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 (M n ) ∞ n=1 is an increasing filtration of von Neumann subalgebras of M then for any non-commutative martingale x = (x n ) ∞ n=1 in L 1 (M), adapted to (M n ) ∞ n=1 , and any sequence of signs (ε n ) ∞ n=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 UMD-constants of the Schatten class S p when p → ∞. Similarly, we prove that the UMD-constant of the finite dimensional Schatten class S 1 n is of order log(n+ 1). We also discuss the Pisier-Xu non-commutative Burkholder-Gundy inequalities.1991 Mathematics Subject Classification. [. Xu[25] considered the non-tracial case of the main result of [37] along with several related inequalities such as non-commutative analogue of the classical Burkholder inequalities on the conditioned square functions among others. Junge proved in [24] non-commutative versions of Doob's maximal inequalities. We remark that most inequalities considered in the aforementioned papers were for p > 1. We continue this line of research by studying martingale transforms of non-commutative bounded L 1 -martingales. In the classical probability, the theory of martingale transforms is well-established and has been proven to be a very powerful tool not only in probabilistic situations but also in several parts of analysis. We refer to the survey [7] for discussions on this classical topic. For instance, Burkholder [6] proved that classical martingale transforms are of weak type (1,1). Our main result (see Theorem 3.1 below) is a non-commutative analogue of this classical fact: non-commutative martingale transforms are bounded as maps from non-commutative L 1 -spaces into the corresponding non-commutative weak-L 1 -spaces. We should point out that this question was explicitly raised by Pisier and Xu in the recent survey [38] (Problem 7.5) as it is closely related to the main result of [37]. Indeed, combined with general theory of interpolations of operators of weak types, our main result implies that for p > 1, martingale difference sequences in non-commutative L p -spaces are unconditional which in turn imply the non-commutative Burkholder-Gundy inequalities. This alternative approach yields constants which are O(p) when p → ∞. This is explained in Sect. 5. Another application of the main result is on UMDconstants of non-commutative L p -spaces. It is now a well known fact that non-commutative L p -spaces on semi-finite von Neumann algebras are UMD-spaces. The UMD-constants of these spaces recorded in the literature thus far seems to be of order O(p 2 ) when p → ∞. Using the estimates on the constant of unconditionality of non-commutative martingale difference sequences, we can deduce that the UMD-constants for non-commutative L p -spaces are of order O...

We provide an analogue of Gundy's decomposition for L 1 -bounded non-commutative martingales. An important difference from the classical case is that for any L 1 -bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type (1, 1) boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder's weak type inequality for square functions. A sequence (xn) n≥1 in a normed space X is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of (xn) n≥1 to l 2 taking each xn to the n-th vector basis of l 2 . We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in L 1 (M, τ ) whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative L 1 -spaces.

We prove a weak-type (1, 1) inequality involving conditioned versions of square functions for martingales in noncommutative $L^p$-spaces associated with finite von Neumann algebras. As application, we determine the optimal orders for the best constants in the noncommutative Burkholder/Rosenthal inequalities from [Ann. Probab. 31 (2003) 948--995]. We also discuss BMO-norms of sums of noncommuting order-independent operators.Comment: Published at http://dx.doi.org/10.1214/009117906000000656 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in L 2 and L 1 . More precisely, the following non-commutative analogue of a classical result of Burkholder holds: there exists an absolute constant K > 0 such that if M is a seminite von Neumann algebra and ðM n Þ 1 n¼1 is an increasing ltration of von Neumann subalgebras of M then for any given martingale x ¼ ðx n Þ 1 n¼1 that is bounded in L 2 ðMÞ \ L 1 ðMÞ, adapted to ðM n Þ 1

Abstract. We provide generalizations of Burkholder's inequalities involving conditioned square functions of martingales to the general context of martingales in noncommutative symmetric spaces. More precisely, we prove that Burkholder's inequalities are valid for any martingale in noncommutative space constructed from a symmetric space defined on the interval (0, ∞) with Fatou property and whose Boyd indices are strictly between 1 and 2. This answers positively a question raised by Jiao and may be viewed as a conditioned version of similar inequalities for square functions of noncommutative martingales. Using duality, we also recover the previously known case where the Boyd indices are finite and are strictly larger than 2.