Open subgroups of G and almost periodic functionals on A(G)

Let G be a locally compact group and let C-delta*(G) denote the C*-algebra generated by left translation operators on L-2(G). Let AP((G) over cap) and WAP((G) over cap) be the spaces of almost periodic and weakly almost perio dic functionals on the Fourier algebra A(G), respectively. It is shown that if G contains an open abelian subgroup, then (1) AP((G) over cap) = C-delt a*(G) if and only if AP((G) over cap)(c) is norm dense in AP((G) over cap); (2) WAP((G) over cap) is a C*-algebra if WAP((G) over cap)(c) is norm dens e in WAP((G) over cap), where X-c denotes the set of elements in X with com pact support. In particular, for any amenable locally compact group G which contains an open abelian subgroup, G has the dual Bohr approximation prope rty and WAP((G) over cap) is a C*-algebra.