  Let $A$ be a commutative semisimple Arens regular unital Banach algebra. The
correlation between the BSE-property of the Banach algebra $A$ and its second
duals are assessed. It is found and approved that if $A$ is a BSE-algebra, then
so is $A^{**}$. The opposite correlation will hold in certain conditions. The
correlation of the BSE-norm property of the Banach algebra and its second dual
are assessed and examined. It is revealed that, if $A$ is a commutative Arens
regular unital Banach algebra where $A^{**}$ is semisimple, then, $A$ and
$A^{**}$ are BSE-norm algebra.
