## 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 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.

Original language | English (US) |
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Pages (from-to) | 35-67 |

Number of pages | 33 |

Journal | Studia Mathematica |

Volume | 219 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

## Keywords

- C-algebra
- Domination problem
- Non-commutative function space
- Operator ideal
- Ordered Banach spaces

## ASJC Scopus subject areas

- Mathematics(all)