On the Noncommutative Residue for Projective Pseudodifferential Operators

TL;DR

This paper proves that the noncommutative residue of projective pseudodifferential projections vanishes by connecting algebraic properties with twisted K-theory, extending classical results and providing a shorter proof.

Contribution

It introduces a general algebraic framework showing the residue depends only on the principal part, leading to a new proof that the residue vanishes for projective pseudodifferential projections.

Findings

Noncommutative residue depends only on the principal part in a filtered algebra.

Residue map factors through twisted K-theory of the co-sphere bundle.

Residue of projective pseudodifferential projections vanishes.

Abstract

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.