Remark 1

Let A be a normed algebra. If an element $x\in A$ satisfies $\varrho (x) =\|x\|$, one has $\|x^n\|=\|x\|^n$ for every $n\in \mathbf{N}$, by (3) and (4).

Conversely, suppose that $\|x^2\|=\|x\|^2$ for every $x\in A$. Then, for every integer $n\geqslant 0$, one has the equality $\|x^{2^n}\|=\|x\|^{2^n}$, hence $\|x\|=$ $\|x^{2^n}\|^{2^{-n}}$; when $n$ tends to $+\infty$, one obtains $\|x\|=\varrho (x)$.