A Vector-Valued Extension of a Theorem of Marciszewski

We establish a vector-valued extension of a theorem of Marciszewski. Let $K$ be a compact Hausdorff space, let $\mathrm{\Gamma }$ be a set, and let $X$ be a nonzero Banach space containing no isomorphic copy of $c_0$. We prove that if $C(K)$ embeds isomorphically into $c_0(\mathrm{\Gamma },X)$, then $C(K)$ is isomorphic to $c_0(\mathrm{\Lambda })$ for some set $\mathrm{\Lambda }$. Moreover, when $\mathrm{\Gamma }$ is infinite, the optimal distortion of embeddings of $C(K)$ into $c_0(\mathrm{\Gamma },X)$ equals that of embeddings into $c_0(\mathrm{\Gamma })$. The classification result answers a question posed in an earlier paper.