Noncommutative maximal ergodic theorems (original) (raw)
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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.
arXiv (Cornell University), 2020
The Department of Mathematics, IISER Bhopal organised an international conference '16 th Discussion Meeting in Harmonic Analysis' during December 16-19, 2019. This conference is the most important activity in the area of harmonic analysis in India and is conducted once in every two years. The main goal of the conference is to provide the young researchers a platform to interact with the leading experts in the area from around the world. There were around 95 participants in this conference from around the world. Prof. Malabika Pramanik (University of British Columbia, Canada) delivered the plenary lecture series on the topic 'Directional Operators in Harmonic Analysis and Configuration of Sets'. The conference featured about 25 research talks by invited speakers and participants along with a poster session by young researchers. The event was funded by National Board for Higher Mathematics (NBHM), Science and Engineering Research Board (SERB), International Mathematical Union (CDC-IMU), and IISER Bhopal. The details of the talks are given in the attached annexure.
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 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).
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 few remarks in non-commutative ergodic theory
2005
Individual ergodic theorems for free group actions and Besicovitch weighted ergodic averages are proved in the context of the bilateral almost uniform convergence in the L 1-space over a semifinite von Neumann algebra. Some properties of the non-commutative counterparts of the pointwise convergence and the convergence in measure are discussed.
Maximal inequalities in noncommutative probability spaces
Stochastics, 2021
We employ some techniques involving projections in a von Neumann algebra to establish some maximal inequalities such as the strong and weak symmetrization, Lévy, Lévy-Skorohod, and Ottaviani inequalities in the realm of quantum probability spaces.
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.
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