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 Sqn in Sp(ℓqm) with optimal order m∼n2. 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.