超有限II_1型因子是一类很重要的von Neumann代数，它的一类算子的谱来源于物理学家Kac提出问题并为此悬赏10瓶马蒂尼酒。本项目着重研究超有限II_1型因子中的一类算子uf(v)的下列理论问题：利用K理论的知识来计算K_1(uf(v)),K_0(uf(v))，并且深入研究这类算子生成C*-代数分类问题；利用不变子空间相关知识来进一步研究这类算子满足一定条件下非平凡不变子空间的存在性；对于单个元生成问题，利用函数演算和谱投影分解定理等理论来研究一类von Neumann代数是由三个投影生成的，进一步研究非超有限II_1型von Neumann代数由单个元生成问题。这些问题是超有限II_1型因子非常重要问题，故本课题的研究具有非常好的理论价值和实际价值，并可进一步完善超有限II_1型因子的相关体系。

The hyperfinite type II_1 factor is a kind of important von Neumann algebra. The spectrum of its operator comes from the physicist Kac who put forward the question and offered 10 bottles of martinis for it. This project focuses on the following theoretical problems of operator uf(v) in hyperfinite II_1 type factor: using the knowledge of K theory to calculate K_1(uf(v)), K_0(uf(v)), and deeply studying the classification of such operators; The existence of nontrivial invariant subspace is studied by using the knowledge of invariant subspace. For the generation of single element, the theory of function calculus and spectral projection decomposition theorem is used to study the generation of a type of von Neumann algebra by three projections, and the generation of non-hyperfinite II_1 type von Neumann algebra by single element is further studied. These problems are very important for hyperfinite II_1 type factors, so the research of this topic has very good theoretical value and practical value, and can further improve the related system of hyperfinite II_1 type factors.