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

Definition

Let $A$ be a unital complex $C^*$ -algebra, let $(H,\pi,F)$ be a normalized odd Fredholm module, and set $P=(I+F)/2$ . For a unitary $u\in M_n(A)$ representing a $K_1(A)$ -class, the compression

P_n\pi_n(u)P_n:P_nH^n\longrightarrow P_nH^n

is Fredholm. The odd index pairing, with the Fredholm-index sign convention used here, is

\langle[u],[H,\pi,F]\rangle =\operatorname{ind}(P_n\pi_n(u)P_n).

It depends only on the classes in $K_1(A)$ and $K^1(A)$ , and extends bilinearly to a pairing $K_1(A)\times K^1(A)\to\mathbb Z$ .

Why the compression is Fredholm

The compact commutator $[P,\pi(a)]$ implies that

P_n\pi_n(u^{-1})P_n

is a two-sided inverse to $P_n\pi_n(u)P_n$ modulo compact operators. Atkinson's characterization therefore makes the compression Fredholm. Homotopy of $u$ , stabilization, compact perturbation of $F$ , and stable homotopy of the cycle preserve its index. Hence the formula descends to $K_1(A)$ and odd analytic K-homology. Connes, Chapter IV, Section 1, Proposition 2(b).

The same construction works with an invertible representative. For a $C^*$ -algebra, polar decomposition lets one use unitary representatives without changing the $K_1$ -class.

Circle and Toeplitz example

Take $A=C(S^1)$ , $H=L^2(S^1)$ , and let $P$ be the Hardy projection. With $F=2P-I$ , multiplication by the coordinate function $u(z)=z$ has compact commutator with $P$ . Its compression is the Toeplitz operator $T_z$ , the unilateral shift on the Hardy space, so

\langle[u],[H,\pi,F]\rangle=\operatorname{ind}(T_z)=-1.

This sign uses the convention $\operatorname{ind}T=\dim\ker T-\dim\ker T^*$ . Replacing $u$ by $u^{-1}$ gives $+1$ .

Conventions and scope

For nonunital $A$ , use a unitary in a matrix algebra over the unitization whose scalar image is the identity. The resulting class lies in $K_1(A)$ .

Some authors define the odd pairing with an overall minus sign, often to align a chosen orientation or boundary-map convention. A stated sign convention is therefore part of a numerical computation. The formula in the core agrees with Connes's Proposition 2(b).

References