How do the positive embeddings of C 0 ( 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 L p space. Suppose that T is a positive Banach lattice isomorphism fromC O ( K )intoC O ( S , X ) . Then for each ordinal α the cardinalities of the αth derivativesK ( α )andS ( α )satisfy the following inequalityK ( α )1 / p ≤ ∥ T ∥T − 1S ( α )1 / p . Moreover, if∥ T ∥T − 1< 2 1 / 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 number2 1 / pis sharp in these vector‐valued theorems.