2012/09/04 by Timur Oikhberg, Oikhberg, Timur, Eugeniu Spinu +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.1209.0699
arxiv created 2013/10/17 · arxiv updated 2013/10/18
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 :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.