  We derive the generalized second law (GSL) for arbitrary cuts of Killing
horizons from the perspective of crossed-product gravitational algebras, making
use of a recent proposal by one of us for the construction of local
gravitational algebras. This construction relies on the existence of a state
whose modular flow is geometric on the horizon. In both free and interacting
quantum field theories, such states are guaranteed to exist by the properties
of half-sided translations on the horizon. Using geometric identities derived
from the canonical analysis of general relativity on null surfaces, we show
that the crossed product entropy agrees with the generalized entropy of the
horizon cut in a semiclassical limit, and further reproduce Wall's result
relating the GSL to monotonicity of relative entropy of the quantum field
algebras. We also give a novel generalization of the GSL for interacting
theories in asymptotically flat spacetimes involving the concept of an algebra
at infinity for a half-sided translation, which accounts for triviality of the
algebra of fields smeared only on the horizon. Going beyond the semiclassical
limit, we compute subleading corrections to the crossed product entropy, but
are unable to determine if the GSL continues to hold after accounting for
these. We speculate that an improved GSL could follow from a hidden subalgebra
structure of the crossed products, assuming the existence of an operator-valued
weight between horizon cut algebras.
