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<p>We provide a method for reconstructing the underlying locallycompact Hausdorff space of an algebra of continuous functions vanishing at infinity, using the norm attainment sets of continuous functions. As an application,it is demonstrated that an order isomorphism of norm attainment sets betweenspaces of continuous functions induces a homeomorphism between the underlying topological spaces, even without linearity. Moreover, it turns out that a linearmap between algebras of continuous functions is an order isomorphism of normattainment sets if and only if it is a scalar multiple of an isometric isomorphismprovided that the underlying topological spaces are not two-point sets. We alsopresent a counterexample to the above statement in the two-point setting.</p>

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On order isomorphisms of norm attainment sets of continuous functions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月24日

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  • K. Igarashi,
  • J. Nakamura,
  • S. Roy,
  • R. Tanaka

摘要

We provide a method for reconstructing the underlying locallycompact Hausdorff space of an algebra of continuous functions vanishing at infinity, using the norm attainment sets of continuous functions. As an application,it is demonstrated that an order isomorphism of norm attainment sets betweenspaces of continuous functions induces a homeomorphism between the underlying topological spaces, even without linearity. Moreover, it turns out that a linearmap between algebras of continuous functions is an order isomorphism of normattainment sets if and only if it is a scalar multiple of an isometric isomorphismprovided that the underlying topological spaces are not two-point sets. We alsopresent a counterexample to the above statement in the two-point setting.