Domination of operators in the non-commutative setting

We consider majorization problems in the non-commutative setting. More specifically, suppose E and F are ordered normed spaces (not necessarily lattices), and 0 T 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 A and B, the following are equivalent: (1) at least one of the two conditions holds: (i) A is scattered, (ii) B is compact; (2) if 0 T S : A ! B, and S is compact, then T is compact.