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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm219-1-3,
author = {Timur Oikhberg and Eugeniu Spinu},
title = {Domination of operators in the non-commutative setting},
journal = {Studia Mathematica},
volume = {215},
year = {2013},
pages = {35-67},
zbl = {1296.47033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm219-1-3}
}
Timur Oikhberg; Eugeniu Spinu. Domination of operators in the non-commutative setting. Studia Mathematica, Tome 215 (2013) pp. 35-67. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm219-1-3/