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!



K-Theory of Algebra of Zeroth Order Pseudo differential operators
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 $M$.


