Analytic K-Homology · Oxford University Press (2000), Ch. 8–9: Fredholm modules, the index pairing, Kasparov product
Atiyah, M. F. — Global theory of elliptic operators · Proc. Int. Symp. Functional Analysis (Tokyo 1969), 21–30 — abstract elliptic operators and K-homology
Connes, A. & Moscovici, H. — The local index formula in noncommutative geometry · Geom. Funct. Anal. 5 (1995) 174–243 — the residue cocycle computing the index pairing

Estimated time

beginner: 18m
intermediate: 55m
master: 95m