Entanglement entropy in QFT

Well, after doing some reading on entropy for states on C^* algebras, I am still puzzled whether it can be connected to the recent interest in entanglement entropy for a vacuum state in Quantum Field Theory. That entropy is, properly speaking, infinite, but after regularization and subtraction of a piece which diverges like the surface area of the spatial region whose entanglement one is computing, one manages to get a sensible and finite result, which appears to be independent of the details of regularization. On the other hand, the algebra of observables localized in such a spatial region is a hyperfinite factor of Type III. Such operator algebras do not admit any pure normal states at all, so their entropy is infinite, by definition. This sort of agrees with the fact that the regularized vacuum entanglement entropy is UV-divergent, but does not help us to define the "renormalized" vacuum entanglement entropy.