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A solution to the Cambern problem for finite-dimensional Hilbert spaces

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Abstract

Let X be a Hilbert space of real dimension n ≥ 2, and δ > 0 satisfying

$$n - 1 < \frac{{2 - \delta }}{{10\delta + 8\sqrt {2\delta + {\delta ^2}} }}.$$

In this paper, it is proven that if K and S are locally compact Hausdorff spaces and T is an isomorphism from C0(K,X) onto C0(S,X) satisfying

$$||T||||{T^{ - 1}}|| < \sqrt {2 + \delta }, $$

then K and S are homeomorphic.

This solves a long-standing open problem posed by Cambern on Hilbert-valued Banach–Stone theorems via isomorphisms T with distortion ||T|| ||T−1|| strictly greater than \(\sqrt 2 \).

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Authors and Affiliations

  1. Department of Mathematics, IME, University of São Paulo, Rua do Matão 1010, São Paulo, Brazil

    Elói Medina Galego & André Luis Porto da Silva

Authors
  1. Elói Medina Galego
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  2. André Luis Porto da Silva
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Correspondence to Elói Medina Galego.

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Galego, E.M., Porto da Silva, A.L. A solution to the Cambern problem for finite-dimensional Hilbert spaces. Isr. J. Math. 231, 419–436 (2019). https://doi.org/10.1007/s11856-019-1858-6

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  • DOI: https://doi.org/10.1007/s11856-019-1858-6

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