Deformation quantizations with separation of variables on a K\"ahler manifold

We give a simple geometric description of all formal deformation quantizations on a K\"ahler manifold $M$ which enjoy the following property of separation of variables into holomorphic and antiholomorphic ones. For each open subset $U\subset M$, $\star$-multiplication from the left by a holomorphic function and from the right by an antiholomorphic function on $U$ coincides with the pointwise multiplication by these functions. These quantizations are in 1-1 correspondence with formal deformations of the original K\"ahler metrics on $M$. It has been shown in [Ka] that the formal deformation quantizations obtained from the full asymptotic expansion of Berezin's $*$-product on the orbits of a compact semisimple Lie group in [Mo2] and [CGR1] and on bounded symmetric domains in [Mo1] and [CGR2] are those with separation of variables and correspond to the trivial deformation of the original K\"ahler metrics.