  We construct a Type II$_\infty$ von Neumann algebra that describes the large
$N$ physics of single-trace operators in AdS/CFT in the microcanonical
ensemble, where there is no need to include perturbative $1/N$ corrections.
Using only the extrapolate dictionary, we show that the entropy of
semiclassical states on this algebra is holographically dual to the generalized
entropy of the black hole bifurcation surface. From a boundary perspective,
this constitutes a derivation of a special case of the QES prescription without
any use of Euclidean gravity or replicas; from a purely bulk perspective, it is
a derivation of the quantum-corrected Bekenstein-Hawking formula as the entropy
of an explicit algebra in the $G \to 0$ limit of Lorentzian effective field
theory quantum gravity. In a limit where a black hole is first allowed to
equilibrate and then is later potentially re-excited, we show that the
generalized second law is a direct consequence of the monotonicity of the
entropy of algebras under trace-preserving inclusions. Finally, by considering
excitations that are separated by more than a scrambling time we construct a
"free product" von Neumann algebra that describes the semiclassical physics of
long wormholes supported by shocks. We compute R\'{e}nyi entropies for this
algebra and show that they are equal to a sum over saddles associated to
quantum extremal surfaces in the wormhole. Surprisingly, however, the saddles
associated to "bulge" quantum extremal surfaces contribute with a negative
sign.
