Doob’s Inequality for Non-commutative Martingales

Introduction: Inspired by quantum mechanics and probability, non-commutative probability has become an independent field of mathematical research. We refer to P.A. Meyer’s exposition [Me], the successive conferences on quantum probability [AvW], the lecture notes by Jajte [Ja1, Ja2] on almost sure and uniform convergence and finally the work of Voiculescu, Dykema, Nica [VDN] and of Biane, Speicher [BS] concerning the recent progress in free probability and free Brownian motion. Doob’s inequality is a classical tool in probability and analysis. Transferring classical inequalities into the non-commutative setting theory often requires an additional insight. Pisier, Xu [PX, Ps3] use functional analytic and combinatorial methods to establish the non-commutative versions of the Burkholder-Gundy square function inequality. The absence of stopping time arguments, at least until the time of this writing, imposes one of the main difficulties in this recent branch of martingale theory.