Lie Triple Derivations Research Articles

Generalized Lie triple derivations of Dihedron algebras

Generalized Lie triple derivations of Dihedron algebras

Nonlinear Mixed Lie Triple Derivations by Local Actions on Von Neumann Algebras

Nonlinear Mixed Lie Triple Derivations by Local Actions on Von Neumann Algebras

Characterization of Nonlinear Mixed Bi-Skew Lie Triple Derivations on ∗-Algebras

This paper concentrates on examining the characterization of nonlinear mixed bi-skew Lie triple ∗- derivations within an ∗-algebra denoted by A which contains a nontrivial projection with a unit I. Additionally, we expand this investigation to applications by describing these derivations within prime ∗-algebras, von Neumann algebras, and standard operator algebras.

The second nonlinear mixed Lie triple derivations on standard operator algebras

Abstract Let 𝒜 {\mathcal{A}} be a standard operator algebra containing the identity operator I on an infinite dimensional complex Hilbert space ℋ {\mathcal{H}} which is closed under adjoint operation. Suppose that ϕ : 𝒜 → 𝒜 {\phi:\mathcal{A}\to\mathcal{A}} is the second nonlinear mixed Lie triple derivation. Then ϕ is an additive ∗ {\ast} -derivation.

Lie triple derivations of dihedron algebra

Let K be a 2-torsion free unital ring and D(K) be dihedron algebra over K. In the present article, we prove that every Lie triple derivation of D(K) can be written as the sum of the Lie triple derivation of K, Jordan triple derivation of K, and some inner derivation of D. We also prove that a generalized Lie triple derivation ϱ:D(K)→D(K) associated with the Lie triple derivation h:D(K)→D(K) exists if ϱ can be represented in the form ϱ(τ) = h(τ) + λτ, where λ lies in the center of D(K). We finally conclude that to obtain the complete algebra of the Lie triple derivation and generalized Lie triple of D(K), we first need to find the Lie triple derivation and Jordan triple derivation of K.

The nonlinear mixed bi-skew lie triple derivations on *-algebras

Let A be a unital *-algebra. In this paper, under some mild conditions on A, it is shown that a map ? : A ? Ais a nonlinear mixed bi-skew Lie triple derivation if and only if ? is an additive*-derivation. As applications, nonlinear mixed bi-skew Lie triple derivations on prime *-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.

Maps on $ C^\ast $-algebras are skew Lie triple derivations or homomorphisms at one point

<abstract><p>In this paper, we show that every continuous linear map between unital $ C^\ast $-algebras is skew Lie triple derivable at the identity is a $ \ast $-derivation and that every continuous linear map between unital $ C^\ast $-algebras which is a skew Lie triple homomorphism at the identity is a Jordan $ \ast $-homomorphism.</p></abstract>

Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective

Let R be a commutative ring with unity and T be a triangular algebra over R. Let a sequence Δ={δn}n∈N of nonlinear mappings δn:T→T is a Lie triple higher derivation by local actions satisfying the equation. Under some mild conditions on T, we prove in this paper that every Lie triple higher derivation by local actions on the triangular algebras is proper. As an application, we shall give a characterization of Lie triple higher derivations by local actions on upper triangular matrix algebras and nest algebras, respectively.

Lie triple derivations of standard operator algebras

Lie triple derivations of standard operator algebras

Lie triple centralizers on generalized matrix algebras

In this article, we introduce the notion of Lie triple centralizer as follows. Let be an algebra, and be a linear mapping. We say that ϕ is a Lie triple centralizer whenever for all . Then we characterize the general form of Lie triple centralizers on a generalized matrix algebra and under some mild conditions on we present the necessary and sufficient conditions for a Lie triple centralizer to be proper. As an application of our results, we characterize generalized Lie triple derivations on generalized matrix algebras.

Multiplicative Lie triple higher derivations on generalized matrix algebras

Multiplicative Lie triple higher derivations on generalized matrix algebras

Nonlinear mixed Lie triple derivations on prime *-rings

Nonlinear mixed Lie triple derivations on prime *-rings

Multiplicative Lie triple derivations on standard operator algebras

Abstract Let χ be a Banach space of dimension n > 1 and 𝔘 ⊂ 𝔅(χ) be a standard operator algebra. In the present paper it is shown that if a mapping d : 𝔘 → 𝔘 (not necessarily linear) satisfies d ( [ [ U , V ] , W ] ) = [ [ d ( U ) , V ] , W ] + [ [ U , d ( V ) , W ] ] + [ [ U , V ] , d ( W ) ] d\left( {\left[ {\left[ {U,V} \right],W} \right]} \right) = \left[ {\left[ {d\left( U \right),V} \right],W} \right] + \left[ {\left[ {U,d\left( V \right),W} \right]} \right] + \left[ {\left[ {U,V} \right],d\left( W \right)} \right] for all U, V, W ∈ 𝔘, then d =ψ + τ, where ψ is an additive derivation of 𝔘 and τ : 𝔘 → 𝔽I vanishes at second commutator [[U, V ], W ] for all U, V, W ∈ 𝔘. Moreover, if d is linear and satisfies the above relation, then there exists an operator S ∈ 𝔘 and a linear mapping τ from 𝔘 into 𝔽I satisfying τ ([[U, V ], W ]) = 0 for all U, V, W ∈ 𝔘, such that d(U) = SU − US + τ (U) for all U ∈ 𝔘.

Nonlinear Lie triple derivations by local actions on triangular algebras

Let [Formula: see text] be a triangular algebra over a commutative ring [Formula: see text]. In this paper, under some mild conditions on [Formula: see text], we prove that if [Formula: see text] is a nonlinear map satisfying [Formula: see text] for any [Formula: see text] with [Formula: see text]. Then [Formula: see text] is almost additive on [Formula: see text], that is, [Formula: see text] Moreover, there exist an additive derivation [Formula: see text] of [Formula: see text] and a nonlinear map [Formula: see text] such that [Formula: see text] for [Formula: see text], where [Formula: see text] for any [Formula: see text] with [Formula: see text].

Generalized Lie triple derivations on generalized matrix algebras

Let be a commutative ring with identity, be -algebras, be an -bimodule and be a -bimodule. The -algebra is a generalized matrix algebra defined by the Morita context In this article, we give the structure of generalized Lie triple derivations GL on generalized matrix algebras and prove that under certain restrictions GL can be written as where Δ is a generalized derivation and χ is a central valued mapping.

Generalized Lie Triple Derivations of Lie Color Algebras and Their Subalgebras

Consider a Lie color algebra, denoted by L. Our aim in this paper is to study the Lie triple derivations TDer(L) and generalized Lie triple derivations GTDer(L) of Lie color algebras. We discuss the centroids, quasi centroids and central triple derivations of Lie color algebras, where we show the relationship of triple centroids, triple quasi centroids and central triple derivation with Lie triple derivations and generalized Lie triple derivations of Lie color algebras L. A classification of Lie triple derivations algebra of all perfect Lie color algebras is given, where we prove that for a perfect and centerless Lie color algebra, TDer(L)=Der(L) and TDer(Der(L))=Inn(Der(L)).

On the Lie triple derivations

This article aims to interpret some Lie triple derivations of Tensor algebras , using generalized quaternion algebra over a field F and assuming T as an F-linear associative algebra. The study concludes with complete description of all Lie triple derivations of A ⊗ F L. It is pertinent to mention here that A is assumed to be a finite-dimensional commutative algebra and L is a simple Lie algebra over an algebraically closed field.

Generalized Lie (Jordan) Triple Derivations on Arbitrary Triangular Algebras

Generalized Lie (Jordan) Triple Derivations on Arbitrary Triangular Algebras

Characterizations of Lie triple derivations on generalized matrix algebras

Let be a commutative ring with unity and be a generalized matrix algebra. In this article, we give the structure of Lie triple derivation on a generalized matrix algebra and prove that under certain appropriate assumptions on is proper, i.e., where δ is a derivation on and χ is a mapping from into its center which annihilates all second commutators in i.e., for all

The Second Nonlinear Mixed Lie Triple Derivations on Finite Von Neumann Algebras

Let $${\mathcal {M}}$$ be a finite von Neumann algebra with no central summands of type $${I}_{1}$$ . Suppose that $$L: {\mathcal {M}}\rightarrow {\mathcal {M}}$$ is the second nonlinear mixed Lie triple derivation. Then L is an additive $$*$$ -derivation. We also show that each local and 2-local second Lie triple derivation on finite von Neumann algebras with no central summands of type $${I}_{1}$$ is a $$*$$ -derivation. Besides, each local and 2-local second Lie triple derivation on factor von Neumann algebras $${\mathcal {M}}$$ with dim $${\mathcal {M}}>1$$ is also a $$*$$ -derivation.