Lattice renormings of C0(X) spaces

Suppose X is a locally compact Polish space, and G is a group of lattice isometries of C₀(X) which satisfies certain conditions. Then we can equip C₀(X) with an equivalent lattice norm |||·||| so that G is the group of lattice isometries of (C₀(X),|||·|||). As an application, we show that for any locally compact Polish group G there exists a locally compact Polish space X, and a lattice norm |||·||| on C₀(X), so that G is the group of lattice isometries of (C₀(X),|||·|||).