Boutet de Monvel’s calculus and groupoids I

Abstract

Can Boutet de Monvel’s algebra on a compact manifold with boundary be obtained as the algebra ${\Psi }^0(G)$ of pseudodifferential operators on some Lie groupoid $G$? If it could, the kernel $\mathcal{G}$ of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra $C^{\ast }(G)$. While the answer to the above question remains open, we exhibit in this paper a groupoid $G$ such that $C^{\ast }(G)$ possesses an ideal $\mathcal{I}$ isomorphic to $\mathcal{G}$. In fact, we prove first that $\mathcal{G}\simeq \Psi \otimes \mathcal{K}$ with the C*-algebra $\Psi$ generated by the zero order pseudodifferential operators on the boundary and the algebra $\mathcal{K}$ of compact operators. As both $\Psi \otimes \mathcal{K}$ and $\mathcal{I}$ are extensions of $C(S^{\ast }Y)\otimes \mathcal{K}$ by $\mathcal{K}$ ($S^{\ast }Y$ is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.