Domination of operators in the non-commutative setting
Volume 219 / 2013
Abstract
We consider majorization problems in the non-commutative setting. More specifically, suppose 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.
