Derivations in Commutative Regular Algebras

Ber, A. F.; Sukochev, Fedor; Chilin, Vladimir

doi:10.1023/b:matn.0000023321.64184.5a

Cited by 34 publications

(108 citation statements)

References 3 publications

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“…The algebra LS(Z) = S(Z) is isomorphic to the algebra L 0 (Ω) = L(Ω, Σ, μ) of all measurable functions on a measure space (see Section 2) and therefore it admits (in nonatomic cases) non-zero derivations (see [3,13]). …”

Section: Derivations On the Algebra Ls(m)mentioning

confidence: 99%

“…It has been proved in [3,13] that the algebra LS(N ) = S(N) ∼ = L 0 (Ω) admits nontrivial derivations if and only if the measure space (Ω, Σ, μ) is not atomic.…”

Section: Derivations On the Algebra S(m τ )mentioning

confidence: 99%

“…Sukochev, V.I. Chilin [3] obtained necessary and sufficient conditions for the existence of nontrivial derivations on commutative regular algebras. In particular they have proved that the algebra L 0 (0, 1) of all (classes of equivalence of) complex measurable functions on the interval (0, 1) admits nontrivial derivations.…”

Section: Introductionmentioning

confidence: 99%

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Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

Albeverio

Ayupov

Kudaybergenov

2009

Journal of Functional Analysis

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Given a von Neumann algebra M denote by S(M) and LS(M) respectively the algebras of all measurable and locally measurable operators affiliated with M. For a faithful normal semi-finite trace τ on M let S(M, τ ) be the algebra of all τ -measurable operators from S(M). We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra M. In particular, we prove that if M is of type I ∞ then every derivation on LS(M) (resp. S(M) and S(M, τ )) is inner.

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Section: Derivations On the Algebra Ls(m)mentioning

confidence: 99%

“…It has been proved in [3,13] that the algebra LS(N ) = S(N) ∼ = L 0 (Ω) admits nontrivial derivations if and only if the measure space (Ω, Σ, μ) is not atomic.…”

Section: Derivations On the Algebra S(m τ )mentioning

confidence: 99%

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Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

Albeverio

Ayupov

Kudaybergenov

2009

Journal of Functional Analysis

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“…As it was noted above the commutative algebra L 0 (0, 1) of all complex measurable functions on (0, 1) admits non-zero derivations (see [6,13]). On the other hand the Corollary 3.5 shows that the Arens algebra L ω (0, 1) admits only zero derivations (similar to the algebra L ∞ (0, 1)), though it contains unbounded elements.…”

Section: Remarkmentioning

confidence: 96%

“…In the particular commutative case where M = L ∞ (0; 1) is the algebra of all essentially bounded measurable complex functions on (0; 1), the algebra L(M) is isomorphic to the algebra L 0 (0; 1) of all measurable functions on (0; 1). Recent results of [6] (see also [13]) show that L 0 (0; 1) admits non-zero (and hence discontinuous) derivations. Therefore the properties of derivations on the unbounded operator algebra L(M) are very far from being similar to those on C * -or von Neumann algebras.…”

Section: Introductionmentioning

confidence: 99%

Non-commutative Arens algebras and their derivations

Albeverio

Ayupov

Kudaybergenov

2007

Journal of Functional Analysis

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Given a von Neumann algebra M with a faithful normal semi-finite trace τ , we consider the non-commutative Arens algebra L ω (M, τ ) = p 1 L p (M, τ ) and the related algebras L ω 2 (M, τ ) = p 2 L p (M, τ ) and M +L ω 2 (M, τ ) which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra M + L ω 2 (M, τ ) is inner and all derivations of the algebras L ω (M, τ ) and L ω 2 (M, τ ) are spatial and implemented by elements of M + L ω 2 (M, τ ). In particular we obtain that if the trace τ is finite then any derivation on the noncommutative Arens algebra L ω (M, τ ) is inner.

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Lateral order on complex vector lattices and narrow operators

Dzhusoeva

Huang

Pliev

et al. 2023

Mathematische Nachrichten

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In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification 𝐸 ℂ of a real vector lattice 𝐸 and introduce the lateral order on 𝐸 ℂ . Our first main result asserts that the set of all fragments 𝔉 𝑣 of an element 𝑣 ∈ 𝐸 ℂ of the complexification of an uniformly complete vector lattice 𝐸 is a Boolean algebra. Then, we study narrow operators defined on the complexification 𝐸 ℂ of a vector lattice 𝐸, extending the results of articles [22,27,28] to the setting of operators defined on complex vector lattices.We prove that every order-to-norm continuous linear operator  ∶ 𝐸 ℂ → 𝑋 from the complexification 𝐸 ℂ of an atomless Dedekind complete vector lattice 𝐸 to a finite-dimensional Banach space 𝑋 is strictly narrow. Then, we prove that every 𝐶-compact order-to-norm continuous linear operator  from 𝐸 ℂ to a Banach space 𝑋 is narrow. We also show that every regular order-no-norm continuous linear operator from 𝐸 ℂ to a complex Banach lattice (𝓁 𝑝 () ℂ is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.

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Derivations in Commutative Regular Algebras

Cited by 34 publications

References 3 publications

Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

Non-commutative Arens algebras and their derivations

Lateral order on complex vector lattices and narrow operators

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