Hostname: page-component-6d856f89d9-4thr5 Total loading time: 0 Render date: 2024-07-16T06:37:49.371Z Has data issue: false hasContentIssue false

How Lipschitz Functions Characterize the Underlying Metric Spaces

Published online by Cambridge University Press:  20 November 2018

Lei Li
School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China e-mail:
Ya-Shu Wang
Department of Mathematics, University of Alberta, Edmonton, AB, T6G 2G1 e-mail:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $X$ and $Y$ be metric spaces and $E$, $F$ be Banach spaces. Suppose that both $X$ and $Y$ are realcompact, or both $E$, $F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z\left( f \right)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if

$$z\left( f \right)\,\subseteq \,z\left( g \right)\,\,\,\,\Leftrightarrow \,\,\,\,z\left( Tf \right)\,\subseteq \,z\left( Tg \right),\,\,\,\,\,\text{or}\,\,\,\,z\left( f \right)\,=\,\varnothing \,\,\,\Leftrightarrow \,\,\,z\left( Tf \right)\,=\,\varnothing ,$$

respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim\,E\,=\,\dim\,F\,<\,+\infty$, is a weighted composition operator $\left( Tf \right)\left( y \right)\,=\,{{J}_{y}}\left( f\left( \tau \left( y \right) \right) \right)$. We show that the map $\tau \,:\,Y\,\to \,X$ is a locally (little) Lipschitz homeomorphism.

Research Article
Copyright © Canadian Mathematical Society 2014


Li is supported by The National Natural Science Foundation of China (11271199)


[1] Abramovich, Yu. A., Multiplicative representation of disjointness preserving operators. Nederl. Akad. Wetensch. Indag. Math. 45 (1983, no. 3, 265279. CrossRefGoogle Scholar
[2] Abramovich, Yu. A., Veksler, A. I., and Kaldunov, A. V., On operators preserving disjointness. Soviet Math. Dokl. 20 (1979, 10891093.Google Scholar
[3] Araujo, J., Separating maps and linear isometries between some spaces of continuous functions. J. Math. Anal. Appl. 226 (1998, no. 1, 2339. CrossRefGoogle Scholar
[4] Araujo, J., Realcompactness and spaces of vector-valued functions. Fund. Math. 172 (2002, no. 1, 2740. CrossRefGoogle Scholar
[5] Araujo, J., Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity. Adv. Math. 187 (2004, no. 2, 488520. Google Scholar
[6] Araujo, J., Beckenstein, E., and Narici, L., Biseparating maps and homeomorphic real-compactifications. J. Math. Anal. Appl. 192 (1995, no. 1, 258265. Google Scholar
[7] Araujo, J. and Dubarbie, L., Biseparating maps between Lipschitz function spaces. J. Math. Anal. Appl. 357 (2009, no. 1, 191200. Google Scholar
[8] Dubarbie, L., Separating maps between spaces of vector-valued absolutely continuous functions. Canad. Math. Bull. 53 (2010, no. 3, 466474.Google Scholar
[9] Garrido, M. I. and Jaramillo, J. A., Homomorphisms on function lattices. Monatsh. Math. 141 (2004, no. 2, 127146. CrossRefGoogle Scholar
[10] Gau, H.-L., Jeang, J.-S., and Wong, N.-C., Biseparating linear maps between continuous vector valued function spaces. J. Aust. Math. Soc. 74 (2003, no. 1, 101109. CrossRefGoogle Scholar
[11] Gillman, L. and Jerison, M., Rings of continuous functions. The University Series in Higher Mathematics, D. Van Nostrand Co., Princeton, NJ, 1960.Google Scholar
[12] Hernandez, S., Beckenstein, E., and Narici, L., Banach-Stone theorems and separating maps. Manuscripta Math. 86 (1995, no. 4, 409416. Google Scholar
[13] Jimènez-Vargas, A., Villegas-Vallecillos, M., and Wang, Y.-S., Banach-Stone theorems for vector-valued little Lipschitz functions. Publ. Math. Debrecen 74 (2009, no. 12, 81100.Google Scholar
[14] Jimènez-Vargas, A. and Wang, Y.-S., Linear biseparating maps between vector-valued little Lipschitz function spaces. Acta Math. Sin. (Engl. Ser.) 26 (2010, no. 6, 10051018. Google Scholar
[15] Leung, D. H., Biseparating maps on generalized Lipschitz function spaces. Studia Math. 196 (2010, no. 1, 2340. Google Scholar
[16] Li, L. and Wong, N.-C., Kaplansky theorem for completely regular spaces. Proc. Amer. Math. Soc., to appear. [17] L. Li and N.-C.Wong, Banach-Stone theorems for vector valued functions on completely regular spaces. J. Math. Anal. Appl. 395 (2012, no. 1, 265274. CrossRefGoogle Scholar
[18] McShane, E. J., Extension of range of functions. Bull. Amer. Math. Soc. 40 (1934, no. 12, 837842. CrossRefGoogle Scholar
[19] Scanlon, C. H., Rings of functions with certain Lipschitz properties. Pacific J. Math. 32 (1970, 197201. Google Scholar