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Noncommutative strong maximals and almost uniform convergence in several directions

Part of: Probability theory on algebraic and topological structures Measure-theoretic ergodic theory Individual linear operators as elements of algebraic systems Harmonic analysis in several variables Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  20 November 2020

José M. Conde-Alonso Open the ORCID record for José M. Conde-Alonso [Opens in a new window] ,
Adrián M. González-Pérez  and
Javier Parcet
José M. Conde-Alonso
Affiliation:
UAM - Departamento de Matemáticas, 7 Francisco Tomás y Valiente, 28049Madrid, Spain; E-mail: jose.conde@uam.es
Adrián M. González-Pérez
Affiliation:
IMPAN (Instytut Matematyczny Polskiej Akademii Nauk), ul. Sniadekich 8, 00-656 Warsaw; E-mail: agonzalez-perez@impan.pl
Javier Parcet
Affiliation:
Consejo Superior de Investigaciones Científicas - ICMAT, 23 Nicolás Cabrera, 28049Madrid, Spain; E-mail: parcet@icmat.es

Abstract

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Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$ -norm of the $\limsup $ of a sequence of operators as a localized version of a $\ell _\infty /c_0$ -valued $L_p$ -space. In particular, our main result gives a strong $L_1$ -estimate for the $\limsup $ —as opposed to the usual weak $L_{1,\infty }$ -estimate for the $\mathop {\mathrm {sup}}\limits $ —with interesting consequences for the free group algebra.

Let $\mathcal{L} \mathbf{F} _2$ denote the free group algebra with $2$ generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside $L_1(\mathcal{L} \mathbf{F} _2)$ for which the free Poisson semigroup converges to the initial data. Currently, the best known result is $L \log ^2 L(\mathcal{L} \mathbf{F} _2)$ . We improve this result by adding to it the operators in $L_1(\mathcal{L} \mathbf{F} _2)$ spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative $\limsup $ together with new transference techniques.

We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak $(\Phi ,\Phi )$ inequality—as opposed to weak $(\Phi ,1)$ —for noncommutative multiparametric martingales and $\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$ . This logarithmic power is an $\varepsilon $ -perturbation of the expected optimal one. The proof combines a refinement of Cuculescu’s construction with a quantum probabilistic interpretation of M. de Guzmán’s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu’s projections.

Keywords

von Neumann algebrasnoncommutative Lp-martingalesergodic meanmaximal operators

MSC classification

Primary: 42B99: None of the above, but in this section 47C15: Operators in $C^*$- or von Neumann algebras
Secondary: 28D99: None of the above, but in this section 47D07: Markov semigroups and applications to diffusion processes 60B99: None of the above, but in this section
Type
Analysis
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e57
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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