Lattice renormings of $C_0(X)$ spaces

Suppose $X$ is a locally compact Polish space, and $G$ is a group of lattice isometries of $C_0(X)$ which satisfies certain conditions. Then we can equip $C_0(X)$ with an equivalent lattice norm $\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\cdot \mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }$ so that $G$ is the group of lattice isometries of $(C_0(X),\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\cdot \mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid })$. As an application, we show that for any locally compact Polish group $G$ there exists a locally compact Polish space $X$, and an lattice norm $\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\cdot \mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }$ on $C_0(X)$, so that $G$ is the group of lattice isometries of $(C_0(X),\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\cdot \mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid }\text{\hspace{0.17em}⁣}\mathrm{\mid })$.