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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2020-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509420000377/type/journal_article |
| _version_ | 1811156173554974720 |
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| author | José M. Conde-Alonso Adrián M. González-Pérez Javier Parcet |
| author_facet | José M. Conde-Alonso Adrián M. González-Pérez Javier Parcet |
| author_sort | José M. Conde-Alonso |
| collection | DOAJ |
| description | 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.
|
| first_indexed | 2024-04-10T04:47:07Z |
| format | Article |
| id | doaj.art-58ac1139974349629928c54ad3585356 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:07Z |
| publishDate | 2020-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-58ac1139974349629928c54ad35853562023-03-09T12:34:47ZengCambridge University PressForum of Mathematics, Sigma2050-50942020-01-01810.1017/fms.2020.37Noncommutative strong maximals and almost uniform convergence in several directionsJosé M. Conde-Alonso0https://orcid.org/0000-0002-6614-9807Adrián M. González-Pérez1Javier Parcet2UAM - Departamento de Matemáticas, 7 Francisco Tomás y Valiente, 28049 Madrid, Spain; E-mail:IMPAN (Instytut Matematyczny Polskiej Akademii Nauk), ul. Sniadekich 8, 00-656 Warsaw; E-mail:Consejo Superior de Investigaciones Científicas - ICMAT, 23 Nicolás Cabrera, 28049 Madrid, Spain; E-mail: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. https://www.cambridge.org/core/product/identifier/S2050509420000377/type/journal_articlevon Neumann algebrasnoncommutative Lp-martingalesergodic meanmaximal operators42B9947C1528D9947D0760B99 |
| spellingShingle | José M. Conde-Alonso Adrián M. González-Pérez Javier Parcet Noncommutative strong maximals and almost uniform convergence in several directions Forum of Mathematics, Sigma von Neumann algebras noncommutative Lp-martingales ergodic mean maximal operators 42B99 47C15 28D99 47D07 60B99 |
| title | Noncommutative strong maximals and almost uniform convergence in several directions |
| title_full | Noncommutative strong maximals and almost uniform convergence in several directions |
| title_fullStr | Noncommutative strong maximals and almost uniform convergence in several directions |
| title_full_unstemmed | Noncommutative strong maximals and almost uniform convergence in several directions |
| title_short | Noncommutative strong maximals and almost uniform convergence in several directions |
| title_sort | noncommutative strong maximals and almost uniform convergence in several directions |
| topic | von Neumann algebras noncommutative Lp-martingales ergodic mean maximal operators 42B99 47C15 28D99 47D07 60B99 |
| url | https://www.cambridge.org/core/product/identifier/S2050509420000377/type/journal_article |
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