On the distortion of a linear embedding of \(C(K)\) into a \(C_{0}(\varGamma,X)\) space (Q1684824)

The distortion \(\| T\| \| T^{-1}\|\) of a linear embedding \(T: C(K) \to C_{0}(\Gamma, X)\) is studied, where \(K\) is a compact Hausdorff space, \(\Gamma\) is an infinite discrete topological space and \(X\) is a Banach space. The author proves that, if \(X\) contains no subspace isomorphic to \(c_{0}\) and, for some positive integer \(n\), the \(n\)th derivative of \(K\) is non-empty, then \(\| T\| \| T^{-1}\| \geq 2n+1\). This theorem extends a result of \textit{L. Candido} and \textit{E. M. Galego} [Fundam. Math. 218, No. 2, 151--163 (2012; Zbl 1258.46002)] and answers a question from [\textit{L. Candido} and \textit{E. M. Galego}, J. Math. Anal. Appl. 402, No. 1, 185--190 (2013; Zbl 1271.46006)].