  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$_\infty$
with a trivial center. As a result, density matrices and entropies on the
boundary algebra are uniquely defined up to, 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
previous arguments that focused on $\mathcal{O}(1)$ fluctuations to a black
hole of specified mass, this Type II$_\infty$ algebra describes states at all
temperatures or energies. We also consider the role of spacetime wormholes. One
can try to define operators associated with wormholes that commute with the
boundary algebra, but this fails in an instructive way. In a regulated version
of the theory, wormholes and topology change can be incorporated
perturbatively. The bulk Hilbert space $\mathcal{H}_\mathrm{bulk}$ that
includes baby universe states is then much bigger than the space of states
$\mathcal{H}_\mathrm{bdry}$ accessible to a boundary observer. However, to a
boundary observer, every pure or mixed state on $\mathcal{H}_\mathrm{bulk}$ is
equivalent to some pure state in $\mathcal{H}_\mathrm{bdry}$.
