Symmetries and Modular Intersections of Von Neumann Algebras

Let be von Neumann algebras acting on a Hilbert space and let be a common cyclic and separating vector. We say that have the modular intersection property with respect to if(1) -half-sided modular inclusions,(2)(If (1) holds the strong limit exists.) We show that under these conditions the modular groups of and generate a 2-dim. Lie group.This observation is the basis for obtaining group representations of Sl(2,)/Z₂ generated by modular groups.