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R.M. Aron et al. proved that the Cluster Value Theorem in the infinitedimensional Banach space setting holds for the Banach algebra$\mathcal{H}^\infty (B_{c_0})$. On the other hand, B.J. Cole and T.W. Gamelinshowed that $\mathcal{H}^\infty (\ell_2 \cap B_{c_0})$ is isometricallyisomorphic 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 aBanach space $X$ for which $\mathcal{H}^\infty (U)$ is isometrically isomorphicto $\mathcal{H}^\infty (B_{c_0})$. We prove that there exist polydisk typedomains $U$ of any infinite dimensional Banach space $X$ with a Schauder basissuch that $\mathcal{H}^\infty (U)$ is isometrically isomorphic to$\mathcal{H}^\infty (B_{c_0})$, which generalizes the result by Cole andGamelin. Furthermore, we study the analytic and algebraic structure of thespectrum of $\mathcal{H}^\infty (U)$ and show that the Cluster Value Theorem istrue for $\mathcal{H}^\infty (U)$.

The spectra of Banach algebras of holomorphic functions on polydisk type domains