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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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