Bounded cohomology of lattices in higher rank Lie groups

We prove that the natural map H^2_b(\Gamma)\to H^2(\Gamma) from bounded to usual cohomology is injective if \Gamma is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for \Gamma : the stable commutator length vanishes and any C^1 -action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating H^\bullet_b(\Gamma) to the continuous bounded cohomology of the ambient group with coefficients in some induction module.