Banach代数之间保单位线性映射的若干性质  

Some properties of the unit-preserving linear mapping between Banach algebras

作　　者：曹怀信[1] 戴时勋[2] 

机构地区：[1]西安交通大学理学院,陕西西安710049 [2]陕西师范大学数学与信息科学学院,陕西西安710062

出　　处：《西北大学学报（自然科学版）》2001年第6期461-464,共4页Journal of Northwest University（Natural Science Edition）

基　　金：国家自然科学基金资助项目 (197710 5 6 )

摘　　要：引入了代数的复同态分离性质 ,证明如果Φ是从有单位 Banach代数 A到有单位且具有复同态分离性质的 Banach代数 B中的保单位线性映射 ,则以下等价 :1Φ是保可逆映射 ;2Φ是保乘法映射 ;3Φ是保逆运算映射 ;4Φ是保平方映射 ;5Φ是谱压缩映射 ;6Φ是 Jordan同态。作为应用 ,证明了从 Banach代数到半单交换 Banach代数的保单位且保可逆的线性映射是自动连续的代数同态。最后 ,还证明了当 n不小于 2时 ,从矩阵代数 Mn(C)到任一具有复同态分离性质的代数的任一代数同态必为零。A separating property called complex-homormophism-separating-property (CHomSP) of an algebra is introduced. It is proved that if A and B are unital Banach algebras,B has CHomSP,Φ is a unit-preserving linear mapping from A into B,then the following four properties are equivalent:(a) Φ is invertibility-preserving; (b) Φ is multiplicativity-preserving; (c) Φ is inverse-preserving; (d) Φ is sqare-preserving; (e) Φ is spectrum-compressing; (f) Φ is a Jordan homomorphism. As a result,it is showed that a unit-preserving and invertibility-preserving linear mapping from a unital Banach algebra into a unital semi-simple commutative Banach algebra is a automatic continuous homormophism.It is proved that every homomorphism from the matrix algebra M n(C)(n≥2) into a algebra with CHomSP must be zero.

关 键 词：BANACH代数 保单位线性映射 保可逆映射 保乘法映射 保逆运算映射 保平方映射 谱压缩映射 Jordan同态 

分 类 号：O177.5[理学—数学]