Journal of Noncommutative Geometry

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Volume 4, Issue 3, 2010, pp. 313–329
DOI: 10.4171/JNCG/57

Published online: 2010-07-26

Boutet de Monvel’s calculus and groupoids I

Johannes Aastrup[1], Severino T. Melo[2], Bertrand Monthubert[3] and Elmar Schrohe[4]

(1) University of Hannover, Germany
(2) Universidade de Sao Paulo, Brazil
(3) Université Paul Sabatier, Toulouse, France
(4) University of Hannover, Germany

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.

Keywords: Boundary value problems on manifolds, index theory, groupoids, KK-theory, extension theory

Aastrup Johannes, Melo Severino, Monthubert Bertrand, Schrohe Elmar: Boutet de Monvel’s calculus and groupoids I. J. Noncommut. Geom. 4 (2010), 313-329. doi: 10.4171/JNCG/57