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

Abstract

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