Gravity and the crossed product

A bstract Recently Leutheusser and Liu [1, 2] identified an emergent algebra of Type 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 III 1 algebra becomes an algebra of Type II ∞ . The Type II ∞ algebra is the crossed product of the Type III 1 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.