Fredholm module - Imported Knowlpedia Content

Definition

Let $A$ be a complex $C^*$ -algebra. A Fredholm module over $A$ is a complex Hilbert space $H$ , a representation $\pi:A\to\mathcal B(H)$ , and a bounded operator $F\in\mathcal B(H)$ such that, for every $a\in A$ ,

[F,\pi(a)],\qquad (F^2-I)\pi(a),\qquad (F-F^*)\pi(a)

are compact operators. This is the locally compact, Kasparov-cycle convention. A module is normalized when $F=F^*$ and $F^2=I$ , leaving only compactness of the commutators. It is degenerate when all three displayed expressions vanish. Parity is supplied by an additional grading or by declaring the module ungraded.

Normalized and unnormalized conventions

Connes's bounded convention begins with a self-adjoint involution $F$ and requires $[F,\pi(a)]$ to be compact. The more flexible Kasparov convention used in the core permits self-adjointness and involutivity to fail locally by compact operators. These conventions give the same K-homological cycles after the standard normalization and stabilization procedures; one should nevertheless say which convention a formula assumes. Connes, Chapter IV, Section 1, Definition 1 and the paragraph following it records both forms.

Nondegeneracy of $\pi$ is commonly imposed for $C^*$ -algebras. A degenerate zero summand can be removed, so allowing arbitrary representations does not change the resulting stable theory.

Structure and consequences

For a unital algebra represented unitally, the compactness conditions say that the image of $F$ in the Calkin algebra is a self-adjoint involution commuting with the image of $A$ . In the general nonunital convention, self-adjointness and involutivity hold only locally after multiplication by $\pi(a)$ ; the defects of $F$ need not themselves be compact. Thus a Fredholm module is not merely a Fredholm operator: it is a Fredholm-type operator together with an algebra action that it intertwines modulo compact error.

Direct sums of modules are formed componentwise. Unitary equivalence, norm-continuous operator homotopy, and addition or removal of degenerate modules generate the stable equivalence used in analytic K-homology. Under these operations, the index pairings remain unchanged. Connes, Chapter IV, Section 1 and Appendix A develops this passage from modules to $KK(A,\mathbb C)$ .

Examples and non-examples

For a compact smooth manifold $M$ , an order-zero elliptic pseudodifferential operator between Hermitian bundles gives, after adjoining a parametrix in the opposite direction, an even Fredholm module over $C(M)$ . Compactness of the commutators with multiplication operators is the analytic shadow of pseudolocality. Connes, Chapter IV, Section 1, example following Definition 1 gives this construction.

By contrast, a self-adjoint involution $F$ and a representation $\pi$ do not form a Fredholm module when some $[F,\pi(a)]$ is noncompact. The failed axiom is compatibility of the operator with the algebra action modulo compact operators.

References