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

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