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Non-Commutative Martingale Transforms

✍ Scribed by Narcisse Randrianantoanina

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
275 KB
Volume
194
Category
Article
ISSN
0022-1236

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✦ Synopsis

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 ðM n Þ 1 n¼1 is an increasing filtration of von Neumann subalgebras of M; then for any non-commutative martingale

for every N52: 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 S p when p ! 1: Similarly, we prove that the UMD-constant of the finite-dimensional Schatten class S 1 n is of order logðn þ 1Þ: We also discuss the Pisier-Xu non-commutative Burkholder-Gundy inequalities.

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