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Archiv der Mathematik

, Volume 101, Issue 6, pp 541–548 | Cite as

A remark on maximal functions for noncommutative martingales

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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.

Mathematics Subject Classification (2010)

Primary 46L52 46L53 Secondary 46L51 60G42 

Keywords

Noncommutative Lp-spaces Martingale theory 

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsMiami UniversityOxfordUSA

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