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. Schmiidgen 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.