  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.
