Local Ergodic Theorems in Symmetric Spaces of Measurable Operators

Chilin, Vladimir; Litvinov, Semyon

doi:10.1007/s00020-019-2519-1

Cited by 5 publications

(6 citation statements)

References 25 publications

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“…The following lemma was known before for the special case z ∈ (L 1 + L ∞ )(M) (see [16,Lemma 4.4] for a similar result). where the last equality follows immediately from the definition of singular value functions (see also [25, Chapter III, Proposition 2.10]).…”

Section: Order-preserving Isometries Into ∆-Normed Spacesmentioning

confidence: 79%

“…For any x ∈ M 1 ∩ E(M 1 , τ 1 ), let J(x) * = u|J(x) * | be polar decomposition. By (16), we have that…”

Section: Hencementioning

confidence: 98%

“…where p is a central projection in M 1 defined as in (16). Since p|x * |+(1 M1 −p)|x| ∈ M 1 ∩ E(M 1 , τ 1 ), it follows from Proposition 3.9 that BJ(p|x * | + (1 M1 − p)|x|) ∈ S(M 2 , τ 2 ).…”

Section: Hencementioning

confidence: 99%

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Logarithmic submajorisation and order-preserving linear isometries

Huang

Sukochev

Zanin

2020

Journal of Functional Analysis

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Let E and F be noncommutative operator spaces affiliated with semifinite von Neumann algebras M 1 and M 2 , respectively. We establish a noncommutative version of Abramovich's theorem [1], which provides the general form of normal order-preserving linear operators T : E into −→ F having the disjointness preserving property. As an application, we obtain a noncommutative Huijsmans-Wickstead theorem [43].By establishing the disjointness preserving property for an order-preserving isometry T : E → F from a noncommutative symmetrically ∆-normed (in particular, quasi-normed) space into another, we obtain the existence of a Jordan *monomorphism from M 1 into M 2 and the general form of this isometry, which extends and complements a number of existing results such as [12, Theorem 1], [64, Corollary 1], [69, Theorem 2] and [15, Theorem 3.1]. In particular, we fully resolve the case when F is the predual of M 2 and other untreated cases in [75].2010 Mathematics Subject Classification. 46B04, 46L52, 46A16.

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Section: Order-preserving Isometries Into ∆-Normed Spacesmentioning

confidence: 79%

“…For any x ∈ M 1 ∩ E(M 1 , τ 1 ), let J(x) * = u|J(x) * | be polar decomposition. By (16), we have that…”

Section: Hencementioning

confidence: 98%

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Logarithmic submajorisation and order-preserving linear isometries

Huang

Sukochev

Zanin

2020

Journal of Functional Analysis

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“…It is clear that R µ admits a more direct description (2). Note that if µ(Ω) < ∞, then R µ is simply L 1 (Ω).…”

Section: Preliminariesmentioning

confidence: 99%

“…convergence in Dunford-Schwartz individual ergodic theorem holds. Let (2) R µ = {f ∈ L 1 + L ∞ : µ{|f | > λ} < ∞ for all λ > 0}.…”

Section: Introductionmentioning

confidence: 99%

Almost uniform and strong convergences in ergodic theorems for symmetric spaces

Chilin

Litvinov

2018

Acta Math. Hungar.

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Let (Ω, µ) be a σ-finite measure space, and let X ⊂ L 1 (Ω)+L ∞ (Ω) be a fully symmetric space of measurable functions on (Ω, µ). If µ(Ω) = ∞, necessary and sufficient conditions are given for almost uniform convergence in X (in Egorov's sense) of Cesàro averages Mn(T )(f ) = 1 n n−1 k=0 T k (f ) for all Dunford-Schwartz operators T in L 1 (Ω) + L ∞ (Ω) and any f ∈ X. Besides, it is proved that the averages Mn(T ) converge strongly in X for each Dunford-Schwartz operator T in L 1 (Ω) + L ∞ (Ω) if and only if X has order continuous norm and L 1 (Ω) is not contained in X.

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