Duality and Black Hole Evaporation

We construct a model that describes the quantum black hole evaporation unitarily in a Hilbert space of infinite dimension. This construction generalizes Page's finite dimensional approach to infinite dimensions. The basic ingredient is the Murray-von Neumann coupling for finite type II factors. This coupling measures, at each time of the evaporation, the relative continuous dimension of the radiation and the black hole subspaces. The unitary transformation, implementing the quantum evaporation and thus determining the time dependence of the coupling, is identified with the dual modular automorphism. In the appendix we sketch, using von Neumann construction of infinite tensor products of EPR pairs of q-bits, some q-bit holographic correspondences as well as an algebraic definition of ER bridges.