Exercise 3, § 6

Let $A$ be an involutive unital algebra. Suppose that for every $x$ in $A$, the spectrum of $x$ does not contain $-1$.

a) For every idempotent element $x$ of $A$, now prove that there exists an idempotent hermitian element $y$ of $A$ such that $xA =yA$. (Consider the inverse $z$ of $1 + (x-x^*)(x^*-x)$ ; now prove that $z$ commutes with $x$ and put $y=xx^*z.$) b) For every idempotent element $x$ of $A$, now prove that there exists an idempotent hermitian element $y$ of $A$ and an invertible $u$ in $A$ such that $x=uyu^{-1}$. (If $xA =yA$, put $u= (1-(y-x))^{-1}.$)