A remark on maximal functions for noncommutative martingales

Abstract

Let ${\mathcal{M}}$ be a finite von Neumann algebra equipped with a normal tracial state τ. It is shown that if ${\{x_n\}_{n\geq1}}$ is a sequence of positive marginales that is bounded in ${L^1(\mathcal{M},\mathcal{T})}$, then for every 0 < p < 1, there exists ${y \in L^p(\mathcal{M},\mathcal{T})}$ satisfying the property that ${x_n \leq y}$ for all ${n\geq 1}$. Thus we obtain a noncommutative analogue of a maximal function theorem from classical martingale theory.