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 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. *

Fuzzy Complex Grassmannian Spaces and Their Star Products

“…The corresponding matrix geometries can be viewed as non-commutative versions of T * S 4 , the co-tangent bundle to S 4 [21]. A star product on the complex Grassmannians as an infinite sum over derivatives is known [22,23]. However, this formula cannot be restricted to finite-dimensional sub-algebras and therefore cannot serve as a star product on a fuzzy approximation to the manifold.…”

States and Representations in Deformation Quantization

“…Here it is the class of * -algebras over C which are non-degenerate and idempotent. 19) such that this diagram commutes.…”

“…In general this turns out to be a rather delicate problem usually depending in a very specific way on the particular example one considers, see e.g. [19,39,80] and references therein. So, unfortunately, not very much can be said about this point in general.…”

Generalized Berezin-Toeplitz quantization of Kähler supermanifolds

“…For earlier discussions of these spaces relying on group theoretic methods, see [13,34]. Another possible approach would be to extend the techniques of [35,36] to the supercase.…”