The centaur-algebra of observables

This letter explores a transition in the type of von Neumann algebra for asymptotically AdS spacetimes from the implementations of the different gravitational constraints. We denote it as the \emph{centaur-algebra} of observables. In the first part of the letter, we employ a class of flow geometries interpolating between AdS${}_2$ and dS${}_2$ spaces, the centaur geometries. We study the type II${}_{\mathrm{\infty }}$ crossed product algebra describing the semiclassical gravitational theory, and we explore the algebra of bounded sub-regions in the bulk theory following $T\bar{T}$ deformations of the geometry and study the gravitational constraints with respect to the quasi-local Brown-York energy of the system at a finite cutoff. In the second part, we study arbitrary asymptotically AdS spacetimes, where we implement the boundary protocol of an infalling observer modeled as a probe black hole proposed by arXiv:2211.16512 to study modifications in the algebra. In both situations, we show how incorporating the constraints requires a type II${}_1$ description.