On the distinction between the classes of Dixmier and Connes-Dixmier traces

TL;DR

This paper demonstrates that the classes of Dixmier and Connes-Dixmier traces are not equivalent on the Dixmier ideal, providing a specific example of an operator measurable by Connes-Dixmier but not by Dixmier.

Contribution

It constructs a Marcinkiewicz space and a positive operator that distinguishes the classes of Dixmier and Connes-Dixmier traces, showing they differ even on the Dixmier ideal.

Findings

Dixmier and Connes-Dixmier trace classes differ on the Dixmier ideal.

A specific operator is Connes-Dixmier measurable but not Dixmier measurable.

The paper provides a concrete example to distinguish the trace classes.

Abstract

In the present paper we prove that the classes of Dixmier and Connes-Dixmier traces differ even on the Dixmier ideal $\mathcal M_{1,\infty}$. We construct a Marcinkiewicz space $\mathcal M_\psi$ and a positive operator $T\in \mathcal M_\psi$ which is Connes-Dixmier measurable but which is not Dixmier measurable.