A Few Remarks in Non-Commutative Ergodic Theory (original) (raw)

On ergodic theorems for free group actions on noncommutative spaces

Probability Theory and Related Fields, 2006

We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s 2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s 2n (x)) for x in noncommutative spaces L p (A). For the Cesàro means 1 n n−1 k=0 s k and p = +∞, this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota "Alternierende Verfahren" theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators.

On noncommutative weighted local ergodic theorems on L p -spaces

Periodica Mathematica Hungarica, 2007

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ , and {αt} a strongly continuous extension to L p (M, τ) of a semigroup of absolute contractions on L 1 (M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and x ∈ L p (M, τ), the averages 1 T Z T 0 b(t)αt(x)dt converge bilateral almost uniform in L p (M, τ) as T → 0.

Noncommutative weighted individual ergodic theorems with continuous time

Infinite Dimensional Analysis, Quantum Probability and Related Topics

We show that ergodic flows in the noncommutative [Formula: see text]-space (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford–Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also proved. We then extend these results to arbitrary noncommutative fully symmetric spaces and present applications to noncommutative Orlicz (in particular, noncommutative [Formula: see text]-spaces), Lorentz, and Marcinkiewicz spaces. The commutative counterparts of the results are derived.

Individual ergodic theorems for semifinite von Neumann algebras

2016

It is known that, for a positive Dunford-Schwartz operator in a noncommutative L^p-space, 1≤ p<∞ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space E such that μ_t(x) → 0 as t → 0 for every x ∈ E, where μ_t(x) is a non-increasing rearrangement of x. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmetric spaces for which Dunford-Schwartz-type individual ergodic theorems hold is outlined. Also, almost uniform convergence in noncommutative Wiener-Wintner theorem is proved.

On multiparameter weighted ergodic theorem for noncommutative -spaces

Journal of Mathematical Analysis and Applications, 2008

In the paper we consider T 1 , . . . , T d absolute contractions of von Neumann algebra M with normal, semi-finite, faithful trace, and prove that for every bounded Besicovitch weight {a(k)} k∈N d and every x ∈ Lp(M), (p > 1) the averages

On noncommutative ergodic theorems for semigroup and free group actions

arXiv (Cornell University), 2023

In this article, we consider actions of Z d + , R d + and finitely generated free groups on a von Neumann algebras M and prove a version of maximal ergodic inequality. Additionally, we establish non-commutative analogues of pointwise ergodic theorems for associated actions in the predual when M is finite.

On multiparameter Weighted ergodic theorem for Noncommutative LpL_{p}Lp-spaces

2006

In the paper we consider T1,...,TdT_{1}, ..., T_{d}T1,...,Td absolute contractions of von Neumann algebra M\MM with normal, semi-finite, faithful trace, and prove that for every bounded Besicovitch family a(kb)kbinbnd\{a(\kb)\}_{\kb\in\bn^d}a(kb)kbinbnd and every xinLp(M)x\in L_{p}(\M)xinLp(M) the averages ANb(x)=frac1∣Nb∣sumlimitskb=1Nba(kb)Tbkb(x)A_{\Nb}(x) = \frac{1}{|\Nb|} \sum\limits_{\kb=1}^{\Nb}a(\kb)\Tb^{\kb}(x)ANb(x)=frac1∣Nb∣sumlimitskb=1Nba(kb)Tbkb(x) converge bilaterally almost uniformly in Lp(M)L_{p}(\M)Lp(M).

Uniform equicontinuity of sequences of measurable operators and non-commutative ergodic theorems

Proceedings of the American Mathematical Society, 2012

The notion of uniform equicontinuity in measure at zero for sequences of additive maps from a normed space into the space of measurable operators associated with a semifinite von Neumann algebra is discussed. It is shown that uniform equicontinuity in measure at zero on a dense subset implies the uniform equicontinuity in measure at zero on the entire space, which is then applied to derive some non-commutative ergodic theorems.

Individual ergodic theorems in noncommutative symmetric spaces

2016

It is known that, for a positive Dunford-Schwartz operator in a noncommutative L^p-space, 1≤ p<∞ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge bilaterally almost uniformly in each noncommutative symmetric space E such that μ_t(x) → 0 as t → 0 for every x ∈ E, where μ_t(x) is a non-increasing rearrangement of x. In particular, these averages converge bilaterally almost uniformly in all noncommutative symmetric spaces with order continuous norm.

A mean ergodic theorem in von-Neumann algebras

Cornell University - arXiv, 2012

We explore a duality between von-Neumann's mean ergodic theorem in von-Neumann algebra and Birkhoff's mean ergodic theorem in the pre-dual Banach space of von-Neumann algebras. Besides improving known mean ergodic theorems on von-Neumann algebras, we prove Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space.

On the non-commutative Neveu decomposition and stochastic ergodic theorems

arXiv (Cornell University), 2023

In this article, we prove Neveu decomposition for the action of locally compact amenable semigroup of positive contractions on semifinite von Neumann algebras and thus, it entirely resolves the problem for the actions of arbitrary amenable semigroup on semifinite von Neumann algebras. We also prove it for amenable group actions by Markov automorphisms on any σ-finite von Neumann algebras. As an application, we obtain stochastic ergodic theorem for actions of Z d + and R d + for d ∈ N by positive contractions on L 1-spaces associated with a finite von Neumann algebra. It yields the first ergodic theorem for positive contraction on non-commutative L 1-spaces beyond the Danford-Schwartz category.

Weak ergodicity of nonhomogeneous Markov chains on noncommutative L1L^1L1-spaces

Banach Journal of Mathematical Analysis, 2013

In this paper we study certain properties of Dobrushin's ergodicity coefficient for stochastic operators defined on noncommutative L 1 -spaces associated with semi-finite von Neumann algebras. Such results extends the well-known classical ones to a noncommutative setting. This allows us to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. We provide a sufficient conditions for such processes to satisfy the weak ergodicity. Moreover, a necessary and sufficient condition is given for the satisfaction of the L 1 -weak ergodicity of NDMP. It is also provided an example showing that L 1 -weak ergodicity is weaker that weak ergodicity. We applied the main results to several concrete examples of noncommutative NDMP.

Noncommutative maximal ergodic theorems

Journal of the American Mathematical Society

This paper is devoted to the study of various maximal ergodic theorems in noncommutative L p L_p -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on L p 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.

Local Ergodic Theorems in Symmetric Spaces of Measurable Operators

Integral Equations and Operator Theory

Local mean and individual (with respect to almost uniform convergence in Egorov's sense) ergodic theorems are established for actions of the semigroup R d + in symmetric spaces of measurable operators associated with a semifinite von Neumann algebra.

Ergodic theorems in fully symmetric spaces of tau−\tau-tau−measurable operators

arXiv (Cornell University), 2014

In [11], employing the technique of noncommutative interpolation, a maximal ergodic theorem in noncommutative Lp−spaces, 1 < p < ∞, was established and, among other things, corresponding maximal ergodic inequalities and individual ergodic theorems were derived. In this article, we derive maximal ergodic inequalities in noncommutative Lp−spaces directly from [25] and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these results to noncommutative fully symmetric Banach spaces with Fatou property and non-trivial Boyd indices, in particular, to noncommutative Lorentz spaces Lp,q. Norm convergence of ergodic averages in noncommutative fully symmetric Banach spaces is also studied.