# $K$-Theory of operators I, Higson notes

I am having trouble understanding the following statement:

3.20 Proposition, pg44: Let $$D$$ be a symmetric, odd graded elliptic operator on a graded vector bundle $$S$$ over a compact manifold. The element $$[\phi_D] \in K(\Bbb C) \cong \Bbb Z$$ is the Fredholm index of the operator $$D$$.

What exactly are used in making sense of this statement:

1. Def 2.10, page 19. A Graded $$*$$-homomoprhism $$\phi_D:\mathcal{S} \rightarrow \mathcal{K}(H)$$ from spectral theorem.

2. Prop 3.17. There is an isomoprhism $$\Phi:K(A) \rightarrow [\mathcal{S}, A \otimes \mathcal{K}(H)]$$

What the above two means is explain in my other post.

The proof goes as follows:

We have a homotopy of $$*$$-homomoprhisms $$\phi_{s^{-1}D}(f) = f(s^{-1}D)$$. At $$s=1$$ we have $$\phi_D$$ at $$s=0$$ we have the homomorphism of $$f \mapsto f(0)P$$ where $$P$$ is projection onto the kernel of $$D$$.

How does one obtain continuity at $$s=0$$ - why is the resulting map as described?

This corresponds to the integer $$\dim(\ker D \cap H_+) - \dim (\ker D \cap H_-)$$.

How does one make this computation?

The operator $$D$$ here is special: If $$D$$ is elliptic on a compact manifold, it will have compact resolvant by Rellich's lemma: $$(1+D^2)^-\frac{1}{2}$$ is compact. Therefore the spectrum of $$D$$ are eigenvalues, they have no limit point except $$\lambda=+\infty$$. If $$f(x)$$ is a continuous function of spectrum of $$D$$, $$f(D)$$ is well-defined. This will contain something like $$\delta_0(x)$$ who is 1 when $$x=0$$ and is 0 elsewhere. This is not continuous on $$\mathbb{R}$$, but is continuous on the spectrum of $$D$$!
If $$f$$ vanishes at the infinity, $$f(D)$$ is bounded. The continuity of the operators $$f(s^{-1}D)$$ are considered under the norm topology. Since $$\lim_{s\to 0} f(s^{-1}x)\to \delta_0(x)f(0)$$ As continuous functions over the spectrum of $$D$$. Also $$\delta_0(D)$$ is the projection $$P$$ to the eigenspace for $$\lambda=0$$. We have the first limit of your question (1).
Also by the result of compact resolvant, the dimension of the eigenspaces are finite dimensional, Let $$P_+ ,P_-$$ be the projection on $$\ker D\cap H_+$$, $$\ker D\cap H_-$$ respectivly. They will be trace-class operators(as operators on $$L^2(M)$$. We have $$\dim(\ker D\cap H_+)=Tr(P_+),\quad \dim(\ker D\cap H_-)=Tr(P_-)$$
We define a "supertrace" $$Str(P):=Tr(P_+)-Tr(P_-)$$.
We have $$Index(D)=Str(P)$$