Classification of traces and hypertraces on spaces of classical pseudodifferential operators

  • Matthias Lesch

    University of Bonn, Germany
  • Carolina Neira Jiménez

    University of Regensburg, Germany


Let MM be a closed manifold and let CL(M)\operatorname{CL}^{\bullet}(M) be the algebra of classical pseudodifferential operators. The aim of this note is to classify trace functionals on the subspaces CLa(M)CL(M)\operatorname{CL}^a(M)\subset \operatorname{CL}^{\bullet}(M) of operators of order aa. CLa(M)\operatorname{CL}^a(M) is a CL0(M)\operatorname{CL}^0(M)-module for any real aa; it is an algebra only if aa 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 CLa(M)\operatorname{CL}^a(M) for any aRa\in\mathbb{R}, as well as the traces on CLa(M)\operatorname{CL}^a(M) for aZa\in\mathbb{Z}, a0a\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.

Cite this article

Matthias Lesch, Carolina Neira Jiménez, Classification of traces and hypertraces on spaces of classical pseudodifferential operators. J. Noncommut. Geom. 7 (2013), no. 2, pp. 457–498

DOI 10.4171/JNCG/123