Weak forms of Banach–Stone theorem for C0(K,X) spaces via the αth derivatives of K

Let X be a Banach space and S   be a locally compact Hausdorff space. By C0(S,X)C0(S,X) we will stand the Banach space of all continuous X-valued functions on S endowed with the supremum norm.Suppose that C0(S,X)C0(S,X) contains a copy of some C0(K)C0(K) space with K infinite. Does it follow that the cardinality of the αth derivative of K is less than or equal to the αth derivative of S, for every ordinal number α  ? In general the answer is no, even when α=0α=0.In the present paper we prove that the answer is yes whenever X   contains no copy of c0c0 and α=0α=0. Moreover, in the case where α>0α>0 and the αth derivative of S   is infinite, we show that the existence an isomorphism from C0(K)C0(K) into C0(S,X)C0(S,X) with distortion ‖T‖‖T−1‖‖T‖‖T−1‖ strictly less than 3 provides also a positive answer to this question.As a consequence, we improve a classical Cengiz theorem and a recent result on isomorphisms between spaces of vector-valued continuous functions by obtaining two weak forms of Banach–Stone theorem for C0(S,X)C0(S,X) spaces via the αth derivatives of S.