On the isomorphic classification of C(K, X) spaces

We provide isomorphic classifications of some C(K, X) spaces, the Banach spaces of all continuous X-valued functions defined on infinite compact metric spaces K, equipped with the suprernum norm. We first introduce the concept of w(1)-quotient of Banach spaces X. Thus, we prove that if X has some w(1)-quotient which is uniformly convex, then for all K-1 and K-2 the following statements are equivalent: (a) C(K-1, X) is isomorphic to C(K-2, X). (b) C(K-1) is isomorphic to C(K-2). This allows us to classify, up to an isomorphism, some C(K,Y (1) circle plus l(p) (Gamma)) spaces, 1 < p <= infinity, and certain C(S) spaces involving large compact Hausdorff spaces S. (C) 2015 Elsevier Inc. All rights reserved.