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

Litvinov, Semyon

doi:10.1090/s0002-9939-2011-11483-7

Cited by 22 publications

(28 citation statements)

References 6 publications

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“…. At this moment we apply inequality (6) to the sequence {a k } ⊂ M yielding in view of (1) and (2) that sup λ∈C1 a n (x, λ)p Thus, we conclude that (7) lim n→∞ sup λ∈C1 a n (x, λ)p ∞ = 0, whence x ∈ W W .…”

Section: Proof Of the Main Resultsmentioning

confidence: 96%

“…Therefore α(a n (x)) → α(x) in L 2 , so α(x) =x, and the ergodicity of α implies thatx = c(x) · I. Then, since τ is also continuous in L 2 , we have τ (a n (x)) → τ (x) = c(x), hence c(x) = τ (x) because τ (a n (x)) = τ (x) for each n. It is known ( [4], [6]) that a n (x) →x ∈ L 2 a.u., which implies that a n (x) →x in measure. Since · 2 −convergence entails convergence in measure, we conclude thatx =x = τ (x) · I.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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A non-commutative Wiener–Wintner theorem

Litvinov¹

2014

Illinois J. Math.

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Section: Proof Of the Main Resultsmentioning

confidence: 96%

Section: Proof Of the Main Resultsmentioning

confidence: 99%

A non-commutative Wiener–Wintner theorem

Litvinov¹

2014

Illinois J. Math.

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“…(respectively, a.u.). Later, in [10] (see also [2]), it was shown that this result can be obtained directly from Yeadon's maximal inequality for L 1 (M, τ ) established in [21]. In particular, it was shown that a.u.…”

Section: Introductionmentioning

confidence: 89%

“…The main finding of the article is Lemma 3.2 where the matrix {e k,n } of projections in M is constructed. Also, the notion of (bilaterally) uniform equicontinuity in measure at zero of a family of maps from a normed space into the space of τ −measurable operators (see [10]) plays an important role.…”

Section: Introductionmentioning

confidence: 99%

Almost uniform convergence in the noncommutative Dunford–Schwartz ergodic theorem

Litvinov

2017

Comptes Rendus. Mathématique

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Abstract. This article gives an affirmative solution to the problem whether the ergodic Cesáro averages generated by a positive Dunford-Schwartz operator in a noncommutative space L p (M, τ ), 1 ≤ p < ∞, converge almost uniformly (in Egorov's sense). This problem goes back to the original paper of Yeadon [21], published in 1977, where bilaterally almost uniform convergence of these averages was established for p = 1. IntroductionAlmost uniform (a.u.) convergence in Egorov's sense in a von Neumann algebra M was considered by Lance [9], where a breakthrough noncommutative individual ergodic theorem was established for a positive state preserving automorphism of M. Later Lance's result was generalized, while the proofs were simplified; see [8,4,6].For a semifinite von Neumann algebra M with a faithful normal semifinte trace τ , Yeadon [21] introduced the so-called bilaterally almost uniform (b.a.u.) convergence in Egorov's sense to prove a noncommutative individual ergodic theorem for a positive Dunford-Schwartz operator in the space L 1 (M, τ ) of τ −integrable operators affiliated with M.Since b.a.u. convergence is generally weaker than a.u. convergence, serious attempts have been made to show that there is a.u. convergence in Yeadon's seminal result. But the problem persisted, and a significant number of noncommutative individual ergodic theorems concerning b.a.u. convergence of ergodic averages have been established; see, for example, [15,5,3,11,7,18,12,2].It was derived in [7] that if 1 < p < 2 (2 ≤ p < ∞), then for a positive DunfordSchwartz operator in a noncommutative space L p (M, τ ), the corresponding ergodic Cesáro averages converge b.a.u. (respectively, a.u.). Later, in [10] (see also [2]), it was shown that this result can be obtained directly from Yeadon's maximal inequality for L 1 (M, τ ) established in [21]. In particular, it was shown that a.u. convergence for p ≥ 2 follows easily due to Kadison's inequality. But the case 1 ≤ p < 2 still remained open.The aim of this article is to prove that there is a.u. convergence for all 1 ≤ p < ∞, which is given in Theorem 2.3. Note that this result was not known even for a finite trace. Lemma 3.1 is crucial. The main finding of the article is Lemma 3.2 where the matrix {e k,n } of projections in M is constructed. Also, the notion of (bilaterally) Date: June 28, 2017. 2010 Mathematics Subject Classification. 47A35(primary), 46L52(secondary). Key words and phrases. Semifinite von Neumann algebra; Dunford-Schwartz operator; individual ergodic theorem; bilaterally uniform equicontinuity in measure at zero; almost uniform convergence.

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“…The asymmetric result can be seen as a generalization of [31] (see also [19,18]). We will need the following proposition, whose proof is immediate (and thus we omit it).…”

Section: The Jessen-marcinkiewicz-zygmund Inequalitymentioning

confidence: 98%

Noncommutative strong maximals and almost uniform convergence in several directions

Conde-Alonso¹,

González-Pérez

Parcet

2020

Forum of Mathematics, Sigma

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Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$ -norm of the $\limsup $ of a sequence of operators as a localized version of a $\ell _\infty /c_0$ -valued $L_p$ -space. In particular, our main result gives a strong $L_1$ -estimate for the $\limsup $ —as opposed to the usual weak $L_{1,\infty }$ -estimate for the $\mathop {\mathrm {sup}}\limits $ —with interesting consequences for the free group algebra. Let $\mathcal{L} \mathbf{F} _2$ denote the free group algebra with $2$ generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside $L_1(\mathcal{L} \mathbf{F} _2)$ for which the free Poisson semigroup converges to the initial data. Currently, the best known result is $L \log ^2 L(\mathcal{L} \mathbf{F} _2)$ . We improve this result by adding to it the operators in $L_1(\mathcal{L} \mathbf{F} _2)$ spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative $\limsup $ together with new transference techniques. We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak $(\Phi ,\Phi )$ inequality—as opposed to weak $(\Phi ,1)$ —for noncommutative multiparametric martingales and $\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$ . This logarithmic power is an $\varepsilon $ -perturbation of the expected optimal one. The proof combines a refinement of Cuculescu’s construction with a quantum probabilistic interpretation of M. de Guzmán’s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu’s projections.

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

Cited by 22 publications

References 6 publications

A non-commutative Wiener–Wintner theorem

A non-commutative Wiener–Wintner theorem

Almost uniform convergence in the noncommutative Dunford–Schwartz ergodic theorem

Noncommutative strong maximals and almost uniform convergence in several directions

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