How does the distortion of linear embedding of \(C_0(K)\) into \(C_0(\Gamma ,X)\) spaces depend on the height of \(K\)? (Q1947276)

The authors consider the classical questions of the `Banach-Stone' type for the space of vector-valued continuous functions \(C(K, X)\), when the Banach space \(X\) has non-trivial cotype. It is shown that for an infinite discrete set \(\Gamma\) and a locally compact space \(K\), if there is an into isomorphism \(T: C_0(K) \rightarrow C_0(\Gamma, X)\), then \(K\) has finite height and \(\|T^{-1}\|\|T\| \geq 2 ht(K)-1\). See also [the authors, Fundam. Math. 220, No. 1, 83--92 (2013; Zbl 1271.46005)] for related results.