Abstract: It was recently shown that the von Neumann algebras of observables dressed to the mass of a Schwarzschild-AdS black hole or an observer in de Sitter are Type II, and thus admit well-defined traces. The von Neumann entropies of "semi-classical" states were found to be generalized entropies. However, these arguments relied on the existence of an equilibrium (KMS) state and thus do not apply to, for example, black holes formed from gravitational collapse, Kerr black holes, or black holes in asymptotically de Sitter space.

In this talk, I will show that the algebra of observables in the “exterior” of any Killing horizon always contains a Type II factor "localized" on the horizon and, consequently, the entropy of semi-classical states is the generalized entropy. I will illustrate this with two examples of (1) a black hole in asymptotically flat spacetime and (2) black holes in asymptotically de Sitter. In all cases, the von Neumann entropy for semiclassical states is given by the generalized entropy.

More generally, our results suggest that in all cases where there exists another "boundary structure" (e.g., an asymptotic boundary or another Killing horizon) the algebra of observables has no maximum entropy state and in the absence of such structures (e.g., de Sitter) the algebra contains a maximum entropy state.