Classification of traces and hypertraces on spaces of classical pseudodifferential operators

Let $M$ be a closed manifold and let $\operatorname{CL}^{\bullet}(M)$ be the algebra of classical pseudodifferential operators. The aim of this note is to classify trace functionals on the subspaces $\operatorname{CL}^a(M)\subset \operatorname{CL}^{\bullet}(M)$ of operators of order $a$. $\operatorname{CL}^a(M)$ is a $\operatorname{CL}^0(M)$-module for any real $a$; it is an algebra only if $a$ is a non-positive integer. Therefore, it turns out to be useful to introduce the notions of pretrace and hypertrace. Our main result gives a complete classification of pre- and hypertraces on $\operatorname{CL}^a(M)$ for any $a\in\mathbb{R}$, as well as the traces on $\operatorname{CL}^a(M)$ for $a\in\mathbb{Z}$, $a\le 0$. We also extend these results to classical pseudodifferential operators acting on sections of a vector bundle. As a by-product we give a new proof of the well-known uniqueness results for the Guillemin–Wodzicki residue trace and for the Kontsevich–Vishik canonical trace. The novelty of our approach lies in the calculation of the cohomology groups of homogeneous and log-polyhomogeneous differential forms on a symplectic cone. This allows to give an extremely simple proof of a generalization of a theorem of Guillemin about the representation of homogeneous functions as sums of Poisson brackets.