Completeness of symmetric $\varDelta $-normed spaces of $\tau $-measurable operators

Huang, Jinghao; Levitina, Galina; Sukochev, Fedor

doi:10.4064/sm8349-10-2016

Cited by 14 publications

(16 citation statements)

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“…There exists a strong connection between symmetric function spaces and operator spaces exposed in [48] (see also [10,40,73]). The operator space E(M, τ ) defined by…”

Section: 2mentioning

confidence: 99%

“…is a complete symmetrically ∆-normed space whenever (E(0, ∞), · E ) is a complete symmetrically ∆-normed function space on (0, ∞) [40] (see also [48,73]). Proof.…”

Section: 2mentioning

confidence: 99%

“…[50]). We define In particular, Λ p w (M, τ ) is a quasi-Banach space equipped with quasi-norm X Λ p w = µ(X) Λ p w , X ∈ Λ p w (M, τ ) [40,73]. Assume that w is a strictly positive decreasing function on (0, ∞) such that W (∞) = ∞.…”

Section: Order-preserving Isometries Onto ∆-Normed Spacesmentioning

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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“…There exists a strong connection between symmetric function spaces and operator spaces exposed in [48] (see also [10,40,73]). The operator space E(M, τ ) defined by…”

Section: 2mentioning

confidence: 99%

“…is a complete symmetrically ∆-normed space whenever (E(0, ∞), · E ) is a complete symmetrically ∆-normed function space on (0, ∞) [40] (see also [48,73]). Proof.…”

Section: 2mentioning

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 be a linear subspace in Sp0, 1q equipped with a ∆-norm }¨} E . We say that E is a symmetrically ∆-normed space if for x P E, y P Sp0, 1q and µpyq ď µpxq imply that y P E and }y} E ď }x} E [3,20,21]. Here, µpf q stands for the decreasing rearrangement of f P Sp0, 1q [16,32].…”

Section: 2mentioning

confidence: 99%

Non-existence of translation-invariant derivations on algebras of measurable functions

Ber¹,

Huang²,

Kudaybergenov³

et al. 2020

Preprint

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Let Sp0, 1q be the ˚-algebra of all classes of Lebesgue measurable functions on the unit interval p0, 1q and let pA, }¨} A q be a complete symmetric ∆-normed ˚-subalgebra of Sp0, 1q, in which simple functions are dense, e.g., L8p0, 1q, L log p0, 1q, Sp0, 1q and the Arens algebra L ω p0, 1q equipped with their natural ∆-norms. We show that there exists no non-trivial derivation δ : A Ñ Sp0, 1q commuting with all dyadic translations of the unit interval. Let M be a type II (or I8) von Neumann algebra, A be its abelian von Neumann subalgebra, let SpMq be the algebra of all measurable operators affiliated with M. We show that any non-trivial derivation δ : A Ñ SpAq can not be extended to a derivation on SpMq. In particular, we answer an untreated question in [8].

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“…Criterion 4.2],[11, Lemma 3.7].Lemma 5.8. Let (Z, • ) be a complete ∆-normed space with constant C Z .…”

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confidence: 99%

Pietsch correspondence for symmetric functionals on Calkin operator spaces associated with semifinite von Neumann algebras

Levitina¹,

Usachev²

2020

Preprint

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In this paper we extend the Pietsch correspondence for ideals of compact operators and traces on them to the semifinite setting. We prove that a shift-monotone space E(Z) of sequences indexed by Z defines a Calkin space E(M, τ ) of τ -measurable operators affiliated with a semifinite von Neumann algebra M equipped with a faithful normal semifinite trace τ . Furthermore, we show that shift-invariant functionals on E(Z) generate symmetric functionals on E(M, τ ). In the special case, when the algebra M is atomless or atomic with atoms of equal trace, the converse also holds and we have a bijective correspondence between all shift-monotone spaces E(Z) and Calkin spaces E(M, τ ) as well as a bijective correspondence between shift-invariant functionals on E(Z) and symmetric functionals on E(M, τ

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Completeness of symmetric $\varDelta $-normed spaces of $\tau $-measurable operators

Cited by 14 publications

References 0 publications

Logarithmic submajorisation and order-preserving linear isometries

Logarithmic submajorisation and order-preserving linear isometries

Non-existence of translation-invariant derivations on algebras of measurable functions

Pietsch correspondence for symmetric functionals on Calkin operator spaces associated with semifinite von Neumann algebras

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