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Zbl 1018.46005
Gau, Hwa-Long; Jeang, Jyh-Shyang; Wong, Ngai-Ching
An algebraic approach to the Banach-Stone theorem for separating linear bijections.
(English)
[J] Taiwanese J. Math. 6, No.3, 399-403 (2002). ISSN 1027-5487

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 2000:
*46B04 Isometric theory of Banach spaces
47B38 Operators on function spaces

Keywords: ideal structures of $C(X)$; Banach-Stone theorem

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