Addendum to “Amenability and acyclicity in bounded cohomology”

We show that a surjective homomorphism (Formula presented) of (discrete) groups induces an isomorphism (formula presented) in bounded cohomology for all dual normed K-modules V if and only if the kernel of (formula presented) is boundedly acyclic. This complements a previous result by the authors that characterized this class of group homomorphisms as bounded cohomology equivalences with respect to(formula presented)generated Banach K-modules. We deduce a characterization of the class of maps between pathconnected spaces that induce isomorphisms in bounded cohomology with respect to coefficients in all dual normed modules, complementing the corresponding result shown previously in terms of R-generated Banach modules. The main new input is the proof of the fact that every boundedly acyclic group Γ has trivial bounded cohomology with respect to all dual normed trivial Γ-modules.