Theory and applications of functional analysis

算子代数理论是本研究的主要课题。(1) II_1因子子因子的Jones指数与Pimsner和Popa的相对熵。这些因素相互之间有着密切的关系，也与原始因素的结构有着密切的关系。我们研究了这些相互关系，并特别注意当群作为一组外部自同构作用于因子时，因子及其与有限群的交叉积的情况。(2) C^代数间的完全正映射。C^代数间同态的概念被推广为完全正映射。这是近年来研究的一个重要概念。在由C^代数间有界线性算子组成的有序巴拿赫空间中，可以引入完全正映射的概念。那么问题来了：什么时候仅仅假设积极性就意味着完全的积极性？我们注意到，对于这个问题的肯定答案，C^代数分量的阿贝尔特征起着一定的作用。(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。原代数状态对叉积的提升问题。等等......。(4)所谓的相似性问题，它问一个C^代数的有界表示是否等价于一个^表示。它与完全有界映射的导数和范数有关。当所讨论的C^代数是有限维时，我们精确地确定了这些范数，并将这一结果应用于一般情况。

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.