Theory and applications of functional analysis

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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.