Noncommutative good-$\lambda$ inequalities

We propose a novel approach in noncommutative probability, which can be regarded as an analogue of good- \lambda inequalities from the classical case due to Burkholder and Gundy (Acta Math. 124, 249–304, 1970). This resolves a longstanding open problem in noncommutative realm. Using this technique, we offer a new, simpler and unified approach to fundamental results in noncommutative martingale theory, obtained earlier by Junge, Pisier, Randrianantoanina and Xu. We also present some new applications of the good- \lambda approach to noncommutative probability and noncommutative harmonic analysis. These include inequalities for noncommutative martingales with tangent difference sequences; estimates for sums of tangent positive operators; bounds for differentially subordinate operators, originating in the directional Hilbert transforms on free group von Neumann algebras; and estimates for the j -th Riesz transform on group von Neumann algebras. Finally, we provide a simple proof of a bound for a certain mean-oscillation space, which addresses a problem raised by Junge and Xu. We emphasize that all the constants obtained in this paper are of optimal orders.