Can Boutet de Monvel’s algebra on a compact manifold with boundary be obtained as the algebra of pseudodifferential operators on some Lie groupoid ? If it could, the kernel of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra . While the answer to the above question remains open, we exhibit in this paper a groupoid such that possesses an ideal isomorphic to . In fact, we prove first that with the C*-algebra generated by the zero order pseudodifferential operators on the boundary and the algebra of compact operators. As both and are extensions of by ( is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.
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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