  We consider majorization problems in the non-commutative setting. More
specifically, suppose $E$ and $F$ are ordered normed spaces (not necessarily
lattices), and $0 \leq T \leq S :E \to F$. 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 $C^*$-algebras, preduals of von Neumann algebras, or
non-commutative function spaces. In particular, we show that, for
$C^*$-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.
