Assume that we are in the non-Archimedean case. For each $\gamma\in\operatorname{Hom}_{\operatorname{sg}}(M_{\sigma},\mathbb{R}_{\geq 0})$, the seminorm that, to a function $\sum\alpha_{m}\chi^{m}\in K[M_{\sigma}]$ assigns the value $\sup_{m}(|\alpha_{m}|\gamma(m))$, is a multiplicative seminorm on $K[M_{\sigma}]$ that extends the norm of $K$. Therefore it determines a point of $X^{{\text{\rm an}}}_{\sigma}$ that we denote as $\theta_{\sigma}(\gamma)$. The maps $\theta_{\sigma}$ are injective, continuous and proper. Moreover, they glue together to define a map

$$\theta_{\Sigma}\colon X_{\Sigma}(\mathbb{R}_{\geq 0})\longrightarrow X_{\Sigma}^{{\text{\rm an}}}$$

that is injective, continuous and proper. Every point in the image of $\theta_{\Sigma}$ is fixed under the action of $\mathbb{S}^{{\text{\rm an}}}$.