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- W2949626913 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 $(epsilon_n)_{n=1}^infty$, $$leftVert epsilon_1 x_1 + sum_{n=2}^N epsilon_n(x_n -x_{n-1}) rightVert_{1,infty} leq C leftVert x_N rightVert_1 $$ for $Ngeq 2$. 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." @default.
- W2949626913 created "2019-06-27" @default.
- W2949626913 creator A5068059294 @default.
- W2949626913 date "2001-11-26" @default.
- W2949626913 modified "2023-09-27" @default.
- W2949626913 title "Non-commutative martingale transforms" @default.
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