The spectral algebra seminorm $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ of an algebra seminorm $\vvvert\mathord{\cdot}\vvvert$ on a $k$-algebra $A$ is the one defined by

$$\forall f\in\mathcal{A},\quad\vvvert f\vvvert_{\mathrm{sp}}:=\lim_{\begin{subarray}{c}n\to\infty\end{subarray}}\vvvert f^{n}\vvvert^{\frac{1}{n}}.$$

Note that the triangle inequality for $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ follows from sub-multiplicativity of $\vvvert\mathord{\cdot}\vvvert$. In general, $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is only a seminorm even if $\vvvert\mathord{\cdot}\vvvert$ is a norm.