  Let $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the
open unit ball $B_X$ of a complex Banach space $X$. Considering the Gelfand
transform image $\widehat{\mathcal{A}}$ of the Banach algebra $\mathcal{A}$,
which is a uniform algebra on the spectrum of $\mathcal{A}$, we obtain an
explicit description of the Shilov boundary for $\widehat{\mathcal{A}}$ for
classical Banach spaces $X$ in the case where $\mathcal{A}$ is a certain Banach
algebra, for instance, $\mathcal{A}_\infty (B_X)$, $\mathcal{A}_u (B_X)$ or
$\mathcal{A}_{wu} (B_X)$. Some possible application of our result to the famous
Corona theorem is also briefly discussed.
