Hamiltonian Carleman Approximation of the Calogero-Moser Space

We derive the Hamiltonian tangential Carleman approximation for the complex Calogero-Moser space Cₙ with a non-compact real form Cₙ^R, using the symplectic density property of Cₙ. The Hamiltonian tangential Carleman approximation means to approximate symplectic diffeomorphisms of Cₙ^R which are smoothly isotopic to the identity by symplectic holomorphic automorphisms of Cₙ which in addition preserve the real form Cₙ^R. This appproximation is in the strongest topology, in the fine Whitney topology.