Pseudodifferential analysis on continuous family groupoids

By Robert Lauter, Bertrand Monthubert, Victor Nistor. Documenta Mathematica, 2000

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We study properties and representations of the convo- lution algebra and the algebra of pseudodieren tial operators asso- ciated to a continuous family groupoid. We show that the study of representations of the algebras of pseudodieren tial 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 pseudodieren tial operators of order zero. We prove a structure theorem for the norm completions of these algebras asso- ciated to groupoids with invariant ltrations. As a consequence, we obtain criteria for an operator to be compact or Fredholm. We end with a discussion of the signicance 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

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