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

We investigate the geometry of $C(K,X)$ and $\ell _{\infty }(X)$ spaces through complemented subspaces of the form $(\bigoplus _{i\in \varGamma }X_i)_{c_0}$. For Banach spaces $X$ and $Y$, we prove that if $\ell _{\infty }(X)$ has a complemented subspace isomorphic to $c_0(Y)$, then, for some $n \in \mathbb 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 $C(K)\sim c_0(C(K))$ and $C(L)\sim c_0(C(L))$ and $\ell _{\infty }(C(K,X))\sim \ell _{\infty }(C(L,Y))$, then $K$ and $L$ have the same cardinality.

(2) If $K$ and $L$ are infinite and metrizable and $\ell _{\infty }(C(K,X))\sim \ell _{\infty }(C(L,Y))$, then $C(K)$ is isomorphic to $C(L)$.