I propose that integrality obstructions to geometric quantization can be circumvented
for some compact Kaehler manifolds by passing to the universal covering
space. This can be done if the lift of the symplectic form to the universal covering
space is cohomologically trivial. I prove that this construction does give a strict
quantization. The construction is related to the Baum-Connes assembly map. I
also propose a type of further generalized Toeplitz construction, classify the structure
involved, and give a simple construction for the resulting algebras. These
constructions involve twisted group C*-algebras of the fundamental group. These
are determined by a group cocycle constructed from the cohomology class of the
symplectic form.
