Zero-separating algebras of continuous functions

Let A be a lattice-ordered algebra endowed with a topology compatible with the structure of algebra. We provide internal conditions for A to be isomorphic as lattice-ordered algebras and homeomorphic to Ck(X), the lattice-ordered algebra C(X) of real continuous functions on a completely regular and Hausdorff topological space X, endowed with the topology of uniform convergence on compact sets. As a previous step, we determine this topology among the locally m-convex topologies on C(X) with the property that each order closed interval is bounded.