On subspaces of non-commutative Lp-spaces

We study some structural aspects of the subspaces of the non-commutative (Haagerup) L_p-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 S_pⁿ, it contains their ?_p-direct sum too. We obtain a version of the classical Kadec-Pe?czyński dichotomy theorem for L_p-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 L_p 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 L_p-spaces based on finite von Neumann algebras concerning subspaces of containing ?_p are extended to the general case.