Catalysis in the trace class and weak trace class ideals

Given operators $A,B$ in some ideal $\mathcal{I}$ in the algebra $\mathcal{L}(H)$ of all bounded operators on a separable Hilbert space $H$, can we give conditions guaranteeing the existence of a trace-class operator $C$ such that $B\otimes C$ is submajorized (in the sense of Hardy–Littlewood) by $A\otimes C$ ? In the case when $\mathcal{I}={\mathcal{L}}_1$, a necessary and almost sufficient condition is that the inequalities ${\rm Tr} (B^p) \leq {\rm Tr} (A^p)$holdforevery$p \in [1,\infty]$. We show that the analogous statement fails for $\mathcal{I}={\mathcal{L}}_{1,\mathrm{\infty }}$ by connecting it with the study of Dixmier traces.