P_{1,\operatorname{D}})$ does indeed have a value modulo local terms that is independent of $P_1$ modulo local terms. The commutator $[P_+,P_+']$ of two pseudodifferential operators $P$ and $P'$ truncated to $X$, however, will be of the form $P_+''+G''$, where, in general, the singular Green operator $G''$ will be of class $r>0$ unless both $P$ and $P'$ are of normal order $\le 0$, cf. [@G2; @Section; @2.6]. Hence we can only hope to show the commutator property for this class.

Recall (cf. e.g. [@G2; @Lemma; @1.3.1]) that any $\psi $do $P$ having the transmission property at $X'$ can be written as a sum $$P=P^{(1)}+P^{(2)}+P^{(3)},\tag 2.6$$ where $P^{(1)}$ is a differential operator, $P^{(2)}$ is a $\psi $do whose symbol has normal order $-1$, and the symbol of $P^{(3)}$ has, near $X'$, a large power of $x_n$ as a factor to the left or right.

For a [*differential*]{} operator $P^{(1)}$, the expansion in (1.10) (with $A=P^{(1)}$) is valid without the second series over $k$; all terms are locally determined. Therefore $C_{-1}(P^{(1)},P_{1,\operatorname{D}})$ vanishes, and $C_0(P^{(1)},P_{1,\operatorname{D}})$ is local in this case, and so is $C_0(P^{(1)},P_{1,\operatorname{D}})-C_0(P^{(1)},P_{2,\operatorname{D}})$ for another choice of auxiliary operator $P_2$. Thus $C_0(P^{(1)},P_{1,\operatorname{D}})$ satisfies the first part of the requirement for being a quasi-trace (with 