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How do the positive embeddings of C0(K,X) Banach lattices depend on the αth derivatives of K?

Let K and S be locally compact Hausdorff spaces and X be an abstract Lp space. Suppose that T is a positive Banach lattice isomorphism from CO(K) into CO(S,X) . Then for each ordinal α the cardinalities of the αth derivatives K(α) and S(α) satisfy the following inequality K(α)1/p≤∥T∥T−1S(α)1/p.Moreover, if ∥T∥T−1<21/p,then K(α) is a continuous image of a subset of S(α) which can be taken closed when K is compact. The first statement of this result for p=1 is a vector‐valued extension of a Cengiz's theorem and the second one is vector‐valued version of a Holsztyński's theorem. A simple example shows that the number 21/p is sharp in these vector‐valued theorems.