  In the setting of geometric quantization, we associate to any prequantum
bundle automorphism a unitary map of the corresponding quantum space. These
maps are controlled in the semiclassical limit by two invariants of symplectic
topology: the Calabi morphism and a quasimorphism of the universal cover of the
Hamiltonian diffeomorphism group introduced by Entov, Py, Shelukhin.
