Domination of operators in the non-commutative setting

TL;DR

This paper investigates how operator domination affects membership in operator ideals within non-commutative spaces like $C^*$-algebras, establishing conditions under which properties like compactness are preserved.

Contribution

It characterizes when domination preserves operator ideal membership in non-commutative settings, especially for $C^*$-algebras and von Neumann algebra preduals.

Findings

Domination preserves compactness under specific conditions.

Characterization of $C^*$-algebras where domination preserves operator ideals.

Equivalence between algebraic properties and ideal-preservation results.

Abstract

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