Pseudodifferential analysis on continuous family groupoids

Abstract

We study properties and representations of the convolution algebra and the algebra of pseudodifferential operators associated to a continuous family groupoid. We show that the study of representations of the algebras of pseudodifferential operators of order 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 associated 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 theory 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.