Domination of operators in the non-commutative setting

Timur Oikhberg ¹   ; Eugeniu Spinu ²

¹ Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A.
² Department of Mathematical and Statistical Sciences University of Alberta Edmonton, AB T6G 2G1, Canada

Studia Mathematica, Tome 219 (2013) no. 1, pp. 35-67

Résumé

We consider majorization problems in the non-commutative setting. More specifically, suppose $E$ and $F$ are ordered normed spaces (not necessarily lattices), and $0 \leq T \leq S$ 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.

Détail

DOI : 10.4064/sm219-1-3

Keywords: consider majorization problems non commutative setting specifically suppose ordered normed spaces necessarily lattices leq leq belongs certain ideal instance ideal compact dunford pettis operators does follow belongs ideal concentrate * algebras preduals von neumann algebras non commutative function spaces particular * algebras mathcal mathcal following equivalent least conditions holds mathcal scattered mathcal compact nbsp leq leq mathcal mathcal compact compact