We prove that quasi-morphisms and quasi-states on a closed rational symplectic manifold descend under symplectic reduction to symplectic hyperplane sections. Along the way we show that quasi-morphisms that arise from spectral invariants are the Calabi homomorphism when restricted to Hamiltonians supported on stably displaceable sets.
"Symplectic reduction of quasi-morphisms and quasi-states." J. Symplectic Geom. 10 (2) 225 - 246, June 2012.