Non-commutative martingale transforms

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 $(\epsilon_n)_{n=1}^\infty$, $$\left\Vert \epsilon_1 x_1 + \sum_{n=2}^N \epsilon_n(x_n -x_{n-1}) \right\Vert_{1,\infty} \leq C \left\Vert x_N \right\Vert_1$$ for $N\geq 2$. 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.Comment: 31 page

Factorization of operators on C∗C^*-algebras

Let $A$ be a $C^*$-algebra. It is shown that every absolutely summing operator from $A$ into $\ell_2$ factors through a Hilbert space operator that belongs to the 4-Schatten- von Neumann class. We also provide finite dimensinal examples that show that one can not improve the 4-Schatten-von Neumann class to $p$-Schatten von Neumann class for any $p<4$. As application, we prove that there exists a modulus of capacity $\epsilon \to N(\epsilon)$ so that if $A$ is a $C^*$-algebra and $T \in \Pi_1(A,\ell_2)$ with $\pi_1(T)\leq 1$, then for every $\epsilon >0$, the $\epsilon$-capacity of the image of the unit ball of $A$ under $T$ does not exceed $N(\epsilon)$. This aswers positively a question raised by Pe\l czynski