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 π of G, let B_π(G) denote the w*-closed linear subspace of B(G) generated by all coefficient functions of π, and B⁰_π(G) the closure of B_π(G) ∩ A_c(G), where A_c(G) consists of all functions in A(G) with compact support. In this paper we present descriptions of B⁰_π(G) and its orthogonal complement B^s_π(G) in B_π(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 π, either B⁰_π(G) = {0} or B⁰_π(G) = A(G). We also characterize functions in B⁰_π(G) = A_c(G) + B⁰_π(G) and study the question of whether B⁰_π (G) = A(G) implies that π weakly contains the regular representation.