Nonlinear Lie-type derivations of von Neumann algebras and related topics

Fošner, Ajda; Wei, Feng; Xiao, Zhankui

doi:10.4064/cm132-1-5

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Introduction3

The Finite Case1

Characterisations Of Standard Forms Of Lie N-higher Derivations1

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Cited by 43 publications

(20 citation statements)

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“…and C ii ik + C kk ik = 0 for i < k. Recall that C ii ii = C ii jj for i < j by (4) and C ii ii = C ii kk for k < i by (7). Combining these facts with the identity (15) or (16), we have…”

Section: The Finite Casementioning

confidence: 90%

Lie triple derivations of incidence algebras

Wang

Xiao

2019

Communications in Algebra

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“…and C ii ik + C kk ik = 0 for i < k. Recall that C ii ii = C ii jj for i < j by (4) and C ii ii = C ii kk for k < i by (7). Combining these facts with the identity (15) or (16), we have…”

Section: The Finite Casementioning

confidence: 90%

Lie triple derivations of incidence algebras

Wang

Xiao

2019

Communications in Algebra

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“…In [2,4,18], the authors discussed sufficient conditions for Lie n-derivations to be standard on A, where A is a triangular ring, a von Neumann algebra without abelian central summands of type I 1 or a unital algebra with a wide idempotent. From Theorem 3.1, we can obtain sufficient conditions for Lie n-higher derivations to be standard on A.…”

Section: Characterisations Of Standard Forms Of Lie N-higher Derivationsmentioning

confidence: 99%

“…He described the form of Lie n-derivations of a certain von Neumann algebra (or of its skew-adjoint part). Lie n-derivations on various unital algebras are considered in [2,4,15,18]. Let R A be a nonempty subset of A n .…”

Section: Introductionmentioning

confidence: 99%

Lie -Higher Derivations and Lie -Higher Derivable Mappings

Ding

2017

Bull. Aust. Math. Soc.

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Let ${\mathcal{A}}$ be a unital torsion-free algebra over a unital commutative ring ${\mathcal{R}}$. To characterise Lie $n$-higher derivations on ${\mathcal{A}}$, we give an identity which enables us to transfer problems related to Lie $n$-higher derivations into the same problems concerning Lie $n$-derivations. We prove that: (1) if every Lie $n$-derivation on ${\mathcal{A}}$ is standard, then so is every Lie $n$-higher derivation on ${\mathcal{A}}$; (2) if every linear mapping Lie $n$-derivable at several points is a Lie $n$-derivation, then so is every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points; (3) if every linear mapping Lie $n$-derivable at several points is a sum of a derivation and a linear mapping vanishing on all $(n-1)$th commutators of these points, then every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points is a sum of a higher derivation and a sequence of linear mappings vanishing on all $(n-1)$th commutators of these points. We also give several applications of these results.

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“…Abdullaev [11] discussed the form of Lie n-derivations on a certain von Neumann algebra. After that, the problem how to characterize the structure of Lie n-derivations has been investigated for triangular rings, von Neumann algebras, generalized matrix algebras and full matrix algebras, respectively (see [12][13][14][15]). …”

Section: Introductionmentioning

confidence: 99%