AN UNBOUNDED GENERALIZATION OF TOMITA's OBSERVABLE ALGEBRAS II

In a previous paper [4] we tried to build the basic theory of unbounded Tomita's observable algebras called T^†-algebras which are related to unbounded operator algebras, especially unbounded Tomita-Takesaki theory, operator algebras on Krein spaces, studies of positive linear functionals on *-algebras and so on. And we defined the notions of regularity, semisimplicity and singularity of T^†-algebras and characterized them. In this paper we shall proceed further with our studies of T^†-algebras and investigate whether a T^†-algebra is decomposable into a regular part and a singular part.