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On Perturbed Isometries Between the Positive Cones of Certain Continuous Function Spaces

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Abstract

Let X, Y be two compact Hausdorff perfectly normal spaces (in particular, compact metrizable spaces), C(X) be the real Banach space of all continuous functions on X, and \(C_+(X)\) be the positive cone of C(X). In this paper, we show that if there exists a \(\delta \)-surjective \(\varepsilon \)-isometry \(F: C_+(X)\rightarrow C_+(Y)\), then X and Y are homeomorphic. Moreover, we show that there exists a unique additive surjective isometry \(V:C_+(X)\rightarrow C_+(Y)\) (the restriction of a linear surjective isometry \(U:C(X)\rightarrow C(Y)\) induced by the homeomorphism) such that

$$\begin{aligned} \Vert F(f)-V(f)\Vert \le 2\varepsilon ,\quad \textrm{for}\; \textrm{all}\;\;f\in C_+(X). \end{aligned}$$

This can be regarded as a localized generalization of the Banach–Stone theorem for compact Hausdorff perfectly normal spaces.

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Acknowledgements

The authors are grateful to the referee and the editor for their constructive comments and helpful suggestions. The first author also thanks Professor Lixin Cheng for his invaluable encouragement and advice.

Funding

Longfa Sun was supported by the National Natural Science Foundation of China (Grant no. 12101234) and the Natural Science Foundation of Hebei Province (Grant no. A2022502010).

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  1. Hebei Key Laboratory of Physics and Energy Technology, School of Mathematics and Physics, North China Electric Power University, Baoding, 071003, China

    Longfa Sun, Yinghua Sun & Shenghua Wang

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  1. Longfa Sun
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Longfa Sun was supported by the National Natural Science Foundation of China (Grant No. 12101234) and the Natural Science Foundation of Hebei Province (Grant No. A2022502010).

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Sun, L., Sun, Y. & Wang, S. On Perturbed Isometries Between the Positive Cones of Certain Continuous Function Spaces. Results Math 78, 63 (2023). https://doi.org/10.1007/s00025-023-01844-3

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