  We study the vector-valued spectrum
$\mathcal{M}_{u,\infty}(B_{\ell_2},B_{\ell_2})$ which is the set of nonzero
algebra homomorphisms from $\mathcal{A}_u(B_{\ell_2})$ (the algebra of
uniformly continuous holomorphic functions on $B_{\ell_2}$) to $\mathcal
{H}^\infty(B_{\ell_2})$ (the algebra of bounded holomorphic functions on
$B_{\ell_2}$). This set is naturally projected onto the closed unit ball of
$\mathcal {H}^\infty(B_{\ell_2}, \ell_2)$ giving rise to an associated
fibering. Extending the classical notion of cluster sets introduced by I. J.
Schark (1961) to the vector-valued spectrum we define vector-valued cluster
sets. The aim of the article is to look at the relationship between fibers and
cluster sets obtaining results regarding the existence of analytic balls into
these sets.
