Exercise 23, § 2

Soit A a normed algebra and R its radical.

a) If $x\in R$, then $x$ is quasi-nilpotent. (One has Sp$'_A(x) =\{0\}$ and fortiori Sp$'_{\widehat{A}}(x) =\{0\}.$)

b) Suppose that A is a Banach algebra. Let I be a left ideal of A all of whose elements are quasi-nilpotent. Then $I\subset R$. (One may suppose that A admits a unit element. If $x\in R$ and $a\in A$, one has Sp$_A(ax) =\{0\}$, hence $1-ax$ is invertible; hence $x\in R$.)