Subalgebras with Converging Star Products in Deformation Quantization: An Algebraic Construction for \complex \mbox{\LARGE P}^n

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.Comment: 16 pages, LaTeX with AMS Font

Phase Space Reduction for Star-Products: An Explicit Construction for CP^n

We derive a closed formula for a star-product on complex projective space and on the domain $SU(n+1)/S(U(1)\times U(n))$ using a completely elementary construction: Starting from the standard star-product of Wick type on $C^{n+1} \setminus \{ 0 \}$ and performing a quantum analogue of Marsden-Weinstein reduction, we can give an easy algebraic description of this star-product. Moreover, going over to a modified star-product on $C^{n+1} \setminus \{ 0 \}$, 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 $SU(n+1)/S(U(1)\times U(n))$.Comment: LaTeX, 17 page