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 {em 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 $alpha$: the resulting associative algebras are infinite-dimensional except for the case $alpha=1/K$, $K$ a positive integer, where they turn out to be isomorphic to the finite-dimensional algebra of linear operators in the $K$th energy eigenspace of an isotropic harmonic oscillator with $n+1$ degrees of freedom. Other examples like the $2n$-torus and the Poincare disk are discussed.
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