  We investigate the geometry of $C(K,X)$ and $\ell_{\infty}(X)$ spaces through
complemented subspaces of the form $\left(\bigoplus_{i\in
\varGamma}X_i\right)_{c_0}$. Concerning the geometry of $C(K,X)$ spaces we
extend some results of D. Alspach and E. M. Galego from \cite{AlspachGalego}.
On $\ell_{\infty}$-sums of Banach spaces 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 $c_0(Y)$. We further prove the
following:
  (1) If $C(K)\sim c_0(C(K))$ and $C(L)\sim c_0(C(L))$ and
$\ell_{\infty}(C(K))\sim \ell_{\infty}(C(L))$, then $K$ and $L$ have the same
cardinality.
  (2) If $K_1$ and $K_2$ are infinite metric compacta, then
$\ell_{\infty}(C(K_1))\sim \ell_{\infty}(C(K_2))$ if and only if $C(K_1)$ is
isomorphic to $C(K_2)$.
