The spaces for compact metric spaces K and X with a uniformly convex maximal factor

In this paper, we prove that if a Banach space X contains some uniformly convex subspace in certain geometric position, then the C ( K , X ) spaces of all X-valued continuous functions defined on the compact metric spaces K have exactly the same isomorphism classes that the C ( K ) spaces. This provides a vector-valued extension of classical results of Bessaga and Pełczyński (1960) [2] and Milutin (1966) [13] on the isomorphic classification of the separable C ( K ) spaces. As a consequence, we show that if 1 < p < q < ∞ then for every infinite countable compact metric spaces K 1 , K 2 , K 3 and K 4 are equivalent:(a) 1 , l p ) ⊕ C ( K 2 , l q ) is isomorphic to C ( K 3 , l p ) ⊕ C ( K 4 , l q ) . 1 ) is isomorphic to C ( K 3 ) and C ( K 2 ) is isomorphic to C ( K 4 ) .