Surjectivity of the comparison map in bounded cohomology for Hermitian Lie groups

We prove surjectivity of the comparison map from continuous bounded cohomology to continuous cohomology for Hermitian Lie groups with finite center. For general semisimple Lie groups with finite center, the same argument shows that the image of the comparison map contains all the even generators. Our proof uses a Hirzebruch type proportionality principle in combination with Gromov's results on boundedness of primary characteristic classes and classical results of Cartan and Borel on the cohomology of compact homogeneous spaces.