Gravity and the crossed product

Abstract

Recently Leutheusser and Liu [1, 2] identified an emergent algebra of Type III1 in the operator algebra of mathcal{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.