Study on free probability and operator algebras

我学习了算子代数和与自由概率论相关的主题。更准确地说，我得到了以下结果：(1)我给出了任意自由积von Neumann代数是因子的一个充分必要条件。更强的是，我明确地描述了它的中心分解，给出了I型部分的明确描述，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明确算法。此外，我证明了该因子总是满的。(2)我计算了任何iii型因子的sd -和τ-不变量作为自由积冯·诺伊曼代数的非I型部分产生。(3)证明了非交换有界Hardy空间的前似中弱紧子集的一个非常特殊的性质。(4)研究了混合自由积冯·诺伊曼代数的析因性、类型分类和充分性问题，并对这些问题给出了部分答案。

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.