Exercise 7, § 6

Let $A$ be a commutative unital involutive Banach algebra. In order that every character of $A$ be hermitian, it is necessary and sufficient that $1+xx^*$ be invertible for every $x\in A$. (If every character of $A$ is hermitian, $\mathscr{G}(1+xx^*)$ is everywhere $>0$ on $\mathsf{X}(A)$. If a character $\chi$ of $A$ is non-hermitian, there exists a hermitian $x\in A$ such that $\chi (x) =i$, whence $\chi (1+xx^*) = 0$ and $1+xx^*$ is not invertible.)