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

Abstract

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:(a) C(K1, X) is isomorphic to C(K2, X).(b) C(K1) is isomorphic to C(K2). This allows us to classify, up to an isomorphism, some C(K, Y ⊕ lp(Γ)) spaces, 1 < p ≤ ∞, and certain C(S) spaces involving large compact Hausdorff spaces S.