Nonlinear ∗-Jordan triple derivations on von Neumann algebras

Zhao, Fangfang; Li, Changjing

doi:10.1515/ms-2017-0089

Cited by 40 publications

(9 citation statements)

References 16 publications

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“…Therefore, studying Lie triple derivations enables us to treat both important class of Jordan derivations and Lie derivations simultaneously. More recently, Jordan-type derivations on triangular algebras, prime rings, matrix algebras, nest algebras and von Neumann algebras are considered by Lin, Qi and Zhao et al, see [33,53,56]. An R-bilinear mapping ϕ : A × A −→ A is a Jordan biderivation if it is a Jordan derivation with respect to both components, implying that ϕ(x • y, z) = ϕ(x, z) • y + x • ϕ(y, z) and ϕ(x, y • z) = ϕ(x, y) • z + y • ϕ(x, z) for all x, y ∈ A.…”

Section: Related Topics For Further Researchmentioning

confidence: 99%

Nonlinear Lie-Type Derivations of finitary Incidence Algebras and Related Topics

Yang,

Wei

2021

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This is a continuation of our earlier works [29,51,52] with respect to (non-)linear Lie-type derivations of finitary incidence algebras. Let X be a pre-ordered set, R be a 2-torsionfree and (n − 1)torsionfree commutative ring with identity, where n ≥ 2 is an integer. Let F I(X, R) be the finitary incidence algebra of X over R. In this paper, a complete clarification is obtained for the structure of nonlinear Lie-type derivations of F I(X, R). We introduce a new class of derivations on F I(X, R) named inner-like derivations, and prove that each nonlinear Lie n-derivation on F I(X, R) is the sum of an inner-like derivation, a transitive induced derivation and a quasi-additive induced Lie n-derivation. Furthermore, if X is finite, we show that a quasi-additive induced Lie n-derivation can be expressed as the sum of an additive induced Lie derivation and a central-valued map annihilating all (n − 1)-th commutators. We also provide a sufficient and necessary condition such that every nonlinear Lie n-derivation of F I(X, R) is of proper form. Some related topics for further research are proposed in the last section of this article.Let A be an associative algebra over a commutative ring R. Let [ , ] and • be the Lie product and Jordan product respectively, i.e., [x, y] = xy − yx and x • y = xy + yx for all x, y ∈ A. Then (A, [ , ]) is a Lie algebra and (A, •) is a Jordan algebra. It is a fascinating topic to study the connection between the associative, Lie and Jordan structures of A. In this field, there are two classes of important algebraic maps: algebra homomorphisms and differential operators. For example Jordan homomorphisms, Lie homomorphisms, Jordan derivations and Lie derivations. In the AMS Hour Talk of 1961, Herstein proposed many problems

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Section: Related Topics For Further Researchmentioning

confidence: 99%

Nonlinear Lie-Type Derivations of finitary Incidence Algebras and Related Topics

Yang,

Wei

2021

Preprint

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“…Note that, unlike von Neumann algebras which are always weakly closed, a standard operator algebra is not necessarily closed. The current work together with [7,[10][11][12][13][23][24][25][26] indicates that it is feasible to investigate * -Jordan-type derivations and * -Lie-type derivations on operator algebras under a unified framework-η- * -Jordan-type derivations. We have good reasons to believe that characterizing η- * -Jordan-type derivations on operator algebras is also of great interest.…”

Section: Related Topics For Future Researchmentioning

confidence: 99%

“…* -Jordan-type derivations on operator algebras have been studied by several authors. Let H be a complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. Li et al in [10] showed that if [26]. Taghavi et al [23] and Zhang [25] independently investigate * -Jordan derivations on factor von Neumann algebras, respectively.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear $\ast$-Jordan-Type Derivations on von Neumann Algebras

Lin

2018

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Let H be a complex Hilbert space, B(H) be the algebra of all bounded linear operators on H and A ⊆ B(H) be a von Neumann algebra without central summands of type I 1 . For arbitrary elements A, B ∈ A, one can define their * -Jordan product in the sense of A ⋄ B = AB + BA * . Let pn(x 1 , x 2 , • • • , xn) be the polynomial defined by n indeterminates x 1 , • • • , xn and their * -Jordan products. In this article, it is shown that a mapping δ : A −→ B(H) satisfies the conditionand only if δ is an additive * -derivation.

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“…Similar conclusion have been obtained in [11] by Bell and Kappe. Zhao and Li, in [12], proved that every nonlinear * -Jordan triple derivation on von Neumann algebras is an additive * -derivation. For other similar results about Jordan triple derivations (nonlinear Jordan triple derivable mappings), we refer the readers to [13][14][15] and references therein, for more details.…”

Section: Introductionmentioning

confidence: 99%