On order isomorphisms of norm attainment sets of continuous functions

We provide a method for reconstructing the underlying locally compact Hausdorff space of an algebra of continuous functions vanishing at infinity, using the norm attainment sets of continuous functions. As an application, it is demonstrated that an order isomorphism of norm attainment sets between spaces of continuous functions induces a homeomorphism between the underlying topological spaces, even without linearity. Moreover, it turns out that a linear map between algebras of continuous functions is an order isomorphism of norm attainment sets if and only if it is a scalar multiple of an isometric isomorphism provided that the underlying topological spaces are not two-point sets. We also present a counterexample to the above statement in the two-point setting.