Generalized Jordan Derivations Research Articles

Centrally-extended generalized Jordan derivations in rings

Abstract In this article, we introduce the notion of centrally-extended generalized Jordan derivations and characterize the structure of a prime ring (resp. *-prime ring) R that admits a centrally-extended generalized Jordan derivation F satisfying [F(x), x] ∈ Z(R) (resp. [F(x), x*] ∈ Z(R)) for all x ∈ R.

Generalized Derivations and Generalized Jordan Derivations on C ∗ -Algebras through Zero Products

Let A be a unital C ∗ -algebra and X be a unitary Banach A-bimodule. In this paper, we characterize continuous generalized derivations and generalized Jordan derivations as form D : A ⟶ X through the action on zero product. In other words, we show that under some conditions on elements of A, a linear map on A can be a generalized Jordan derivation.

Generalized Jordan N-Derivations of Unital Algebras with Idempotents

Let A be a unital algebra with idempotent e over a 2-torsionfree unital commutative ring ℛ and S : A ⟶ A be an arbitrary generalized Jordan n-derivation associated with a Jordan n-derivation J . We show that, under mild conditions, every generalized Jordan n-derivation S : A ⟶ A is of the form S x = λ x + J x in the current work. As an application, we give a description of generalized Jordan derivations for the condition n = 2 on classical examples of unital algebras with idempotents: triangular algebras, matrix algebras, nest algebras, and algebras of all bounded linear operators, which generalize some known results.

Generalized Derivations and Generalized Jordan Derivations of Quaternion Rings

Generalized Derivations and Generalized Jordan Derivations of Quaternion Rings

Generalized Jordan Derivations of Incidence Algebras

For a given ring $\Re$ and a locally finite pre-ordered set $(X, \leq)$, consider $I(X, \Re)$ to be the incidence algebra of $X$ over $\Re$. Motivated by a Xiao’s result which states that every Jordan derivation of $I(X, \Re)$ is a derivation in the case $\Re$ is 2-torsion free, one proves that each generalized Jordan derivation of $I(X, \Re)$ is a generalized derivation provided $\Re$ is 2-torsion free, getting as a consequence the above mentioned result.

GENERALIZED JORDAN DERIVATIONS ON SEMIPRIME RINGS

The purpose of this note is to prove the following. Suppose $\mathfrak{R}$ is a semiprime unity ring having an idempotent element $e$ ($e\neq 0,~e\neq 1$) which satisfies mild conditions. It is shown that every additive generalized Jordan derivation on $\mathfrak{R}$ is a generalized derivation.

Continuity of Generalized Jordan Derivations on Semisimple Banach Algebras

A generalized Jordan derivation D on a Banach algebra $${\mathcal {A}}$$ is a linear mapping $$D:{\mathcal {A}}\rightarrow {\mathcal {A}}$$ for which there exists a positive $$\varepsilon $$ satisfying $$\begin{aligned}\Vert D(a\circ b)-a\circ D(b)-D(a)\circ b\Vert \le \varepsilon \Vert a\Vert \Vert b\Vert \;\;(a, b \in {\mathcal {A}}),\end{aligned}$$ where $$\circ $$ denotes the Jordan product on $${\mathcal {A}}$$ . We study generalized Jordan derivations on Banach algebras and prove that every generalized Jordan derivation on a semisimple Banach algebra is automatically continuous.

Prime gamma rings with centralizing and commuting generalized Jordan derivations

Prime gamma rings with centralizing and commuting generalized Jordan derivations

Left Multiplicative Generalized Jordan Derivations of Semiprime Rings

In this paper we prove that left multiplicative generalized Jordan derivation and left multiplicative generalized Jordan triple derivation of 2-torsion free semiprime rings are left multiplicative generalized derivation.

Continuity of generalized derivations on JB*-algebras

We prove that every generalized Jordan derivation D from a JB∗-algebra 𝒜 into itself or into its dual space is automatically continuous. In particular, we establish that every generalized Jordan derivation from a C∗-algebra to a Jordan Banach module is continuous. As a consequence, every generalized derivation from a C∗-algebra to a Banach bimodule is continuous.

Characterizations of generalized derivations associated with hochschild 2-cocycles and higher derivations

Let be a unital Banach algebra and be a unital -bimodule. A bilinear mapping α : is called a Hochschild 2-cocycle if xα(y, z) − α(xy, z) + α(x, yz) − α(x, y)z = 0 for any . We show that if δ is a linear mapping from into satisfying δ(xy) = δ(x)y + xδ(y) + α(x, y) for any with xy = W, where is a left or right separating point of , then δ is a generalized Jordan derivation associated with a Hochschild 2-cocycle α. We also find the relation of higher derivations and generalized derivations associated with Hochschild 2-cocycles.

Characterizations of linear mappings through zero products or zero Jordan products

Let $\mathcal{A}$ be a unital algebra and $\mathcal{M}$ be a unital $\mathcal{A}$-bimodule. A characterization of generalized derivations and generalized Jordan derivations from $\mathcal{A}$ into $\mathcal{M}$, through zero products or zero Jordan products, is given. Suppose that $\mathcal{M}$ is a unital left $\mathcal{A}$-module. It is investigated when a linear mapping from $\mathcal{A}$ into $\mathcal{M}$ is a Jordan left derivation under certain conditions. It is also studied whether an algebra with a nontrivial idempotent is zero Jordan product determined, and Jordan homomorphisms, Lie homomorphisms and Lie derivations on zero Jordan product determined algebras are characterized.

A Note on Generalized Jordan Derivations in Semiprime Rings

The main purpose of this paper is to study and investigate some results concerning generalized Jordan derivation and generalized derivation on semiprime ring R, where D an additive mapping on R such that for all and D acts as left centralizer.

Jordan weak amenability and orthogonal forms on JB$^*$-algebras

We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$. We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all anti-symmetric orthogonal forms on $\mathcal{J}$, and of all Lie Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$.

A characterization of generalized Jordan derivations on Banach algebras

Generalized Jordan left derivations and generalized \((\sigma ,\tau )\)-Jordan derivations on Banach algebras are characterized and some results of [6] and [7] are extended.

Almost partial generalized Jordan derivations: a fixed point approach

Abstract Using fixed point method, we investigate the Hyers-Ulam stability and the superstability of partial generalized Jordan derivations on Banach modules related to Jensen type functional equations. Mathematics Subject Classification 2010: Primary, 39B52; 47H10; 47B47; 13N15; 39B72; 17C50; 39B82; 17C65.

Generalized Jordan derivations on Lie ideals associate with Hochschild 2-cocycles of rings

Let R be a 2-torsion free ring and L be a Lie ideal of R. In this paper we initiate the study of generalized derivations on Lie ideals associate with Hochschild 2-cocycles and prove that every generalized derivation associate with a Hochschild 2-cocycle on L is a generalized derivation on L under certain conditions.

Nearly generalized Jordan derivations

Abstract Let A be an algebra and let X be an A-bimodule. A ∂-linear mapping d: A → X is called a generalized Jordan derivation if there exists a Jordan derivation (in the usual sense) δ: A → X such that d(a 2) = ad(a)+δ(a)a for all a ∈ A. The main purpose of this paper is to prove the Hyers-Ulam-Rassias stability and superstability of the generalized Jordan derivations.

Generalized derivations associated with Hochschild 2-cocycles on some algebras

We investigate a new type of generalized derivations associated with Hochschild 2-cocycles which was introduced by A. Nakajima. We show that every generalized Jordan derivation of this type from CSL algebras or von Neumann algebras into themselves is a generalized derivation under some reasonable conditions. We also study generalized derivable mappings at zero point associated with Hochschild 2-cocycles on CSL algebras.

On generalized Jordan derivations of Lie triple systems

Under some conditions we prove that every generalized Jordan triple derivation on a Lie triple system is a generalized derivation. Specially, we conclude that every Jordan triple θ-derivation on a Lie triple system is a θ-derivation.