Jordan generalized derivations on triangular algebras

Let 𝒜 be a unital algebra and let ? be a unitary 𝒜-bimodule. We consider Jordan generalized derivations mapping from 𝒜 into ?. Our results on unitary algebras are applied to triangular algebras. In particular, we prove that any Jordan generalized derivation of a triangular algebra is a generalized derivation.     

A characterization of generalized derivations of JSL algebras

Let AlgL be a J-subspace lattice algebra on a Banach space X and M be an operator in AlgL. We prove that if δ : AlgL → B(X) is a linear mapping satisfying δ(AB) = δ(A)B + Aδ(B)for all A, B ∈ AlgL with AMB = 0, then δ is a generalized derivation. This result can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.     

A note on generalized derivations in Banach algebras

In this note we characterize a complex Banach algebra A admitting a generalized derivation g such that the cardinality of the spectrum σ(g(x)) is exactly one for all x∈A.     

On generalized left derivations in rings and Banach algebras

The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let \({G: R \longrightarrow R}\) be a generalized Jordan left derivation with associated Jordan left derivation \({\delta: R \longrightarrow R}\). Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253–261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer (or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous.     

On generalized Lie derivations of Lie ideals of prime algebras

Let A be a prime algebra of characteristic not 2 with extended centroid C, let R be a noncentral Lie ideal of A and let B be the subalgebra of A generated by R. If f,d:R→A are linear maps satisfying that

     

Cohomology and Deformations of 3-Dimensional Heisenberg Hom-Lie Superalgebras

Czechoslovak Mathematical Journal - We study Hom-Lie superalgebras of Heisenberg type. For 3-dimensional Heisenberg Hom-Lie superalgebras we describe their Hom-Lie super structures, compute the...     

Cohomology and deformations for the Heisenberg Hom-Lie algebras

ABSTRACT

In this work, we consider the Heisenberg Lie algebra with all its Hom-Lie structures. We completely characterize the cohomology and deformations of any order of all Heisenberg Hom-Lie algebras of dimension 3.     

Characterizing centralizers and generalized derivations on triangular algebras by acting on zero product

Let U = Tri(A,M,B) be a triangular ring, where A and B are unital rings, and M is a faithful (A, B)-bimodule. It is shown that an additive map Φ on U is centralized at zero point (i.e., Φ(A)B = AΦ(B) = 0 whenever AB = 0) if and only if it is a centralizer. Let δ : U → U be an additive map. It is also shown that the following four conditions are equivalent: (1) δ is specially generalized derivable at zero point, i.e., δ(AB) = δ(A)B + Aδ(B) Aδ(I)B whenever AB = 0; (2) δ is generalized derivable at zero point, i.e., there exist additive maps τ1 and τ2 on U derivable at zero point such that δ(AB) = δ(A)B + Aτ1 (B) = τ2 (A)B + Aδ(B) whenever AB = 0; (3) δ is a special generalized derivation; (4) δ is a generalized derivation. These results are then applied to nest algebras of Banach space.     

Left-symmetric algebras of derivations of free algebras

A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric algebra of derivations of a free nonassociative symmetric m-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described.     

Hom-Lie algebra structures on semi-simple Lie algebras

Hom-Lie algebras can be considered as a deformation of Lie algebras. In this note, we prove that the hom-Lie algebra structures on finite-dimensional simple Lie algebras are trivial. We find when a finite-dimensional semi-simple Lie algebra admits non-trivial hom-Lie algebra structures and the isomorphic classes of non-trivial hom-Lie algebras are determined.     

Quaternion algebras with derivations

We study derivations on quaternion algebras that stabilise quadratic subfields. Following the work of L. Juan and A. Magid [10], we provide an explicit construction of a differential splitting field for a given differential quaternion algebra. We also examine the presence and impact of those derivations on quaternion algebras that admit new constants.     

Convexoid and generalized derivations

Let E,F be two Banach spaces and let S be a symmetric norm ideal of L(E,F). For AL(F) and BL(E) the generalized derivation δ_S,A,B is the operator on S that sends X to AX−XB. A bounded linear operator is said to be convexoid if its (algebraic) numerical range coincides with the convex hull of its spectrum. We show that δ_S,A,B is convexoid if and only if A and B are convexoid.     

Jordan derivations of triangular algebras

In this note, it is shown that every Jordan derivation of triangular algebras is a derivation.     

An identity with generalized derivations on lie ideals, right ideals and Banach algebras

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, F a non-zero generalized derivation of R. Suppose that [F(u), u]F(u) = 0 for all u ε L, then one of the following holds:

1. there exists α ε C such that F(x) = α x for all x ε R
2. R satisfies the standard identity s ₄ and there exist a ε U and α ε C such that F(x) = ax + xa + αx for all x ε R.

We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.