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

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 $$\sqrt 2 $$ .