  We present a procedure for quantizing complex projective spaces
$\mathbb{CP}^{p,q}$, $q\ge 1$, 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 $AdS_2$, 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 $AdS_2$ can be extended to quantized $\mathbb{CP}^{p,q}$, i.e.,
noncommutativity is restricted to a limited neighborhood of some origin, and
these quantum spaces approach $\mathbb{CP}^{p,q}$ in the asymptotic limit.
