The noncommutative residue and canonical trace in the light of Stokes' and continuity properties

Abstract

We show that the noncommutative residue density, resp. the cut-off regularised integral are the only closed linear, resp. continuous closed linear forms on certain classes of symbols. This leads to alternative proofs of the uniqueness of the noncommutative residue, resp. the canonical trace as linear, resp. continuous linear forms on certain classes of classical pseudodifferential operators which vanish on brackets. The uniqueness of the canonical trace actually holds on classes of classical pseudodifferential with vanishing residue density which include non integer order operators in all dimensions and odd-class (resp. even-class) operators in odd (resp. even) dimensions. The description of the canonical trace for non integer order operators as an integrated global density on the manifold is extended to odd-class (resp. even-class) operators in odd (resp. even) dimensions on the grounds of defect formulae for regularised traces of classical pseudodifferential operators.