Exercise 14, § 2

Let A be a complex unital normed algebra. If $\|x^{-1}\|=\|x\|^{-1}$ for every invertible element $x$ of A, then $A =\mathbf{C}\cdot 1$. (Reduce to the case where A is complete. Let $x\in A$, at a distance $\alpha >0$ from $\mathbf{C}\cdot 1$. If $\lambda_0\in \mathbf{C}$ is such that $x-\lambda_0$ is invertible, then $x-\lambda$ is invertible for $|\lambda -\lambda_0|< \alpha$. Deduce that the spectrum of $x$ would be empty.)