Traces of compact operators and the noncommutative residue

We extend the noncommutative residue of M. Wodzicki on compactly supported classical pseudo-differential operators of order −d−d and generalise A. Connes’ trace theorem, which states that the residue can be calculated using a singular trace on compact operators. Contrary to the role of the noncommutative residue for the classical pseudo-differential operators, a corollary is that the pseudo-differential operators of order −d−d do not have a ‘unique’ trace; pseudo-differential operators can be non-measurable in Connes’ sense. Other corollaries are given clarifying the role of Dixmier traces in noncommutative geometry, including the definitive statement of Connes’ original theorem.