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1 February 2021 Noncommutative maximal ergodic inequalities associated with doubling conditions
Guixiang Hong, Ben Liao, Simeng Wang
Duke Math. J. 170(2): 205-246 (1 February 2021). DOI: 10.1215/00127094-2020-0034

Abstract

We study noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group G of polynomial growth with a symmetric compact subset V. Let α be a continuous action of G on a von Neumann algebra M by trace-preserving automorphisms. We then show that the operators defined by Anx=1m(Vn)Vnαgxdm(g),xLp(M),nN,1p, are of weak type (1,1) and of strong type (p,p) for 1<p<. Consequently, the sequence (Anx)n1 converges almost uniformly for xLp(M) for 1p<. Also, we establish the noncommutative maximal and individual ergodic theorems associated with more general doubling conditions, and we prove the corresponding results for general actions on one fixed noncommutative Lp-space which are beyond the class of Dunford–Schwartz operators considered previously by Junge and Xu. As key ingredients, we also obtain the Hardy–Littlewood maximal inequality on metric spaces with doubling measures in the operator-valued setting. After the groundbreaking work of Junge and Xu on the noncommutative Dunford–Schwartz maximal ergodic inequalities, this is the first time that more general maximal inequalities are proved beyond Junge and Xu’s setting. Our approach is based on quantum probabilistic methods as well as random-walk theory.

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Guixiang Hong. Ben Liao. Simeng Wang. "Noncommutative maximal ergodic inequalities associated with doubling conditions." Duke Math. J. 170 (2) 205 - 246, 1 February 2021. https://doi.org/10.1215/00127094-2020-0034

Information

Received: 2 February 2018; Revised: 22 December 2019; Published: 1 February 2021
First available in Project Euclid: 30 October 2020

Digital Object Identifier: 10.1215/00127094-2020-0034

Subjects:
Primary: 46L53
Secondary: 37A55, 46L51, 46L55

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 2 • 1 February 2021
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