K-Theory of Algebra of Zeroth Order Pseudo differential operators

Question

Any one knows a reference for computing K_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators?

Thanx!

Answer 1

I do not have a reference, except that Higson-Roe's book ''Analytic K-homology'', p. 47ff should give enough information.

Let $P(M)$ be the algebra of order zero scalar pseudodifferential operators on the closed manifold $M$. There is a short exact sequence

$$ K(L^2(M)) \to P(M) \to C(SM) $$

where $C(SM)$ are the functions on the unit sphere tangent bundle. The long exact sequence in $K$-theory is

$$0=K_1 (K(L^2(M))) \to K_1 (P(M)) \to K_1 (C(SM)) \to K_0 (K(L^2 (M)))=\mathbb{Z} \to K_0 (P(M)) \to K_0 (C(SM)) \to 0.$$

I claim that the map $K_1 (C(SM)) \to \mathbb{Z}$ is surjective. The map sends a unitary matrix $s$ with entries in $C(SM)$ to the index of a pseudodifferential operator with symbol $s$, so the surjectivity amounts to saying that on each clsed manifold, there is a pseudo-DO with index $1$ and unitary symbol. Such an operator is constructed in Atiyah-Singer ''Index of elliptic operators I'', using the excision principle for the index.

The answer is that $K_0 (P(M)) \to K_0 (C(SM))$ is an isorphism; the group $K_0 (C(SM))$ of course depends heavily on $M$.