We analyze the algebra of boundary observables in canonically quantised JT gravity with or without matter. In the absence of matter, this algebra is commutative, generated by the ADM Hamiltonian. After coupling to a bulk quantum field theory, it becomes a highly noncommutative algebra of Type II infinity with a trivial center. As a result density matrices and entropies on the boundary algebra are uniquely defined upto, respectively, a rescaling or shift. We show that this algebraic definition of entropy agrees with the usual replica trick definition computed using Euclidean path integrals. Unlike in the previous arguments that focussed on O(1) fluctuations to a black hole of specified mass, this type II infinity algebra describes states at all temperatures or energies.
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