PreprintCC BY 4.0arXiv
Noncommutative extensions of an ergodic theorem of Breiman
Abstract
Let (ℳ,τ) be a tracial von Neumann algebra, T:L1(ℳ,τ)→ L1(ℳ,τ) be induced by a normal τ-preserving *-automorphism of ℳ, and x∈ L1(ℳ,τ). We will prove the bilaterally almost uniform convergence of averages of the form 1/n∑k=0n−1Tk(xk), generalizing an ergodic theorem proven by Breiman in \cite{Breiman57}, where (xk)k=0∞ is an element of the noncommutative vector-valued Lp-space Lp(ℳ,τ;ℓ∞) such that there exists x∈ Lp(ℳ,τ) with (xk−x)k=0∞∈ Lp(ℳ,τ;c0). A generalization to semifinite (ℳ,τ) and general positive Dunford-Schwartz operators T∈ DS+(ℳ,τ) is shown as well when 1<p<∞. In addition to these, we also prove a number of generalizations of the results to weighted, subsequential, and other types of averages.
Related
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BibTeX
@misc{obrien2026noncommutative,
title = {{Noncommutative extensions of an ergodic theorem of Breiman}},
author = {Morgan O'Brien},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02535},
eprint = {2611.02535},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02535},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Noncommutative extensions of an ergodic theorem of Breiman
AU - Morgan O'Brien
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02535
UR - https://arxiv.org/abs/2611.02535
AB - Let (ℳ,τ) be a tracial von Neumann algebra, T:L_1(ℳ,τ)→ L_1(ℳ,τ) be induced by a normal τ-preserving *-automorphism of ℳ, and x∈ L_1(ℳ,τ). We will prove the bilaterally almost uniform convergence of averages of the form 1/n∑_k=0^n−1T^k(x_k), generalizing an ergodic theorem proven by Breiman in \cite{Breiman57}, where (x_k)_k=0^∞ is an element of the noncommutative vector-valued L_p-space L_p(ℳ,τ;ℓ_∞) such that there exists x∈ L_p(ℳ,τ) with (x_k−x)_k=0^∞∈ L_p(ℳ,τ;c_0). A generalization to semifinite (ℳ,τ) and general positive Dunford-Schwartz operators T∈ DS^+(ℳ,τ) is shown as well when 1<p<∞. In addition to these, we also prove a number of generalizations of the results to weighted, subsequential, and other types of averages.
ER -