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Domination of operators in the non-commutative setting

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Oikhberg T., Spinu E.

Studia Mathematica

2013

tom 219

nr 1

35-67

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.

Ograniczanie wyników

1 Studia Mathematica

1 Oikhberg T.

1 Spinu E.

1 2013

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Domination of operators in the non-commutative setting