完全保持斜Jordan零积和斜交换性映射的研究

发布时间：2018-01-28 18:34

本文关键词： 标准算子代数不定内积空间因子von Neumann代数斜Jordan零积斜交换性完全保持问题　出处：《太原科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：算子之间的斜Jordan零积、斜交换性等性质特征在数学领域、量子力学和密码学等领域中都有着广泛的实际应用背景.因此,越来越多的学者在保持问题框架下对算子之间的斜交换性等性质进行了研究.对保持问题的研究主要是对算子代数或算子空间上保持某种不变量(某种性质、子集或关系等)的映射进行研究,从而刻画出该映射的具体结构形式.在不同的算子空间或算子代数上对保持问题进行讨论,成为了泛函分析和算子代数理论上非常活跃的研究课题,且获得了一系列深刻的成果.近年来,随着研究的不断深入,完全保持问题的思想被众多的学者深入探讨.在已有的研究成果的基础上,本文以算子的斜Jordan零积和斜交换性作为不变量,分别在标准算子代数、不定内积空间的标准算子代数和因子von Neumann代数上进行研究.从而对保持这些不变量的一般映射进行刻画.本文的主要研究结果如下:1.讨论了无限维复的Hilbert空间上的*-标准算子代数之间双边完全保斜Jordan零积和斜交换性的一般映射,并证明了此类映射是同构或者是共轭同构的常数倍;2.研究了不定内积空间上的?-标准算子代数之间双边完全保不定斜Jordan零积和不定斜交换性的一般映射,结果表明此类映射是同构或者是共轭同构的常数倍;3.刻画了无限维复的Hilbert空间上的因子von Neumann代数之间双边完全保斜Jordan零积和斜交换性的一般映射,并且给出了映射的具体结构.
[Abstract]:The properties of oblique Jordan zero product and skew commutativity between operators have a wide range of practical applications in the fields of mathematics quantum mechanics and cryptography. More and more scholars have studied the properties of oblique commutativity between operators under the framework of maintenance problems. The study of conservation problems is mainly to preserve some invariants (some properties) on operator algebra or operator space. In this paper, we study the mapping of subsets or relations, so as to characterize the specific structural form of the mapping, and discuss the preserving problem in different operator spaces or operator algebras. It has become a very active research topic in functional analysis and operator algebra theory, and has obtained a series of profound results. On the basis of existing research results, this paper takes the oblique Jordan zero product and skew commutativity of operators as invariants, respectively, in the standard operator algebra. In this paper, we study the algebras of standard operators and factor von Neumann algebras of indefinite inner product spaces and characterize the general mappings that preserve these invariants. The main results of this paper are as follows:. 1. In this paper, we discuss the general mappings of two-sided completely skew preserving Jordan zero product and skew commutativity between two-sided completely skew Jordan algebras on infinite dimensional complex Hilbert spaces. It is proved that this kind of mapping is constant times of isomorphism or conjugate isomorphism. 2. In this paper, we study the indeterminate inner product space? General mappings between standard operator algebras which are completely indeterminate Jordan zero product and indeterminate commutativity. The results show that such mappings are constant times of isomorphism or conjugate isomorphism. 3. In this paper, we characterize the general mappings of two-sided completely oblique Jordan zero product and oblique commutativity between factor von Neumann algebras on infinite dimensional complex Hilbert spaces. The specific structure of the mapping is also given.
【学位授予单位】：太原科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177

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