  For an infinitesimal symplectic action of a Lie algebra ${\goth g}$ on a
symplectic manifold, we construct an infinitesimal crossed product of
Hamiltonian vector fields and Lie algebra ${\goth g}$. We obtain its second
crossed product in case ${\goth g}=R$ and show an infinitesimal version for a
theorem type of Takesaki duality.
