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

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Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces Mϕ,w and mϕ,w respectively, introduced earlier in [3] for describing the Köthe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases (ϕ is an Orlicz function and w a decreasing weight). We study these Mϕ,w classes in the most general setting, where they may even not be linear, and identify their Köthe duals with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces Mϕ,w which proves to be the Köthe biduals of the Mϕ,w classes. In the case when the class Mϕ,w is a separable quasi-Banach space, Mϕ,w is its Banach envelope. t 0 g * ≤ t 0 f * for all t ∈ I. Similarly for sequences x = {x(n)}, y = {y(n)} we write y ≺ x, if m n=1 y * (n) ≤ m n=1 x * (n) for all m ∈ N.

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Given 1 ≤ p, q < ∞, let BLpLq be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some Lp(Lq)-Banach lattice. We show that the range of a positive contractive projection on any BLpLq-Banach lattice is itself in BLpLq. It is a consequence of this theorem and previous results that BLpLq is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BLpLqBanach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BLpLq. We also consider the class of all sublattices of Lp(Lq)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine's well-known axioms for the subspaces of Lp-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞. Mathematics Subject Classification (2000). Primary: 46B42, 46E30; Secondary: 03C65, 46M07.