Dixmier traces generated by exponentiation invariant generalised limits

TL;DR

This paper introduces a new class of singular positive traces on a specific operator ideal, generated by exponentiation invariant generalized limits, and establishes their properties and relation to Dixmier traces.

Contribution

It defines a novel class of traces on $\,\mathcal M_{1, abla}$ generated by exponentiation invariant limits, expanding the understanding of singular traces.

Findings

The new class is strictly contained within all Dixmier traces.

A Lidskii-type formula is proved for this class of traces.

The class is generated by exponentiation invariant generalized limits.

Abstract

We define a new class of singular positive traces on the ideal $\mathcal M_{1,\infty}$ of $B(H)$ generated by exponentiation invariant generalized limits. We prove that this new class is strictly contained in the class of all Dixmier traces. We also prove a Lidskii-type formula for this class of traces.