Von Neumann代数理论是泛函分析的重要研究对象，而因子是构成von Neumann代数的基石。与群相关的群von Neumann代数是因子理论的基本例子，也是其研究的中心。本项目主要研究与自由Burnside群相关的群von Neumann代数。本项目将首先用群论的方法和技巧确定自由Burnside群是否是ICC群，然后再综合利用中心序列代数、自由概率论、von Neumann代数理论的技巧和方法来研究自由Burnside群的群von Neumann代数的Connes嵌入问题，讨论其Laplacian子代数是否是一个MASA，刻画这类群von Neumann代数与自由群因子的关系，判断这类群von Neumann代数是否是素因子，刻画具有相同个数生成元但不同阶的自由Burnside群von Neumann代数之间的关系。

The theory of von Neumann algebras is the center of functional analysis, and the theory of factors is the fundamental of the theory of von Neumann algebras. The basic and most important examples of von Neumann algebras are the group von Neumann algberas. In this project, we mainly study the group von Neumann algebras associated with the free Burnside groups. Firstly, we will try to answer the basic question: is the free Burnside group an ICC group? Then by combining the theory of central sequence algebras and free probabity theory and the theory of von Neumann algebras, we will investigate the Connes' immbedding problem associated with the group von Neumann algebras of the free Burnside group, try to know whether the Laplacian subalgebra is a MASA of the group von Neumann algebra, describe the relationship between the group von Neumann algebras and the free group factors, answer the question whether the group von Neumann is prime and discuss the automorphism problem of the group von Neumann algebras on the same numbers of generators but with different orders.