Additive conjugacy and the Bohr compactification of orthogonal representations

We say that two unitary or orthogonal representations of a finitely generated group G are additive conjugates if they intertwined by an map, which need not be continuous. associate to each representation topological action is complete conjugacy invariant: the automorphisms on Bohr compactification underlying Hilbert space. Using this construction we show property having almost invariant vectors invariant. As application amenable and only there nonzero homomorphism from \(L^2(G)\) into \(\mathbb {R}/\mathbb {Z}\) G-action.