  In this paper we give an example of a Banach algebra $A$ and a closed ideal
$I$ of $A$ such that the multiplier algebra of $I$ is equal to $A$ but $I$ does
not have any bounded approximate identity. In the case that $I$ has an
approximate identity, we give a necessary condition on $I$ for which
$A=\mathcal{M}(I)$, where $\mathcal{M}(I)$ denotes the multiplier algebra of
$I$. Finally, as a corollary of our results, we show that the Fourier algebra
of an amenable group is strictly dense in the Fourier-Stieltjes algebra.
