  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, e.g., black holes formed from gravitational collapse, Kerr black
holes, or black holes in asymptotically de Sitter space. In this paper, we
present a general framework for obtaining the algebra of dressed observables
for linear fields on any spacetime with a Killing horizon. We prove, assuming
the existence of a stationary (but not necessarily KMS) state and suitable
decay of solutions, a structure theorem that the algebra of dressed observables
always contains a Type II factor "localized" on the horizon. These assumptions
have been rigorously proven in most cases of interest. Applied to the algebra
in the exterior of an asymptotically flat Kerr black hole, where the fields are
dressed to the black hole mass and angular momentum, we find a product of a
Type II$_{\infty}$ algebra on the horizon and a Type I$_{\infty}$ algebra at
past null infinity. In Schwarzschild-de Sitter, despite the fact that we
introduce an observer, the quantum field observables are dressed to the
perturbed areas of the black hole and cosmological horizons and is the product
of Type II$_{\infty}$ algebras on each horizon. In all cases, the von Neumann
entropy for semiclassical states is given by the generalized entropy. 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 is Type II$_{\infty}$ and in the absence of such
structures (e.g., de Sitter) the algebra is Type II$_{1}$.
