  Given a noncommutative Hamiltonian space $A$, we show that the conjecture
``{\it quantization commutes with reduction}'' holds on $A$. We also construct
a semi-product algebra $A \rtimes \mG^A$, equivariant sheaves on the
representation space are related to left $A \rtimes \mG^A$-modules. In the
quiver setting, via the quantum and classical trace maps, we establish the
explicit correspondence between quantizations on a preprojective algebra and
those on a quiver variety.
