Introduction Probabilistic methods play an important role in harmonic analysis and Banach space theory. Let us just mention the relevance of sums of independent random variables, p-stable processes or martingale inequalities in both fields. The analysis of subspaces of the classical Lp spaces is specially benefited from such probabilistic notions. Viceversa, Burkholder’s martingale inequality for the conditional square function has been discovered in view of Rosenthal’s inequality for the norm in Lp of sums of independent random variables. This is only one example of the fruitful interplay between harmonic analysis, probability theory and Banach space geometry carried out mostly in the 70’s by Burkholder, Gundy, Kwapie´ n, Maurey, Pisier, Rosenthal and many others. More recently it became clear that a similar endeavor for noncommutative Lp spaces requires an additional insight from quantum probability and operator space theory [15, 17, 50]. A noncommutative theory of martingale inequalities finds its beginnings in the work of Lust-Piquard [36] and Lust-Piquard/Pisier [37] on the noncommutative Khintchine inequality. The seminal paper of Pisier and Xu on the noncommutative analogue of Burkholder-Gundy inequality [51] started a new trend in quantum probability. Nowadays, most classical martingale inequalities have a satisfactory noncommutative analogue, see [16, 28, 41, 53]. In the proof of these results the classical stopping time arguments are no longer available, essentially because point sets disappear after quantization. These arguments are replaced by functional analytic or combinatorial arguments. In the functional analytic approach we often encounter new spaces. Indeed, maximal functions in the noncommutative context can only be understood and defined through analogy with vector-valued Lp spaces. A careful analysis of these spaces is crucial in establishing basic results such as Doob’s inequality [16] for noncommutative martingales and the noncommutative maximal theorem behind Birkhoff’s ergodic theorem [29]. The proof of maximal theorems and noncommutative versions of Rosenthal’s inequality often uses square function and conditioned square function estimates, see [26] and the references therein. These are examples of more general classes of noncommutative function spaces to be defined below. However, all of them illustrate our main motto in this paper. Namely, certain problems can be solved by finding and analyzing the appropriate class of Banach spaces. We shall develop in this paper a new theory of generalized noncommutative Lp spaces with three problems in mind for a given von Neumann algebra A. Problem 1. Calculate the Lp(A q ) norm for sums of free random variables. Problem 2. If 1 ≤ p q ≤ 2, find a cb-embedding of Lq(A) into some Lp space. 1

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