The Role of Type $II_{\infty}$ v.Neumann Algebras and their Tensor Structure in Quantum Gravity}}

Abstract

We will argue in this paper that the type classification of v.Neumann algebras play an important role in a theory of quantum gravity and quantum space-time physics. We provide arguments that type $II_{\infty}$ and its representation as a tensor product of an ordinary (exterior) Hilbert space algebra $\cB(\cH_I)$ and an (internal) type $II_1$ algebra, encoding, in our view, the hidden microscopic gravitational degrees of freedom, do represent the first step away from the semiclassical picture towards a full theory of quantum gravity.