【作者】 马飞；

【导师】 吉国兴；

【作者基本信息】 陕西师范大学 ， 基础数学， 2007， 硕士

【摘要】 算子代数理论产生于20世纪30年代，随着这一学科的迅速发展，它已成为现代数学中的一个热门分支，它与量子力学，非交换几何，线型系统和控制理论，甚至数论以及其他一些重要数学分支都有着出人意料的联系和相互渗透。为了进一步探讨算子代数的结构，近年来，国内外许多学者对算子代数上的线性映射进行了深入研究，并不断提出新的思路。例如：局部映射，双局部映射，线性保持问题等概念先后被引入和研究。目前这些映射已经成为研究算子代数不可或缺的重要工具。而在众多代数中，套代数是一类非常重要的非自伴非半素的算子代数，它的无限维情形比较复杂，但其有限维模型是上三角矩阵块代数。本文主要对上三角矩阵代数到其双模上的广义Jordan导子，广义导子和广义反导子的一些性质进行了研究，具体内容如下：第一章主要介绍了本文中要用到的一些符号，定义以及本文要用到的一些已知结论和定理。第二节我们介绍了广义Jordan导子，广义导子，素环，半素环等概念。第三节主要给出了本文中所需的一个命题。第二章首先对一个半素的复的赋范~*-代数A上的线性映射δ进行了研究，证明了如果对任意的投影p∈P_A，都有δ(p)=pδ(p)+δ(p)p-pδ(I)p，那么对于任意的ω∈D_A，有δ(ω)=ωδ(ω)+δ(ω)ω—ωδ(I)ω。另外，如果D_A在H_A中是稠密的，且δ是连续的，那么δ是A上的一个广义Jordan导子，进而是一个广义导子。其次证明了从一个非交换的含单位元I的单环到其2和3非挠的代数双模上的广义Jordan导子是一个广义导子。第三章主要证明了从2-非挠的上三角矩阵代数到其2-非挠的双模上的广义Jordan导子可以分解成一个广义导子和反导子的和，并且在定义了对角线上的反导子为零的情况下，分解是唯一的。第四章中我们首先引入了广义反导子的概念，并且对广义反导子的一些性质进行了讨论，证明了从上三角矩阵代数到其一些矩阵双模上不存在真的广义反导子。更多还原

【Abstract】 The study of operator algebra theory began in 1930s. With thefast development of the theory, now it has become a hot branch playing the roleof an initiator in modern mathematics. It has unexpected relations and mutualpenetration with quantum mechanics, noncommutative geometry, linear system andcontrol theory, indeed number theory as well as some other important branches ofmathematics. In order to discuss the structure of operator algebras, in recent years,a lot of scholars both here and abroad have focused on linear mappings of operatoralgebras and have been introduced more and more new methods. For instance, localmappings, bilocal mappings, linear preserving problems and so on were introducedsuccessively, at present time these mappings have become important tool in studyingoperator algebras. Among all kinds of operator algebras, nest algebras is a kindof very important non-selfadjoint and non-semiprime algebra. When the algebra isinfinity dimention, it is very complicated. The model of finity dimentional is trianglematrix algebra. In this paper, we mainly dicuss some characteristics of generalizedJordan derivation, generalized derivation and generalized antiderivation. The detailsas following:In Chapter 1, some notations, definitions are introduced and some theoremsare given. In section 2, some concopts are introduced, such as the definition ofgeneralized Jordan derivation, generalized derivation, prime and semiprime ring. Insection 3, we give a given theorem that we Will use in this paper.In Chapter 2, we first discuss a linear mapping on semiprime normed complex~*-algebra, and we have proved that if a linear mappingδfrom A into its bimodulesatisfiedδ(p)= pδ(p)+δ(p)p-pδ(I)p, for all p∈P_A, thenδ(ω) =ωδ(ω)+δ(ω)ω-ωδ(I)ω, for allω∈D_A. Moreover, if D_A dense in H_A, andδis continuous, thenδisa generalized derivation. Moreover, we proved that a generalized Jordan derivationfrom nonconmunity simple ring with identity operator I into its 2-torsion free and3-torsion free bimodule is a generalized derivation.In Chapter 3, we discuss generalized Jordan derivation from 2-torsion free tri-angular matrix into its 2-torsion free bimodule is the sum of a generalized derivaton and an antiderivation, and proved that if we define antidervation is zero on diagonalmatrics, then the factorzation is unquieness.In Chapter 4, we first present the notion of generalized antiderivation anddiscuss some result of generalized antiderivation, and prove that there is no propergeneralized antiderivation from upper triangular matrix algebra into its some matrixbimodule.更多还原