Theory and applications of functional analysis

算子代数理论是本研究的主要内容。(1)II_1因子的子因子的Jones指数和Pimsner和Popa的相对熵。它们之间既有密切的关系，又与原要素的结构密切相关。我们研究了它们之间的相互关系，特别是当一个因子及其与有限群的交叉积作为一个外自同构群作用于该因子时。(2)C^代数之间的完全正映射。将C^代数之间的同态概念在某种程度上推广为完全正映射。这是S近年来研究中的一个重要概念.在由C^代数之间的有界线性算子组成的序Banach空间中，可以引入完全正映射的概念。然后是一个问题：什么时候仅仅是对积极的假设就意味着完全的积极？观察到，对于这个问题的肯定回答，分量C^代数的交换性起到了一定的作用。(3)C^动力系统。研究是从应用的角度进行的。固定子代数。谱，即不可约表示的空间。原代数的状态到交叉积的提升问题。(4)所谓的相似性问题，即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.