Study on free probability and operator algebras

我学习了算子代数和与自由概率理论相关的话题。更确切地说，我得到了以下结果：(1)给出了任意自由积von Neumann代数为因子的充要条件。较强地刻画了它的中心分解，给出了I型部分的显式描述，证明了非I型部分总是出现并成为II1或III型因子，并给出了确定该因子类型的显式算法。(2)计算了作为自由积von Neumann代数的非I型部分的任何III1因子的SD-和τ-不变量。(3)证明了非交换有界Hardy空间的乘积中弱紧子集的一个非常特殊的性质。(4)研究了合并自由积von Neumann代数的阶乘性、类型分类和满足性问题，并给出了这些问题的几个部分答案。

I studied operator algebras and related topics in relation to free probability theory. More precisely, I obtained the following results :(1) I gave a necessary and sufficient condition for an arbitrary free product von Neumann algebra to be a factor. More strongly, I described its central decomposition explicitly, gave an explicit description of the type I part, proved that the non-type I part always appears and becomes a type II1 or III factor, and gave an explicit algorithm for determining the type of that factor. Moreover I proved that that factor is always full.(2) I computed the Sd-andτ-invariants of any type III1 factor arising as the non-type I part of a free product von Neumann algebra.(3) I proved a very special property for weakly compact subsets in the preduals of non-commutative bounded Hardy spaces.(4) I studied the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras, and gave several partial answers to those questions.