齐性空间是在李群作用下具有迁移性的微分流形，其核心是与微分流形间的微分同胚，李群作用提供了微分同胚的构造方法。本项目拟在算子代数的背景下，应用算子矩阵分块、算子逼近等技巧，借助谱理论和再生核理论，探讨新的研究思路和方法去构造无限维齐性空间，以及其上具有不变性的微分几何结构。主要内容包括: (1) 拟研究算子代数中基于李群作用下无限维齐性空间上的微分几何，重点考虑不同类型的Grassmann流形以及与其相关的代数量集合，在算子代数中具有重要结构或在其他学科领域有重要应用的代数量集合的微分几何结构。(2) 在(1)的研究基础上进一步考虑这类无限维齐性空间上的Hopf-Rinow定理。(3) 拟研究态空间的几何特征，同时给出GNS表示限制在算子代数李群上的几何构造。(4) 拟研究齐性空间的代数结构。本项目的研究希望揭示算子代数中无限维齐性空间的微分结构，为算子代数结构和分类研究提供新的信息。

A homogeneous space is a manifold on which a Lie group acts transitively. Its core is differential homeomorphism between it and differential manifold, and the action of Lie group provides a construction method for this differential homeomorphism. By use of block operator matrices technique and operator approximation, we shall explore new ideas and methods to construct infinite dimensional homogeneous spaces and their invariant differential geometric structures in operator algebras according to spectral theory and the theory of reproducing kernel. The main contents include: (1) We shall study differential geometry of infinite dimensional homogeneous spaces in operator algebras under the action of Lie groups, mainly consider the differential geometric structures of different types Grassmannnians and their associated algebraic elements, and algebraic elements having important structures in operator algebras or important applications in other areas. (2) Base on (1), we further consider Hopf-Rinow theorem on these infinite dimensional homogeneous spaces. (3) We shall focus on the geometric characteristics of state space, and give the geometric constructions of GNS representations restricted in Lie groups of operator algebras. (4) We shall study the algebraic structures of homogeneous spaces. The purpose of this project is to reveal the differential structures of infinite dimensional homogeneous spaces in operator algebras, and to provide new informations for the study of the structures and classification of operator algebras.