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Invariance spectrale des algèbres d'opérateurs pseudodifférentiels
Abstract
We construct and study several algebras of pseudodifferential operators that are closed under holomorphic functional calculus. This leads to a better understanding of the structure of inverses of elliptic pseudodifferential operators on certain non-compact manifolds. It also leads to decay properties for the solutions of these operators. To cite this article: R. Lauter et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1095–1099.
We develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids (also called Lie groupoids) with compact basis. We analyze these operators as unbounded operators acting on Hilbert modules over C∗(G), and we show in particular that elliptic operators are regular. We construct a scale of Sobolev modules which are the abstract analogues of the ordinary Sobolev spaces, and analyze their properties. Furthermore, we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense.
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincare-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A* of a Lie algebroid.
We study properties and representations of the convo-lution algebra and the algebra of pseudodifferential operators asso-ciated to a continuous family groupoid. We show that the study of representations of the algebras of pseudodifferential operators of or-der zero completely reduces to the study of the representations of the ideal of regularizing operators. This recovers the usual boundedness theorems for pseudodifferential operators of order zero. We prove a structure theorem for the norm completions of these algebras asso-ciated to groupoids with invariant filtrations. As a consequence, we obtain criteria for an operator to be compact or Fredholm. We end with a discussion of the significance of these results to the index the-ory of operators on certain singular spaces. For example, we give a new approach to the question of the existence of spectral sections for operators on coverings of manifolds with boundary. We expect that our results will also play a role in the analysis on more general singular spaces.
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincar\'e-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual of a Lie algebroid.
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 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 b -calculus.
For purely mathematical reasons it is necessary to consider spaces which cannot be represented as point set sand where the coordinates describing the space do not commute.In other words,spaces which are described by algebras of coordinates which arenot commutative.If you conside rsuch spaces,then it is necessary to rethink most of the notions of classical geometry and redefine them. Motivated from pure mathematics it turns out that there are very striking parallels to what is done in quantum physics In the following lectures, I hope to discuss some of these parallels.
Nous construisons une KK-théorie pour les algèbres de Banach équivariante par l’action d’un groupoïde et nous montrons la conjecture de Baum–Connes à coefficients commutatifs pour les groupes hyperboliques et pour les groupoïdes de Poincaré des feuilletages à base compacte qui peuvent être munis d’une métrique riemannienne longitudinale à courbure sectionnelle strictement négative.
Let G be a Lie groupoid. The pseudodifferential calculus over G defines the analytic index. The tangent groupoid induces a morphism in K-theory; we prove that it coincides with the analytic index. Thus the latter may be defined without considering the pseudodifferential calculus.
We associate to any manifold with corners (even with non-embedded hyperfaces) a (non-Hausdorff) longitudinally smooth Lie groupoid, on which we define a pseudodifferential calculus. This calculus generalizes the b-calculus of Melrose, defined for manifolds with embedded corners. The groupoid of a manifold with corners is shown to be unique up to equivalence for manifolds with corners of same codimension. Using tools from the theory of -algebras of groupoids, we also obtain new proofs for the study of b-calculus.