  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 Poincar\'e disk are discussed.
