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

  • Elói Medina GalegoEmail author
  • André Luis Porto da Silva


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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Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Elói Medina Galego
    • 1
    Email author
  • André Luis Porto da Silva
    • 1
  1. 1.Department of Mathematics, IMEUniversity of S˜ao PauloS˜ao PauloBrazil

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