a banach algebra $\frak{A}$ is called an annihilator algebra if, for arbitrary closed left ideal $\frak{L}$ and closed right adeal $\frak{R}$ in $\frak{A}$, both of the following conditions are satisfied:
$(1):\mathcal{A}_l(\frak{L})=\{0\}$ if and only if ${\frak{L}}=\frak{A}$.
$(2):\mathcal{A}_r(\frak{R})=\{0\}$ if and only if ${\frak{R}}=\frak{A}$.
and $\frak{A}$ is a dual algebra if
$(1):\mathcal{A}_l[\mathcal{A}_r(\frak{L})]=\frak{L}$
$(2):\mathcal{A}_r[\mathcal{A}_l(\frak{R})]=\frak{R}$
where for arbitrary set $E\subseteq A$, we have
$\mathcal{A}_l(E)=\{x\in A\;|\;xE=\{0\}\}$
$\mathcal{A}_r(E)=\{x\in A\;|\;Ex=\{0\}\}$
then $\mathcal{A}_l(E)$ is called left annihilator and $\mathcal{A}_r(E)$ the right annihilator of $E$.
it is easily proved that an annihilator algebra is dual and even if $\frak{A}$ be a semisimple annihilator banach algebra, then it is not necessary that $\frak{A}$ be a dual algebra. now my question is that
under what conditions an annihilator banach algebra will be a dual algebra?