Main part of this note is a summary of my studies during several years of
the Tomita-Takesaki theory in O*-algebra. In 1995 I began this work for the
preparation of the seminars during the summer of 1996 at the Mathematics
Institute of Leipzig university. I wish to thank Professor K. D. Kiirsten and
K. Sehmfidgen for their warm hospitality, for their interest in this work and
for their encouragement. Further, I wish to thank them for many helpful discussions
and suggestions when they visited the Department of our university
in 1996, 1997. I also aeknowlege Professor J. P. Antoine (Louvain Catholique
University), Van Daele (Leuven Katholieke University), W. Karwowski (Wroclaw
University), G. Epifanio and C. Trapani (Palermo University) and A.
Arai and M. Kishimoto (Hokkaido University) for giving me opportunities
of the seminars and lectures about this work for their colleagues and graduate
students of their Mathematical Departments and for many invaluable
discussions and many helpful suggestions. I should like to thank Professor
H. Kurose and Dr. Ogi for their encouragement and many helpful conversations.
It remains for me to express my gratitude to M. Takakura for typing
this mannuseript in TeX. This work was supported in part by Japan Society
for the Promotion Science and Japan Private School Promotion Foundation.