Pseudodifferential Operators and Regularized Traces

This is a survey on trace constructions on various operator algebras with an emphasis on regularized traces on algebras of pseudodifferential operators. For motivation our point of departure is the classical Hilbert space trace which is the unique semifinite normal trace on the algebra of bounded operators on a separable Hilbert space. Dropping the normality assumption leads to the celebrated Dixmier traces. Then we give a leisurely introduction to pseudodifferential operators. The parameter dependent calculus is emphasized and it is shown how this calculus leads naturally to the asymptotic expansion of the resolvent trace of an elliptic differential operator. The Hadamard partie finie regularization of an integral is explained and used to extend the Hilbert space trace to the Kontsevich-Vishik canonical trace on pseudodifferential operators of non–integral order. Then the stage is well prepared for the residue trace of WodzickiGuillemin and its purely functional analytic interpretation as a Dixmier trace by Alain Connes. We also discuss existence and uniqueness of traces for the algebra of parameter dependent pseudodifferential operators; the results are surprisingly different. Finally, we will discuss the analogue of the regularized traces on the symbolic level and study the de Rham cohomology of R with coefficients being symbol functions. This generalizes a recent result of S. Paycha concerning the characterization of the Hadamard partie finie integral and the residue integral in light of the Stokes property.