An Amir–Cambern theorem for subspaces of Banach lattice-valued continuous functions

Abstract

For \(i=1,2\), let \(E_i\) be a reflexive Banach lattice over \({\mathbb {R}}\) with a certain parameter \(\lambda ^+(E_i)>1\), let \(K_i\) be a locally compact (Hausdorff) topological space and let \({\mathcal {H}}_i\) be a closed subspace of \({\mathcal {C}}_0(K_i, E_i)\) such that each point of the Choquet boundary \({\text {Ch}}_{{\mathcal {H}}_i} K_i\) of \({\mathcal {H}}_i\) is a weak peak point. We show that if there exists an isomorphism \(T:{\mathcal {H}}_1\rightarrow {\mathcal {H}}_2\) with \(\left\| T\right\| \cdot \left\| T^{-1}\right\| <\min \lbrace \lambda ^+(E_1), \lambda ^+(E_2) \rbrace\) such that T and \(T^{-1}\) preserve positivity, then \({\text {Ch}}_{{\mathcal {H}}_1} K_1\) is homeomorphic to \({\text {Ch}}_{{\mathcal {H}}_2} K_2\).

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Acknowledgements

The first author was supported by the Research grant GAUK 864120.

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Correspondence to Jakub Rondoš.

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Communicated by Maria Joita.

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Rondoš, J., Spurný, J. An Amir–Cambern theorem for subspaces of Banach lattice-valued continuous functions. Banach J. Math. Anal. 15, 30 (2021). https://doi.org/10.1007/s43037-020-00112-8

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Keywords

  • Function space
  • Vector-valued Banach–Stone theorem
  • Amir–Cambern theorem
  • Banach lattice

Mathematics Subject Classification

  • 47B38
  • 46A55