套子代数的Hochschild上同调及套的分类

项目名称： 套子代数的Hochschild上同调及套的分类

项目编号： No.11471199

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 张建华

作者单位： 陕西师范大学

项目金额： 70万元

中文摘要： 本项目主要研究因子von Neumann代数中套子代数的Hochschild上同调及套的分类问题。以Haagerup张量积和算子空间理论为基础，利用Stinespring表示定理，给出完全有界 n-上循环和完全有界模线性映射的结构，解决套子代数的完全有界Hochschild上同调群。通过完全有界与有界Hochschild上同调群之间关系的研究，得到套子代数的有界Hochschild上同调群，并应用于其代数结构的稳定性研究之中。以Ⅱ无限型因子中套子代数到其相对紧理想的一阶Hochschild上同调群的研究为基础，探讨Ⅱ无限型因子中套的相对紧扰动的特征，建立套的相对紧扰动与近似酉等价以及套的相似性之间的关系，进而获得Ⅱ无限型因子von Neumann代数中套的相似分类的特征。通过本项目的研究，以期能充实von Neumann代数中套子代数的理论，并对非自伴算子代数的研究产生积极的影响。

中文关键词： 算子代数；算子理论；希尔伯特空间

英文摘要： In the project, we mainly study Hochschild cohomology of nest subalgebras and classification nests in factor von Neumann algebras. Base on the theories of Haagerup tensor product and operator space, applying Stinespring representation theorem, we give the structures of completely bounded n-cocycles and completely bounded modular linear maps on nest subalgebras of factor von Neumann algebras, and then solve their completely bounded Hochschild cohomology groups. From the study of the relation between completely bounded Hochschild cohomology groups and bounded Hochschild cohomology groups, we obtain bounded Hochschild cohomology groups of nest subalgebras of factor von Neumann algebras, and discuss the stablity of algebraic structures of nest subalgebras of von Neumann algebras. Base on the study of one-order Hochschild cohomology groups of nest subalgebras with coefficients in the compact ideal relatived to a type Ⅱ infinite factor von Neumann algebra, we find a characterization of relative compact perturbation of two nests in the factor von Neumann algebra, build the relation between relative compact perturbation and approximately unitarily equivalence as well as similarity of nests in a type Ⅱ infinite factor von Neumann algebra，and then obtain a characterization of similarity of nests in a type Ⅱ infinite factor von Neumann algebra．We wish the present project can enrich the theory of nest subalgebras of von Neumann algebras, and can bring active effect for non-self-adjoint operator algebras.

英文关键词： operator algebra;operator theory;Hilbert space