On the Lattice of Subalgebras Associated with the Principle of Half-sided Modular Inclusion

Abstract

Given a von Neumann algebra M with cyclic and separating vector Ω and with the modular group δit which is associated with the pair (M, Ω) we will investigate the von Neumann subalgebras N ⊂ M which fulfil the principle of half-sided modular inclusion. We show that this set is almost a lattice with respect to intersection and union. Furthermore, in this set we can introduce an equivalence relation respecting the lattice structure. To every von Neumann subalgebra fulfilling the condition of half-sided modular inclusion is associated a unique one-parametric translation group which fulfils the spectrum condition. Within this setting, one deals with two orders, the order of inclusion of subalgebras, and the order of positive operators between the generators of the translations. The first order implies the reverse of the second but the converse holds only if the corresponding translations commute.