ar X iv : m at h / 03 07 16 9 v 1 [ m at h . FA ] 1 1 Ju l 2 00 3 On Subspaces of Non - commutative L p - Spaces

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 a. If a subspace X of Lp(a) contains uniformly the spaces lnp , n ≥ 1, it contains an almost isometric, almost 1-complemented copy of lp. If X contains uniformly the finite dimensional Schatten classes S p , it contains their lp-direct sum too. We obtain a version of the classical Kadec-Pe lczyński 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 Lp(a), together with a careful analysis of the elements of an ultrapower Lp(a)U which are disjoint from the subspace Lp(a). 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 Lp(a) containing lp are extended to the general case.