An Operator Algebraic Approach To Black Hole Information

We present an operator algebraic perspective on the black hole information problem. For a black hole after Page time that is entangled with the early radiation we formulate a version of the information puzzle that is well-posed in the $G\to 0$ limit. We then give a description of the information recovery protocol in terms of von Neumann algebras using elements of the Jones index theory of type II$_1$ subfactors. The subsequent evaporation and recovery steps are represented by Jones's basic construction, and an operation called the canonical shift. A central element in our description is the Jones projection, which leads to an entanglement swap and implements an operator algebraic version of a quantum teleportation protocol. These aspects are further elaborated on in a microscopic model based on type I algebras. Finally, we argue that in the emergent type III algebra the canonical shift may be interpreted as a spacetime translation and, hence, that at the microscopic level "translation = teleportation".