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We present a procedure for quantizing complex projective spaces$\mathbb{CP}^{p,q}$, $q\ge 1$, as well as construct relevant star products onthese spaces. The quantization is made unique with the demand that it preservesthe full isometry algebra of the metric. Although the isometry algebra, namely$su(p+1,q)$, is preserved by the quantization, the Killing vectors generatingthese isometries pick up quantum corrections. The quantization procedure is anextension of one applied recently to Euclidean $AdS_2$, where it was found thatall quantum corrections to the Killing vectors vanish in the asymptotic limit,in addition to the result that the star product trivializes to pointwiseproduct in the limit. In other words, the space is asymptotically anti-deSitter making it a possible candidate for the $AdS/CFT$ correspondenceprinciple. In this article, we find indications that the results for quantizedEuclidean $AdS_2$ can be extended to quantized $\mathbb{CP}^{p,q}$, i.e.,noncommutativity is restricted to a limited neighborhood of some origin, andthese quantum spaces approach $\mathbb{CP}^{p,q}$ in the asymptotic limit.

Asymptotic commutativity of quantized spaces: the case of $\mathbb{CP}^{p,q}$