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 $C^(G)$. While the answer to the above question remains open, we exhibit in this paper a groupoid $G$ such that $C^(G)$ 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 $C(S^\*Y)\otimes \mathcal{K}$ by $\mathcal{K}$ ($S^\*Y$ is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.