Boutet de Monvel's Calculus and Groupoids I

Details Boutet de Monvel's Calculus and Groupoids I Boutet de Monvel's Calculus and Groupoids I

2006-11-11

arxiv.org/abs/math/0611336v1

Mathematics

K-Theory and Homology

17 pages

Scientific paper

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 ${\mathcal G}$. %ES, the kernel of the principal symbol homomorphism on Boutet de Monvel's algebra. 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.

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