No abstract

Abstract. We build a longitudinally smooth, differentiable groupoid associated to any manifold with corners. The pseudodifferential calculus on this groupoid coincides with the pseudodifferential calculus of Melrose (also called b-calculus). We also define an algebra of rapidly decreasing functions on this groupoid; it contains the kernels of the smoothing operators of the (small) b-calculus.

Nous construisons un groupoïde différentiable longitudinalement lisse associéà une variétéà coins. Le calcul pseudo-différentiel sur ce groupoïde coïncide avec le calcul pseudo-différentiel de Melrose (aussi appelé b-calculus). Nous définissonségalement une algbre de fonctionsà décroissance rapide sur ce groupoïde ; il contient les noyaux des opérateurs régularisants du (petit) b-calculus.

We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using semi-ideals, one using commutators, and one based on Schwartz spaces on the groupoid.

We define an analytic index and prove a topological index theorem for a non-compact manifold M 0 with poly-cylindrical ends. Our topological index theorem depends only on the principal symbol, and establishes the equality of the topological and analytical index in the group K 0 (C * (M )), where C * (M ) is a canonical C * -algebra associated to the canonical compactification M of M 0 . Our topological index is thus, in general, not an integer, unlike the usual Fredholm index appearing in the Atiyah-Singer theorem, which is an integer. This will lead, as an application in a subsequent paper, to the determination of the K-theory groups K 0 (C * (M )) of the groupoid C * -algebra of the manifolds with corners M . We also prove that an elliptic operator P on M 0 has an invertible perturbation P + R by a lower-order operator if and only if its analytic index vanishes.