Non-commutative martingale transforms

Abstract

We prove that non-commutative martingale transforms are of weak type (1,1). More precisely, there is an absolute constant C such that if M is a semi-finite von Neumann algebra and (Mn)n=1∞ is an increasing filtration of von Neumann subalgebras of M, then for any non-commutative martingale x = (Xn)n=1∞ in L1(M), adapted to (Mn)n=1∞, and any sequence of signs (εn)n=1∞, ∥ε1x1 + ∑n=2N εn(xn - xn-1)∥1,∞ ≤ C∥xN∥1 for every N ≥ 2. This generalizes a result of Burkholder from classical martingale theory to non-commutative setting and answers positively a question of Pisier and Xu. As applications, we get the optimal order of the unconditional Martingale differences (UMD)-constants of the Schatten class Sp when p → ∞. Similarly, we prove that the UMD-constant of the finite-dimensional Schatten class Sn1 is of order log(n + 1). We also discuss the Pisier-Xu non-commutative Burkholder-Gundy inequalities. © 2002 Elsevier Science (USA).