  We use operator algebras and operator theory to obtain new result concerning
Berezin quantization of compact K\"ahler manifolds. Our main tool is the notion
of subproduct systems of finite-dimensional Hilbert spaces, which enables all
involved objects, such as the Toeplitz operators, to be very conveniently
expressed in terms of shift operators compressed to a subspace of full Fock
space. This subspace is not required to be contained in the symmetric Fock
space, so from finite-dimensional matrix algebras we can construct
noncommutative manifolds with extra structure generalizing that of a projective
variety endowed with a positive Hermitian line bundle and a canonical K\"ahler
metric in the class of the line bundle. Even in the commutative setting these
constructions are very fruitful. Firstly, we show that the algebra of smooth
functions on any smooth projective variety can be quantized in a strong sense
of inductive limits, as was previously only accomplished for homogeneous
manifolds. In this way the K\"ahler manifold is recovered exactly from
quantization and not just approximately. Secondly, we obtain a strict
quantization also for singular varieties. Thirdly, we show that the Arveson
conjecture is true in full generality for shift operators compressed to the
subspace of symmetric Fock space associated with any homogeneous ideal. For
noncommutative examples we consider homogeneous spaces for compact matrix
quantum groups which generalize $q$-deformed projective spaces, and we show
that these can be obtained as the cores of Cuntz--Pimsner algebras constructed
solely from the representation theory of the quantum group. We also discuss
interesting connections with noncommutative random walks.
