Non-commutative martingale transforms
| dc.creator | Randrianantoanina, Narcisse | |
| dc.date | 2001-11-26 | |
| dc.date.accessioned | 2026-07-07T04:44:46Z | |
| dc.date.available | 2026-07-07T04:44:46Z | |
| dc.description | 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 $(ε_n)_{n=1}^\infty$, $$\left\Vert ε_1 x_1 + \sum_{n=2}^N ε_n(x_n -x_{n-1}) \right\Vert_{1,\infty} \leq C \left\Vert x_N \right\Vert_1 $$ 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. | |
| dc.description | 31 pages | |
| dc.identifier | https://arxiv.org/abs/math/0111264 | |
| dc.identifier | http://arxiv.org/abs/math/0111264 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62728 | |
| dc.subject | Functional Analysis | |
| dc.subject | Operator Algebras | |
| dc.subject | 46L53, 46L52 | |
| dc.title | Non-commutative martingale transforms | |
| dc.type | text |