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

Weakly almost periodic topologies, idempotents and ideals