We derive a closed formula for a star-product on complex projective space and on the domain SU (n + 1)/S(U (1) × U (n)) using a completely elementary construction: Starting from the standard star-product of Wick type on C n+1 \ {0} and performing a quantum analogue of Marsden-Weinstein reduction, we can give an easy algebraic description of this starproduct. Moreover, going over to a modified star-product on C n+1 \ {0}, obtained by an equivalence transformation, this description can be even further simplified, allowing the explicit computation of a closed formula for the star-product on CP n which can easily transferred to the domain SU (n + 1)/S

Based on a closed formula for a star product of Wick type on CP n , which has been discovered in an earlier article of the authors, we explicitly construct a subalgebra of the formal star-algebra (with coefficients contained in the uniformly dense subspace of representative functions with respect to the canonical action of the unitary group) that consists of converging power series in the formal parameter, thereby giving an elementary algebraic proof of a convergence result already obtained by Cahen, Gutt, and Rawnsley. In this subalgebra the formal parameter can be substituted by a real number α: the resulting associative algebras are infinite-dimensional except for the case α = 1/K, K a positive integer, where they turn out to be isomorphic to the finitedimensional algebra of linear operators in the Kth energy eigenspace of an isotropic harmonic oscillator with n + 1 degrees of freedom. Other examples like the 2n-torus and the Poincaré disk are discussed. *