Lebesgue type decomposition of subspaces of Fourier-Stieltjes algebras

Let $G$ be a locally compact group and let $A(G)$ and $B(G)$ be the Fourier algebra and the Fourier-Stieltjes algebra of $G$, respectively. For any unitary representation $\pi$ of $G$, let $B_\pi (G)$ denote the $w^\ast$-closed linear subspace of $B(G)$ generated by all coefficient functions of $\pi$, and $B_\pi ^0(G)$ the closure of $B_\pi (G) \cap A_c(G)$, where $A_c(G)$ consists of all functions in $A(G)$ with compact support. In this paper we present descriptions of $B_\pi ^0(G)$ and its orthogonal complement $B_\pi ^s(G)$ in $B_\pi (G)$, generalizing a recent result of T. Miao. We show that for some classes of locally compact groups $G$, there is a dichotomy in the sense that for arbitrary $\pi$, either $B_\pi ^0(G) = \{0\}$ or $B_\pi ^0(G) = A(G)$. We also characterize functions in ${\mathcal B}_\pi ^0(G) = A_c(G) + B_\pi ^0(G)$ and study the question of whether ${\mathcal B}_\pi ^0(G) = A(G)$ implies that $\pi$ weakly contains the regular representation.