2 INTRODUCTION Problem 3. Any reflexive subspace of A∗ embeds into some Lp for certain p 1. Our main contribution in this paper is the calculation of mixed norms for sums of free random variables and its application to construct a complete embedding of Lq into Lp. Unfortunately, the generalization of Rosenthal’s theorem [57] to noncommutative Lp spaces –whose simplest version is the content of Problem 3 above– is beyond the scope of this paper and we analyze it in [23]. We should nevertheless note that its solution is also deeply related to the main results in this paper. Namely, the interplay of interpolation and intersection is at the heart of both results. On the other hand, operator space Lp embedding theory is motivated by the classical notion of q-stable variables and norm estimates for sums of independent random variables. Let us briefly explain this. The construction of the cb-embedding for (p, q) = (1, 2) was obtained in [17]. The simplest model of 2-stable variables is provided by normalized gaussians (gk). In this particular case and after taking operator coefficients (ak) in some noncommutative L1 space, the noncommutative Khintchine inequality [37] tells us that (1) E k akgk 1 inf ak=rk+ck ( k rkrk ) 1 2 1 + ( k ckck ) 1 2 1 . Let us point out that operator spaces are a very appropriate framework for analyzing noncommutative Lp spaces and linear maps between them. Indeed, inequality (1) describes the operator space structure of the subspace spanned by the gk’s in L1 as the sum R + C of row and column subspaces of B( 2 ). We refer to [11] and [47] for background information on operator spaces. In the language of noncommutative probability many operator space inequalities translate into module valued versions of scalar inequalities, this will be further explained below. The only drawback of inequality (1) is that it does not coincide with Pisier’s definition of the operator space 2 (2) k ak δk L1(M 2 ) = inf ak=αγkβ α L4(M) k γk 2 L2(M) 1 2 β L4(M) . However, it was proved by Pisier that the right side in (2) is obtained by complex interpolation between the row and the column square functions appearing on the right of (1). One of the main results in this article is a far reaching generalization of this observation. In fact, the solution of Problem 2 in full generality is closely related to this analysis. Following our guideline we will now introduce and discuss the new class of spaces relevant for these problems and martingale theory. These generalize Pisier’s theory of Lp(Lq) spaces over hyperfinite von Neumann algebras. We begin with a brief review of some noncommutative function spaces which have lately appeared in the literature, mainly in noncommutative martingale theory. We refer to Chapter 1 below for a more detailed exposition. 0.1. Noncommutative function spaces Inspired by Pisier’s theory [46], several noncommutative function spaces have been recently introduced in quantum probability. The first motivation comes from some of Pisier’s fundamental equalities, which we briefly review. Let N1 and N2 be two hyperfinite von Neumann algebras. Then, given 1 p, q and defining 1/r = |1/p 1/q|, we have
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