Strong Limit Theorems in Noncommutative L2-Spaces

Jajte, Ryszard

doi:10.1007/bfb0098424

Cited by 24 publications

(35 citation statements)

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“…iii) Jajte [Ja2] states a multiple individual ergodic theorem in L 2 (M) (Theorem 2.3.4 there), which corresponds to the first part of Corollary 7.12 in the case of p = 2. His proof uses in an essential way his previous Theorem 2.2.4.…”

Section: Maximal Ergodic Inequalitiesmentioning

confidence: 88%

“…To this end we first need an appropriate analogue for the noncommutative setting of the usual almost everywhere convergence. This is the almost uniform convergence introduced by Lance [L] (see also [Ja1]). …”

Section: Individual Ergodic Theoremsmentioning

confidence: 97%

“…There are several such generalizations. Here we adopt the almost sure convergence introduced by Jajte [Ja2] (following ideas from [DJ2]). In the L ∞ -case, we continue to use Lance's almost uniform convergence.…”

Section: Maximal Ergodic Inequalitiesmentioning

confidence: 99%

“…At the early stage of noncommutative ergodic theory, only mean ergodic theorems have been obtained (cf., e.g., [Ja1,Ja2] for more information). The study of individual ergodic theorems really took off with Lance's pioneering work [L].…”

Section: Introductionmentioning

confidence: 99%

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Noncommutative maximal ergodic theorems

Junge¹,

Xu²

2006

J. Amer. Math. Soc.

175

245

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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.

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Section: Maximal Ergodic Inequalitiesmentioning

confidence: 88%

Section: Individual Ergodic Theoremsmentioning

confidence: 97%

Section: Maximal Ergodic Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Noncommutative maximal ergodic theorems

Junge¹,

Xu²

2006

J. Amer. Math. Soc.

175

245

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“…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].…”

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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Beppo levi and Lebesgue type theorems for bundle convergence in noncommutative L 2-spaces

Gac¹,

Móricz

2001

Recent Advances in Operator Theory and Related Topics

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Strong Limit Theorems in Noncommutative L2-Spaces

Cited by 24 publications

References 0 publications

Noncommutative maximal ergodic theorems

Noncommutative maximal ergodic theorems

Almost uniform convergence in the noncommutative Dunford–Schwartz ergodic theorem

Beppo levi and Lebesgue type theorems for bundle convergence in noncommutative L 2-spaces

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