A vector-valued Banach-Stone theorem with distortion \(\sqrt{2}\) (Q2398625)

Let \(K, S\) be locally compact Hausdorff spaces and let \(X\) be a real Hilbert space of dimension greater than or equal to \(2\). Let \(T: C(K,X) \rightarrow C(S,X)\) be a linear isomorphism. A classical result of \textit{M. Cambern} [Proc. Am. Math. Soc. 18, 1062--1066 (1967; Zbl 0165.47402)] says that when \(X\) is the scalar field and if \(\|T\|\|T^{-1}\|<2\), then \(K\) and \(S\) are homeomorphic. This conclusion can fail if \(T\) is such that \(\|T\|\|T^{-1}\| = 2\). Since then there have been a lot of efforts for obtaining vector-valued analogues of this result as well as to improve on the bound \(2\). In this interesting paper, the author improves on \textit{M. Cambern}'s result in the vector-valued case with \(\|T\|\|T^{-1}\| < \sqrt{2}\) [Ill. J. Math. 20, 1--11 (1976; Zbl 0317.46030)] by showing that the topological spaces are homeomorphic even when \(\|T\|\|T^{-1}\| = \sqrt{2}\).