On left Jordan derivations of rings and Banach algebras
On left Jordan derivations of rings and Banach algebras
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DOI：
10.1007/s00010-007-2872-z
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发表时间：
2008-06
影响因子：
0.8
通讯作者：
J. Vukman
中科院分区：
文献类型：
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作者：
J. Vukman
众所周知，复交换半单Banach代数上不存在非零线性导子。在本文中，我们证明了这一结果的下列推广。设A是复半单Banach代数，D：A→是满足关系式D(X2)=2xD(X)的线性映射，其中∈R满足D(X2)=2xD(X)。如果nx=0，x∈Rimpliesx=0，则环Ris-扭转自由，其中>；1是一个整数。通常，交换式−yx将由[x，y]表示。我们将对所有x，y，z∈R使用交换子恒等式[xy，z]=[x，z]y+x[y，z]和[x，yz]=[x，y]z+y[x，z]回想一下，如果a，b∈R，arb=(0)意味着a=0 orb=0，则环R是素数，并且在aRa=(0)的情况下是半素数。如果D(Xy)=D(X)y+xD(Y)对所有配对x，y∈R成立，则称加法映射Dd为导子，且称为若当(X2)=D(X)x+xD(X)对所有x∈R成立。显然，任何导子都是Jordan导子。通常情况下，情况并非如此。Herstein([8])证明了2-无挠素环上的任何Jordan导子都是导子(另见[1])。Cusack([5])将Herstein的结果推广到2-无挠半质环(另一种证明见[2])。若D(Xy)=yD(X)+xD(Y)对所有配对x都成立，则称D：R→R为左导子，y Banach随机数在Case(X2)=2xD(X)中称为左∈导子(或∈左导子)。
It is well known that there are no nonzero linear derivations on complex commutative semisimple Banach algebras. In this paper we prove the following extension of this result. LetAbe a complex semisimple Banach algebra and letD:A→Abe a linear mapping satisfying the relationD(x2) = 2xD(x) for allx∈R. In this caseD= 0.Throughout,Rwill represent an associative ring with centerZ(R). A ringRisn-torsion free, wheren> 1 is an integer, ifnx= 0,x∈Rimpliesx= 0. As usual the commutatorxy−yxwill be denoted by [x,y]. We shall use the commutator identities [xy,z] = [x,z]y+x[y,z] and [x,yz] = [x,y]z+y[x,z] for allx,y,z∈R. Recall that a ringRis prime if fora,b∈R,aRb= (0) implies that eithera= 0 orb= 0, and is semiprime in caseaRa= (0) implies thata= 0. An additive mappingDis called a derivation ifD(xy) =D(x)y+xD(y) holds for all pairsx,y∈R, and is called a Jordan derivation in caseD(x2) =D(x)x+xD(x) is fulfilled for allx∈R. Obviously, any derivation is a Jordan derivation. The converse is in general not true. Herstein ([8]) has proved that any Jordan derivation on a 2-torsion free prime ring is a derivation (see also [1]). Cusack ([5]) has generalized Herstein’s result to 2-torsion free semiprime rings (see [2] for an alternative proof). An additive mappingD:R→Ris called a left derivation ifD(xy) =yD(x) +xD(y) holds for all pairsx,y∈Rand is called a left Jordan derivation (or Jordan left derivation) in caseD(x2) = 2xD(x) is fulfilled for allx∈R. In this paper by a Banach algebra we mean a Banach algebra over the complex field.