  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.
