  R.M. Aron et al. proved that the Cluster Value Theorem in the infinite
dimensional Banach space setting holds for the Banach algebra
$\mathcal{H}^\infty (B_{c_0})$. On the other hand, B.J. Cole and T.W. Gamelin
showed that $\mathcal{H}^\infty (\ell_2 \cap B_{c_0})$ is isometrically
isomorphic to $\mathcal{H}^\infty (B_{c_0})$ in the sense of an algebra.
Motivated by this work, we are interested in a class of open subsets $U$ of a
Banach space $X$ for which $\mathcal{H}^\infty (U)$ is isometrically isomorphic
to $\mathcal{H}^\infty (B_{c_0})$. We prove that there exist polydisk type
domains $U$ of any infinite dimensional Banach space $X$ with a Schauder basis
such that $\mathcal{H}^\infty (U)$ is isometrically isomorphic to
$\mathcal{H}^\infty (B_{c_0})$, which generalizes the result by Cole and
Gamelin. Furthermore, we study the analytic and algebraic structure of the
spectrum of $\mathcal{H}^\infty (U)$ and show that the Cluster Value Theorem is
true for $\mathcal{H}^\infty (U)$.
