How do the positive embeddings of Banach lattices depend on the αth derivatives of K?

Abstract

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 COfalse(Kfalse) into COfalse(S,Xfalse). Then for each ordinal α the cardinalities of the αth derivatives Kfalse(αfalse) and Sfalse(αfalse) satisfy the following inequality ||K(α)1/p≤false∥Tfalse∥T−1||S(α)1/p.Moreover, if false∥Tfalse∥T−1<21/p,then Kfalse(αfalse) is a continuous image of a subset of Sfalse(αfalse) 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.