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Domination of operators in the non-commutative setting

Tom 219 / 2013

Timur Oikhberg, Eugeniu Spinu Studia Mathematica 219 (2013), 35-67 MSC: Primary 47B60; Secondary 46B42, 46L05, 46L52, 47L20. DOI: 10.4064/sm219-1-3

Streszczenie

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.

Autorzy

  • Timur OikhbergDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    Urbana, IL 61801, U.S.A.
    e-mail
  • Eugeniu SpinuDepartment of Mathematical and
    Statistical Sciences
    University of Alberta
    Edmonton, AB T6G 2G1, Canada
    e-mail

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