Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]