Let $\mathcal{A}$ be a $C^*$ -algebra. Let $R_{\sigma}(a)$ denote the spectral radius of the element $a \in \mathcal{A}$ .

Theorem - For every $a \in \mathcal{A}$ we have that $\|a\| = \sqrt{R_{\sigma}(a^*a)}$ .

This result shows that the norm in a $C^*$ -algebra has a purely algebraic nature. Moreover, the norm in a $C^*$ -algebra is unique (in the sense that there is no other norm for which the algebra is a $C^*$ -algebra).