Dual multiplier Banach algebras and Connes-amenability

Following M. Daws, we consider a Banach algebra B for which the multiplier algebra M(B) is a dual Banach algebra in the sense of V. Runde, and show that under certain continuity condition, B is amenable if and only if M(B) is Connes-amenable. As a result, we conclude that for a discrete amenable group G, the Fourier Stieltjes algebra B(G) is Connes-amenable if and only if G is abelian by finite.