Complementations in C(K, X) AND ℓ ∞ (X)

Abstract

We investigate the geometry of C(K, X) and ℓ ∞ (X) spaces through complemented subspaces of the form ( i∈Γ X i ) c 0 . For Banach spaces X and Y , we prove that if ℓ ∞ (X) has a complemented subspace isomorphic to c 0 (Y ), then, for some n ∈ N, X n has a subspace isomorphic to c 0 (Y ). If K and L are Hausdorff compact spaces and X and Y are Banach spaces having no subspace isomorphic to c 0 we further prove the following: (1) If C(K) ∼ c 0 (C(K)) and C(L) ∼ c 0 (C(L)) and ℓ ∞ (C(K, X)) ∼ ℓ ∞ (C(L, Y )), then K and L have the same cardinality. (2) If K and L are infinite and metrizable and ℓ ∞ (C(K, X)) ∼ ℓ ∞ (C(L, Y )), then C(K) is isomorphic to C(L).