Laboratoire de Mathématiques de Besançon - UMR 6623 CNRS

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24 mars 2020: 1 événement

  • Séminaire d’Analyse Fonctionnelle

    Mardi 24 mars 13:45-15:00 - Adrian Gonzalez Perez - Clermont-Ferrand

    Almost uniform convergence and strong maximals on von Neumann algebras

    Résumé : The notion of almost uniform convergence was introduce by Lance in the 70s as a generalization for von Neumann algebras of the classical notion of almost everywhere convergence over measure spaces. This notion was used by both Lance and Yeadon to obtain analogues of the individual ergodic theorems for von Neumann algebras.
    Like in the classical case, the standard tool for proving almost uniform convergence is to control a maximal function. Here, we shall introduce a new type of limit maximal function that recovers the $\limsup$ of a sequence in the commutative case. We will do this by introducing an ad hoc tensor product of $L^p$ and $\ell^\infty/c_0$. As a consequence, we obtain a generalization of a result of Jessen, Marcinkiewicz and Zygmund on strong maximals. We also obtain an improvement on the best know class over the free group algebra for which the Markovian semigroup generated by the group length converges almost uniformly.
    Lastly, we study the maximal inequality in two variables and prove an $\varepsilon$-perturbation of the conjecture optimal weak Orlicz type by improving techniques originally developed by Cuculescu.

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