arXiv subjects

Morgan O'Brien

Publications and source records attributed to Morgan O'Brien.

5 recordsLinked to original sources

arXiv2026

Noncommutative extensions of an ergodic theorem of Breiman

Let $(\mathcal{M},τ)$ be a tracial von Neumann algebra, $T:L_1(\mathcal{M},τ)\to L_1(\mathcal{M},τ)$ be induced by a normal $τ$-preserving $*$-automorphism of $\mathcal{M}$, and $x\in L_1(\mathcal{M},τ)$. We will prove the bilaterally almost uniform convergence of averages of the form $$\frac{1}{n}\sum_{k=0}^{n-1}T^k(x_k),$$ generalizing an ergodic theorem proven by Breiman in \cite{Breiman57}, where $(x_k)_{k=0}^{\infty}$ is an element of the noncommutative vector-valued $L_p$-space $L_p(\mathcal{M},τ;\ell_\infty)$ such that there exists $x\in L_p(\mathcal{M},τ)$ with $(x_k-x)_{k=0}^{\infty}\in L_p(\mathcal{M},τ;c_0)$. A generalization to semifinite $(\mathcal{M},τ)$ and general positive Dunford-Schwartz operators $T\in DS^+(\mathcal{M},τ)$ is shown as well when $1<p<\infty$. In addition to these, we also prove a number of generalizations of the results to weighted, subsequential, and other types of averages.

math.OA↗

arXiv2026

$T$-admissible processes and noncommutative weighted ergodic theorems

In this article, we study the bilaterally almost uniform (b.a.u.) convergence of weighted averages of a positive Dunford-Schwartz operator $T$ acting on the noncommutative $L_p$-spaces $L_p(\mathcal{M},τ)$ associated to a semifinite von Neumann algebra $(\mathcal{M},τ)$ for every $1\leq p<\infty$ for a variety of sequences which includes those generated by bounded i.i.d. sequences, the Möbius function, the Liouville function, and totally balanced automatic sequences. If $1<p<\infty$, then sequences generated by i.i.d. sequences with finite $q$-th moment for $q=\frac{p}{p-1}$ being the Hölder conjugate of $p$ are also shown to generate good weights for $L_p(\mathcal{M},τ)$. We also study similar problems for $T$-admissible processes on a semifinite von Neumann algebra, showing that if a Wiener-Wintner type ergodic theorem holds for a class $\mathcal{W}\subset W_q$ of weights, then it also holds for strongly $p$-bounded $T$-admissible processes assuming that the duality $\frac{1}{p}+\frac{1}{q}=1$ holds and that $T$ is a normal $τ$-preserving $*$-automorphism. A version of this result for arbitrary positive Dunford-Schwartz operators is also considered.

math.OA↗

arXiv2023

A noncommutative weak type maximal inequality for modulated ergodic averages with general weights

In this article, we prove a weak type $(p,p)$ maximal inequality, $1<p<\infty$, for weighted averages of a positive Dunford-Schwarz operator $T$ acting on a noncommutative $L_p$-space associated to a semifinite von Neumann algebra $\mathcal{M}$, with weights in $W_q$, where $\frac{1}{p}+\frac{1}{q}=1$. This result is then utilized to obtain modulated individual ergodic theorems with $q$-Besicovitch and $q$-Hartman sequences as weights. Multiparameter versions of these results are also investigated.

math.OA↗

arXiv2021

Noncommutative Wiener-Wintner type ergodic theorems

In this article, we obtain a version of the noncommutative Banach Principle suitable to prove Wiener-Wintner type results for weights in W1-space. This is used to obtain noncommutative Wiener-Wintner type ergodic theorems for various types of weights for certain types of positive Dunford-Schwartz operators. We also study the b.a.u. (a.u.) convergence of some subsequential averages and moving averages of such operators

math.OA↗

arXiv2021

Some noncommutative subsequential weighted individual ergodic theorems

This article is devoted to studying individual ergodic theorems for subsequential weighted ergodic averages on the noncommutative Lp-spaces associated to a semifinite von Neumann algebra M. In particular, we establish the convergence of these averages along sequences with density one and certain types of block sequences with positive lower density, and we extend known results along uniform sequences in the sense of Brunel and Keane.

math.OA↗