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


Can Boutet de Monvel’s algebra on a compact manifold with boundary be obtained as the algebra Ψ0(G)\Psi^0(G) of pseudodifferential operators on some Lie groupoid GG? If it could, the kernel G{\mathcal G} of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C(G)C^*(G). While the answer to the above question remains open, we exhibit in this paper a groupoid GG such that C(G)C^*(G) possesses an ideal I\mathcal{I} isomorphic to G{\mathcal G}. In fact, we prove first that GΨK{\mathcal G}\simeq\Psi\otimes{\mathcal K} with the C*-algebra Ψ\Psi generated by the zero order pseudodifferential operators on the boundary and the algebra K\mathcal K of compact operators. As both ΨK\Psi\otimes \mathcal K and I\mathcal{I} are extensions of C(SY)KC(S^*Y)\otimes \mathcal{K} by K\mathcal{K} (SYS^*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