Vanishing up to the rank in bounded cohomology (Q2470961)

In the paper under review the vanishing for the non-trivial unitary representations of the bounded cohomology \(H^n_b\) of \(SL_d (k)\) up to degree \(d-1\) is considered. Main results of the paper: 1. Let \(k\) be a local field, \(G=SL_d (k)\) and \(V\) a unitary representation of \(G\) not containing the trivial one. Then \(H^n_b (G,V)=0\) for all \(0\leq n\leq d-1 \). 2. Let \(k\) be a local field, \(G=SL_d (K), \Gamma <G\) a lattice and \(W\) a unitary representation of \(\Gamma\) not containing the trivial one. Then \(H^n_b (\Gamma ,W)=0\) for all \(0\leq n\leq d-1\). 3. The preceding results are true for uniformly bounded representations of superreflexive spaces. 4. Let \(k\) be a local field, \(G=SL_d (K)\) and \(\Gamma <G\) a lattice. Then the restriction map \(H^n_b (G,R)\rightarrow H^n_b (\Gamma ,R)\) is an isomorphism for all \(0\leq n\leq d-1\).