Domination of operators in the non-commutative setting

Timur Oikhberg, and Eugeniu Spinu. "Domination of operators in the non-commutative setting." Studia Mathematica 219.1 (2013): 35-67. <http://eudml.org/doc/286670>.

@article{TimurOikhberg2013,
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 ≤ 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.},
author = {Timur Oikhberg, Eugeniu Spinu},
journal = {Studia Mathematica},
keywords = {ordered Banach spaces; -algebra; noncommutative function space; domination problem; operator ideal},
language = {eng},
number = {1},
pages = {35-67},
title = {Domination of operators in the non-commutative setting},
url = {http://eudml.org/doc/286670},
volume = {219},
year = {2013},
}

TY - JOUR
AU - Timur Oikhberg
AU - Eugeniu Spinu
TI - Domination of operators in the non-commutative setting
JO - Studia Mathematica
PY - 2013
VL - 219
IS - 1
SP - 35
EP - 67
AB - 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.
LA - eng
KW - ordered Banach spaces; -algebra; noncommutative function space; domination problem; operator ideal
UR - http://eudml.org/doc/286670
ER -