Direct Products for the Hamiltonian Density Property

Abstract We show that the direct product of two Stein manifolds with the Hamiltonian density property enjoys the Hamiltonian density property as well. We investigate the relation between the Hamiltonian density property and the symplectic density property. We then establish the Hamiltonian and the symplectic density property for $$(\mathbb {C}^*)^{2n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> and for the so-called traceless Calogero–Moser spaces. As an application we obtain a Carleman-type approximation for Hamiltonian diffeomorphisms of a real form of the traceless Calogero–Moser space.