Abstract
In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let \({(\mathcal{M}, \tau)}\) be a noncommutative probability space equipped with a filtration of von Neumann subalgebras \({(\mathcal{M}_n)_{n \ge 1}}\), whose union \({\bigcup_{n\geq1}\mathcal{M}_n}\) is weak-* dense in \({\mathcal{M}}\). Let \({\mathcal{E}_n}\) denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for \({1 < p < 2}\) and \({x \in L_p(\mathcal{M},\tau)}\) one can find \({a, b \in L_p(\mathcal{M},\tau)}\) and contractions \({u_n, v_n \in \mathcal{M}}\) such that
Moreover, it turns out that \({a u_n}\) and \({v_n b}\) converge in the row/column Hardy spaces \({\mathcal{H}_p^r(\mathcal{M})}\) and \({\mathcal{H}_p^c(\mathcal{M})}\) respectively. In particular, this solves a problem posed by the Defant and Junge in 2004. In the case p = 1, our results establish a noncommutative form of the Davis celebrated theorem on the relation betwe en martingale maximal and square functions in L 1, whose noncommutative form has remained open for quite some time. Given \({1 \le p \le 2}\), we also provide new weak type maximal estimates, which imply in turn left/right almost uniform convergence of \({\mathcal{E}_n(x)}\) in row/column Hardy spaces. This improves the bilateral convergence known so far. Our approach is based on new forms of Davis martingale decomposition which are of independent interest, and an algebraic atomic description for the involved Hardy spaces. The latter results are new even for commutative von Neumann algebras.
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Communicated by Y. Kawahigashi
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Hong, G., Junge, M. & Parcet, J. Algebraic Davis Decomposition and Asymmetric Doob Inequalities. Commun. Math. Phys. 346, 995–1019 (2016). https://doi.org/10.1007/s00220-016-2581-3
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DOI: https://doi.org/10.1007/s00220-016-2581-3