Even K-theory/K-homology index pairing - Imported Knowlpedia Content

Definition

Let $A$ be a unital complex $C^*$ -algebra, let $(H^+\oplus H^-,\pi,F)$ be a normalized even Fredholm module, and write $F^+:H^+\to H^-$ for its positive-to-negative part. For a projection $p\in M_n(A)$ representing a $K_0(A)$ -class, put $p^\pm=\pi_n^\pm(p)$ . The compression

p^-F_n^+p^+:p^+H^{+n}\longrightarrow p^-H^{-n}

is Fredholm, and the even index pairing is

\langle[p],[H,\pi,F]\rangle=\operatorname{ind}(p^-F_n^+p^+).

It extends additively to differences of projections, giving a bilinear pairing $K_0(A)\times K^0(A)\to\mathbb Z$ .

Why the compression is Fredholm

Because $[F,\pi(a)]$ is compact, $p^-F_n^+p^+$ has $p^+F_n^-p^-$ as an inverse modulo compact operators. The compressed operator is therefore Fredholm by Atkinson's characterization. Its index is unchanged under stabilization of $p$ , homotopy of the projection, compact perturbation of $F$ , and stable homotopy of the Fredholm module. These facts make the formula descend to $K_0(A)$ and analytic K-homology. Connes, Chapter IV, Section 1, Proposition 2(a).

For a class $[p]-[q]$ , the value is

\operatorname{ind}(p^-F_n^+p^+)-\operatorname{ind}(q^-F_m^+q^+).

This subtraction is essential: the pairing is defined on the Grothendieck group, not only on individual projections.

Geometric example

Let $M$ be a closed even-dimensional spin manifold and take the even Fredholm module obtained from its Dirac operator. A projection $p\in M_n(C^\infty(M))$ determines a vector bundle $E$ . The compression represents the Dirac operator twisted by $E$ , so the pairing is its Fredholm index. This is the operator-theoretic bridge from a $K_0$ -class to an integer index. Connes, Chapter IV, Section 1, example and Proposition 2.

For $A=\mathbb C$ , pairing the class of a finite-rank projection with an even cycle scales the cycle's basic index by the rank.

Conventions and scope

For nonunital $A$ , projections are taken in matrix algebras over the unitization, with the scalar projection subtracted so that the class lies in $K_0(A)$ . An unnormalized Fredholm module may be normalized or handled directly modulo compact operators before compression.

Switching which grading summand is called positive, or replacing the Fredholm-index convention $\dim\ker T-\dim\ker T^*$ by its negative, reverses the displayed sign. The convention in the core agrees with Connes's Proposition 2(a).

References