  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.
