An algebraic approach to the Banach-Stone theorem for separating linear bijections. (English) Zbl 1018.46005

Gau, Hwa-Long; Jeang, Jyh-Shyang; Wong, Ngai-Ching

Taiwanese J. Math. 6, No. 3, 399-403 (2002).

Summary: 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 \(C(X)\).

MSC:

46B04	Isometric theory of Banach spaces
47B38	Linear operators on function spaces (general)

Keywords:

ideal structures of \(C(X)\); Banach-Stone theorem