  The algebra of observables of an N-body Calogero model is represented on the
S_N-symmetric subspace of the positive definite Fock space. We discuss some
general properties of the algebra and construct four different realizations of
the dynamical symmetry algebra of the Calogero model. Using the fact that the
minimal algebra of observables is common to the Calogero model and the finite
Chern-Simons (CS) matrix model, we extend our analysis to the CS matrix model.
We point out the algebraic similarities and distinctions of these models.
