Jordan Derivations Research Articles

Nonlinear $\ast$-Jordan triple derivation on prime $\ast$-algebras

Let 𝒜 be a prime ∗ -algebra and suppose that Φ preserves triple ∗ -Jordan derivation on 𝒜 , that is, for every A , B ∈ 𝒜 , Φ ( A ◇ B ◇ C ) = Φ ( A ) ◇ B ◇ C + A ◇ Φ ( B ) ◇ C + A ◇ B ◇ Φ ( C ) , where A ◇ B = A B + B A ∗ . Then Φ is additive. Moreover, if Φ ( α I ) is selfadjoint for α ∈ { 1 , i } , then Φ is a ∗ -derivation.

F-Biderivations and Jordan biderivations of unital algebras with idempotents

The notion of [Formula: see text]-derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper, we introduce the notion of [Formula: see text]-biderivations as a natural “biderivation” counterpart of the notion of “[Formula: see text]-derivations”. We first show, under some conditions, that any [Formula: see text]-biderivation is a Jordan biderivation. Then, we turn to study [Formula: see text]-biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibiderivation and an extremal biderivation. As a consequence, we show that every Jordan biderivation on a triangular algebra is a biderivation.

Characterizing Jordan n-Derivations of Unital Rings Containing Idempotents

Assume that $${\mathcal {R}}$$ is a unital ring containing a nontrivial idempotent. By introducing the concept of Jordan n-derivations (n is any positive integer), it is shown that, under certain conditions, every multiplicative (without any linearity and additivity) Jordan n-derivation $$\delta $$ on $${\mathcal {R}}$$ is additive; and furthermore, $$\delta $$ is a Jordan derivation if the characteristic of $${\mathcal {R}}$$ is not $$n-1$$ and $$\delta $$ is a generalized Jordan derivation if the characteristic of $${\mathcal {R}}$$ is $$n-1$$ $$(n>3)$$ . Based on these results, it turns out that a map on $${\mathcal {R}}$$ is a multiplicative Jordan n-derivation if and only if it is an additive Jordan derivation. As applications, multiplicative Jordan n-derivations on triangular rings, prime rings, matrix algebras, nest algebras and von Neumann algebras are, respectively, completely characterized, which generalize some known related results.

Linear maps on block upper triangular matrix algebras behaving like Jordan derivations through commutative zero products

Linear maps on block upper triangular matrix algebras behaving like Jordan derivations through commutative zero products

A new proof of Singer-Wermer Theorem with some results on {g, h}-derivations.

Singer and Wermer proved that if A is a commutative Banach algebra and d: A → A is a continuous derivation, then d(A) ⊆ rad(A), where rad(A) denotes the Jacobson radical of A. In this paper, we establish a new proof of that theorem. Moreover, we prove that every continuous Jordan derivation on a finite dimensional Banach algebra, under certain conditions, is identically zero. As another objective of this article, we study {g, h}-derivations on algebras. In this regard, we prove that if f is a {g, h}-derivation on a unital algebra, then f, g and h are generalized derivations. Additionally, we achieve some results concerning the automatic continuity of {g, h}-derivations on Banach algebras. In the last section of the article, we introduce the concept of a {g, h}-homomorphism and then we present a characterization of it under certain conditions.

Multiplicative Jordan derivations on triangular n-matrix rings

Multiplicative Jordan derivations on triangular n-matrix rings

A note on Jordan derivations of triangular rings

In this note we prove that every Jordan derivation on a triangular ring is a derivation. Moreover, we show that, under some conditions, every Jordan derivation on a 2-torsion free ring is a derivation.

Jordan derivations of alternative rings

Let be a unital alternative ring with nontrivial idempotent and be a Jordan derivation. Then is of the form , where d is a derivation of and δ is a singular Jordan derivation of . Moreover, d and δ are uniquely determined. This extends the main result of Benkovič and Širovnik’s to the case of alternative 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.

Weak Jordan derivations of prime rings

Let R be a prime ring with extended centroid C and with maximal left ring of quotients . An additive map is called a weak Jordan derivation if for all . Applying the theory of functional identities and dealing with the low dimensional cases, we give a complete characterization of weak Jordan derivations of prime rings. Moreover, we generalize Brešar's theorem concerning additive maps satisfying for all .

A note on non-linear ∗-Jordan derivations on ∗-algebras

Abstract Taghavi et al. in [TAGHAVI, A.—ROHI, H.—DARVISH, V.: Non-linear ∗-Jordan derivations on von Neumann algebras, Linear Multilinear Algebra 64 (2016), 426–439] proved that the map Φ: 𝓐 → 𝓐 which satisfies the following condition Φ ( A ⋄ B ) = Φ ( A ) ⋄ B + A ⋄ Φ ( B ) $$\begin{array}{} \Phi(A\diamond B)=\Phi(A)\diamond B+A\diamond \Phi(B) \end{array} $$ where A ⋄ B = AB+BA * for every A, B ∈ 𝓐 is an additive ∗-derivation. In this short note, we prove that when A is a prime ∗-algebras and Φ: 𝓐 → 𝓐 satisfies the above condition, then Φ is ∗-additive. Moreover, if Φ(iI) is self-adjoint then Φ is derivation.

Jordan Derivations of Special Subrings of Matrix Rings

Let K be a 2-torsion free ring with identity and Rn(K, J) be the ring of all n × n matrices over K such that the entries on and above the main diagonal are elements of an ideal J of K. We describe all Jordan derivations of the matrix ring Rn(K, J) in this paper. The main result states that every Jordan derivation Δ of Rn(K, J) is of the form Δ = D + Ω, where D is a derivation of Rn(K, J) and Ω is an extremal Jordan derivation of Rn(K, J).

Zero Lie product determined Banach algebras, II

A Banach algebra A is said to be zero Lie product determined if every continuous bilinear functional φ : A × A → C satisfying φ ( a , b ) = 0 whenever a b = b a is of the form φ ( a , b ) = ω ( a b − b a ) for some ω ∈ A ⁎ . We prove that A has this property provided that any of the following three conditions holds: (i) A is a weakly amenable Banach algebra with property B and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from A into A ⁎ is an inner derivation, (iii) A is the algebra of all n × n matrices, where n ≥ 2 , over a cyclically amenable Banach algebra with a bounded approximate identity.

Lie (Jordan) derivations of arbitrary triangular algebras

In this paper we construct a triangular algebra from a given triangular algebra, using the notion of maximal left (right) ring of quotients. As an application we give a description of Lie (Jordan) derivations of arbitrary triangular algebras through the constructed triangular algebra.

A Review on Orthogonal Derivations in Rings

This paper presents a brief review of derivations used in rings such as orthogonal derivation, orthogonal generalized derivation, orthogonal Jordan derivation, orthogonal symmetric derivation, and orthogonal semiderivation.

Characterizations of (m,n)-Jordan Derivations and (m,n)-Jordan Derivable Mappings on Some Algebras

Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n ≠ 0. An additive mapping δ from R into M is called an (m, n)-Jordan derivation if (m + n)δ(A2) = 2mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every (m, n)-Jordan derivation with m ≠ n from a C* -algebra into its Banach bimodule is zero. An additive mapping δ from R into M is called a (m, n)-Jordan derivable mapping at W in R if (m + n)δ(AB + BA) = 2mδ(A)B + 2mδ(B)A + 2nAδ(B) + 2nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left (right) separating set generated algebraically by all idempotents in A, then every (m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital (A, B)-bimodule and $$\mathcal{U}=\begin{bmatrix}\mathcal{A} & \mathcal{M} \\\mathcal{N} & \mathcal{B} \end{bmatrix}$$ is a generalized matrix algebra, then every (m, n)-Jordan derivable mapping at zero from U into itself is equal to zero.

Centralizers and Jordan triple derivations of semiprime rings

Let R be a semiprime ring with extended centroid C and with maximal left ring of quotients . An additive map is called a Jordan triple derivation if for all . In 1957, Herstein proved that a Jordan triple derivation, which is also a Jordan derivation, of a noncommutative prime ring of characteristic 2, must be a derivation. In 1989, Brešar proved that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation. In the article, we give a complete characterization of Jordan triple derivations of arbitrary semiprime rings. To get such a characterization we first show that, in some sense, an additive map satisfying for all can be realized as a centralizer with only an exceptional case that and R is commutative.

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.

Jordan Ideals and Derivations Satisfying Algebraic Identities

Let $$\mathcal {N}$$ be a 3-prime near-ring with center $$Z(\mathcal {N})$$ and $$\mathcal {J}$$ a nonzero Jordan ideal of $$\mathcal {N}$$ . The aim of this paper is to prove some theorems showing that $$\mathcal {N}$$ must be commutative if it admits a left multiplier F satisfying any one of the following properties: $$(i)\,F(\mathcal {J})\subseteq Z(\mathcal {N})$$ , $$(ii)\,F(\mathcal {J}^{2})\subseteq Z(\mathcal {N})$$ , $$(iii)\,F(ij)+[i, j]\in Z(\mathcal {N})$$ , $$(vi)\,F(ij)-ij+ji\in Z(\mathcal {N})$$ , $$(v)\,F(i\circ j)\in Z(\mathcal {N})$$ and $$(vi)\,F(i)G(j)\in Z(\mathcal {N}),$$ for all $$i, j\in \mathcal {J}.$$ Moreover, we give some examples which show that the hypotheses placed in our results are not superfluous.