Banach-lattice isomorphisms of $C_0(K,X)$ spaces which determine the locally compact spaces $K$

An Amir-Cambern theorem for subspaces of Banach lattice-valued continuous functions

“…The next lemma describes an important property of the parameter λ + . For the proof see [13,Lemma 5.1]. In our case, actually, we need to use this lemma only for r = 1, in which case its conclusion follows easily from the definition.…”

“…In [13] it was shown how the constant λ(E) can be improved under aditional assumptions on the isomorphism T . More specifically, the authors assume that E is a real Banach lattice and T : C 0 (K 1 , E) → C 0 (K 2 , E) is a Banach lattice isomorphism.…”

“…It is easily seen that λ(E) ≤ λ + (E) for each Banach lattice E, and in [13] it is shown that for E = ℓ p , where 1 ≤ p < 2, the inequality is strict, and moreover, that the constant λ + (E) is the best possible for classical spaces E = ℓ p , where p ∈ [1, ∞). Also, from the definition it follows that 1 ≤ λ + (E) ≤ 2 for each Banach space E.…”

“…The aim of this paper is to present an analogue of the result of [13] in the context of subspaces. However, we are able to extend this result to this general setting only in the case where the Banach lattices are reflexive, which is not needed in [13].…”

“…The aim of this paper is to present an analogue of the result of [13] in the context of subspaces. However, we are able to extend this result to this general setting only in the case where the Banach lattices are reflexive, which is not needed in [13]. The reason for this is that meanwhile operators from C(K, E 1 ) spaces to E 2 may be represented by Borel measures on K with values in L(E 1 , E * * 2 ), the space of bounded linear operators from E 1 to E * * 2 (see Remark 5.4), such a representation is not available for operators from H 1 ⊂ C(K, E 1 ) to E 2 .…”