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.