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Kaplansky Theorem for completely regular spaces

Authors: Lei Li and Ngai-Ching Wong
Journal: Proc. Amer. Math. Soc. 142 (2014), 1381-1389
MSC (2000): Primary 46E40, 54D60; Secondary 46B42, 47B65
Published electronically: January 30, 2014
MathSciNet review: 3162258
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Abstract: Let $ X, Y$ be realcompact spaces or completely regular spaces consisting of $ G_\delta $-points. Let $ \phi $ be a linear bijective map from $ C(X)$ (resp. $ C^b(X)$) onto $ C(Y)$ (resp. $ C^b(Y)$). We show that if $ \phi $ preserves nonvanishing functions, that is,

$\displaystyle f(x)\neq 0,\forall \, x\in X, \quad \Longleftrightarrow \quad \phi (f)(y)\neq 0, \forall \, y\in Y, $

then $ \phi $ is a weighted composition operator

$\displaystyle \phi (f)=\phi (1)\cdot f\circ \tau , $

arising from a homeomorphism $ \tau $ from $ Y$ onto $ X$. This result is applied also to other nice function spaces, e.g., uniformly or Lipschitz continuous functions on metric spaces.

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Additional Information

Lei Li
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

Ngai-Ching Wong
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Keywords: Nonvanishing preservers, disjointness preservers, common zero preservers, realcompact spaces, Banach-Stone theorems
Received by editor(s): October 4, 2011
Received by editor(s) in revised form: May 5, 2012
Published electronically: January 30, 2014
Additional Notes: This work was supported by The National Natural Science Foundation of China (grants 11071129, 11301285) and Taiwan NSC (grants 098-2811-M-110-039, 99-2115-M-110-007-MY3).
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.