Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 192

Symmetries and Modular Intersections of von-Neumann-Algebra

H.-W. Wiesbrock

Let N,M be von-Neumann-algebras acting on a Hilbert space H and let $Omega$ in H be a common cyclic and separating vector. We say that (N,M) have the {em modular intersection property} with respect to $ Omega$ if egin{enumerate} item $( N cap M subset M,Omega ) ((N cap M )subset M, Omega) mbox{ - half-sided modular inclusions,}$ item $J_N (s-lim_{t ightarrow infty} Delta_N^{it} Delta_M^{-it}) J_N = s-lim_{t ightarrow infty} Delta_M^{it} Delta_N^{-it}.$ end{enumerate} ( If 1. holds the strong limit exists, see below. ) We show that under these conditions the modular groups of N and M generate a 2-dim. Lie group. This observation is the basis for getting group representations of the 1+2 dim. Lorentz group generated by modular groups.

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