Emergent States and Algebras from the Double-Scaling limit of Pure States in SYK

Recent work has emphasized a subtlety of large- $N$ limits in AdS/CFT: a sequence of pure states in the microscopic theory need not remain pure with respect to the emergent algebra of observables. We study this phenomenon for Kourkoulou-Maldacena (KM) states in the double-scaling limit of the SYK model, and show that their ensemble-averaged algebraic description depends crucially on which observables survive the limit. For fermionic operators of size $N^{1/2}$, generic operators converge to the usual chord operators of double-scaled SYK. The resulting von Neumann algebra is the standard Type II$_1$ factor, and the KM pure states at infinite temperature converge to the tracial state, so generic probes lose access to microscopic purity. We then identify a class of operators adapted to the KM state that also survives the double-scaling limit. Since the KM state may be viewed as a projection inside the tracial state, these become dressed chord creation and annihilation operators. Once included, the limiting algebra becomes Type I$_\infty$ and the limiting state becomes pure. This gives a concrete example in which adding a sufficiently state-adapted operator to the emergent algebra restores access to the purity of the underlying state. We further show that correlators of the dressed operators admit exact modified chord-diagram rules, derive analytic expressions for uncrossed $2n$-point and crossed four-point functions, analyze their finite-temperature semiclassical and Schwarzian limits, study a deformation of the chord Hamiltonian that produces bound states and extends the correspondence with JT gravity plus an EOW brane to general brane tension, and identify an emergent $U(1)$ symmetry together with its finite-$N$ violation. Finally, we discuss analogies with boundary algebras proposed for black hole interiors and closed universes, and suggest lessons from our construction for both.