On subspaces of non-commutative Lp-spaces |
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Authors: | Yves Raynaud Quanhua Xu |
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Affiliation: | a Equipe d'Analyse Fonctionnelle, Institut de Mathématiques de Jussieu (CNRS), Case 186, 4, Place Jussieu, 75252 Paris Cedex 05, France b Laboratoire de Mathématiques, Université de Franche-Comté, Route de Gray, 25030 Besançon Cedex, France |
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Abstract: | We study some structural aspects of the subspaces of the non-commutative (Haagerup) Lp-spaces associated with a general (non-necessarily semi-finite) von Neumann algebra . If a subspace X of contains uniformly the spaces ?pn, n?1, it contains an almost isometric, almost 1-complemented copy of ?p. If X contains uniformly the finite dimensional Schatten classes Spn, it contains their ?p-direct sum too. We obtain a version of the classical Kadec-Pe?czyński dichotomy theorem for Lp-spaces, p?2. We also give operator space versions of these results. The proofs are based on previous structural results on the ultrapowers of , together with a careful analysis of the elements of an ultrapower which are disjoint from the subspace . These techniques permit to recover a recent result of N. Randrianantoanina concerning a subsequence splitting lemma for the general non-commutative Lp spaces. Various notions of p-equiintegrability are studied (one of which is equivalent to Randrianantoanina's one) and some results obtained by Haagerup, Rosenthal and Sukochev for Lp-spaces based on finite von Neumann algebras concerning subspaces of containing ?p are extended to the general case. |
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Keywords: | Primary: 46B20, 46L52 Secondary: 47M07 |
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