Exercise 26, § 2

Let $A$ be a Banach algebra and $R$ its radical.

(a) Prove that, if $r\in R$, the series

converges to an element $x\in R$ such that $x^2-x+r= 0$.

(b) Let $u$ be an element of $A$ whose class in $A/R$ is idempotent. There exists an idempotent of $A$ congruent to $u$ modulo $R$. (One may suppose that $A$ admits a unit element. Let $q=u-u^2\in R$ and $x\in R$ be a solution, constructed with the aid of a), of $x^2-x-q(1-4q)^{-1}= 0$. Then $u-(2u-1)x$ answers the question.)