AN ALGEBRAIC APPROACH TO THE BANACH-STONE THEOREM FOR SEPARATING LINEAR BIJECTIONS

Abstract. Let X be a compact Hausdorff space and C(X) the space of continu-ous functions defined on X. There are three versions of the Banach-Stone theorem. They assert that the Banach space geometry, the ring structure, and the lattice structure of C(X) determine the topological structure of X, respectively. In par-ticular, the lattice version states that every disjointness preserving linear bijection T from C(X) onto C(Y) is a weighted composition operator Tf = h · f ◦ ϕ which provides a homeomorphism ϕ from Y onto X. In this note, we manage to use basically algebraic arguments to give this lattice version a short new proof. In this way, all three versions of the Banach-Stone theorem are unified in an algebraic framework such that different isomorphisms preserve different ideal structures of C(X). Let X be a compact Hausdorff space and C(X) the vector space of continuous (real or complex) functions on X. It is a common interest to see how the topological structure of X can be recovered from C(X). If we look at C(X) as a Banach space then the classical Banach-Stone theorem states that whenever there is a surjectiv