Noncommutative strong maximals and almost uniform convergence in several directions
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...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/711134 |
| Acceso en línea: | http://hdl.handle.net/10486/711134 https://dx.doi.org/10.1017/fms.2020.37 |
| Access Level: | acceso abierto |
| Palabra clave: | Ergodic mean Maximal operators Noncommutative Lp-martingales Von Neumann algebras Matemáticas |
| Sumario: | 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 Lp-norm of the lim sup of a sequence of operators as a localized version of a ℓ∞/c0-valued Lp-space. In particular, our main result gives a strong L1-estimate for the lim sup—as opposed to the usual weak L1,∞-estimate for the sup—with interesting consequences for the free group algebra. Let LF2 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 L1(LF2) for which the free Poisson semigroup converges to the initial data. Currently, the best known result is L log2 L(LF2). We improve this result by adding to it the operators in L1(LF2) 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 lim sup together with new transference techniques. We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak (Φ, Φ) inequality—as opposed to weak (Φ, 1)—for noncommutative multiparametric martingales and Φ(S) = S(1 + log+ S)2+ε. This logarithmic power is an ε-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 |
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