Search

On an infinite dimensional complex Hilbert space $ \mathcal{H} $, we consider a standard operator algebra $ \mathcal{S} $ with an identity operator $ I $ that is closed with respect to adjoint operation. $ P_{n}\left(\mathcal{X}_{1}, \mathcal{X}_{2}, \mathcal{X}_{3}, \ldots, \mathcal{X}_{n}\right) $ is set of polynomials defined under indeterminates $ \mathcal{X}_1, \mathcal{X}_2, \cdots, \mathcal{X}_n $ by $ n $ with multiplicative Lie products with set of positive integers $ \mathbb{N}. $ It is shown that a map $ \Theta: \mathcal{S} \rightarrow \mathcal{S} $ satisfying$ \begin{eqnarray*} \Theta\left(P_{n}\left(\mathcal{D}_{1}, \mathcal{D}_{2}, \mathcal{D}_{3}, \ldots, \mathcal{D}_{n}\right)\right) = \sum\limits_{i = 1}^{n} P_{n}\left(\mathcal{D}_{1}, \ldots, \mathcal{D}_{i-1}, \Theta\left(\mathcal{D}_{i}\right), \mathcal{D}_{i+1}, \ldots, \mathcal{D}_{n}\right), \end{eqnarray*} $for any $ \mathcal{D}_{1}, \mathcal{D}_{2}, \mathcal{D}_{3}, \ldots, \mathcal{D}_{n} \in \mathcal{S} $ with $ \mathcal{D}_{1} \mathcal{D}_{2} \mathcal{D}_{3}\ldots \mathcal{D}_{n} = 0 $ can be represented as $ d(x)+\tau(x) $ for every $ x \in \mathcal{S} $, where $ d: \mathcal{S} \rightarrow \mathcal{S} $ is an additive derivation with another map $ \tau: \mathcal{S} \rightarrow \mathcal{Z}(\mathcal{S}) $ that vanishes on each $ (n-1)^{th} $ commutator $ P_{n}\left(\mathcal{D}_{1}, \mathcal{D}_{2}, \mathcal{D}_{3}, \ldots, \mathcal{D}_{n}\right) $ with $ \mathcal{D}_{1} \mathcal{D}_{2} \mathcal{D}_{3}\cdots \mathcal{D}_{n} = $ 0.

Let ℛ be primitive ring of characteristic different from two with nontrivial idempotents. Then every Lie-type derivation can be written in the standard form under certain assumptions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define Bi- Hom Γ-Lie algebra, Lie triple Hom Γ-derivation on Bi- Hom Γ-Lie algebra (g,[.,.]λ,θ1,θ2)and Jordan triple Hom Γ-derivation on Bi-Hom Γ-Lie algebra (g,[.,.]λ,θ1,θ2 ), Also to study the relation between Lie triple Hom Γ-derivation and Jordan triple Hom Γ-derivation on Bi-Hom Γ-Lie algebra (g,[.,.]λ,θ1,θ2 ). [ABSTRACT FROM AUTHOR]

Let N be a non-trivial nest on a Hilbert space H and alg N be the associated nest algebra. Let G ∈ alg N be an operator with ran (G) ¯ ∈ N \ { H } . In this note, we give a description of Lie derivable maps and generalized Lie 2-derivable maps at G of nest algebra alg N . [ABSTRACT FROM AUTHOR]

Let be a nontrivial nest on a separable Hilbert space and be the associated nest algebra. In this note, we prove that, if contains an atom of infinite rank or is uncountable, then the unit operator I is a Lie all-derivable point of. [ABSTRACT FROM AUTHOR]

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I 1 , and L m : A → A is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ϕ m : A → A and an additive higher map ζ m : A → Z (A) , which annihilates every (n − 1) t h commutator p n (S 1 , S 2 , ⋯ , S n) with S 1 S 2 = 0 such that L m (S) = ϕ m (S) + ζ m (S) f o r a l l S ∈ A. We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results. [ABSTRACT FROM AUTHOR]

This paper surveys the theory of functional identities and its applications. No prior knowledge of the theory is required to follow the paper. [ABSTRACT FROM AUTHOR]

In this article, we study Lie biderivations on triangular ring under certain restrictions and prove that every Lie bideritvation can be written as a sum of an extremal biderivation, an inner biderivation and a central map vanishing at commutators. As an application, we study Lie biderivations on upper triangular matrix ring. [ABSTRACT FROM AUTHOR]

Let 프 be a standard operator algebra on a complex Banach space 픛 , dim ⁡ 픛 > 1 , and p n ⁢ (T 1 , T 2 , ... , T n) the (n - 1) th-commutator of elements T 1 , T 2 , ... , T n ∈ 프 . Then every map ξ : 프 → 프 (not necessarily linear) satisfying ξ ⁢ (p n ⁢ (T 1 , T 2 , ... , T n)) = ∑ i = 1 n p n ⁢ (T 1 , T 2 , ... , T i - 1 , ξ ⁢ (T i) , T i + 1 , ... , T n) for all T 1 , T 2 , ... , T n ∈ 프 is of the form ξ = Ω + Γ , where Ω : 프 → 프 is an additive derivation and Γ : 프 → ℂ ⁢ I is a map that vanishes at each (n - 1) th-commutator p n ⁢ (T 1 , T 2 , ... , T n) for all T 1 , T 2 , ... , T n ∈ 프 . In addition, if the map ξ is linear and satisfies the above relation, then there exist an operator S ∈ 프 and a linear map Γ : 프 → ℂ ⁢ I satisfying Γ ⁢ (p n ⁢ (T 1 , T 2 , ... , T n)) = 0 for all T 1 , T 2 , ... , T n ∈ 프 , such that ξ ⁢ (T) = [ T , S ] + Γ ⁢ (T) for all T ∈ 프 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate local Lie derivations of a certain class of generalized matrix algebras and show that, under certain conditions, every local Lie derivation of a generalized matrix algebra is a sum of a derivation and a linear central-valued map vanishing on each commutator. The main result is then applied to full matrix algebras and unital simple algebras with nontrivial idempotents.

Let 프 be a triangular ring and let p n ⁢ (U 1 , U 2 , ... , U n) denote the (n - 1) th commutator of elements U 1 , U 2 , ... , U n ∈ 프 . Suppose that ℕ is the set of nonnegative integers and 픏 = { ξ r } r ∈ ℕ is a sequence of additive mappings on 프 such that ξ 0 = i ⁢ d 프 , the identity mapping on 프 , and for each r ∈ ℕ , ξ r ⁢ (p n ⁢ (U 1 , U 2 , ... , U n)) = ∑ i 1 + i 2 + ⋯ + i n = r p n ⁢ (ξ i 1 ⁢ (U 1) , ξ i 2 ⁢ (U 2) , ... , ξ i n ⁢ (U n)) for all U 1 , U 2 , ... , U n ∈ 프 with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 . In this paper, it is shown that under certain conditions 픏 = { ξ r } r ∈ ℕ has the standard form, that is, there exist a higher derivation { d r } r ∈ ℕ on 프 and a family { h r } r ∈ ℕ of additive mappings h r : 프 → 풵 ⁢ (프) satisfying h r ⁢ (p n ⁢ (U 1 , U 2 , ... , U n)) = 0 for all U 1 , U 2 , ... , U n ∈ 프 with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 such that for each r ∈ ℕ , ξ r ⁢ (U) = d r ⁢ (U) + h r ⁢ (U) for all U ∈ 프 . [ABSTRACT FROM AUTHOR]

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I1, and Lm:A→A is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ϕm:A→A and an additive higher map ζm:A→Z(A), which annihilates every (n−1)th commutator pn(S1,S2,⋯,Sn) with S1S2=0 such that Lm(S)=ϕm(S)+ζm(S)forallS∈A. We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results.

In this paper, we prove that every local Lie derivation on von Neumann algebras is a Lie derivation; and we show that if M is a type I von Neumann algebra with an atomic lattice of projections, then every local Lie derivation on LS(M) is a Lie derivation.

Let T = T r i (A , M , B) be a triangular algebra where A is a unital algebra, B is an algebra which is not necessarily unital, and M is a faithful (A , B )-bimodule which is unital as a left A -module. In this paper, under some mild conditions on T , we show that if ϕ : T → T is a linear map satisfying A , B ∈ T , A B = P ⟹ ϕ ([ A , B ]) = [ A , ϕ (B) ] = [ ϕ (A) , B ] , where P is the standard idempotent of T , then ϕ = ψ + γ where ψ : T → T is a centralizer and γ : T → Z (T) is a linear map vanishing at commutators [A, B] with A B = P whrere Z (T) is the center of T . Applying our result, we characterize linear maps on T that behave like generalized Lie 2-derivations at idempotent products as an application of above result. Our results are applied to upper triangular matrix algebras and nest algebras. [ABSTRACT FROM AUTHOR]

This article is about the study of n-Lie derivations on triangular algebras under certain restrictions and it is shown that every n-Lie derivation apparently split into an extremal nderivation and an n-linear central map on triangular algebras. In addition, we implement this result to the classical examples of triangular algebras. [ABSTRACT FROM AUTHOR]

Let U be a unital algebra over a unital commutative ring R and M be a U -bimodule. A trivial extension algebra U ⋉ M is defined as an R -algebra with the usual operations of the R -linear space U ⊕ M and the multiplication defined by (u 1 , m 1) (u 2 , m 2) = (u 1 u 2 , u 1 m 2 + m 1 u 2) for all u 1 , u 2 ∈ U , m 1 , m 2 ∈ M. In this article, the structure of Lie triple derivations on U ⋉ M are given, and it is shown that under some mild restrictions every Lie triple derivation on U ⋉ M is of the form δ + τ , where δ : U ⋉ M → U ⋉ M is a derivation and τ : U ⋉ M → Z (U ⋉ M) is an R -linear mapping. [ABSTRACT FROM AUTHOR]

Let R be a ring with the center Z (R) containing a nontrivial idempotent. Suppose p n (X 1 , X 2 , ⋯ , X n) is the polynomial defined by n noncommuting indeterminates X 1 , ⋯ , X n and their multiple Lie products. In this article, under a lenient condition on R , it is shown that if a mapping L : R → R satisfies L (p n (A 1 , A 2 , ⋯ , A n)) = ∑ k = 1 n p n (A 1 , ⋯ , A k - 1 , L (A k) , A k + 1 , ⋯ , A n) , for all A 1 , A 2 , ⋯ , A n ∈ R and n ≥ 2 be a fixed positive integer, then for all A , B ∈ R , there exists Z A , B (depending on A and B) in Z (R) such that L (A + B) = L (A) + L (B) + Z A , B . [ABSTRACT FROM AUTHOR]

Let N be the set of nonnegative integers and A be a (n - 1) -torsion free triangular algebra over a commutative ring R . In the present paper, under some mild assumptions on A , it is prove that if δ : A → A is an R -linear mapping satisfying δ (p n (X 1 , X 2 , ⋯ , X n)) = ∑ i = 1 i = n p n (X 1 , X 2 , ⋯ , X i - 1 , δ (X i) , X i + 1 , ⋯ , X n) for all X 1 , X 2 , ⋯ , X n ∈ A with X 1 X 2 = 0 (resp. X 1 X 2 = P , where P is a nontrivial idempotent of A ), then δ = d + τ ; where d : A → A is a derivation and τ : A → Z (A) (where Z (A) is the center of A ) is an R -linear map vanishing at every (n - 1) -th commutator p n (X 1 , X 2 , ⋯ , X n) with X 1 X 2 = 0 (resp. X 1 X 2 = P ). [ABSTRACT FROM AUTHOR]

Let $ \aleph $ be a factor von Neumann algebra with $ dim > 1 $ that operates on a Hilbert space. Within the manuscript, we let out the characterization of Lie type derivation on factor von Neumann algebra of zero product as well as at projection product and notice that it has standard form.

Let N be a nest on a complex separable Hilbert space H and A l g N be the associated nest algebra. In this paper, we prove that every bilocal Lie derivation from A l g N into itself is of the form A → [ A , T ] + λ A + f (A) , where T ∈ A l g N , λ ∈ C and f : A l g N → C I is a linear map vanishing on each commutator. Moreover, we show that every bilocal Lie derivation from A l g N into itself is a Lie derivation if N is a non-atomic nest or there exists an atom E of N with dim ⁡ E > 1. [ABSTRACT FROM AUTHOR]

This investigation is carried out by the inspiration of noteworthy results pertaining to approximate Lie ⋆ -derivations connected with various additive, quadratic, cubic, quartic functional equations. For the first time, we study the stabilities of reciprocal Lie ⋆ -derivations and reciprocal-quadratic Lie ⋆ -derivations in the setting of normed ⋆ -algebras through direct method. The stabilities associated with different upper bounds are also discussed. In addition, we present the relationship of reciprocal derivations dealt in this study with automorphism. The comparative study of stability results obtained in this investigation is also discussed at the end of this paper. [ABSTRACT FROM AUTHOR]

Let T = T 3 (R i , M ij) be a triangular 3-matrix ring. In the present paper, we study of multiplicative Lie triple derivation on triangular 3-matrix rings and prove that every multiplicative Lie triple derivation on triangular 3-matrix rings can be written as a sum of an additive derivation and a center valued map vanishing at each second commutator. [ABSTRACT FROM AUTHOR]

Let A be a factor von Neumann algebra acting on a complex separable Hilbert space H with dim (A) > 9. In this paper, we prove that every bilocal Lie derivation from A into itself is a Lie derivation. [ABSTRACT FROM AUTHOR]

In this paper we initiate the study of mapping problems on triangular algebras without assuming unity. We give the definitions of strong faithful bimodules, extreme centres, and extreme Lie derivations. Our main result is to give a description of Lie derivations on triangular algebras without assuming unity. As an application we shall give a characterization of Lie derivations on upper triangular matrix algebras over a faithful algebra. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity and G = G (A , M , N , B) be a generalized matrix algebra. In this article, we give the structure of Lie triple derivation L on a generalized matrix algebra G and prove that under certain appropriate assumptions L on G is proper, i.e., L = δ + χ , where δ is a derivation on G and χ is a mapping from G into its center Z (G) which annihilates all second commutators in G , i.e., χ ([ [ x , y ] , z ]) = 0 for all x , y , z ∈ G. [ABSTRACT FROM AUTHOR]

We investigate functions that satisfy the condition defining generalized Lie derivations only asymptotically approximately, in a neighbourhood of the origin in a Banach algebra. We show that, under suitable assumptions, they are close to generalized Lie derivations. [ABSTRACT FROM AUTHOR]

In this paper, using a decomposition formula for any derivations in the coframe bundle, a complete and horizontal lifts of derivations from manifold to its coframe bundle are determined and their properties are studied. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let K be a two-torsion free associative ring with identity. We give a description of the Lie derivations of R = R n (K , J) = N T n (K) + M n (J) , 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. [ABSTRACT FROM AUTHOR]

Let S be a unital ring in which 2 is invertible, and let be the quaternion ring over S. In this paper, we describe the Lie derivations and generalized Lie derivations of R, we show that if S is commutative or semiprime, then every Lie derivation (resp. generalized Lie derivation) of R decomposes into the sum of a derivation (resp. generalized derivation) and an additive central map that vanishes on all commutators of R. [ABSTRACT FROM AUTHOR]

In this paper, we investigate homomorphisms in proper Lie CQ*-algebras and derivations on proper Lie CQ*-algebras associated with the following Pexiderized functional equation f(x + y) = f0(x) + f1(y). Moreover, we prove the Hyers-Ulam stability of homomorphisms in proper Lie CQ*-algebras and of derivations on proper Lie CQ*-algebras. [ABSTRACT FROM AUTHOR]

Let be an associative ring. We characterize rings with simple Lie ring of all Lie derivations, reduced noncommutative Noetherian ring with the simple Lie ring of all derivations and obtain some properties of -torsion-free rings with the simple Lie ring of all Jordan derivations. [ABSTRACT FROM AUTHOR]

Without the faithful assumption, we prove that every Jordan derivation on a class of path algebras of quivers without oriented cycles is a derivation and that every Lie derivation on such kinds of algebras is of the standard form. [ABSTRACT FROM AUTHOR]

Our main purpose in this paper is to resolve, in a rational homotopy theory context, the following open question asked by S. Theriaul: given a topological space X, what one may say about the nilpotency of aut1.(X) when the cocategory of its classifying space Baut1.(X) is finite? Here aut1.(X) denotes the path component of the identity map in the set of self homotopy equivalences of X. More precisely, we prove that HnilQ.aut1.(X) 6 cocatQ.Baut1.(X); when X is a simply connected CW-complex of finite type and that the equality holds when Baut1.(X) is coformal.Many intersections with other popular open questions will be discussed. [ABSTRACT FROM AUTHOR]

Letbe a field andQa finite simply laced quiver without oriented cycles. Firstly, we prove that each Lie derivation of the generalized one-point extension of path algebrais of the standard form, and the standard decomposition is unique. Secondly, we calculated the-dimensions of all the Hochschild cohomology of the generalized one-point extension oftype path algebra, and gave a description of the cup product in the Hochschild cohomology ring. [ABSTRACT FROM PUBLISHER]

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie⁻Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky⁻Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler⁻Kostant⁻Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky⁻Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky⁻Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky⁻Novikov algebras, including their fermionic version and related multiplicative and Lie structures.

Let X be a locally finite preordered set, R a commutative ring with identity and I ( X , R ) the incidence algebra of X over R . In this note we prove that each Lie derivation of I ( X , R ) is proper, provided that R is 2-torsion free. [ABSTRACT FROM AUTHOR]

We prove a result on hyperstability (in normed spaces) of the equation that defines the p-Wright affine functions and show that it yields a simple characterization of complex inner product spaces. We also obtain in this way some inequalities describing derivations, Lie derivations and Lie homomorphisms. [ABSTRACT FROM AUTHOR]

Let N be a nest on a Hilbert space H and Alg N be the associated nest algebra. We prove that every local Lie derivation from Alg N into B ( H ) is a Lie derivation. [ABSTRACT FROM AUTHOR]

Letbe a unital algebra with a nontrivial idempotentover a unital commutative ring. We show that under suitable assumptions, every Lie triple derivationonis of the form, whereis a derivation of,is a singular Jordan derivation ofandis a linear mapping fromto its centrethat vanishes on. As an application, we characterize Lie triple derivations and Lie derivations on triangular algebras and on matrix algebras. [ABSTRACT FROM AUTHOR]

Let X be a Banach space of dimension > 2. We show that every local Lie derivation of B( X) is a Lie derivation, and that a map of B( X) is a 2-local Lie derivation if and only if it has the form $${A \mapsto AT - TA + \psi(A)}$$ , where $${T \in B(X)}$$ and ψ is a homogeneous map from B( X) into $${\mathbb{F}I}$$ satisfying $${\psi(A + B) = \psi(A)}$$ for $${A, B \in B(X)}$$ with B being a sum of commutators. [ABSTRACT FROM AUTHOR]

Let p: E . B be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut1 (p) denote the identity component of the space of all fibre-homotopy self-equivalences of p. Let Baut1 (p) denote the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut1(p), connecting the results of recent work by the authors and others. We use this model to give classification results for the rational homotopy types represented by Baut11(p) and also to obtain conditions under which the monoid aut11(p) is a double loop-space after rationalization. [ABSTRACT FROM AUTHOR]

Let ℛ be a factor von Neumann algebra with dimension > 4. In this article, we prove that if δ : ℛ → ℛ is a linear map satisfying for any x, y ∈ ℛ with xy = 0 (resp., xy = p, where p is a fixed nontrivial projection of ℛ), then δ = d + τ, where d is a derivation of ℛ and τ : ℛ → 𝒞I (where 𝒞 is the field of complex numbers) is a linear map vanishing at commutators [x, y] with xy = 0 (resp., xy = p). [ABSTRACT FROM PUBLISHER]

To give a Z3-graded Cartan calculus on the extended quantum plane, the noncommutative differential calculus on the extended quantum plane is extended by introducing inner derivations and Lie derivatives. [ABSTRACT FROM AUTHOR]

Abstract: We introduce the notion of a multiplicative Lie n-derivation of a ring, generalizing the notion of a Lie (triple) derivation. The main goal of the paper is to consider the question of when do all multiplicative Lie n-derivations of a triangular ring have the so-called standard form. The main result is applied to the classical examples of triangular rings: nest algebras and (block) upper triangular matrix rings. [Copyright &y& Elsevier]