Multivariable $({\phi }_q,{\mathcal{O}}_K^{\times })$-modules associated to $p$-adic representations of $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{K}\mathrm{/}K)$

Let $K$ be an unramified extension of ${\mathbb{Q}}_p$, and $E$ a finite extension of $K$ with ring of integers ${\mathcal{O}}_E$. We associate to every finite type continuous ${\mathcal{O}}_E$-representation $\rho$ of $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{K}\mathrm{/}K)$ an étale $({\phi }_q,{\mathcal{O}}_K^{\times })$-module $D_{A_{\mathrm{m}\mathrm{v},E}}^{(0)}(\rho )$ over $A_{\mathrm{m}\mathrm{v},E}$, where $A_{\mathrm{m}\mathrm{v},E}$ is the $p$-adic completion of a completed localization of the Iwasawa algebra ${\mathcal{O}}_E[\text{\hspace{0.17em}⁣}[{\mathcal{O}}_K]\text{\hspace{0.17em}⁣}]$. Furthermore, we prove that the functor $D_{A_{\mathrm{m}\mathrm{v},E}}^{(0)}$ is fully faithful and exact. This functor is a $p$-adic analogue of $D_A^{(0)}$ in the recent work of Breuil, Herzig, Hu, Morra and Schraen.