# Boutet de Monvel’s calculus and groupoids I

### Johannes Aastrup

University of Hannover, Germany### Severino T. Melo

Universidade de Sao Paulo, Brazil### Bertrand Monthubert

Université Paul Sabatier, Toulouse, France### Elmar Schrohe

University of Hannover, Germany

## 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.

## Cite this article

Johannes Aastrup, Severino T. Melo, Bertrand Monthubert, Elmar Schrohe, Boutet de Monvel’s calculus and groupoids I. J. Noncommut. Geom. 4 (2010), no. 3, pp. 313–329

DOI 10.4171/JNCG/57