BSE-property for some certain Segal and Banach algebras

For a commutative semi-simple Banach algebra ${A}$ which is an ideal in its second dual we give a necessary and sufficient condition for an essential abstract Segal algebra in ${A}$ to be a BSE-algebra. We show that a large class of abstract Segal algebras in the Fourier algebra $A(G)$ of a locally compact group $G$ are BSE-algebra if and only if they have bounded weak approximate identities. Also, in the case that $G$ is discrete we show that $A_{\rm cb}(G)$ is a BSE-algebra if and only if $G$ is weakly amenable. We study the BSE-property of some certain Segal algebras implemented by local functions that were recently introduced by J. Inoue and S.-E. Takahasi. Finally we give a similar construction for the group algebra implemented by a measurable and sub-multiplicative function.