Abstract. We show norm estimates for the sum of independent random variables in noncommutative Lp spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal’s inequalities in the commutative case. Among other applications, we derive a formula for p-norm of the eigenvalues for matrices with independent entries, and characterize those symmetric subspaces and unitary ideal spaces which can be realized as subspaces of noncommutative Lp for 2 < p < ∞. 0. Introduction and Notation Martingale inequalities have a long tradition in probability. The applications of the work of Burkholder and his collaborators [B73,?, BDG72, B71a, B71b, BGS71, BG70, B66] ranges from classical harmonic analysis to stochastical differential equations and the geometry of Banach spaces. When proving the estimates for the ‘little square function ’ Burkholder

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 a. If a subspace X of Lp(a) contains uniformly the spaces ℓn p, n ≥ 1, it contains an almost isometric, almost 1-complemented copy of ℓp. If X contains uniformly the finite dimensional Schatten classes Sn p, it contains their ℓp-direct sum too. We obtain a version of the classical Kadec-Pe̷lczyński 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.

Abstract. We provide an analogue of Gundy’s decomposition for L1-bounded non-commutative martingales. An important difference from the classical case is that for any L1-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type (1, 1) boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder’s weak type inequality for square functions. A sequence (xn)n≥1 in a normed space X is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of (xn)n≥1 to l2 taking each xn to the n-th vector basis of l2. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in L1(M, τ) whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative L1-spaces.

Abstract. We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative Lp space for some p> 1. This is a noncommutative version of Rosenthal’s result for commutative Lp spaces. Similarly for 1 ≤ q < 2, an infinite dimensional subspace X of a noncommutative Lq space either contains ℓq or embeds in Lp for some q < p < 2. The novelty in the noncommutative setting is a double sided change of density.

Abstract. We investigate the norm of sums of independent vector-valued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class S n q in Sp(ℓm q) with optimal order m ∼ n 2. Using these embeddings we show the surprising fact that the sharp type (cotype) index in the sense of operator spaces for Lp[0,1] is min(p, p ′ ) (max(p, p ′)). Similar techniques are used to show that the operator space notions of B-convexity and K-convexity are equivalent.

Abstract. We construct classes of von Neumann algebra modules by considering “column sums ” of noncommutative L p spaces. Our abstract characterization is based on an L p/2-valued inner product, thereby generalizing Hilbert C*-modules and representations on Hilbert space. While the (single) representation theory is similar to the L 2 case, the concept of L p bimodule (p ̸ = 2) turns out to be nearly trivial. Noncommutative L p spaces, by now, are standard objects in the theory of operator algebras. Starting with a von Neumann algebra M, there are a variety of equivalent methods for producing the (quasi-)Banach space L p (M). If M is L ∞ (X, µ), the result is (isometric to) L p (X, µ), so this can rightfully be thought

Abstract. We determine the optimal orders for the best constants in the non-commutative Burkholder-Gundy, Doob and Stein inequalities obtained recently in the non-commutative martingale theory. AMS Classification: 46L53, 46L51 Key words: Non-commutative martingale, inequality, optimal order, triangular projection † Marius Junge is partially supported by the NSF 1

Abstract. We prove a weak-type (1,1) inequality for square functions of noncommutative martingales that are simultaneously bounded in L 2 and L 1. More precisely, the following non-commutative analogue of a classical result of Burkholder holds: there exists an absolute constant K> 0 such that if M is a semi-finite von Neumann algebra and (Mn) ∞ n=1 is an increasing filtration of von Neumann that is bounded subalgebras of M then for any given martingale x = (xn) ∞ n=1 in L2 (M) ∩ L1 (M), adapted to (Mn) ∞ n=1, there exist two martingale difference sequences, a = (an) ∞ n=1 and b = (bn) ∞ n=1, with dxn = an + bn for every n ≥ 1, and n=1 n=1 a ∗ n an a ∗ n an

Let A denote the reduced amalgamated free product of a family

We give a systematic study on the Hardy spaces of functions with values in the non-commutative L p-spaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the non-commutative martingale inequalities. Our non-commutative Hardy spaces are defined by the non-commutative Lusin integral function. The main results of this paper include: (i) The analogue in our setting of the classical Fefferman duality theorem between H 1 and BMO. (ii) The atomic decomposition of our non-commutative H 1. (iii) The equivalence between the norms of the non-commutative Hardy spaces and of the noncommutative L p-spaces (1 < p < ∞). (iv) The non-commutative Hardy-Littlewood maximal inequality. (v) A description of BMO as an intersection of two dyadic BMO. (vi) The interpolation results on these Hardy spaces. Plan: