The norm of sums of independent noncommutative random variables in Lp( 1)

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 Sn q in Sp( mq ) 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. © 2004 Elsevier Inc. All rights reserved.