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We prove that the natural map H 2 b ( ) → H 2 ( ) from bounded to usual cohomology is injective if is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for : 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 • b ( ) to the continuous bounded cohomology of the ambient group with coefficients in some induction module.

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