Noncommutative maximal ergodic theorems

Junge, Marius; Xu, Quanhua

doi:10.1090/S0894-0347-06-00533-9

Noncommutative maximal ergodic theorems
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by Marius Junge and Quanhua Xu

J. Amer. Math. Soc. 20 (2007), 385-439

DOI: https://doi.org/10.1090/S0894-0347-06-00533-9

Published electronically: May 18, 2006

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Abstract:

This paper is devoted to the study of various maximal ergodic theorems in noncommutative $L_p$-spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on $L_p$ and the analogue of Stein’s maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

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