Martingales on von Neumann algebras

Weak type estimates associated to Burkholder’s martingale inequality

“…In the case of maximal functions, there exist two natural substitutes. Roughly speaking, we use a construction due to Cuculescu [3] when dealing with weak type inequalities, while for strong inequalities the right notion was formulated by Junge in [10]. We shall use here Cuculescu's construction.…”

“…. Namely, the sequence of q k (λ)'s is adapted for all λ > 0 and satisfies an analogue of the weak type (1, 1) Doob's maximal inequality, see [3,17,20] for more details. There is however one natural property which is not satisfied by Cuculescu's projections.…”

Classical and quantum probability

“…Cuculescu [124] proved a martingale convergence theorem for discrete time. Barnett [123] obtained a martingale theorem for continuous time.…”

Non-Commutative Martingale Transforms

“…Noncommutative martingales have been studied by several authors. For instance, pointwise convergence of non-commutative martingales was considered in [11,12]. In [37], Pisier and Xu proved a non-commutative analogue of the Burkholder-Gundy square function inequalities.…”

“…Our proof is completely self-contained. It is based on a maximal inequality-type result from a paper of Cuculescu [11] (see Proposition 2.3) which allows ones to reduce the case of bounded L 1 -martingales to bounded L 2 -supermartingales. Although, such reduction to supermartingales is standard in classical martingale theory (see for instance [18,Chap.…”