On positive embeddings of \(C(K)\) spaces into \(C(S,X)\) lattices (Q1664499)

A result of \textit{G. Plebanek} [Stud. Math. 216, No. 2, 179--192 (2013; Zbl 1286.46026)] established that a positive isomorphism from $C(K)$ into (not necessarily onto) $C(S)$ for compact \(K\) and \(S\) implies the existence of a~map \(\psi\) from $S$ into the collection of all subsets of \(K\) with cardinally less than or equal to a~number \(n\) which is upper semicontinuous and \(\cup \{\psi (s): s \in S\}=K\). Here, the authors extend this result to maps from \(C(K)\) into \(C(S,X)\) for a~class of Banach-lattices. These Banach-lattices \(X\) have the property that \(\lambda ^+(X) = \operatorname{inf}\{\max ||x-y||,||x+y|| : x \text{ and }y \text{ are both positive with norm }1\}>1\). A~variety of results and examples are discussed in relation to values of \(\lambda^+\). Furthermore, it is shown that, for \(K\) and \(S\) locally compact and \(X\) a Banach lattice with \(\lambda^+ (X) >1\), if a positive isomorphism \(T:C_0(K)\rightarrow C_0(S,X)\) (where the subscript \(0\) denotes functions vanishing at infinity) satisfies \(\|T\| \,\|T^{-1}\|< \lambda ^+(X)\), then for all ordinals \(\alpha\), \(K^{(\alpha)}\) is a continuous image of a subspace of \(S^{(\alpha)}\). The set \(K ^{(\alpha)}\) (or \(S^{(\alpha)}\)) is defined recursively by deleting isolated points. The paper includes several unsolved problems suggested by these results.