Let $\mathcal{A}$ be a Banach $k$-algebra. The radical of $\mathcal{A}$ is the null-space of its spectral seminorm $\mathfrak{n}(\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}})$. A Banach $k$-algebra with radical equal to $\{0\}$ is called semi-simple. Elements in the radical are said to be quasi-nilpotent (or topological nilpotent).