含幂等元的环上的Jordan导子的刻画-在零点Jordan可导的可加映射

设A为包含非平凡幂等元且有单位的环(或代数),δ:A→A是可加(或线性)映射.称δ在零点Jordan可导,若δ(A)B+Aδ(B)+δ(B)A+Bδ(A)=0对任意满足AB+BA=0的A,B∈A成立.在一定条件下,证明了δ在零点Jordan可导当且仪当存在可加Jordan导子τ,使得δ(A)=τ(A)+δ(I)A对任意的A∈A成立.利用此结论,完全刻画了因子von Neumann代数上在零点Jordan可导的可加映射.此外,还刻画了一般von Neumann代数和C*代数上在零点Jordan可导的有界线性映射.     

含幂等元的环上的Jordan导子的刻画—在零点Jordan可导的可加映射

设A为包含非平凡幂等元且有单位的环(或代数),δ:A→A是可加(或线性)映射.称δ在零点Jordan可导,若δ(A)B+Aδ(B)+δ(B)A+Bδ(A)=0对任意满足AB+BA=0的A,B∈A成立.在一定条件下,证明了δ在零点Jordan可导当且仅当存在可加Jordan导子τ,使得δ(A)=τ(A)+δ(I)A对任意的A∈A成立.利用此结论,完全刻画了因子von Neumann代数上在零点Jordan可导的可加映射.此外,还刻画了一般von Neumann代数和C~*代数上在零点Jordan可导的有界线性映射.     

矩阵代数上的拟三重Jordan可导映射

设R是一个含单位元的可交换2-无挠环,且M_n(R)是R上的n×n阶矩阵代数.本文证明了M_n(R)(n≥2)上的满足Φ(ABA)=Φ(A)BA+AΦ(B)A+ABΦ(A)的映射Φ具有形式:存在T∈M_n(R)和R上的一个可加导子φ,使得对任意A= (a_(ij))∈M_n(R),有Φ(A)=AT-TA+A_φ,这里A_φ=(φ(a_(ij))).     

$C^*$-代数上可加Jordan左$*$-导子的刻画

设$\delta$是一个$*$-代数$\mathcal A$到其左$\mathcal A$-模$\mathcal M$的可加映射, 如果对任意$A\in\mathcal A$, 有$\delta(A^2)=A\delta(A)+A^*\delta(A)$, 则称$\delta$~是一个可加Jordan左$*$-导子. 在本文中, 我们证明了复的单位$C^*$- 代数到其Banach左模的每个可加Jordan左$*$-导子都恒等于零. 设$G\in\mathcal A$, 如果对任意$A,B\in \mathcal A$, 当$AB=G$时, 有$\delta(AB)=A\delta(B)+B^*\delta(A)$, 则称$\delta$在$G$处左$*$-可导. 我们证明了复的单位$C^*$-代数到其Banach左模的在单位点处左$*$-可导的连续可加映射恒等于零.     

三角代数上Jordan高阶导子的刻画

设u=Tri(A,M,B)是含单位元I的三角代数,()={()_n}_(n∈N)是u上一簇线性映射.本文证明了:如果对任意U,V∈u且UV=VU=I,有()_n(UV+VU)=∑_(i+j=n)(()_i(U)_(()_j)(V)+()_i(V)()_j(U)),则()={()_n}_(n∈N)是u上高阶导子.作为应用,得到了套代数上Jordan高阶导子的一个刻画.     

Jordan李超代数的一些性质

代数A称为不可分解的,如果A不能分解成理想的直和.证明了满足C(L_o)=C(L)={0}的Jordan李超代数L一些重要性质.     

von Neumann代数上的Lie可导映射

设A是不含交换中心投影的von Neumann代数,投影P∈A使得P=0, P=I.称可加映射δ:A→A在Ω∈A Lie可导,若δ([A,B])=[δ(A,δ(B)],■A,B∈A,AB=Ω.该文证明,若Ω∈A满足PΩ=Ω,则δ在ΩLie可导当且仅当存在导子τ:A→A和可加映射f:A→Z(A)使得δ(A)=τ(A)+f(A),■A∈A其中f([A,B)=0,■A,B∈A,AB=Ω.特别地,若A是因子von Neumann代数,Ω∈A满足ker(Ω)≠0或ran(Ω)≠H,则可加映射δ:A→A在ΩLie可导当且仅当δ有上述形式.     

Jordan李超代数的分解再论

代数A称为不可分解的,如果A不能分解成理想的直和.满足C(Lo)=C(L)={0}的Jordan李超代数L能够分解成不可分解理想的直和,这种分解在不计理想次序的前提下是唯一的.并证明了完备Jordan李超代数的一些性质.     

Banach Jordan代数的有限维性特征

复数域 C 上的 Jordan 代数 A 是一个复向量空间,并具有一个双线性非结合乘法运算“(?)”满足关系式特别地,设 B 是一个结合代数,乘法运算为(a,b)→ab,如果定义a(?)b=1/2(ab ba),则 B~ =[B,(?)]就成为一个 Jordan 代数,称为特殊 Jordan 代数.一个 Banach Jordan 代数 A 是指一个复数域 C 上的 Jordan 代数,并装备了一个完备的范数成为 Banach 空间,其乘积的范数满足条件:     

三角代数上Jordan积为幂等元处的高阶ξ-Lie可导映射

设u=Tri(A,M,B)是三角代数,{φn}n∈N:u→u是一列线性映射.本文利用代数分解的方法,证明了如果对任意U,V∈u且U。V=P为标准幂等元,有φn([U,V]ξ)=Σi+j=n(φi(U)φj(V)-ξφi(V)φj(U))(ξ≠±1),则{φn}n∈N是一个高阶导子,其中φ0=id为恒等映射,UoV=UV+VU为Jordan积,[U,V]ξ=UV-ξVU为ξ-Lie积.     

Characterizations of Jordan derivations and Jordan homomorphisms

Let 𝒜 be a unital Banach algebra and ? be a unital 𝒜-bimodule. We show that if δ is a linear mapping from 𝒜 into ? satisfying δ(ST)?=?δ(S)T?+Sδ(T) for any S, T?∈?𝒜 with ST?=?W, where W is a left or right separating point of ?, then δ is a Jordan derivation. Also, it is shown that every linear mapping h from 𝒜 into a unital Banach algebra ? which satisfies h(S)h(T)?=?h(ST) for any S,?T?∈?𝒜 with ST?=?W is a Jordan homomorphism if h(W) is a separating point of ?.     

Jordan derivations of full matrix algebras

Let A be a unital associative ring and M be a 2-torsion free A-bimodule. Using an elementary and constructive method we show that every Jordan derivation from M_n(A) into M_n(M) is a derivation.     

Nearly generalized Jordan derivations

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.     

Jordan higher all-derivable points on nontrivial nest algebras

Let A be a Banach algebra with unity I and M be a unital Banach A-bimodule. A family of continuous additive mappings D=(δ_i)_i∈N from A into M is called a higher derivable mapping at X, if δ_n(AB)=∑_i+j=nδ_i(A)δ_j(B) for any A,B∈A with AB=X. In this paper, we show that D is a Jordan higher derivation if D is a higher derivable mapping at an invertible element X. As an application, we also get that every invertible operator in a nontrivial nest algebra is a higher all-derivable point.     

Multiplicative Jordan Decomposition in Group Rings of 3-Groups,II

In this paper, we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ?[G] satisfies the multiplicative Jordan decomposition (MJD). In the nonabelian case, we show that ?[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 3³ = 27.     

Positive solutions of nonlinear three-point boundary-value problems

Let a∈C[0,1], b∈C([0,1],(−∞,0]). Let φ₁(t) be the unique solution of the linear boundary value problem

u″(t)+a(t)u′(t)+b(t)u(t)=0,t∈(0,1),u(0)=0,u(1)=1.     

Norms of certain Jordan elementary operators

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. For A,B∈B(H), the Jordan elementary operator UA,B is defined by UA,B(X)=AXB+BXA, ∀X∈B(H). In this short note, we discuss the norm of UA,B. We show that if dimH=2 and ‖UA,B‖=‖A‖‖B‖, then either AB^∗ or B^∗A is 0. We give some examples of Jordan elementary operators UA,B such that ‖UA,B‖=‖A‖‖B‖ but AB^∗≠0 and B^∗A≠0, which answer negatively a question posed by M. Boumazgour in [M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl. 342 (2008) 386-393].     

Jordan zero-product preserving additive maps on operator algebras

Let Φ:A→B be an additive surjective map between some operator algebras such that AB+BA=0 implies Φ(A)Φ(B)+Φ(B)Φ(A)=0. We show that, under some mild conditions, Φ is a Jordan homomorphism multiplied by a central element. Such operator algebras include von Neumann algebras, C^∗-algebras and standard operator algebras, etc. Particularly, if H and K are infinite-dimensional (real or complex) Hilbert spaces and A=B(H) and B=B(K), then there exists a nonzero scalar c and an invertible linear or conjugate-linear operator U:H→K such that either Φ(A)=cUAU⁻¹ for all A∈B(H), or Φ(A)=cUA^∗U⁻¹ for all A∈B(H).     

Generalized Jordan left derivations on semiprime algebras

Let A{\mathcal{A}} be a semiprime algebra of characteristic not 2. Then any generalized Jordan left derivation on A{\mathcal{A}} is a generalized left derivation and is also a generalized derivation. This gives an affirmative answer to a question in Ashraf and Ali (Bull Korean Math Soc 45:253–261, 2008). Moreover, we prove that there are no nonzero generalized Jordan left derivations that take only nilpotent values on A{\mathcal{A}} .     

Jordan triple derivations on J-subspace lattice algebras

Let L be a J-subspace lattice on a Banach space X and Alg L the associated J-subspace lattice algebra. Let A be a standard operator subalgebra (i.e., it contains all finite rank operators in AlgL) of AlgL and M■B(X) the Alg L-bimodule. It is shown that every linear Jordan triple derivation from A into M is a derivation, and that every generalized Jordan (triple) derivation from A into M is a generalized derivation.