Study on algebraic・topological・analytic K theory

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We obtained the following results.1. We determined general results on continuous saps which are honotopic to a fixed map on the finite sketetons.2. We succeeded to generalize the concepts of Gottlieb sets and Varadarajan sets. We studied the actions of the fundamental groups and localizations.3. We obtained some results on the spaces with an indefinite inner-product.4. We obtained a result on a generatization of Tomita-Takesaki theory to a *-algebra.5. We call a family which is closed under a partially defined product a partial O*-algebra, and studied the properties of the algebra. We studied a family of unbounded operators algebraically.6. We studied a projective plane of order 27. We determined a translation complement group by a method of Oyama.7. We also constructed some new translation planes of order 27. Furthermore we obtained a generatization of the Sherk plane of general order.8. We studied the local algebra of homogeneous Banach algebra on 1-dimensioned torus. We determined the structure of type-C to some extent.9. We studied the properties of CCR-algebras. We proved Tomita-Takesaki theorem for a representation which is fundamental for the study.10. We determined a condition for real valued plurisubharmonic function to be the real part of a holomorphic function on a Riemann domain over a flag manifold.11. We obtained a condition for real valued plurisubharmonic function to be the real part of a holomorphic function, using cohomology group.