This is an announcement for the paper "Banach-Stone Theorems for maps preserving common zeros" by Denny H. Leung and Wee-Kee Tang.

Abstract: Let $X$ and $Y$ be completely regular spaces and $E$ and $F$ be Hausdorff topological vector spaces. We call a linear map $T$ from a subspace of $C(X,E)$ into $C(Y,F)$ a \emph{Banach-Stone map} if it has the form $Tf(y) = S_{y}(f(h(y))$ for a family of linear operators $S_{y} : E \to F$, $y \in Y$, and a function $h: Y \to X$. In this paper, we consider maps having the property: \cap^{k}_{i=1}Z(f_{i}) \neq\emptyset\iff\cap^{k}_{i=1}Z(Tf_{i}) \neq \emptyset, where $Z(f) = {f = 0}$. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including $C^{\infty}$), as Banach-Stone maps. In particular, we confirm a conjecture of Ercan and "{O}nal: Suppose that $X$ and $Y$ are realcompact spaces and $E$ and $F$ are Hausdorff topological vector lattices (respectively, $C^{*}$-algebras). Let $T: C(X,E) \to C(Y,F)$ be a vector lattice isomorphism (respectively, $*$-algebra isomorphism) such that Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. Then $X$ is homeomorphic to $Y$ and $E$ is lattice isomorphic (respectively, $C^{*}$-isomorphic) to $F$. Some results concerning the continuity of $T$ are also obtained.

Archive classification: math.FA

Mathematics Subject Classification: 47B38

The source file(s), Banach_Stone_Lattice6.tex: 92258 bytes, is(are) stored in gzipped form as 0906.0219.gz with size 21kb. The corresponding postcript file has gzipped size 140kb.

Submitted from: matlhh@nus.edu.sg

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