Characterizing Lie<i>n</i>-derivations for reflexive algebras

Qi, Xiaofei

doi:10.1080/03081087.2014.968519

Cited by 13 publications

(6 citation statements)

References 18 publications

(15 reference statements)

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“…Recently, many authors studied Lie n-derivation on various kind of algebras [2,4,7,11,13] and references therein. In the year 2014, Wang and Wang [13] studied multiplicative Lie n-derivation on generalized matrix algebras and proved that it has standard form under certain assumptions.…”

Section: For Future Researchmentioning

confidence: 99%

A study on generalized matrix algebras having generalized Lie derivations

Jabeen,

Ahmad,

Abbasi

2023

ADM

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Section: For Future Researchmentioning

confidence: 99%

A study on generalized matrix algebras having generalized Lie derivations

Jabeen,

Ahmad,

Abbasi

2023

ADM

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“…, A n ∈ A, then the mapping L = D + H is a Lie-type derivation that we say is of standard form. Lie-type derivations and their standard decomposition problems in different backgrounds have been extensively studied, see [10,11,19,21,25,27,32].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, if is an associative derivation and is a linear mapping vanishing on each th commutator for all , then the mapping is a Lie-type derivation that we say is of standard form . Lie-type derivations and their standard decomposition problems in different backgrounds have been extensively studied, see [10, 11, 19, 21, 25, 27, 32].…”

Section: Introductionmentioning

confidence: 99%

Lie-Type Derivations of Nest Algebras on Banach Spaces and Related Topics

Wei

Zhang²

2021

J. Aust. Math. Soc.

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Let $\mathcal {X}$ be a Banach space over the complex field $\mathbb {C}$ and $\mathcal {B(X)}$ be the algebra of all bounded linear operators on $\mathcal {X}$ . Let $\mathcal {N}$ be a nontrivial nest on $\mathcal {X}$ , $\text {Alg}\mathcal {N}$ be the nest algebra associated with $\mathcal {N}$ , and $L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ be a linear mapping. Suppose that $p_n(x_1,x_2,\ldots ,x_n)$ is an $(n-1)\,$ th commutator defined by n indeterminates $x_1, x_2, \ldots , x_n$ . It is shown that L satisfies the rule $$ \begin{align*}L(p_n(A_1, A_2, \ldots, A_n))=\sum_{k=1}^{n}p_n(A_1, \ldots, A_{k-1}, L(A_k), A_{k+1}, \ldots, A_n) \end{align*} $$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ if and only if there exist a linear derivation $D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ and a linear mapping $H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$ vanishing on each $(n-1)\,$ th commutator $p_n(A_1,A_2,\ldots , A_n)$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ such that $L(A)=D(A)+H(A)$ for all $A\in \text {Alg}\mathcal {N}$ . We also propose some related topics for future research.

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“…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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Characterizing Lien-derivations for reflexive algebras

Cited by 13 publications

References 18 publications

A study on generalized matrix algebras having generalized Lie derivations

A study on generalized matrix algebras having generalized Lie derivations

Lie-Type Derivations of Nest Algebras on Banach Spaces and Related Topics

Lie -Higher Derivations and Lie -Higher Derivable Mappings

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