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 $AP(\hat {G})$ and $WAP(\hat {G})$ be the spaces of almost periodic and weakly almost periodic functionals on the Fourier algebra $A(G)$, respectively. It is shown that if $G$ contains an open abelian subgroup, then (1) $AP(\hat {G}) = C_{\delta }^{*}(G)$ if and only if $AP(\hat {G})_{c}$ is norm dense in $AP(\hat {G})$; (2) $WAP(\hat {G})$ is a $C^{*}$-algebra if $WAP(\hat {G})_{c}$ is norm dense in $WAP(\hat {G})$, where $X_{c}$ denotes the set of elements in $X$ with compact support. In particular, for any amenable locally compact group $G$ which contains an open abelian subgroup, $G$ has the dual Bohr approximation property and $WAP(\hat {G})$ is a $C^{*}$-algebra.