An algebraic approach to the Banach-Stone theorem for separating linear bijections

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

Functional Analysis mathscidoc:1910.43664

Taiwanese Journal of Mathematics, 399-403, 2002.9
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 . 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 unified in an algebraic framework such that different isomorphisms preserve different ideal structures of C(X).
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@inproceedings{hwa-long2002an,
  title={An algebraic approach to the Banach-Stone theorem for separating linear bijections},
  author={Hwa-Long Gau, Jyh-Shyang Jeang, and Ngai-Ching Wong},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020210004912345193},
  booktitle={Taiwanese Journal of Mathematics},
  pages={399-403},
  year={2002},
}
Hwa-Long Gau, Jyh-Shyang Jeang, and Ngai-Ching Wong. An algebraic approach to the Banach-Stone theorem for separating linear bijections. 2002. In Taiwanese Journal of Mathematics. pp.399-403. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020210004912345193.
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