Theory and applications of functional analysis

算子代数理论是本研究的主要内容。(1)Ⅱ_1因子子因子的Jones指数及Pimsner和Popa的相对熵。这些因素相互之间有着密切的联系，也与原始要素的结构有着密切的关系。我们研究了这些相互关系，并特别注意的情况下，一个因素和它的交叉产品与一个有限的群时，该集团作为一个集团的外自同构的因素。(2)C^代数中的完全正映射C^-代数之间的同态概念在某种程度上被推广为完全正映射。这是最近研究中的一个重要概念。在由C^代数之间的有界线性算子构成的序Banach空间中，可以引入完全正映射的概念。那么问题就来了：单纯的肯定性假设在什么时候意味着完全的肯定性？人们发现，对于这个问题的肯定回答，分支C^代数的交换特征标起了一定的作用。(3)C^动力系统。从应用的角度进行了研究。固定子代数。谱，即不可约表示的空间。原代数的一个状态到交叉积的提升问题。等(4)所谓的相似性问题，它询问C^代数的有界表示是否等价于^表示。它与完全有界映射的导子和范数有关。当所讨论的C^代数是有限维的时候，我们精确地确定了这些范数，并将这个结果应用到一般情况。

The theory of operator algebras was the principal subject of the present research.(1) Jones index of subfactors of II_1 factors and the relative entropy of Pimsner and Popa. These have intimate relations to each other and also have close relations with the structure of the original factor. We investigated these interrelations, and special attentions were put in the case of a factor and its crossed product with a finite group when the group acted on the factor as a group of outer automorphisms.(2) Completely positive maps among C^ algebras. The notion of homomorphism between C^ algebras is extended somehow as completely positive maps. This s an important notion in recent researches. Among ordered Banach spaces composed of bounded linlear operators between C^ algebras, this notion of completely positive maps can be introduced. Then a question : When does the mere assumption of the positivity imply the complete positiviey ? It was observed that, for the affirmative answer to this question, abelian characters of component C^ algebras play some roles.(3) C^ dynamical system. Tesearches were made from the point of view of applications. The fixed subalgebra. The spectrum, i.e. the space of the irreducible representations. The lifting problem of a state of the original algebra to the crossed product. And so on.(4) The so-called similarity problem, which asks if a bounded representation of a C^ algebra is equivalent to a ^ representation. It is related with the derivations and the norms of completely bounded maps. We determined these norms exactly when the C^ algebras in question were of finite dimension and made applications of this result to the general case.