超有限II_1型因子及其算子的若干问题

This project focuses on completely bounded representations from a C* algebra A to the hyperfinite type II_1 factor R, strongly irreducible operators in R and indexes for irreducible subfactors. It has two main topics:.1. The similarity problem for bounded representations on C* algebras proposed by Kadison is an important subject in the field of operator algebras. It has a close relation to several important problems. Inspired by this problem, we find that it is interesting to consider an analogue question for every completely bounded representation φ from a C* algebra A to the hyperfinite type II_1 factor R. That is whether there exists an invertible operator T in R such that φ is similar to a *-representation..2. Subfactor theory initiated by Vaughn Jones in the 1980’s is an important subject in von Neumann algebras. It also has a close relation with several other subjects such as knot theory, conformal field theory, operator theory, Lie algebras, group theory and vertex operator algebras. In the research around the hyperfinite type II_1 factor R, one important question is to characterize the Jones index for irreducible subfactor. We find a class of strongly irreducible operators in R. It is interesting that the von Neumann algebra generated by a strongly irreducible operator in R is automatically an irreducible subfactor. This leads us to a new method to calculate the Jones index for irreducible subfactor in R. We plan to compute the subfactor indexes with respect to strongly irreducible operators and figure out the relation between them. We also consider the stability of finitely strongly irreducible decompositions for operators in R up to similarity. Meanwile we compute the Brown spectra for strongly irreducible operators in R.

本项目研究与超有限II_1型因子R有关的完全有界表示、强不可约算子以及不可约子因子的若干问题。主要内容有两个方面：.1. C*代数上有界表示的相似性问题是Kadison提出的关于算子代数的重要研究内容。受此启发，若φ是从C*代数A映入R的完全有界表示，是否存在R中的可逆算子使φ相似于*-表示？我们希望在这一问题上取得进展。.2. 子因子理论在上世纪八十年代由Vaughn Jones创立，是算子代数目前发展的重要方向之一，该理论与算子理论、低维拓扑、量子场论等数学分支的联系紧密，并与这些分支结合产生了诸多新的研究内容。我们发现R中的强不可约算子生成的von Neumann代数天然是不可约子因子，这为研究不可约子因子的指标提供了新思路。我们拟在R中构造更多的强不可约算子，计算相应的子因子指标，探讨强不可约算子与子因子指标之间的联系，刻画R中算子的有限强不可约分解的相似唯一性及其Brown谱

本项目主要研究算子代数中的表示理论与算子结构理论。我们考虑了从C*代数到von Neumann代数的表示的相关问题，围绕不可约表示与可约表示展开研究。同时考虑了与之密切相关的算子结构问题与分类问题。具体的，我们证明了如下内容:.1. 刻画了II_1 型中一类不可约算子的Brown谱；.2. 证明了几类经典的II_1型因子可由两个满足u^2=v^3=1的酉算子生成；.3. 运用K理论，对B(H)中可写成强不可约算子直接积分的一类算子进行分类；.4. 在I_n型von Neumann代数中，刻画了算子的强不可约分解的存在性的一个自然的充分必要条件，并运用K理论对这类代数中的算子进行相似分类；.5. 研究了一类Banach代数可逆元群之间等距的稳定性问题。.以上结论为我们进行下一阶段的研究工作奠定了基础。