## Taiwanese Journal of Mathematics

### 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 $C(X)$.

#### Article information

Source
Taiwanese J. Math., Volume 6, Number 3 (2002), 399-403.

Dates
First available in Project Euclid: 20 July 2017

https://projecteuclid.org/euclid.twjm/1500558305

Digital Object Identifier
doi:10.11650/twjm/1500558305

Mathematical Reviews number (MathSciNet)
MR1921602

Zentralblatt MATH identifier
1018.46005

#### Citation

Gau, Hwa-Long; Jeang, Jyh-Shyang; Wong, Ngai-Ching. AN ALGEBRAIC APPROACH TO THE BANACH-STONE THEOREM FOR SEPARATING LINEAR BIJECTIONS. Taiwanese J. Math. 6 (2002), no. 3, 399--403. doi:10.11650/twjm/1500558305. https://projecteuclid.org/euclid.twjm/1500558305