Subalgebras with converging star products in deformation quantization: An algebraic construction for C P n

Abstract

Based on a closed formula for a star product of Wick type on CPn, 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 finite-dimensional algebra of linear operators in the Kth energy eigen- space of an isotropic harmonic oscillator with n+1 degrees of freedom. Other examples like the 2n torus and the Poincaré disk are discussed.