Non-commutative martingale transforms

Abstract

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 $(\M_n)_{n=1}^\infty$ is an increasing filtration of von Neumann subalgebras of $\M$ then for any non-commutative martingale $x=(x_n{)}_{n=1}^{\infty }$ in $L^1(\M)$, adapted to $(\M_n)_{n=1}^\infty$, and any sequence of signs $({\epsilon }_n{)}_{n=1}^{\infty }$, $${\Vert {\epsilon }_1{x}_1+\sum _{n=2}^N{\epsilon }_n(x_n-x_{n-1})\Vert }_{1,\infty }\le C{\Vert x_N\Vert }_1$$ for $N\ge 2$. 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 $p\to \infty$. Similarly, we prove that the UMD-constant of the finite dimensional Schatten class $S_n^1$ is of order $\log (n+1)$. We also discuss the Pisier-Xu non-commutative Burkholder-Gundy inequalities.

31 pages