  Over the last 25 years, the notion of "fuzzy spaces" has become ubiquitous in
the high-energy physics literature. These are finite dimensional noncommutative
approximations of the algebra of functions on a classical space. The most well
known examples come from the Berezin quantization of coadjoint orbits of
compact semisimple Lie groups.
  We develop a theory of Berezin quantization for certain quantum homogeneous
spaces coming from ergodic actions of compact quantum groups. This allows us to
construct fuzzy versions of these quantum homogeneous spaces. We show that the
finite dimensional approximations converge to the homogeneous space in a
continuous field of operator systems, and in the quantum Gromov-Hausdorff
distance of Rieffel.
  We apply the theory to construct a fuzzy version of an ellipsoid which is
naturally endowed with an orbifold structure, as well as a fuzzy version of the
$\theta$-deformed coset spaces $C(G/H)_{\hbar \theta}$ of Varilly. In the
process of the latter, we show that our Berezin quantization commutes with
Rieffel's deformation quantization for actions of $\R^d$.

Operator Algebras