Asymptotic commutativity of quantized spaces: The case of C Pp,q

We present a procedure for quantizing complex projective spaces CPp,q, q≥1, as well as construct relevant star products on these spaces. The quantization is made unique with the demand that it preserves the full isometry algebra of the metric. Although the isometry algebra, namely, su(p+1,q), is preserved by the quantization, the Killing vectors generating these isometries pick up quantum corrections. The quantization procedure is an extension of one applied recently to Euclidean two-dimensional anti-de Sitter space (AdS2), where it was found that all quantum corrections to the Killing vectors vanish in the asymptotic limit, in addition to the result that the star product trivializes to pointwise product in the limit. In other words, the space is asymptotically anti-de Sitter, making it a possible candidate for the AdS/CFT correspondence principle. In this article, we find indications that the results for quantized Euclidean AdS2 can be extended to quantized CPp,q, i.e., noncommutativity is restricted to a limited neighborhood of some origin, and these quantum spaces approach CPp,q in the asymptotic limit.