具有一定代数特征的算子微分流形研究

算子理论和算子代数与微分几何的交叉研究吸引着国内外众多学者的广泛关注。本项目将微分几何与算子代数相结合，研究具有一定代数特征的算子之集形成的算子微分流形。从微分几何的视角，考察分析算子代数中Banach-Lie群的群作用，研究Grassmann流形乘积空间中具有一定代数特征的算子对之集的微分流形结构及其上的测地线；进一步，分析该类算子微分流形的拓扑和代数性质，一方面以算子的几何及谱特征研究这类算子微分流形的逼近问题，另一方面，刻画Grassmann流形乘积空间的子流形生成的算子代数结构，从微分几何的角度获得算子代数结构的信息；同时，应用微分流形中的几何特征和性质刻画由其生成算子代数的同构，为算子代数间的同构研究提供新信息。本项目的研究将有效地推动算子代数与微分几何的交叉发展，揭示算子微分流形中的几何特征对算子代数所产生的影响。

The cross-over study on operator theory, operator algebras and differential geometry has attracted much attention of many domestic and foreign experts. This project is based on the differential geometry and operator algebras, mainly studies the differential manifolds of operators with certain algebraic features. From the viewpoint of differential geometry, under the action of Banach-Lie group, we study the differential structures of pairs of the operators with certain algebraic features in the product space of Grassmann manifold, and characterize the geodesics; we further investigate its differential properties from the topology and algebra viewpoint. On the one hand, we study the approximation of differential manifolds by the geometric and spectral properties of operators, on the other hand, we characterize the algebra structures generated by submanifolds of the product space of Grassmann manifold, therefore we will obtain the structures of operator algebras from the viewpoint of the differential geometry; furthermore, we apply some geometric properties of differentiable manifolds to characterize the isomorphisms between the operator algebras, and provide new information for the isomorphisms on the operator algebras. This project will promote the cross development of operator algebras and differential geometry, and explore the effects of differential structures in operator algebras.