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

Let X be a compact Hausdorfi space and C(X) the space of continu- ous functions deflned 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 - ' which provides a homeomorphism ' 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 unifled in an algebraic framework such that difierent isomorphisms preserve difierent ideal structures of C(X). Let X be a compact Hausdorfi 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 surjective linear isometry T between C(X) and C(Y ) for some other compact Hausdorfi space Y , T induces a homeomorphism between X and Y (see e.g. (3, p.172)). Here is a sketch of the proof. The dual map T⁄ of T preserves extreme points of the dual balls, which are exactly those linear functionals in the form of ‚-x for some unimodular scalar ‚ and point mass -x at some point x 2 X. Thus T⁄-y = h(y)-'(y) deflnes a scalar-valued function h on Y and a map ' : Y ! X. In other words,