  Recently Gambaudo and Ghys proved that there exist infinitely many
quasi-morphisms on the group ${\rm Diff}_\Omega^\infty (D^2, \partial D^2)$ of
area-preserving diffeomorphisms of the 2-disk $D^2$. For the proof, they
constructed a homomorphism from the space of quasi-morphisms on the braid group
to the space of quasi-morphisms on ${\rm Diff}_\Omega^\infty (D^2, \partial
D^2)$. In this paper, we study the homomorphism and prove its injectivity.
