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 ...

The first two implications are trivial. In [2], the second author showed that every almost α-continuous function is η-continuous. In [1], Dickman, Porter and Rubin proved that every η-continuous is θ-continuous and hence weakly continuous. Although θ-continuous functions behave, in general, nicely, they may cause some unexpected problems. For example, if f :X → Y is θ-continuous, then f :X → f(...