Gravity and the crossed product

Recently Leutheusser and Liu [1, 2] identified an emergent algebra of Type III₁ in the operator algebra of N = 4 super Yang-Mills theory for large N. Here we describe some 1/N corrections to this picture and show that the emergent Type III₁ algebra becomes an algebra of Type II_∞. The Type II_∞ algebra is the crossed product of the Type III₁ algebra by its modular automorphism group. In the context of the emergent Type II_∞ algebra, the entropy of a black hole state is well-defined up to an additive constant, independent of the state. This is somewhat analogous to entropy in classical physics.