Proposition: A well-defined trace modulo bulk linear kernels

There is a well-defined linear map

It vanishes on ${\mathcal{M}}_1$ and commutes with conjugation.

Proof. If $m+q_a=m^{\prime }+q_{a^{\prime }}$, then $q_{a-a^{\prime }}=m^{\prime }-m\in {\mathcal{M}}_1$. If its advanced trace were nonzero, Lemma lem:noncancellation would give a timelike conormal wavefront covector, forbidden for ${\mathcal{M}}_1$. Thus the trace is independent of the decomposition. Compact trace support follows from finite propagation. Reality of the differential and Green operators proves the conjugation statement.