On the distinction between the classes of Dixmier and Connes-Dixmier   traces
  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. 
Fedor Sukochev, Alexandr Usachev, Dmitriy Zanin
math.OA math.FA