Moment maps of Abelian groups and commuting Toeplitz operators acting on the unit ball

We prove that to every connected Abelian subgroup H of the biholomorphisms unit ball B n we can associate a set bounded symbols whose corresponding Toeplitz operators generate commutative C ? -algebra on weighted Bergman space. These are form ( z ) = f ? , where is moment map for action . show that, this construction, if maximal subgroup, then introduced precisely -invariant symbols. provide explicit computation maps obtain special sets described in terms coordinates. In particular, it proved our symbol have as particular cases all from current literature yield generating -algebras spaces Furthermore, exhibit examples some work not been considered before. Finally, several formulas spectra presented. include spectral integral expressions simplify known subgroups ball.