  A well known result on pseudodifferential operators states that the
noncommutative residue (Wodzicki residue) of a pseudodifferential projection
vanishes. This statement is non-local and implies the regularity of the eta
invariant at zero of Dirac type operators. We prove that in a filtered algebra
the value of a projection under any residual trace depends only on the
principal part of the projection. This general, purely algebraic statement
applied to the algebra of projective pseudodifferential operators implies that
the noncommutative residue factors to a map from the twisted K-theory of the
co-sphere bundle. We use arguments from twisted K-theory to show that this map
vanishes, thus showing that the noncommutative residue of a projective
pseudodifferential projection vanishes. This also gives a very short proof in
the classical setting.
