算子代数上保持高维数值域的映射

The numerical range originated from the study of quadratic form. Since the concept of the numerical range is very simple, it is a powerful tool to study operators and matrices. The subject is related and has applications to many different branches of pure and applied science such as operator theory, functional analysis, C*-algebras, systems theory, numerical analysis, , quantum physics, etc. Moreover, a wide range of mathematical tools including geometry, combinatorial theory, and computer programming are useful in the study. Therefore, it is of great value on theory and applications to study the numerical range of operators. As an extension of the numerical range, the higher dimensional numericcal range has been extensively studied. In the 1940s, L. K. Hua created the study of geometry of matrices. Motivated by the idea of geometry of matrices, we extended such problems to infinite dimensional case, that is, the general preserver problems of operator algebras : how to characterize maps on operator algebras or operator spaces by as few geometry or algebraic invariants as possible. This project will consider the structure of operator algebra and numerical range in the previous basis, through put forward new ideas and methods, in order to reveal the innate properties and of operator algebras and the inherent relations between the innate properties and the maps of operator algebras. We describe linear or nonlinear mappings of operator algebras via the numerical radius、 higher dimensional numerical radius and Lie product, Jordan η-*- product of higher dimensional numerical range.

算子的数值域起源于对二次型的研究，虽然算子的数值域概念简单，但是它是研究算子和矩阵的有力工具，与很多学科分支密切相关，例如算子理论,泛函分析，C*-代数，系统理论，数值分析和量子物理等，并在控制论等学科中得到了广泛的应用。因此研究算子的数值域有着重要的理论价值和应用价值。高维数值域是数值域自然的推广，吸引了一大批学者投身其中。20世纪40年代，华罗庚先生创建了矩阵几何，推广到无限维的情形，就是算子代数上的一般性保持问题。基本问题是寻找尽可能少的不变量来刻画算子代数间映射的结构。本项目将在前人基础上进一步探究算子代数的结构以及高维数值域,通过提出新的思路和方法，以此来揭示算子代数本身固有的性质以及与其上各种映射间的内在联系，主要以算子的数值半径、高维数值半径以及各种乘积，如 Lie 积、Jordan η-*积等的高维数值域为不变量，刻画算子代数间的线性或非线性映射。

算子代数上的乘法运算是种最基本的运算，其乘法保持问题也是算子代数研究中的热点问题。保持问题就是刻画算子代数间保持某一性质、函数、关系、子集等不变的线性映射或非线性映射，有助于了解算子代数的内部结构。它具有重要的理论价值，许多问题在量子力学、微分几何、微分方程、系统控制和数理统计等领域有着广泛的实际应用背景。本项目主要研究了一秩投影算子的数值域的相关性质以及高维数值域的相关性质；B(H)上保持算子Jordan *积和Lie积的高维数值域的映射，套代数上的Jordan结构。给出映射保持算子Jordan *积的高维数值域的充分必要条件；证明了满足一定条件下，套代数上的Jordan同态是同态或者反同态。