THE NORM OF SUMS OF INDEPENDENT NONCOMMUTATIVE RANDOM VARIABLES IN Lp(l1)

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 q in Sp(lmq ) with optimal order m ∼ n. 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. Introduction Sums of independent random variables have a long tradition both in probability theory and Banach space geometry. More recently, the noncommutative analogs of these probabilistic results have been developed [9, 11, 25] and applied to operator space theory [8, 10, 24]. In this paper, we follow this line of research in studying type and cotype in the sense of operator spaces [19]. This theory is closely connected to the notions of B-convexity and K-convexity. Using embedding results we show that these notions remain equivalent in the category of operator spaces. We recall from [16] that a Banach space X is called K-convex whenever the Gauss projection PG : f ∈ L2(Ω;X) 7−→ ∞ ∑