近十多年来，算子代数上映射的不变量研究得到许多学者的关注。特征值分析在许多学科中有着重要的应用，在应用数学中，研究者更加看重一个矩阵或算子逆的范数，这样pseudo谱应运而生。pseudo谱能否刻画算子？这将是本项目要探索的一个问题，作为应用，进而刻画以算子的各种乘积的pseudo谱为代数不变量的算子代数间非线性映射的结构；探索基本算子代数上保持算子Jordan乘积的peripheral谱的非线性映射的结构；以算子斜乘积的酉（相似）不变范数为几何不变量，刻画算子代数间非线性映射的结构；获得von Neumann代数上强保持斜Lie积的非线性映射的结构；探讨算子代数间保持按照因子交换的非线性映射的结构特征，进而给出保持按照因子乘积的酉相似不变泛函的非线性映射的刻画。应用于量子信息理论，探讨复合态的可分性问题，研究保持简单张量积凸组合的一般映射，进一步解决量子信息理论中提出的几个数学问题。

In recent years, invariants of maps on operator algebras have been extensively studied. Eigenvalue analysis has proved highly successful in many applications. It is well known that in applied mathematics, it is not always enough to ask whether an operator is singular or nonsingular; the norm may be important as well as the fact that it is finite. Thus, the pseudo spectrum is posed. Whether can an operator be characterized by the pseudo-spectrum? We will study such a problem in this project; furthermore, as applications, we will characterize nonlinear maps preserving the pseudo-spectrum of all kinds of operator products; characterize nonlinear maps preserving peripheral spectrum of Jordan products of operators; characterize nonlinear maps preserving unitarily invariant or unitarily similarily invariant of opertor skew products; characterize nonlinear maps preserving strong skew Lie products on von Neumann algebras; characterize nonlinear maps preserving commutativity up to a factor, and moreover, characterize nonlinear maps preserving unitarily invariant of products up to a factor. In applications to quantum information, we will study separablity of composite states, and study nonlinear maps preserving convex combination of simple tensor products, and furthermore, solve several mathematical problems posed in quantum informations.