Domination of operators in the non-commutative setting
Volume 219 / 2013
Abstract
We consider majorization problems in the non-commutative setting. More specifically, suppose $E$ and $F$ are ordered normed spaces (not necessarily lattices), and $0 \leq T \leq S$ in $B(E,F)$. If $S$ belongs to a certain ideal (for instance, the ideal of compact or Dunford–Pettis operators), does it follow that $T$ belongs to that ideal as well? We concentrate on the case when $E$ and $F$ are $C^*$-algebras, preduals of von Neumann algebras, or non-commutative function spaces. In particular, we show that, for $C^*$-algebras $\mathcal {A}$ and ${\mathcal {B}}$, the following are equivalent: (1) at least one of the two conditions holds: (i) $\mathcal {A}$ is scattered, (ii) ${\mathcal {B}}$ is compact; (2) if $0 \leq T \leq S : \mathcal {A}\to {\mathcal {B}}$, and $S$ is compact, then $T$ is compact.
