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 supremum norm. We first introduce the concept of ω1-quotient of Banach spaces X. Thus, we prove that if X has some ω1-quotient which is uniformly convex, then for all K1 and K2 the following statements are equivalent: