Complementations in $C(K,X)$ and $\ell_\infty(X)$

Abstract

We investigate the geometry of C(K, X) and ℓ∞(X) spaces through complemented subspaces of the form i∈Γ Xi c 0 . Concerning the geometry of C(K, X) spaces we extend some results of D. Alspach and E. M. Galego from [1]. On ℓ∞-sums of Banach spaces we prove that if ℓ∞(X) has a complemented subspace isomorphic to c0(Y ), then, for some n ∈ N, X n has a subspace isomorphic to c0(Y ). We further prove the following:(1) If C(K) ∼ c0(C(K)) and C(L) ∼ c0(C(L)) and ℓ∞(C(K)) ∼ ℓ∞(C(L)), then K and L have the same cardinality. (2) if K1 and K2 are infinite metric compacta, then ℓ∞(C(K1)) ∼ ℓ∞(C(K2)) if and only if C(K1)is isomorphic to C(K2).