Half-sided modular inclusions of von-Neumann-algebras

Let N contains or equal to M be von-Neumann-algebras on a Hilbert space H, Ωa common cyclic and separating vector. Denote ΔM,ΔN resp. JM,JN the associated modular operators and conjugations. Assume ΔM-itNΔM+it contains or equal to N for t≥0. We call such an inclusion half-sided modular. Then we prove the existence of a one-parameter unitary group U(a) on H, a element of R, with generator 1/2π(lnΔN-lnΔM)≥0 and relations 1. ΔMitU(a)ΔM-it=ΔNitU(a)ΔN-it=U(e-2πt a) for all a,t element of R, 2. JNJM=U(2), 3. ΔNit=U(1)ΔMitU(-1) for all t element of R, 4. N=U(1)MU(-1). If M is a factor and Ωis also cyclic for N' intersection M, we show that M has to be of type III1. (orig.)