Dixmier traces and extrapolation description of noncommutative Lorentz spaces

TL;DR

This paper explores the connections between Dixmier traces, zeta-functions, and heat semigroup traces within noncommutative Lorentz spaces, extending the framework beyond classical ideals and applying it to pseudo-differential calculus.

Contribution

It introduces the extrapolation description of operator Lorentz spaces as the suitable setting for these investigations, linking Dixmier traces to symbols in pseudo-differential calculus.

Findings

Dixmier traces coincide with the Dixmier integral of symbols in pseudo-differential operators.

The framework extends the analysis of traces beyond the dual of the Macaev ideal.

Application to Hörmander-Weyl calculus demonstrates practical relevance.

Abstract

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.