Notes on noncommutative ergodic theorems
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- by Semyon Litvinov;
- Proc. Amer. Math. Soc. 152 (2024), 3381-3391
- DOI: https://doi.org/10.1090/proc/16807
- Published electronically: June 5, 2024
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Abstract:
Given a semifinite von Neumann algebra $\mathcal M$ equipped with a faithful normal semifinite trace $\tau$, we prove that the spaces $L^0(\mathcal M,\tau )$ and $\mathcal R_\tau$ are complete with respect to pointwise—almost uniform and bilaterally almost uniform—convergences in $L^0(\mathcal M,\tau )$. Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space $L^1(\mathcal M,\tau )$ can be extended to pointwise convergence of such nets in any fully symmetric space $E\subset \mathcal R_\tau$, in particular, in any space $L^p(\mathcal M,\tau )$, $1\leq p<\infty$. Some applications of these results in the noncommutative ergodic theory are discussed.References
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Bibliographic Information
- Semyon Litvinov
- Affiliation: 76 University Drive, The Pennsylvania State University, Hazleton, Pennsylvania 18202
- MR Author ID: 229867
- Email: snl2@psu.edu
- Received by editor(s): July 29, 2023
- Received by editor(s) in revised form: July 30, 2023, and January 1, 2024
- Published electronically: June 5, 2024
- Additional Notes: The author was partially supported by the Pennsylvania State University traveling program.
- Communicated by: Matthew Kennedy
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3381-3391
- MSC (2020): Primary 47A35; Secondary 46L51
- DOI: https://doi.org/10.1090/proc/16807
- MathSciNet review: 4767269