The Making of von Neumann Algebras from Bulk Focusing

The single-trace, infinite-N algebra of an arbitrary region may or may not be a von Neumann algebra depending on the GNS sector. In this paper we identify the holographic dual of this mechanism as a consequence of the focusing of null geodesics; more precisely, this GNS sector-dependence corresponds to the well-known difference between null congruences fired from the bulk and those fired from the boundary. As part of establishing this property, we give a rigorous formulation and proof of causal wedge reconstruction for those general boundary subregions whose single trace algebras support von Neumann algebras at large-N. We discuss a possible finite-N extension and interpretation of our results as an explanation for the Hawking area theorem.