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 continuous 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 particular, the lattice version states that every disjointness preserving linear bijection $T$ from $C(X)$ onto $C(Y)$ is a weighted composition operator $Tf=h\cdot f\circ\varphi$ which provides a homeomorphism $\varphi$ 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 $C(X)$.