Dixmier traces and extrapolation description of noncommutative Lorentz spaces

We study the relationships between Dixmier traces, zeta-functions and traces of heat semigroups beyond the dual of the Macaev ideal and in the general context of semifinite von Neumann algebras. We show that the correct framework for this investigation is that of operator Lorentz spaces possessing an extrapolation description. We demonstrate the applicability of our results to H\"ormander-Weyl pseudo-differential calculus. In that context, we prove that the Dixmier trace of a pseudo-differential operator coincide with the `Dixmier integral' of its symbol.