  It's been argued that a finite effective temperature emerges and
characterizes the thermal property of double-scaled SYK model in the infinite
temperature limit. On the other hand, in static patch of de Sitter, the
maximally entangled state exhibits KMS condition of infinite temperature,
suggesting the Type II$_1$ nature of the algebra formed by operators that are
gravitationally dressed to the static patch observer. In the current work we
study the double-scaled algebra generated by chord operators in double-scaled
SYK model. We show that the algebra exhibits a behavior reminiscent of both
perspectives. In particular, we prove that it's a Type II$_1$ factor, and the
empty state with no chords satisfies the tracial property, thus aligning with
the expectation in previous work. Furthermore, we show it's a cyclic separating
state by exploring the modular structure of the algebra. We then study various
limits of the theory and explore corresponding relations to JT gravity, theory
of baby universe, and Brownian double-scaled SYK. We also present a full
solution to the energy spectrum of $0$- and $1$- particle irreducible
representations.
