有限因子中几何对象的研究

项目名称： 有限因子中几何对象的研究

项目编号： No.11301511

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 袁巍

作者单位： 中国科学院数学与系统科学研究院

项目金额： 22万元

中文摘要： 算子代数的主要研究对象是C*代数和von Neuamnn代数。长期以来，算子代数学家一直将C*代数视为非交换的拓扑空间，而将von Neumann代数理解为非交换的测度空间。本领域中的许多概念都能在其它的数学分支中找到源头。而历史也向我们表明，将经典工具非交换化往往能对算子代数这一学科的发展产生推动。2008年，葛力明和袁巍引入了KS-格的概念，即von Neumann代数的极小生成自反格。他们在研究过程中发现了一类具有自然几何结构的格。本项目旨在通过对此类格的几何性质的探讨来加深对von Neumann代数结构的理解。同时，我们也希望能够将更多的几何概念引入到von Neumann代数的研究之中来。为此，我们将广泛借鉴其它数学领域如：拓扑，几何中的工具及方法来开展研究。

中文关键词： 因子；自由群；KS-代数；KS-格；张量范畴

英文摘要： Operator algebras are mainly about C*-algebras and von Neumann algebras. C*-algebras and von Neumann algebras have long been considered as noncommutative topological spaces and noncommutative measure spaces respectively by operator algebraists. Many concepts in these theories have their origin in other branches of mathematics. And the history also shows us that extending the classical tools to the noncommutative situation can often shed new light on the theory of operator algebras. In 2008, Liming Ge and Wei Yuan introduced KS-lattices which are the minimally generating reflexive subspace lattices of von Neumann algebras. In their study, they find a class of KS-lattices that carry canonical geometrical structures. The major goal in our proposed research is to understand the structure of finite von Neumann algebra by studying these geometric objects in von Neumann algebras. We also hope that we can introduce more geometrical concepts that will help us to understand the structure of von Neumann algebras. In order to achieve this, we shall employ methods and results from other branches of mathematics, especially topology and geometry, in our studies.

英文关键词： factor；free group；KS-algebra；KS-lattice；tensor category