The Crossed Product, Modular (Tomita) Dynamics and its Role in the Transition of Type $III$ to Type $II_{\infty}$ v.Neumann Algebras and Connections to Quantum Gravity

Abstract

We analyse the role of the crossed product and the modular (Tomita) dynamics in the transition of type $III$ to type $II_{\infty}$ v.Neumann algebras which was recently observed in papers by Witten et al. In a preceding paper we argued that type $II_{\infty}$ v.Neumann algebras display certain features which we attributed to quantum gravity effects. We claim that the action of the modular evolution on the quantum fluctuations can be understood as an aspect of quantum gravity. We mention in this context the work of Sakharov on induced gravity. Furthermore we analyse the change of the properties of projectors and partial isometries in the transition from type $III$ to type $II_{\infty}$.