I am (still) working through the paper *Singular semipositive metrics in non-Archimedean geometry* by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. **25** (2016), 77-139, doi:10.1090/jag/656, arXiv:1201.0187).

Here are some other questions i have: In subsection 2.5 they define a model function as a continous $\varphi$ on $X^{\text{an}}$ s.t. there is a vertical Cartier Divisor $D \in \text{Div}_{\mathbb{Q}}(\mathcal{X})$, where $\mathcal{X}$ is a model of $X^{\text{an}}$, with $\varphi_D = \log \vert \mathcal{O}_{\mathcal{X}}(D) \vert = \varphi$. The set of these functions is denoted by $\mathcal{D}(X) = \mathcal{D}(X) _{\mathbb{Q}}$. In Prop. 2.2 they then show, among other things, that $\mathcal{D}(X) _{\mathbb{Z}}$ is stable under max. and seperates points. (I am assuming in the definition of $\mathcal{D}(X) _{\mathbb{Z}}$ one just considers $D \in \text{Div}_{\mathbb{Z}}(\mathcal{X})$?)

Is it clear, that if $\mathcal{D}(X) _{\mathbb{Z}}$ seperates Points and is stable under max. (see Prop.2.2), that this is also true for $\mathcal{D}(X) _{\mathbb{Q}}$?

Is there an isomorphism from $\mathcal{D}(X) _{\mathbb{Q}}$ to $\mathcal{D}(X) _{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes}\mathbb{Q}$?

Is it immediate that the $\mathbb{Q}$-VS $\mathcal{D}(X) _{\mathbb{Q}}$ fulfills the conditions of the Stone-Weierstraß Theorem(see Cor.2.3) and is thus dense in $C^0(X^{\text{an}})$? From my understanding one would require that $\mathbb{Q}$-VS $\mathcal{D}(X) _{\mathbb{Q}}$ is closed under multiplication with elements from $\mathbb{R}$?

Thanks a lot guys.