On the Radical for Some Class of Banach Algebras

Let $A$ be a complex Banach algebra. It is well known that the second dual $A^{**}$ of $A$ can be equipped with a multiplication that extends the original multiplication on $A$ and makes $A^{**}$ a Banach algebra. We show that $\operatorname{Rad}(A)={}^\bot(A^*\cdot A)$ and $\operatorname{Rad}(A^{**})=(A^*\cdot A)^\bot$ for some classes of Banach algebras $A$ with scattered structure space. Some applications of these results are given.