On the Noncommutative Residue for Projective Pseudodifferential Operators

A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a zero order pseudodifferential projection on a closed manifold vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in the odd dimensional case an analogous statement holds for the algebra of projective pseudodifferential operators, i.e. the noncommutative residue of a projective pseudodifferential projection vanishes. Our strategy of proof also simplifies the argument in the classical setting. We show that the noncommutative residue factors to a map from the twisted $K$-theory of the co-sphere bundle and then use arguments from twisted $K$-theory to reduce this question to a known model case. It can then be verified by local computations that this map vanishes in odd dimensions.