Traces of compact operators and the noncommutative residue

We extend the noncommutative residue of M. Wodzicki on compactly supported classical pseudodifferential operators of order -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 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. (C) 2012 Elsevier Inc. All rights reserved.