BSE property of the completion of Fourier algebra in its multiplier algebra

For a locally compact group G, let A(G) be the Fourier algebra and let A_M (G) be the completion of this algebra in its multiplier algebra. In this paper, we show that A(G) is an abstract Segal algebra in A_M (G). Also, a necessary and sufficient condition for equality of these two algebras is given. Then we prove that A_M (G) is an ideal in its second dual if and only if G is discrete. We show that if G is a discrete group, then A_M (G) is a BSE algebra if and only if G is M-weakly amenable. As a corollary, it is proven that A_M (F_2 ) is a BSE algebra while A(F_2 ) is not. Finally, we examine our results for the Lebasque-Fourier algebra and also give a completely new proof for equality of the character space of A(G) and A_M (G).Keywords: Banach algebra, Fourier algebra, Multiplier algebra, BSE property, Locally compact group.