When does the Fourier algebra coincide with the Fourier-Stieltjes algebra?

Question

For a given locally compact group $G$ the Fourier-Stieltjes algebra $B(G)$ is defined as the algebra of matrix coefficients of unitary representations $\pi:G\to B(H)$. Similarly, the Fourier algebra $A(G)$ is defined as the algebra of matrix coefficients of the left regular representation $\lambda:G\to B(L_2(G))$ (see here or here). P.Eymard in his original paper ((3.6), $1^\circ$) says that when $G$ is compact, these algebras coincide: $$ A(G)=B(G). $$ Is this the only case when they coincide?

Answer 1

Yes. Any function in $A(G)$ vanishes at infinity, whereas $B(G)$ contains the constant functions. So whenever $G$ is non-compact, $A(G)\subsetneq B(G)$.

A related question is when $A(G) = B(G)\cap C_0(G)$. See e.g. http://arxiv.org/abs/1311.5400.