A groupoid approach to the Wodzicki residue

Abstract

Originally, the noncommutative residue was studied in the 80's by Wodzicki in his thesis [33] and Guillemin [19]. In this article we give a definition of the Wodzicki residue, using the language of r-fibered distributions from [24], [30], in the context of filtered manifolds. We show that this groupoidal residue behaves like a trace on the algebra of pseudodifferential operators on filtered manifolds and coincides with the usual residue Wodzicki in the case where the manifold is trivially filtered. Moreover, in the context of Heisenberg calculus, we show that the groupoidal residue coincides with Ponge's definition [25] for contact and codimension 1 foliation Heisenberg manifolds.