Nonlinear Jordan Higher Derivations of Triangular Algebras

In this paper, we prove that any nonlinear Jordan higher derivation on triangular algebras is an additive higher derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algebras over infinite dimensional Hilbert suaces is inner.

Nonlinear Jordan Triple Derivations of Triangular Algebras

In this paper, it is proved that every nonlinear Jordan triple derivation on triangular algebra is an additive derivation.

Certain Pair of Derivations on a Triangular Algebra

By using properties of triangular algebra, we prove that if derivations D and G on a triangular algebra T satisfy certain generalized identities, then both D and G are zero mappings. As a corollary we get that if D and G are cocentralizing on T, then both D and G are zero mappings.

Jordan Higher Derivable Maps on Triangular Algebras by Commutative Zero Products

In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained.

CHARACTERIZATION OF DERIVATIONS ON B(X) BY LOCAL ACTIONS

Let A be a unital algebra and M be a unital .A-bimodule. A linear map δ ： A →M is said to be Jordan derivable at a nontrivial idempotent P ∈A if δ（A） o B ＋ A o δ（B） =δ（A o B） for any A,B ∈ .4 with A o B = P, here A o B = AB ＋ BA is the usual Jordan product. In this article, we show that if ,A = AlgAN is a Hilbert space nest Mgebra and M = B（H）, or A =M= B（X）, then, a linear mapδ： A→M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P∈ B（X） if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.

*-Lie Derivable Mappings on Von Neumann Algebras

In this paper,we prove that every*-Lie derivable mapping on a von Neu-mann algebra with no central abelian projections can be expressed as the sum of anadditive*-derivation and a mapping with image in the center vanishing at commuta-tors.

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(A^2) = 2 mAδ(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 mappingδ 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 + 2 mδ(B)A + 2 nAδ(B) + 2 nBδ(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 U = [A M N B] is a generalized matrix algebra, then every(m, n)-Jordan derivable mapping at zero from U into itself is equal to zero.

Characterizing Centralizers and Generalized Derivations on Triangular Algebras by Acting on Zero Product

Let U = Tri（fit, 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 = A,φ（B） = 0 whenever AB = 0） if and only if it is a centralizer. Let 5 ： U →U be an additive map. It is also shown that the following four conditions are equivalent： （1） 5 is specially generalized derivable at zero point, i.e., 5（AB） = δ（A）B ＋ AS（B） - Aδ（I）B whenever AB = 0; （2） 5 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

Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras

The additive （generalized） ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive （generalized） E-Lie derivation with ξ -if and only if it is an additive （generalized） derivation satisfying L（ξA） =- ξL（A） for all A. These results are then used to characterize additive （generalized） ξ-Lie derivations on several operator Mgebras such as Banach space standard operator algebras and yon Neumman algebras.

Local Jordan Derivations and Local Jordan Automorphisms of Upper Triangular Matrix Algebras

Let R be a commutative ring with identity, Tn （R） the R-algebra of all upper triangular n by n matrices over R. In this paper, it is proved that every local Jordan derivation of Tn （R） is an inner derivation and that every local Jordan automorphism of Tn（R） is a Jordan automorphism. As applications, we show that local derivations and local automorphisms of Tn （R） are inner.

Characterization of Lie Higher Derivations on Triangular Algebras

Let A and B be unital rings, and M be an （A, B）-bimodule, which is faithful as a left A-module and also as a right B-module. Let U = Tri（A,M, B） be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on U.

Local derivation on triangular subalgebras of hyperfinite von Neumann algebras

It is proved that every σ-weakly continuous local derivation from triangular subalgebra A of hyperfinite von Neumann algebra B into A (or B) is a derivation. Morevoer, if A is also a σ-Dirichlet subalgebra, each local derivation from A into A is an inner derivation.

Generalized*-Lie Higher Derivable Mappings on*-Rings

Let R be a*-ring with the center Z(R)and N be the set of nonnegative integers.In this paper,it is shown that if R contains a nontrivial self-adjoint idempotent which admits a generalized Lie higher derivable mapping△={G_(n)}_(n∈N)associated with a*-Lie higher derivable mapping L={L_(n)}_(n∈N),then for any X,Y in R and for each n in N there exists an element Z_(X,Y)(depending on X and Y)in the center Z(R)such that G_(n)(X+Y)=G_(n)(X)+G_(n)(Y)+Z_(X,Y).

Lie Triple Derivations of the Lie Algebra of Dominant Block Upper Triangular Matrices

Let N be the Lie algebra of all n x n dominant block upper triangular matrices over a field F. In this paper, we explicitly describe all Lie triple derivations of N when char（F） ≠ 2. As an application, we characterize Lie derivations of N when char（F） ≠ 2.

广义矩阵代数上的一类非线性局部可导映射

设G=G(A,M,N,B)是一个广义矩阵代数,∅:G→G是一个映射(无可加性假设).利用代数分解的方法,证明:如果对任意的X,Y∈G,且X和Y至少有一个是幂等元时,∅(XY)=∅(X)Y+X∅(Y)成立,则∅是G上的可加导子.

三角代数上的零点ξ-Lie可导映射

研究了三角代数上的零点ξ-Lie可导映射,证明了三角代数U上的每一个零点ξ-Lie(ξ≠1)可导映射δ都具有形式T→d(T)+δ(I)T,其中d:U→U是一个可加导子.作为应用,得到:上三角块矩阵代数T上的零点ξ-Lie(ξ≠1)可导映射具有形式T→TS-ST+Td+λT,其中S∈T,λ∈F,d是F上的可加导子且Td=(d(tij));套代数AlgN上的零点ξ-Lie(ξ≠1)可导映射具有形式T→TS-ST+λT,其中S∈AlgN,λ∈F.

三角代数上的ξ-Lie可导映射

设U=Tri(A,M,B)是含单位元I的三角代数并且φ:U→U是线性映射.利用代数分解的方法,证明了当三角代数U满足适当条件时,如果U,V∈U且UV=VU=I,有φ([U,V]ξ)=[φ(U),V]ξ+[U,φ(V)]ξ(ξ≠±1),则φ是导子.并得到了套代数上ξ-Lie可导映射的一个刻画.

三角代数上部分ξ-Lie可导映射的刻画

运用代数分解方法研究了三角代数U=Tri(A,M,B)上的部分ξ-Lie可导映射.证明了如果对任意A∈A存在整数k使得kIA-A可逆,则U上的线性映射为导子当且仅当它是部分ξ-Lie可导映射.作为应用,证明了非平凡套代数上的线性映射是内导子当且仅当其为部分ξ-Lie可导映射.

三角代数上的Jordan零点高阶ξ-Lie可导映射

设U=Tri(A,M,B)是一个2-无扰的三角代数,{φn}n∈N,是U上的一列线性映射.用代数分解方法证明:如果对任意n∈N,U,V∈U且U V=0,有φn([U,V]ξ)=∑i+j=n[φi(U),φj(V)]ξ,ξ≠0,±1,则{φn}n∈N,是一个高阶导子,其中[U,V]ξ=UV-ξVU为ξ-Lie积,U°V=UV+VU为Jordan积.并得到套代数上Jordan零点高阶ξ-Lie可导映射的具体形式.

三角代数上的局部广义李n导子

利用恒等式理论,证明了在一定条件下,三角代数T上的局部广义李n导子δ可以表示为δ=G+h,其中G:T→T为广义导子,h:T→Z(T)满足:对于任意的x_(1),x_(2),…,x_(n)∈T,有h(p_(n)(x_(1),x_(2),…,x_(n)))=0,其中pn为(n-1)-交换子.最后给出了上述结果的一个应用.