Let (G,tau_G) be a topological group. We establish relationships between weakly almost periodic topologies on G coarser than tau_G, central idempotents in the compactification G^W, and certain ideals algebra of functions W(G). gain decompositions representations, generalizing many from literature. look at role pre-locally compact topologies, unitarizable extend or to Fourier-Stieltjes algebras ...