An Unbounded Generalization of the Tomita-Takesaki Theory

Abstract

Algebras of unbounded operators have been studying by many mathematicians (Borchers, Uhlmann, Lassner, Powers, Schmiidgen, Antoine, Gudder, etc... .) from situations of the physical applications as well as the sheer mathematical interest. The study of oneparameter automorphism groups and dynamics in unbounded operator algebras seems to be hardly done except [8, 17], It is well known that the Tomita-Takesaki theory plays an important role for such a study in von Neumann algebras. In this direction we consider an unbounded generalization of the Tomita-Takesaki theory, and treat modular automorphism groups of such algebras. We define the notion of unbounded left Hilbert algebras which is an unbounded generalization of left Hilbert algebras in the sense that the left multiplication is not necessarily bounded. Then a bicommutant 2T of an unbounded left Hilbert algebra 21 is defined and becomes an achieved left Hilbert algebra, and so it induces the fundamental theorem of Tomita for the left von Neumann algebra ^0(2T) and the right von Neumann algebra y0(2T) of 2T: /'^oCSO/'^^oCSO, J^o(aO^~'''=*o(2r) for all t^R, where f is the modular conjugation operator of 2T and A is the modular operator of ST. The first purpose is to extend the above results to an unbounded left Hilbert algebra 2L The following question arises.