Gravity and the Crossed Product

Recently Leutheusser and Liu [1,2] identified an emergent algebra of Type $\mathrm{III}_1$ 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 $\mathrm{III}_1$ algebra becomes an algebra of Type $\mathrm{II}_{\infty}$. The Type $\mathrm{II}_{\infty}$ algebra is the crossed product of the Type $\mathrm{III}_1$ algebra by its modular automorphism group. In the context of the emergent Type $\mathrm{II}_{\infty}$ 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.