Let $G$ be a locally compact group, and let $G_{d}$ denote the same group with the discrete topology.
Let $B(G_{d})$ denote the Fourier-Stieltjes algebra of $G_{d}$.
For a subgroup $H$ of $G$, why is $1_{H}$, the characteristic function of $H$ necessarily in $B(G_{d})$?
I am referred to Eymard's paper for this but do not speak french, and I confess I find the translation paper quite indimidating.
I appreciate any advice!
Update: In the abelian case, I think one answer is that $1_{H}$ is the image of $\hat{H_{d}}$'s left Haar measure under the Fourier-Stieltjes Transform. Is this correct?
(And if so I still do not know the answer in general)
Update 2: I realized afterwards that I overlooked a hypothesis: $H$ is abelian.
I no longer have any reason to believe that my original statement needs to be true for an arbitrary subgroup $H$ of $G$; as a result, this question can be put to rest.