On subspaces of non-commutative Lp-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 . If a subspace X of Lp() contains uniformly the spaces ℓpn, n⩾1, it contains an almost isometric, almost 1-complemented copy of ℓp. If X contains uniformly the finite dimensional Schatten classes Spn, it contains their ℓp-direct sum too. We obtain a version of the classical Kadec–Pełczyński 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(), together with a careful analysis of the elements of an ultrapower Lp()U which are disjoint from the subspace Lp(). 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 Randrianantoaninaʹs one) and some results obtained by Haagerup, Rosenthal and Sukochev for Lp-spaces based on finite von Neumann algebras concerning subspaces of Lp() containing ℓp are extended to the general case