Domination of operators in the non-commutative setting

Abstract

We consider majorization problems in the non-commutative setting. More specifically, suppose $E$ and $F$ are ordered normed spaces (not necessarily lattices), and $0\le T\le S:E\to F$. 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 $C^{\ast }$-algebras, preduals of von Neumann algebras, or non-commutative function spaces. In particular, we show that, for $C^{\ast }$-algebras $\A$ and $\mathcal{B}$, the following are equivalent: (1) at least one of the two conditions holds: (i) $\A$ is scattered, (ii) $\mathcal{B}$ is compact; (2) if $0 \leq T \leq S : \A \to {\mathcal{B}}$, and $S$ is compact, then $T$ is compact.