\section{Introduction}

In 1961, I.J. Schark \cite{S}
proved for each $f \in \mathcal{H}^\infty (\mathbb{D})$, i.e., a bounded holomorphic function on the open disk $\mathbb{D}$, and for each $z \in \overline{\mathbb{D}}$ that the the image of the fiber $\mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D} ))$ under the Gelfand transform $\widehat{f}$ of $f$ coincides with the set of all limits of $(f(x_{\alpha}))$ for the convergent nets $(x_{\alpha})\subset D$ to $z$ (you can find the definitions of the unexplained notations in Section \ref{Background}). This result has been called as the Cluster Value Theorem. It is also known as the weak version of the Corona Theorem, which was proved by L. Carleson \cite{C}. R.M. Aron et al. \cite{ACGLM} presented the first positive results for the Cluster Value Theorem in the infinite dimensional Banach space setting by proving that the theorem is true for the Banach algebra $\mathcal{H}^\infty (B_{c_0})$.
Though they showed that this theorem is true for the closed subalgebra $\mathcal{A}_u (B_{\ell_2})$ of $\mathcal{H}^\infty (B_{\ell_2})$, it is still open if this theorem holds for $\mathcal{H}^\infty (B_{\ell_2})$. Surprisingly, a domain $U$ of a Banach space has not been found for which the Cluster Value Theorem does not hold.

Three years later, W.B. Johanson and S.O. Castillo \cite{JC} proved that the Cluster Value Theorem holds also for $C(K)$-spaces, when $K$ is a scattered compact Hausdorff space by using similar arguments as in \cite[Theorem 5.1]{ACGLM}. However, up to our knowledge, $c_0$ and $C(K)$ with $K$ a compact Hausdorff scattered, are the only known Banach space $X$ for which the Cluster Value Theorem holds for $\mathcal{H}^\infty (B_X)$. For historical background and recent developments on this topic, we refer to the survey article \cite{CGMP}.

On the other hand, we got interested in more general open domains $U$ of a Banach space $X$ than its open unit ball $B_X$ such that the Cluster Value Theorem holds for $\mathcal{H}^\infty (U)$. It was observed by B.J. Cole and T.W. Gamelin \cite{CG} that the Banach algebra $\mathcal{H}^\infty (\ell_2 \cap B_{c_0})$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$. Motivated by this work, we first generalize the open domain $\ell_2\cap B_{c_0}$ to a polydisk type domains $U$ of a Banach space $X$ with a Schauder basis for which the Banach algebra $\mathcal{H}^\infty (U)$ is isometrically isomorphic to the Banach algebra $\mathcal{H}^\infty (B_{c_0})$. Once this is established, we can derive the analytic and algebraic structure of the spectrum $\mathcal{M} (\mathcal{H}^\infty (U))$ from those of $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$ with the aid of the transpose of the aforementioned isometric isomorphism.
For instance, by applying the Cluster Value Theorem for $\mathcal{H}^\infty (B_{c_0})$ (see \cite[Theorem 5.1]{ACGLM}), we would find a new open domain $U$ for which the Cluster Value Theorem holds for $\mathcal{H}^\infty (U)$. We will introduce some polydisk type domains of a Banach space $X$ with a Schauder basis, which will be denoted by $\mathbb{D}_X (\bm{r})$, where $\bm{r}$ is a sequence of positive real numbers (see Definition \ref{def:rD_X}). As a matter of fact, the domain $\ell_2 \cap B_{c_0}$ considered in \cite{CG} is a special case of $\mathbb{D}_X (\bm{r})$.

Let us describe the contents of the paper: Section \ref{Background} is devoted to basic materials and the definition of certain polydisk type domains $\mathbb{D}_X (\bm{r})$, which are of our interest.
In Section \ref{sec:isometric_isomorphism}, We prove that the Banach algebra $\mathcal{H}^\infty (\mathbb{D}_X (\bm{r}))$ of all bounded holomorphic functions on $\mathbb{D}_X (\bm{r})$ (endowed with the supremum norm on $\mathbb{D}_X (\bm{r})$) is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$ as algebras under some assumption on $\bm{r}$, which generalizes the result in \cite{CG}.
As all the algebras $\mathcal{H}^\infty (\mathbb{D}_X (\bm{r}))$ are isometrically isomorphic, without loss of generality, we may fix the sequence $\bm{r}$ to be $(1,1,\ldots)$ and denote the corresponding domain by $\mathbb{D}_X$ in the following sections.
As we commented above, in Section \ref{sec:fiber}, we study the structure of the spectrum $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$ by applying that of the spectrum $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$ through the transpose of the above isometric isomorphism.
Next, in Section \ref{sec:cluster}, we consider the notion of a cluster set of elements in $ \mathcal{H}^\infty (\mathbb{D}_X)$, and examine its relation with that in $\mathcal{H}^\infty (B_{c_0})$. Using this relation, we prove that the corresponding Cluster Value Theorem holds for $\mathcal{H}^\infty (\mathbb{D}_X)$. In Section \ref{sec:Gleason}, we focus on studying the Gleason parts of $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$ by translating Gleason parts of $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$ or $\mathcal{M} (\mathcal{A}_u (B_{c_0}))$ into our case.

\section{Background}\label{Background}

In this section, we provide some necessary notations. Throughout this paper, $X$ will be an infinite dimensional complex Banach space with open unit ball $B_X$. The algebras that we are going to consider are denoted by $\mathcal{H}^\infty (U)$ and $\mathcal{A}_u (U)$, the algebra of all bounded holomorphic functions on an open subset $U$ of $X$ and the algebra of all uniformly continuous holomorphic functions on $U$, respectively, where both are endowed with the supremum norm. For a Banach algebra $\mathcal{A}$, we will denote by $\mathcal{M} (\mathcal{A})$ the spectrum (maximal ideal space) of $\mathcal{A}$, that is, the space of all continuous homomorphisms on $\mathcal{A}$. Notice that the spectrum $\mathcal{M} (\mathcal{A})$ is always a compact set endowed with the restriction of the weak-star topology. For each $a \in \mathcal{A}$, the Gelfand transform of $a$ will be denoted by $\widehat{a}$ that sends $\phi \in \mathcal{M} (\mathcal{A})$ to $\phi (a) \in \mathbb{C}$ for every $\phi \in \mathcal{M} (\mathcal{A})$.

If a Banach algebra $\mathcal{A}$ is taken to be $\mathcal{H}^\infty (B_X)$ or $\mathcal{A}_u (B_X)$, then there is the natural restriction mapping $\pi$ on $\mathcal{M} (\mathcal{A} )$ to $X^{**}$ given as $\pi (\phi) = \phi \vert_{X^*}$ because the dual space $X^*$ is included in $\mathcal{A}$ in this case. By Goldstine's theorem, we have that $\pi (\mathcal{M} (\mathcal{A} )) = \overline{B}_{X^{**}}$ and the \emph{fiber of $\mathcal{M} (\mathcal{A})$ at $z \in \overline{B}_{X^{**}}$} can be well defined as the pre-image of $z$ under the mapping $\pi$, i.e.,
\[
\mathcal{M}_z (\mathcal{A}) = \{ \phi \in \mathcal{A} : \pi (\phi) = z \} = \pi^{-1} (z).
\]
We say that the \emph{Cluster Value Theorem holds for $\mathcal{A}$ at $z \in \overline{B}_{X^{**}}$} when the image of the fiber $\mathcal{M}_z (\mathcal{A})$ under the Gelfand transform $\widehat{f}$ coincides with the \emph{cluster sets $Cl (f, z)$} defined by
\[
\{ \lambda \in \mathbb{C}: \, \text{there exists a net} \,\, (x_{\alpha}) \subset B_X \,\, \text{such that} \,\, x_{\alpha} \xrightarrow{w^*} z \,\, \text{and} \,\, f(x_\alpha) \rightarrow \lambda \},
\]
for every $f \in \mathcal{A}$. If $\widehat{f} (\mathcal{M}_z (\mathcal{A}))$ coincides with $Cl (f,z)$ for every $z \in \overline{B}_{X^{**}}$ and every $f \in \mathcal{A}$, then we simply say that the \emph{Cluster Value Theorem holds for $\mathcal{A}$}.
It is proved in \cite{ACGLM} that the Cluster Value Theorem holds for $\mathcal{H}^\infty (B_{c_0})$ and for $\mathcal{A}_u (B_{\ell_2})$, and that the Cluster Value Theorem holds for $\mathcal{A}_u (B_X)$ at $0$ whenever $X$ is a Banach space with a shrinking $1$-unconditional basis.

However, unlike in the situation when we dealt with $\mathcal{H}^\infty (B_X)$ or with $\mathcal{A}_u (B_X)$, the dual space $X^*$ is not included in $\mathcal{H}^\infty (U)$ in general for an open subset $U$ of a Banach space (see Proposition \ref{Au(D_X)_not_contained_in_Ho(D_X)}).
Thus, `fibers' of $\mathcal{M} (\mathcal{H}^\infty (U))$ cannot be defined in the same manner as before. For this reason, we consider the subspace $X^\sharp$ of $\mathcal{H}^\infty (U)$ defined as
\begin{equation}\label{X_sharp_U}
X^\sharp = \left(\mathcal{H}^\infty (U) \cap X^*, \| \cdot \|_U \right),
\end{equation}
where $\| \cdot \|_U$ is the supremum norm on $U$. In other words, $X^\sharp$ is the space which consists of elements of $\mathcal{H}^\infty (U)$ which are bounded linear functionals on $X$.
Notice that $X^\sharp$ can possibly be the trivial set $\{ 0 \}$. Under the case when $X^\sharp \neq \{ 0 \}$, we can consider the restriction mapping $\pi$ from the spectrum $\mathcal{M}(\mathcal{H}^\infty (U))$ to the closed unit ball $\overline{B}_{(X^\sharp)^*}$ of the dual space $(X^\sharp)^*$. Now, we consider the following definition.

\begin{definition}\label{def:fibers_U}
Let $X$ be a Banach space and $U$ be an open subset of $X$. If the restriction mapping $\pi : \mathcal{M}(\mathcal{H}^\infty (U)) \rightarrow \overline{B}_{(X^\sharp)^*}$ is surjective, then the \emph{fiber of $\mathcal{M} (\mathcal{H}^\infty (U))$ at $z \in \overline{B}_{(X^\sharp)^*}$} is defined as $\{ \phi \in \mathcal{M} (\mathcal{H}^\infty (U)) : \pi (\phi) = z \}$.
\end{definition}

As the above definition might be too general for our purpose, we would like to focus on certain polydisk type domains instead of dealing with an arbitrary open subset of a Banach space. Let us note that the following domain can be seen as a generalization of the domains $B_{c_0}$ (when $X = c_0$) or $\ell_p \cap \mathbb{D}^\mathbb{N}$, $1\leq p < \infty$ (when $X = \ell_p$).

\begin{definition}\label{def:rD_X}
Suppose that $X$ is a Banach space with a normalized Schauder basis and let $\bm{r} = (r_n)_{n =1}^\infty$ be a sequence of positive real numbers. Let us consider the set $\mathbb{D}_X (\bm{r})$ given by
\begin{equation*}
\mathbb{D}_X (\bm{r})= \left\{ x \in X : x = \sum_{j=1}^{\infty} e_j^* (x) e_j \,\, \text{with} \,\, |e_j^*(x)| < r_j \,\, \text{for every} \,\, j \in \mathbb{N} \right\}.
\end{equation*}
In particular, when $\bm{r} = (1, 1, \ldots)$, we simply denote it by $\mathbb{D}_X$.
\end{definition}

Let us make some remarks on $\mathbb{D}_X (\bm{r})$. First, if $\inf\limits_{n\in\mathbb{N}} r_n = 0$, then $\mathbb{D}_X (\bm{r})$ is never open in $X$. For if it were, it would follow that $t B_X \subset \mathbb{D}_X (\bm{r})$ for some $t > 0$. For $N \in \mathbb{N}$ satisfying that $r_N < \frac{t}{2}$, we have that $\frac{t}{2} e_N \in t B_X \subset \mathbb{D}_X (\bm{r})$. This implies that $\frac{t}{2} < r_N < \frac{t}{2}$, a contradiction.
As a matter of fact, we have the following result. Recall that for a Banach space with a normalized Schauder basis $\{e_j\}_{j\in \mathbb{N}}$, we always have $\sup_{j \in \mathbb{N}} \|e_j^* \| \leq 2 \sup_{n \in \mathbb{N}} \|P_n \| < \infty$.

\begin{remark}
Let $X$ be a Banach space with a normalized Schauder basis and $\bm{r} = (r_n)_{n=1}^\infty$ be a sequence of positive real numbers. Then the set $\mathbb{D}_X (\bm{r})$ is an open subset of $X$ if and only if $\inf\limits_{n\in \mathbb{N}} r_n > 0$.
\end{remark}

\begin{proof}
We only need to prove if $\inf\limits_{n\in\mathbb{N}} r_n > 0$ then $\mathbb{D}_X (\bm{r})$ is open. To this end, put $\alpha = \inf\limits_{n\in\mathbb{N}} r_n > 0$ and fix $x \in \mathbb{D}_X (\bm{r})$. Let $N \in \mathbb{N}$ be such that $|e_j^* (x)| < \frac{\alpha}{2}$ for all $j \geq N$. Choose $\varepsilon > 0$ sufficiently small so that $\varepsilon < \frac{\alpha}{2}$ and $\varepsilon < \min\{ r_j - |e_j^* (x)| : 1 \leq j \leq N-1\}$. If $y \in X$ satisfies $\| x - y \| < (\sup\limits_{j \in \mathbb{N}} \|e_j^*\| )^{-1} \varepsilon$, then $|e_j^* (y)| < r_j$ for every $j \in \mathbb{N}$.
\end{proof}

Recall that a \emph{complete Reinhardt domain} $R$ of a Banach space $X$ with a normalized unconditional Schauder basis is a subset such that for every $x \in R$ and $y \in X$ with $|e_j^* (y)| \leq |e_j^* (x)|$ for every $j \in \mathbb{N}$, we have $y \in R$.
Notice that if a basis of a Banach space $X$ is unconditional, then $\mathbb{D}_X (\bm{r})$ becomes a complete Reinhardt domain.
For more theory about holomorphic functions on Reinhardt domains of a Banach space, see \cite{CMV, DeGaMaSe, MN} and the references therein. Nevertheless, most of results in this paper hold for a general normalized Schauder basis.

\section{An isometric isomorphism between $\mathcal{H}^\infty (\mathbb{D}_X (\pmb{r}))$ and $\mathcal{H}^\infty (B_{c_0})$}\label{sec:isometric_isomorphism}

We start with the following main result which can produce many examples of (bounded or unbounded) polydisk type domains $U$ of a Banach space for which $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$.
Using the standard notation, let us denote by $P_n$ the canonical projection from $X$ onto the linear subspace spanned by $\{e_j : 1\leq j \leq n\}$ for each $n\in\mathbb{N}$.

\begin{theorem}\label{Hol:D(X):B_{c_0}} Let $X$ be an infinite dimensional Banach space with normalized Schauder basis, and $\bm{r} = (r_n)_{n=1}^\infty$ be a sequence of positive real numbers with $\inf\limits_{n\in\mathbb{N}} r_n >0$. Then $\mathcal{H}^\infty (\mathbb{D}_X (\bm{r}))$ and $\mathcal{H}^\infty (B_{c_0})$ are isometrically isomorphic.
\end{theorem}

\begin{proof}
	Let us denote by $\iota$ the linear injective mapping from $X$ to $c_0$ defined as
	$\iota (x) = \left( r_j^{-1} e_j^* (x) \right)_{j=1}^{\infty}$ for $x = \sum_{j=1}^{\infty} e_j^* (x) e_j.$
	Then $\iota$ maps the set $\mathbb{D}_X (\bm{r})$ into $B_{c_0}$.
	Consider the mapping $\Psi : \mathcal{H}^\infty ( B_{c_0} ) \rightarrow \mathcal{H}^\infty ( \mathbb{D}_X (\bm{r}))$ defined as
	\[
	\Psi (f) (x) = (f \circ \iota) (x) \quad \text{for every} \,\, f \in \mathcal{H}^\infty (B_{c_0}) \,\, \text{and} \,\, x \in \mathbb{D}_X (\bm{r}).
	\]
	It is clear that $\Psi$ is well-defined.

	To check that $\Psi$ is injective, suppose $\Psi (f) = 0$ for some $f\in \mathcal{H}^\infty (B_{c_0})$. If $z=(z_j)_{j=1}^{N} \in B_{c_{00}}$, then it is obvious that $x'= \sum_{j=1}^{N} r_j z_j e_j \in \mathbb{D}_X (\bm{r})$ and $\iota (x') = z$. So, $f(z) = (f \circ \iota) (x') = \Psi (f) (x') =0$ which implies that $f = 0$ on $B_{c_{00}}$. Noting that $B_{c_{00}}$ is dense in $B_{c_0}$, we have that $f =0$ on $B_{c_0}$.

	We claim that $\Psi$ is surjective. Let us denote by $\kappa$ the natural inclusion mapping from $c_{00}$ into $X$ mapping $z = (z_j)_{j =1}^\infty \in {c_{00}}$ to $\sum_{j=1}^{\infty} r_j z_j e_j$ (a finite sum) and let $g \in \mathcal{H}^\infty (\mathbb{D}_X (\bm{r}))$ be given.
	Notice that the mapping $\kappa$ is not continuous in general, nevertheless, we claim that the function $g \circ \kappa$ is holomorphic on $B_{c_{00}}$. To this end, as $g \circ \kappa$ is bounded on $B_{c_{00}}$, it suffices to check that $g \circ \kappa$ is G\^ateaux holomorphic on $B_{c_{00}}$. Fix $a = (a_j)_{j=1}^\infty \in B_{c_{00}}$ and $w=(w_j)_{j=1}^\infty \in {c_{00}}$, and consider the mapping $\lambda \mapsto (g \circ \kappa )(a + \lambda w)$ on the set $ \Omega := \{ \lambda \in \mathbb{C} : z_0 + \lambda w_0 \in B_{c_{00}} \}$. Note that
	\[
	(g \circ \kappa) (a + \lambda w) = g \left( \sum_{j=1}^m r_j a_j e_j + \lambda \sum_{j=1}^m r_j w_j e_j \right)
	\]
	for some $m \in \mathbb{N}$, and it is holomorphic in the variable $\lambda \in \Omega$ since $g$ is holomorphic.
	Now, consider the holomorphic function $g \circ \kappa \vert_{\mathbb{D}^N} : \mathbb{D}^N \rightarrow \mathbb{C}$ for each $N \in \mathbb{N}$. Note that the domain $\mathbb{D}^N$ is endowed with the supremum norm induced from $c_{00}$.
	 As the function $g \circ \kappa \vert_{\mathbb{D}^N}$ is separately holomorphic, it is holomorphic on $\mathbb{D}^N$ in classical several variables sense due to Hartogs' theorem.
	From this, we have a unique family $(c_{\alpha, N} (g))_{\alpha \in \mathbb{N}_0^N}$ such that
	\[
	\left( g \circ \kappa \vert_{\mathbb{D}^N} \right)(z) = \sum_{\alpha \in \mathbb{N}_0^N} c_{\alpha, N} (g) z^{\alpha} \quad (z \in \mathbb{D}^N).
	\]
	It follows that there exists a unique family $(c_{\alpha}(g))_{\alpha \in \mathbb{N}_0^{(\mathbb{N})}}$ such that
	\[
	(g \circ \kappa)(z) = \sum_{\alpha \in \mathbb{N}_{0}^{(\mathbb{N})}} c_{\alpha} (g) z^{\alpha} \quad (z \in B_{c_{00}}).
	\]
	Also, note that
	\[
	\sup_{N \in \mathbb{N}} \sup_{z \in \mathbb{D}^N} \left| \sum_{\alpha \in \mathbb{N}_0^N} c_{\alpha} (g) z^{\alpha} \right| \leq \sup_{N\in\mathbb{N}} \sup_{z \in \mathbb{D}^N} |\left( g \circ \kappa \vert_{\mathbb{D}^N} \right)(z)| \leq \|g\|_{\mathbb{D}_X (\bm{r})} < \infty.
	\]
	By Hilbert's criterion (see \cite[Theorem 2.21]{DeGaMaSe}), there exists a unique $f \in \mathcal{H}^\infty (B_{c_0})$ such that $c_{\alpha} (f) = c_{\alpha} (g)$ for every $\alpha \in \mathbb{N}_{0}^{(\mathbb{N})}$ with
	\[
	\|f\|_{\infty} := \sup_{x \in B_{c_0}} |f(x)| = \sup_{N \in \mathbb{N}} \sup_{z \in \mathbb{D}^N} \left| \sum_{\alpha \in \mathbb{N}_0^N} c_{\alpha} (g) z^{\alpha} \right|.
	\]
	Hence $f =g \circ \kappa$ on $B_{c_{00}}$ which implies that
	\begin{align*}
	\Psi (f) (x) = f(\iota (x)) &= \lim_{n \rightarrow \infty } f(\iota(P_n (x))) \\
	&= \lim_{n \rightarrow \infty } (g \circ \kappa ) ( ( \iota \circ P_n) (x) ) \quad (\text{since } (\iota \circ P_n )(x) \in B_{c_{00}}) \\
	&= \lim_{n \rightarrow \infty } g(P_n (x)) = g(x)
	\end{align*}
	for every $x \in \mathbb{D}_X (\bm{r})$, i.e., $\Psi (f) = g$.
	As
	\[
	\|f\|_{\infty} \leq \|g\|_{\mathbb{D}_X (\bm{r})} = \| \Psi(f)\|_{\mathbb{D}_X (\bm{r})} \leq \|f\|_{\infty},
	\]
	where the last inequality is obvious by definition of $\Psi$, we see that $\Psi$ is an isometry. It is clear by definition that $\Psi$ is a multiplicative mapping.
\end{proof}

From this moment on, we shall deal only with the domain $\mathbb{D}_X$ since all results on the Banach algebra $\mathcal{H}^\infty (\mathbb{D}_X)$ can be translated to a Banach algebra $\mathcal{H}^\infty (\mathbb{D}_X (\bm{r}))$ with $\inf\limits_{n\in\mathbb{N}} r_n >0$ with the aid of the isometric isomorphism in Theorem \ref{Hol:D(X):B_{c_0}}. Throughout this paper, we denote by $\Psi$ the isometric isomorphism from $\mathcal{H}^\infty (B_{c_0})$ onto $\mathcal{H}^\infty ( \mathbb{D}_X)$ given in the proof of Theorem \ref{Hol:D(X):B_{c_0}}. We also keep the notations $\iota : X \rightarrow c_0$ and $\kappa : c_{00} \rightarrow X$ defined in the above with respect to the sequence $\bm{r} = (1,1,\ldots)$. Let ${\Phi}$ denote the restriction of the adjoint of $\Psi$ to $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X ))$. Note that the mapping ${\Phi}$ is an homeomorphism from $\mathcal{M} (\mathcal{H}^\infty ( \mathbb{D}_X ))$ onto $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$ (actually, an isometry in Gleason sense, see Section \ref{sec:Gleason}).

Recall that every continuous polynomial on $c_0$ is weakly continuous on bounded sets (see \cite[Proposition 1.59]{D} or \cite[Section 3.4]{G}).
From this fact, we shall see that if a continuous polynomial on $X$ is bounded on $\mathbb{D}_X$, then it is weakly continuous on bounded sets.

\begin{prop}
Let $X$ be a Banach space with a normalized Schauder basis and let $P$ be a continuous polynomial on $X$. If $P$ is bounded on $\mathbb{D}_X$, then $P$ is weakly continuous on bounded sets.
\end{prop}

\begin{proof}
Without loss of generality, we may assume that $P$ is an $n$-homogenous polynomial. To prove that $P$ is weakly continuous on bounded sets, it suffices by homogeneity to prove that $P$ is weakly continuous on some ball $r B_X$. Let $(x_\alpha) \subset \big(\sup\limits_{j \in \mathbb{N}} \|e_j^* \| \big)^{-1} B_X$ be a net converging weakly to some $z$. As $P$ is bounded on $\mathbb{D}_X$, it can be observed that $Q:= \Psi^{-1} ( P \vert_{\mathbb{D}_X}) \in \mathcal{H}^\infty (B_{c_0})$ is an $n$-homogeneous polynomial on ${c_0}$. So, $Q$ is weakly continuous on bounded sets. Now,
\begin{align*}
P(x_\alpha) = (P \vert_{\mathbb{D}_X}) (x_\alpha) &= \Psi (Q) (x_\alpha) = Q ( \iota (x_\alpha) ) \rightarrow Q ( \iota(z)) = P (z)
\end{align*}
since $\iota (x_\alpha) \xrightarrow{w(c_0, \,\ell_1)} \iota(z)$.
\end{proof}

Recall that if $U$ is a convex bounded open set in $X$, then $\mathcal{A}_u  (U)$ becomes a subalgebra of $\mathcal{H}^\infty (U)$ as uniformly continuous functions on convex bounded sets of a Banach space are bounded.
Thus, $\mathcal{A}_u  (B_{c_0})$ is a subalgebra of $\mathcal{H}^\infty (B_{c_0})$ and observe that the isometric isomorphism $\Psi$ from $\mathcal{H}^\infty (B_{c_0})$ onto $\mathcal{H}^\infty ( \mathbb{D}_X)$ maps $\mathcal{A}_u  (B_{c_0})$ into the algebra $\mathcal{A}_u  (\mathbb{D}_X)$.
In other words, $\Psi ( \mathcal{A}_u  (B_{c_0})) \subset \mathcal{A}_u  (\mathbb{D}_X)$.
So, one might ask if Theorem \ref{Hol:D(X):B_{c_0}} holds for the Banach algebras $\mathcal{A}_u  (B_{c_0})$ and $\mathcal{A}_u  (\mathbb{D}_X)$, i.e., whether $\mathcal{A}_u  (B_{c_0})$ is isometrically isomorphic to $\mathcal{A}_u  (\mathbb{D}_X)$. The following observations give a negative answer to this question. Moreover, we see that $\mathbb{D}_X$ is not bounded in general and that $\mathcal{A}_u  (\mathbb{D}_X)$ is not a subalgebra of $\mathcal{H}^\infty (\mathbb{D}_X)$.

We would like to thank Daniel Carando for pointing out to us the following result.

\begin{prop}[D. Carando]\label{prop:unbounded_D_X}
If $X$ is a Banach space with a normalized Schauder basis $\{e_j\}_{j \in \mathbb{N}}$, then the set $\mathbb{D}_X$ is bounded if and only if $X$ is isomorphic to $c_0$.
\end{prop}

\begin{proof}
It is clear that if $X$ is isomorphic to $c_0$, then $\mathbb{D}_X$ is bounded. Assume that $\mathbb{D}_X$ is bounded. Then there exists $R > 0$ such that $\mathbb{D}_X \subset R B_X$. This implies that if $x \in X$ satisfies $\sup_{j \in \mathbb{N}} |e_j^* (x)| < 1$ then $\|x\| < R$. It follows that
\[
(\sup_{j \in \mathbb{N}} \|e_j^*\|)^{-1} \sup_{j \in \mathbb{N}} |e_j^* (x)| \leq \| x \| \leq R \sup_{j \in \mathbb{N}} |e_j^* (x)|
\]
for every $x \in X$. Thus, $(e_j)_{j \in \mathbb{N}}$ is equivalent to the canonical basis of $c_0$.
\end{proof}

\begin{prop}\label{Au(D_X)_not_contained_in_Ho(D_X)}
Let $X$ be a Banach space with a normalized Schauder basis, then
$\Psi ( \mathcal{A}_u  (B_{c_0})) \subset \mathcal{A}_u  (\mathbb{D}_X) \cap \mathcal{H}^\infty (\mathbb{D}_X)$. But the space $\mathcal{A}_u  (\mathbb{D}_X)$ is contained in  $\mathcal{H}^\infty (\mathbb{D}_X)$ if and only if $X$ is isomorphic to $c_0$.
\end{prop}

\begin{proof}
The first assertion follows from Theorem \ref{Hol:D(X):B_{c_0}}. By the above proposition, if $X$ is not isomorphic to $c_0$, the set $\mathbb{D}_X$ is not bounded, so not weakly bounded. Thus, there exists $x^* \in X^*$ such that $\sup_{z \in \mathbb{D}_X} |x^* (z)| = \infty$, while $x^*$ is clearly uniformly continuous on $\mathbb{D}_X$. The converse is obvious.
\end{proof}

Note that $\ell_1 = c_0^*$ embeds isometrically in $\mathcal{H}^\infty (B_{c_0})$. As the Banach algebras $\mathcal{H}^\infty (B_{c_0})$ and $\mathcal{H}^\infty (\mathbb{D}_X)$ are isometrically isomorphic, the image of $\ell_1$ under the isometric isomorphism $\Psi$ from $\mathcal{H}^\infty (B_{c_0})$ and $\mathcal{H}^\infty (\mathbb{D}_X)$ forms a closed subspace of $\mathcal{H}^\infty (\mathbb{D}_X)$. As a matter of fact, we will see from the next result that the image $\Psi (\ell_1)$ consists of elements of $\mathcal{H}^\infty (\mathbb{D}_X)$ which are linear on $X$.

\begin{prop}\label{prop:image_of_ell_1}
Let $X$ be a Banach space with a normalized Schauder basis $\{e_j\}_{j\in\mathbb{N}}$. Then
\[
\Psi (\ell_1) =  \mathcal{H}^\infty (\mathbb{D}_X) \cap X^*.
\]
\end{prop}

\begin{proof}
Note that $\Psi$ maps linear functionals on $c_0$ to linear functionals on $X$. For if $y = (y_n)_{n=1}^\infty \in \ell_1$, then
\begin{equation*
| \Psi (y) (x) | =\left| \sum_{n=1}^\infty y_n e_n^* (x) \right| \leq \left(\sup_{j\in\mathbb{N}} \|e_j^* \| \right) \sum_{n=1}^\infty |y_n| = \left(\sup_{j\in\mathbb{N}} \|e_j^* \| \right)\|y\|_{\ell_1},
\end{equation*}
for every $x \in B_X$. This implies that $\Psi (\ell_1) \subset \{ x^* \in \mathcal{H}^\infty (\mathbb{D}_X) : x^* \in X^* \}$.

Conversely, let $x^* \in \mathcal{H}^\infty (\mathbb{D}_X) \cap X^*$.
Choose $\theta_n \in \mathbb{R}$ such that $x^* (e_n) = |x^* (e_n)| e^{i\theta_n} $ for each $n \in \mathbb{N}$. Fix $\varepsilon \in (0,1)$ and let $z_k = \sum_{j=1}^{k} (1-\varepsilon) e^{-i\theta_j} e_j$ for each $k \in \mathbb{N}$. Then $z_k \in \mathbb{D}_X$ for each $k \in \mathbb{N}$ and
	\[
	\sum_{n=1}^{k} (1-\varepsilon ) |x^* (e_n)| = |x^*(z_k)| \leq \|x^*\|_{\mathbb{D}_X}
	\]
	for every $k \in \mathbb{N}$ and $\varepsilon \in (0,1)$. By letting $k \rightarrow \infty$ and $\varepsilon \rightarrow 0$ we obtain that $\| (x^* (e_j) )_{j=1}^{\infty} \|_{\ell_1} \leq \|x^*\|_{\mathbb{D}_X}$.
	Thus, $(x^* (e_j) )_{j=1}^{\infty}$ belongs to $\ell_1$. On the other hand,
	\[
	|x^* (x) | = \left| \sum_{j=1}^\infty x^* (e_j) e_j^* (x) \right| \leq \sum_{j=1}^\infty | x^* (e_j)| = \| (x^* (e_j) )_{j=1}^{\infty} \|_{\ell_1}
	\]
	for every $x \in \mathbb{D}_X$; hence $\| (x^* (e_j) )_{j=1}^{\infty} \|_{\ell_1} = \|x^*\|_{\mathbb{D}_X}$ and $\Psi ((x^* (e_n) )_{n=1}^\infty ) = x^*$.
\end{proof}

	 Let us observe from $\big(\sup\limits_{j\in\mathbb{N}} \|e_j^* \| \big)^{-1} B_X \subset \mathbb{D}_X$ that we have
$\sup_{x \in B_X} |x^* (x)| \leq \left(\sup_{j\in\mathbb{N}} \|e_j^* \| \right) \| x^* \|_{\mathbb{D}_X}$ for every $x^* \in X^\sharp$. Moreover, if we consider the bounded linear operator $T : \mathcal{H}^\infty (\mathbb{D}_X) \rightarrow \mathcal{H}^\infty (B_X)$ defined as $(T f)(x) = f\big( \big(\sup\limits_{j\in\mathbb{N}} \|e_j^* \| \big)^{-1} x \big)$ for every $f \in \mathcal{H}^\infty (\mathbb{D}_X)$ and $x \in B_X$, then $\|T f \|_{B_X} \leq \|f\|_{\mathbb{D}_X}$ for every $f \in \mathcal{H}^\infty (\mathbb{D}_X)$ and $T$ is injective due to the Principle of Analytic Continuation. However, $T$ is not a monomorphism (injection with closed image) in general. For example, let $X = \ell_p$ with $1 < p < \infty$. For each $n \in \mathbb{N}$, put $f_n = \Psi ( (\underbrace{n^{-1}, \ldots, n^{-1}}_{n\text{-many}}, 0,\ldots) ) \in \mathcal{H}^\infty (\ell_p \cap \mathbb{D}^\mathbb{N})$. Then $\|f_n\|_{\ell_p\cap\mathbb{D}^\mathbb{N}} = \| ({n^{-1}, \ldots, n^{-1}}, 0,\ldots)\|_{\ell_1} = 1$ for every $n \in \mathbb{N}$. However,
\begin{align*}
|(T f_n) (y)| =\left| \sum_{j=1}^{n} \frac{1}{n} y_j \right| \leq \left( \sum_{j=1}^{n} \left(\frac{1}{n}\right)^q \right)^{\frac{1}{q}} \left( \sum_{j=1}^n |y_j|^p \right)^{\frac{1}{p}} \leq \frac{1}{n^{1-\frac{1}{q}}}
\end{align*}
for every $y \in B_{\ell_p}$, where $q$ is the conjugate of $p$. Thus, $\|T f_n \|_{B_{\ell_p}} \leq \frac{1}{n^{1-\frac{1}{q}}} \rightarrow 0$ as $n \rightarrow \infty$ while $\|f_n\|_{\ell_p \cap \mathbb{D}^\mathbb{N}} = 1$ for every $n \in \mathbb{N}$. This implies that $T : \mathcal{H}^\infty (\ell_p\cap\mathbb{D}^\mathbb{N}) \rightarrow \mathcal{H}^\infty (B_{\ell_p})$ is not a monomorphism.

\section{Fibers of $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$}\label{sec:fiber}

In this section, we would like to study fibers of the spectrum $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$ which could help to describe the analytic structure of the spectrum. To this end, as we mentioned in Section \ref{Background}, we first consider the subspace $X^\sharp$ of the algebra $\mathcal{H}^\infty (\mathbb{D}_X)$ defined as follows as in \eqref{X_sharp_U}:
	\[
	X^\sharp = \left(\mathcal{H}^\infty (\mathbb{D}_X) \cap X^* , \|\cdot\|_{\mathbb{D}_X} \right).
	\]
	It is clear that $X^\sharp$ is a Banach space endowed with the norm $\| \cdot \|_{\mathbb{D}_X}$ since a limit of linear mappings on $\mathbb{D}_X$ is again linear.
	Moreover, the structure of the space $X^\sharp$ is not as rare as it might seem and Proposition \ref{prop:image_of_ell_1} proves that $X^\sharp$ is actually nothing but $\Psi (\ell_1)$; hence isometrically isomorphic to $\ell_1$.

\begin{prop}\label{ell_1:X^dagger}
	If $X$ is a Banach space with a normalized basis $(e_j)_{j=1}^{\infty}$, then
	\[
	\| x^* \|_{\mathbb{D}_X} = \sum_{j=1}^{\infty} |x^* (e_j)|
	\]
	for every $x^* \in X^\sharp$.
In particular, the mapping $\tau : X^\sharp \rightarrow \ell_1$ defined as $\tau (x^*) = (x^* (e_j))_{j=1}^{\infty}$ is an isometric isomorphism.
\end{prop}

To define fibers of $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$ as in Definition \ref{def:fibers_U}, we need to check that the restriction mapping $\pi : \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X)) \rightarrow X^\sharp$ is surjective in this case. Before proceeding further, let us clarify first some relationship between the mappings $\iota : X \rightarrow c_0$, $\kappa : c_{00} \rightarrow X$ and the isometry $\tau: X^\sharp \rightarrow \ell_1$.

\begin{remark}\label{relation_tau_iota'}
Let $X$ be a Banach space with a normalized Schauder basis.
\[
	\begin{tikzcd}
{c_{00}}\arrow{r}{\kappa} &X \arrow{r}{\iota} \arrow[d,-, dashed]{} & c_0 \arrow[d,-, dashed]{}& \, \, \\
\, & X^* & \ell_1 \arrow{l}{\iota^*} \arrow[d,-, dashed]{} & X^\sharp \arrow{l}{\tau} \arrow[d,-, dashed]{} & \Psi (\ell_1) \ar[equal]{l} \arrow[d,-, dashed]{}\\
\, & \, & \ell_\infty \arrow{r}{\tau^*} & (X^\sharp)^* & \Psi (\ell_1)^* \ar[equal]{l} \\
\end{tikzcd}
	\]
We have that
(a) $\iota \circ \kappa = \text{Id}_{c_{00}}$; (b) $\tau = (\Psi \vert_{\ell_1})^{-1}$; (c) $\iota^* = \tau^{-1}$; (d) $\kappa = \tau^* \vert_{c_{00}}$.
\end{remark}

Note that an element $x$ in $\mathbb{D}_X$ can be naturally considered as an element in $(X^\sharp)^*$ which sends $x^* \in X^\sharp$ to $x^* (x)$ and we observe from (c) of Remark \ref{relation_tau_iota'} that the image of $B_{c_{00}}$ under the mapping $\tau^*$ is contained in $\mathbb{D}_X$. One might ask if $\tau^*$ sends elements in $B_{c_0}$ to elements in $X$ as it does for elements in $B_{c_{00}}$. However, this is not the case when $X$ is $\ell_p$ with $1 \leq p < \infty$. Indeed, consider $u = \left( \frac{1}{2^{1/p}}, \frac{1}{3^{1/p}}, \ldots \right) \in B_{c_0}$. Then
\[
(\tau^* (u) )(e_n^*) = \langle u, \tau (e_n^*) \rangle = u_n = \frac{1}{(n+1)^{1/p}}
\]
for every $n \in \mathbb{N}$. It follows that if $\tau^* (u) \in \ell_p$, then $\tau^* (u)$ coincides with $u$. It contradicts to that $u$ does not belong to $\ell_p$.

Now, we concern with the restriction mapping $\pi : \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X) ) \rightarrow \overline{B}_{(X^\sharp)^*}$, i.e., $\pi (\phi) = \phi \vert_{X^\sharp}$ for every $\phi \in \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X) )$. Since $\phi \in \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X) ) $ is a homomorphism, for $x^* \in X^\sharp$ with $\|x^*\|_{\mathbb{D}_X} \leq 1$, we have $| \pi (\phi) (x^*) | = | \phi (x^*) | \leq 1.$
	In other words, $\pi \left(\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X) ) \right)\subset \overline{B}_{(X^\sharp)^*}$.
The following remark proves that the restriction mapping $\pi: \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X) ) \rightarrow \overline{B}_{(X^\sharp)^*}$ is surjective by presenting a relationship between the mapping $\pi $ and the restriction mapping $\pi_{c_0} : \mathcal{M} (\mathcal{H}^\infty (B_{c_0}) ) \rightarrow \overline{B}_{\ell_\infty}$.

\begin{remark}\label{relation_through_tau} Let $X$ be a Banach space with a normalized Schauder basis.
	Then the action between elements in the space $X^\sharp$ and the spectrum $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$ is related to the one between elements in $\ell_1$ and $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$. Indeed,
	given $x^* \in X^\sharp$ and $\phi \in \mathcal{H}^\infty (\mathbb{D}_X)$, we have
\begin{align*}
\Phi (\phi) ( \tau (x^*)) &= \phi ( x \in \mathbb{D}_X \leadsto (\tau(x^*) \circ \iota) (x) ) \\
&= \phi \left( x \in \mathbb{D}_X \leadsto \sum_{j=1}^{\infty} x^*(e_j) e_j^* (x) \right) = \phi (x^*).
\end{align*}
In other words, if we denote by $\pi_{c_0}$ the natural surjective mapping from $\mathcal{M} (\mathcal{H}^\infty ( B_{c_0}))$ to $\overline{B}_{\ell_\infty}$, then $	\pi (\phi) (\cdot) = \pi_{c_0} (\Phi (\phi)) (\tau (\cdot) ) \, \text{ on } X^\sharp.$
	Therefore, the following diagram commutes:
	\[
	\begin{tikzcd}[row sep=large]
\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X) ) \arrow{r}{\Phi} \arrow[swap]{d}{\pi} & \mathcal{M}(\mathcal{H}^\infty ( B_{c_0})) \arrow{d}{\pi_{c_0}} \\
\overline{B}_{(X^\sharp)^*} & \overline{B}_{\ell_\infty} \arrow{l}{\tau^* \vert_{\overline{B}_{\ell_\infty}} }
\end{tikzcd}
	\]
	In particular, the restriction mapping $\pi$ is surjective. Moreover, if $(\phi_\alpha)$ is a net in $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$ that converges to some $\phi$ in $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$, i.e., $\phi_\alpha (f) \rightarrow \phi(f)$ for every $f \in \mathcal{H}^\infty (\mathbb{D}_X)$, then $\pi (\phi_\alpha) \rightarrow \pi (\phi)$ in the $w ( (X^\sharp)^*, X^\sharp )$-topology.
\end{remark}

As the space $X^\sharp$ is not trivial and the restriction mapping $\pi : \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X)) \rightarrow \overline{B}_{(X^\sharp)^*}$ is surjective, we can restate Definition \ref{def:fibers_U} in the case of the Banach algebra $\mathcal{H}^\infty (\mathbb{D}_X)$ as follows.

\begin{definition}\label{def:fibers}
Let $X$ be a Banach space with a normalized basis. For $z \in \overline{B}_{(X^\sharp)^*}$, the \emph{fiber} $\mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X ))$ of the spectrum $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X ))$ at $z$ is defined as
\[
\mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X)) = \{ \phi \in \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X)) : \pi (\phi) = z \} = \pi^{-1} (z).
\]
\end{definition}

On the other hand, we can view $\mathbb{D}_X$ as a subset of $\overline{B}_{(X^\sharp)^*}$ as each $x \in \mathbb{D}_X$ induces a linear functional with norm less than or equal to one on $X^\sharp$ which maps $x^* \in X^\sharp \mapsto x^* (x)$. From this point of view, we can consider the $w ((X^\sharp)^*, X^\sharp )$-closure of $\mathbb{D}_X$ in the dual space $(X^\sharp)^*$.
For convenience, we shall simply denote $w ((X^\sharp)^*, X^\sharp )$ by $\sigma$ throughout this paper.
In the following result, we show that the $\sigma$-closure of $\mathbb{D}_X$ coincides with the closed ball $\overline{B}_{(X^\sharp)^*}$; hence it is isometrically isomorphic to $\overline{B}_{\ell_\infty}$ as well.

\begin{prop}\label{pi:projection}
	Let $X$ be a Banach space with a normalized basis $(e_j)_{j=1}^{\infty}$, then
	\begin{equation}\label{the_image_of_pi_projection}
	\overline{B}_{(X^\sharp)^*} = \{ z \in (X^\sharp)^* : |z(e_j^*)| \leq 1 \,\, \text{for every} \,\, j \in \mathbb{N} \} = \overline{\mathbb{D}}_X^{\sigma}.
	\end{equation}
\end{prop}

\begin{proof}
	The first equality in \eqref{the_image_of_pi_projection} is clear by definition of the isometry $\tau^*: \ell_\infty \rightarrow (X^\sharp)^*$.
	We claim that $\overline{B}_{(X^\sharp)^*} \subset \overline{\mathbb{D}}_X^{\sigma}$.
	Let $z \in \overline{B}_{(X^\sharp)^*}$ be given. Then there exists $u = (u_n) \in \overline{B}_{\ell_{\infty}}$ such that $\tau^* (u) = z$. That is,
	\[
	z(x^*) = \sum_{j=1}^{\infty} u_j x^* (e_j) \quad (x^* \in X^\sharp).
	\]
	Let $x^* \in X^\sharp$ be fixed. Then
	\[
	x^* \left( \sum_{j=1}^{m} \left(1-\frac{1}{m}\right) u_j e_j \right) = \sum_{j=1}^{m} \left(1-\frac{1}{m}\right) u_j x^* (e_j ) \rightarrow \sum_{j=1}^\infty u_j x^* (e_j) = z(x^*),
	\]
	as $m \rightarrow \infty$.
	This implies that the sequence $\big( \sum_{j=1}^{m} \left(1-\frac{1}{m}\right) u_j e_j \big)_{m=1}^\infty \subset \mathbb{D}_X$ converges in $\sigma$-topology to $z$.
	Thus, $\overline{B}_{(X^\sharp)^*} \subset \overline{\mathbb{D}}_X^{\sigma}$.

	Next, it is clear that $\mathbb{D}_X \subset \pi (\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X)))$. As $\pi$ is $w^*$-$\sigma$-continuous and $\mathcal{M}(\mathcal{H}^\infty (\mathbb{D}_X))$ is compact, we get $	\overline{\mathbb{D}}_X^{\sigma} \subset \pi (\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X)))$.
	Therefore, we obtain that $\overline{B}_{(X^\sharp)^*} \subset \overline{\mathbb{D}}_X^{\sigma} \subset \pi \left(\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X) ) \right)= \overline{B}_{(X^\sharp)^*}$.
\end{proof}

Given $z \in \mathbb{D}_X$, the most simple examples of homomorphisms which belong to fiber $\mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$ would be the point evaluation homomorphism at $z$.
As $\mathbb{D}_X$ is convex and balanced, the result from \cite[Theorem 1.3]{GGM} guarantees that there exists an isometric multiplicative extension operator $AB : \mathcal{H}^\infty (\mathbb{D}_X) \rightarrow \mathcal{H}^\infty ( \accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}} )$,
where $\accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}}$ denotes the $\| \cdot \|_{X^{**}}$-interior of $\overline{\mathbb{D}}_X^{w^*}$ in $X^{**}$ and $\overline{\mathbb{D}}_X^{w^*}$ denotes the $w(X^{**},X^*)$-closure of $\mathbb{D}_X$ in $X^{**}$.
Thus, any $z \in \accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}}$ induces a point evaluation homomorphism $\widetilde{\delta}_z$ in $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$;
\[
\widetilde{\delta}_z : f \leadsto AB(f) (z) \quad (f \in \mathcal{H}^\infty (\mathbb{D}_X)).
\]

\begin{prop}\label{prop:point_evaluation}
Given $z$ in $\accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}}$, we have that $\Phi (\widetilde{\delta_z})$ is the point evaluation at $(\tau^{-1})^* (z) \in B_{\ell_\infty}$ in $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$. Conversely, if $u \in B_{\ell_{\infty}}$, then $\Phi^{-1} (\widetilde{\delta_u} )$ is the point evaluation at $\tau^* (u) \in \accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}}$ in $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$.
\end{prop}

\begin{proof}
As both of them can be proved very similarly, we only prove the direction: Let $f \in \mathcal{H}^\infty (B_{c_0})$ and $z \in \accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}}$ be given. Then
\begin{align*}
\Phi (\widetilde{\delta_z}) (f) = \widetilde{\delta_z} (f \circ \iota) = (AB (f \circ \iota) ) (z) = AB(f) (AB(\iota) (z)),
\end{align*}
where the last equality holds due to \cite[Corollary 2.2]{CGKM} since $c_0$ is symmetric regular. As the mapping $\iota : X \rightarrow c_0$ is linear, we have that $AB(\iota) = \iota^{**}$;
hence
\[
\Phi (\widetilde{\delta_z}) (f) = AB(f) (\iota^{**} (z)) = AB(f) ((\tau^{-1})^{*} (z))
\]
for every $f \in \mathcal{H}^\infty (B_{c_0})$ and the proof is completed.
\end{proof}

The following result gives a different description of the fibers of $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$ by revealing its relation with fibers defined from $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$ in terms of the isometric isomorphism $\Phi$ between $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X) )$ and $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$.

\begin{prop}\label{prop:fiber_representation}
Let $z \in \overline{\mathbb{D}}_X^{\sigma}$. Then we have
\begin{equation*}
\mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X)) = \Phi^{-1} \left( \mathcal{M}_{(\tau^{-1})^* (z)} (\mathcal{H}^\infty (B_{c_0})) \right) = \Phi^{-1} \left( \mathcal{M}_{(z(e_1^*), z(e_2^*), \ldots)} (\mathcal{H}^\infty (B_{c_0})) \right).
\end{equation*}
\end{prop}

\begin{proof}
Let $\phi \in \mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$. For $y \in \ell_1$, choose $x^* \in X^\sharp$ so that $\tau (x^*) = y$. Observe that
\begin{align*}
\Phi (\phi) (y) = \Phi (\phi) ( \tau (x^*)) &= \phi (x^*) \quad (\text{by Remark \ref{relation_through_tau}}) \\
&= (\tau^{-1})^* (z) (\tau (x^*)) = (\tau^{-1})^* (z) (y);
\end{align*}
thus, $\Phi (\phi) \in \mathcal{M}_{(\tau^{-1})^* (z)} (\mathcal{H}^\infty (B_{c_0}))$.
\end{proof}

Let us finish this section by showing that the size of fibers over points of $\overline{\mathbb{D}}_X^{\sigma}$ can be quite big as in the case for the algebra $\mathcal{H}^\infty (B_X)$. As a matter of fact, it is well known that the fiber $\mathcal{M}_u (\mathcal{H}^\infty (B_{c_0}))$ contains a copy of $\beta \mathbb{N} \setminus \mathbb{N}$ for every $u \in \overline{B}_{\ell_{\infty}}$ where $\beta \mathbb{N}$ is the Stone-\v{C}ech compactification of $\mathbb{N}$ \cite[Theorem 11.1]{ACG91}. So, one direct consequence of Proposition \ref{prop:fiber_representation} is the following.

\begin{prop}
For any $z \in \overline{\mathbb{D}}_X^{\sigma}$, the set $\beta \mathbb{N} \setminus \mathbb{N}$ can be injected into the fiber $\mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$.
\end{prop}

Moreover, it is recently proved in \cite{CFGJM} and in \cite{DS} independently that for each $u \in \overline{B}_{\ell_\infty}$, there exists a Gleason isometric analytic injection of $B_{\ell_\infty}$ into the fiber $\mathcal{M}_u (\mathcal{H}^\infty (B_{c_0}))$ (see \cite{ADLM} or Section \ref{sec:Gleason} for the definition of Gleason metric). Combining this result with Proposition \ref{prop:fiber_representation}, we can notice the following result.

\begin{prop}\label{cor:injection_into_the_fiber_2}
For any $z \in \overline{\mathbb{D}}_X^{\sigma}$, there exists a Gleason isometric analytic injection $\Xi$ of $B_{\ell_{\infty}}$ into the fiber $\mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$.
\end{prop}

\section{Cluster values of $\mathcal{H}^\infty (\mathbb{D}_X)$}\label{sec:cluster}

Although it is mentioned in Section \ref{Background}, we start this section by recalling some definitions. For a point $z \in \overline{B}_{X^{**}}$ and $f \in \mathcal{H}^\infty (B_X)$, the \emph{cluster set} $Cl_{B_X} (f,z)$ is defined as the set of all limits of values of $f \in \mathcal{H}^\infty (B_X)$ along nets in $B_X$ converging weak-star to $z$, i.e.,
\[
\{ \lambda \in \mathbb{C}: \, \text{there exists a net} \,\, (x_{\alpha}) \subset B_X \,\, \text{such that} \,\, x_{\alpha} \xrightarrow{w^*} z \,\, \text{and} \,\, f(x_\alpha) \rightarrow \lambda \}.
\]
It is well known that we always get the following inclusion:
\begin{equation}\label{CVT_inclusion_0}
Cl_{B_X} (f,z) \subset \widehat{f} ( \mathcal{M}_{z} (\mathcal{H}^\infty (B_X)) )
\end{equation}
for every $f \in \mathcal{H}^\infty (B_X)$ and $z \in \overline{B}_{X^{**}}$, where $\widehat{f} : \mathcal{M}( \mathcal{H}^\infty (B_X))\rightarrow \mathbb{C}$ is the Gelfand transform of $f$. When the both sets in \eqref{CVT_inclusion_0} coincide for every $f \in \mathcal{H}^\infty (B_X)$, it is referred to as the \emph{Cluster Value Theorem holds for $\mathcal{H}^\infty (B_X)$ at $z$}.
In the similar manner, we can define the cluster set for $f \in \mathcal{H}^\infty (\mathbb{D}_X)$ and $z \in \overline{\mathbb{D}}_X^{\sigma}$:

\begin{definition}
Let $X$ be a Banach space with a normalized basis. For each $z \in \overline{\mathbb{D}}_X^{\sigma}$ and $f \in \mathcal{H}^\infty (\mathbb{D}_X)$, the \emph{cluster set $Cl_{\mathbb{D}_X} (f, z)$ of $f$ at $z$} is defined as
\[
\{ \lambda \in \mathbb{C} : \, \text{there exists a net} \,\, (x_{\alpha}) \subset \mathbb{D}_X \,\, \text{such that} \,\, x_{\alpha} \xrightarrow{\,\sigma\,} z \,\, \text{and} \,\, f(x_\alpha) \rightarrow \lambda \}.
\]
\end{definition}
One can observe that the following inclusion holds:
\begin{equation}\label{CVT_inclusion_for_D_X}
Cl_{\mathbb{D}_X} (f,z) \subset \widehat{f} ( \mathcal{M}_{z} (\mathcal{H}^\infty (\mathbb{D}_X)) )
\end{equation}
for every $f \in \mathcal{H}^\infty (\mathbb{D}_X)$ and $z \in \overline{\mathbb{D}}_X^{\sigma}$. Indeed, if $\lambda \in Cl_{\mathbb{D}_X} (f, z)$, there exists a net $(x_\alpha) \subset \mathbb{D}_X$ such that $x_\alpha \xrightarrow{\,\sigma\,} z$ and $f(x_\alpha) \rightarrow \lambda$. Passing to a subnet, if necessary, we may assume that $\delta_{x_\alpha} \rightarrow \phi$ in $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$. Thus, $\widehat{f} (\phi) = \phi (f) = \lambda$. For $x^* \in X^\sharp$, note that $\phi (x^*) = \lim_\alpha \delta_{x_\alpha} (x^*) = z(x^*)$, which implies that $\pi (\phi) = z$.

Now, we are ready to define the Cluster Value Theorem for the Banach algebra $\mathcal{H}^\infty (\mathbb{D}_X)$.

\begin{definition}\label{def:CVT_D_X}
Let $X$ be a Banach space with a normalized basis. We shall say that the \emph{Cluster Value Theorem holds for $\mathcal{H}^\infty (\mathbb{D}_X)$ when the both sets in \eqref{CVT_inclusion_for_D_X} coincide for every $f \in \mathcal{H}^\infty (\mathbb{D}_X)$ and $z \in \overline{\mathbb{D}}_X^{\sigma}$}.
\end{definition}

It is natural to investigate the relation between cluster sets from $\mathcal{H}^\infty (B_{c_0})$ and the ones from $\mathcal{H}^\infty (\mathbb{D}_X)$.

\begin{prop}\label{prop:same_cluster_sets}
Let $X$ be a Banach space with a normalized basis. Given $f \in \mathcal{H}^\infty (B_{c_0})$ and $u \in \overline{B}_{\ell_\infty}$, we have
\begin{equation}
Cl_{B_{c_0}} (f, u) = Cl_{\mathbb{D}_X} (\Psi(f), \tau^*(u)).
\end{equation}
\end{prop}

\begin{proof}
Let $\lambda \in Cl_{B_{c_0}} (f, u)$ be given. By the separability of $\ell_1$, we may choose a sequence $(x_n) \subset B_{c_0}$ such that $x_n \xrightarrow{w(\ell_\infty, \,\,\ell_1)} u$ and $f(x_n) \rightarrow \lambda$. Taking $w_n \in B_{c_{00}}$ for each $n \in \mathbb{N}$ so that $\|x_n - w_n \| \rightarrow 0$ and $|f(x_n)-f(w_n)| \rightarrow 0$ as $n \rightarrow \infty$, we have that $w_n \xrightarrow{w(\ell_\infty, \,\,\ell_1)} u$ and $f(w_n) \rightarrow \lambda$. By Remark \ref{relation_tau_iota'} and the continuity in the $w(\ell_\infty,\ell_1)$-$\sigma$ topology of the mapping $\tau^*$,
\[
\kappa (w_n) = \tau^* (w_n) \xrightarrow{\,\sigma\,} \tau^* (u) \,\text{ and }\, \Psi (f) (\kappa (w_n) ) = f ( \iota (\kappa (w_n))) = f(w_n) \rightarrow \lambda.
\]
It follows that $\lambda \in Cl_{\mathbb{D}_X} (\Psi(f), \tau^*(u))$; thus $Cl_{B_{c_0}} (f, u) \subset Cl_{\mathbb{D}_X} (\Psi(f), \tau^*(u))$.

Conversely, let $\lambda \in Cl_{\mathbb{D}_X} (\Psi(f), \tau^*(u))$. By \eqref{CVT_inclusion_for_D_X}, there exists $\phi \in \mathcal{M}_{\tau^* (u)} (\mathcal{H}^\infty (\mathbb{D}_X))$ such that $\lambda = \phi (\Psi (f))$. Note that $\lambda = \Phi (\phi ) (f)$ and from Proposition \ref{prop:fiber_representation} that
\[
\Phi (\phi) \in \Phi \left( \mathcal{M}_{\tau^* (u)} (\mathcal{H}^\infty (\mathbb{D}_X)) \right) = \mathcal{M}_u (\mathcal{H}^\infty ( B_{c_0})).
\]
This implies that $\lambda \in \widehat{f} ( \mathcal{M}_u (\mathcal{H}^\infty ( B_{c_0})) )$. As the Cluster Value Theorem holds for $\mathcal{H}^\infty (B_{c_0})$ \cite[Theorem 5.1]{ACGLM}, we obtain that $\lambda \in Cl_{B_{c_0}} (f, u)$. Hence, we conclude that $Cl_{\mathbb{D}_X} (\Psi(f), \tau^*(u)) \subset Cl_{B_{c_0}} (f, u)$.
\end{proof}

Note that Proposition \ref{prop:same_cluster_sets} and \cite[Lemma 2.1]{ACGLM} imply that each cluster set $Cl_{\mathbb{D}_X} (f, z)$ is a compact connected set. Now, we are ready to prove the Cluster Value Theorem for $\mathcal{H}^\infty (\mathbb{D}_X)$ provided that a Banach space $X$ has a normalized basis.

\begin{theorem}
Let $X$ be a Banach space with a normalized basis. Then Cluster Value Theorem holds for $\mathcal{H}^\infty (\mathbb{D}_X)$.
\end{theorem}

\begin{proof} Let $z \in \overline{\mathbb{D}}_X^{\sigma}$, $f \in \mathcal{H}^\infty (\mathbb{D}_X)$ and $\phi \in \mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$ be given.
Thanks to the inclusion \eqref{CVT_inclusion_for_D_X}, it suffices to prove $\lambda := \phi (f) \in Cl_{\mathbb{D}_X} (f,z)$. Note from Proposition \ref{prop:fiber_representation} that $\Phi (\phi) \in \mathcal{M}_{(\tau^{-1})^* (z)} (\mathcal{H}^\infty (B_{c_0}))$. From the Cluster Value Theorem for $\mathcal{H}^\infty (B_{c_0})$ at every $u \in \overline{B}_{\ell_{\infty}}$,
we get that
\begin{align*}
\lambda = \phi(f) = \Phi (\phi) ( \Psi^{-1} (f) ) \in Cl_{B_{c_0}} ( \Psi^{-1} (f), (\tau^{-1})^* (z)).
\end{align*}
By Proposition \ref{prop:same_cluster_sets}, we have that
\[
Cl_{B_{c_0}} ( \Psi^{-1} (f), (\tau^{-1})^* (z)) = Cl_{\mathbb{D}_X} (f, z).
\]
Thus, $\lambda \in Cl_{\mathbb{D}_X} (f,z)$ and it completes the proof.
\end{proof}

Recently, T. R. Alves and D. Carando \cite{AC} studied algebraic structure in the set of holomorphic functions which have large cluster sets at every point. Given a Banach space $X$ and a set $M \subset \overline{B}_{X^{**}}$, they defined the following sets:
\begin{align*}
\mathcal{F}_M (B_X) &= \{f \in \mathcal{H}^\infty (B_X) : \cap_{x \in M} Cl_{B_X} (f, x) \text{ contains a disk centered at } 0\}, \\
\mathcal{E}_M (B_X) &= \{ f \in \mathcal{H}^\infty (B_X) : Cl_{B_X} (f, x) \text{ contains a disk for each } x \in M \}.
\end{align*}
Notice that the difference between $\mathcal{F}_M (B_X) $ and $\mathcal{E}_M (B_X)$ is that we fix a single disk in the definition of $\mathcal{F}_M (B_X)$ while radii of disks in two different cluster sets can be different in $\mathcal{E}_M (B_X)$.
The authors proved that if $X$ is a separable Banach space, then $\mathcal{F}_{\overline{B}_{X^{**}}} (B_X)$ is strongly $\mathfrak{c}$-algebrable and that for $0 < \rho < 1$, the set $\mathcal{E}_{\rho B_{c_0}} (B_{c_0})$ is strongly $\mathfrak{c}$-algebrable and contains an almost isometric copy of $\ell_1$.

Similarly, we can define such sets of holomorphic functions in $\mathcal{H}^\infty (\mathbb{D}_X)$ whose cluster sets are large.

\begin{definition}
Let $X$ be a Banach space with a normalized basis and $M \subset \overline{\mathbb{D}}_X^{\sigma}$. We define sets $\mathcal{F}_{M} (\mathbb{D}_X)$ and $\mathcal{E}_{M} (\mathbb{D}_X)$ as
\begin{align*}
\mathcal{F}_M (\mathbb{D}_X) &= \{f \in \mathcal{H}^\infty (\mathbb{D}_X) : \cap_{x \in M} Cl_{\mathbb{D}_X} (f, x) \text{ contains a disk centered at } 0\}, \\
\mathcal{E}_M (\mathbb{D}_X) &= \{ f \in \mathcal{H}^\infty (\mathbb{D}_X) : Cl_{\mathbb{D}_X} (f, x) \text{ contains a disk for each } x \in M \}.
\end{align*}
\end{definition}

We can describe these sets as images of $\mathcal{F}_M (B_{c_0})$ and $\mathcal{E}_M (B_{c_0})$ under the isometric isomorphism $\Psi$.

\begin{prop}\label{prop:F_M_and_E_M}
Let $X$ be a Banach space with a normalized basis, and let $M \subset \overline{\mathbb{D}}_X^{\sigma}$.
Then
\begin{enumerate}	\setlength\itemsep{0.4em}
\item[(a)] $\mathcal{F}_{M} (\mathbb{D}_X) = \Psi ( \mathcal{F}_{(\tau^* \vert_{\overline{B}_{\ell_{\infty}}} )^{-1} (M)} (B_{c_0}) )$
\item[(b)] $\mathcal{E}_{M} (\mathbb{D}_X) = \Psi ( \mathcal{E}_{(\tau^* \vert_{\overline{B}_{\ell_{\infty}}} )^{-1} (M)} (B_{c_0}) )$.
\end{enumerate}
\end{prop}

\begin{proof}
As (a) and (b) can be proved in a similar way, we here give a proof of (a): Let $f \in \mathcal{F}_{(\tau^* \vert_{\overline{B}_{\ell_{\infty}}} )^{-1} (M)} (B_{c_0})$ be given. Then there exists $r >0$ such that
 \[
 \mathbb{D} (0, r) = \{ \lambda\in \mathbb{C} : |\lambda| < r \} \subset \bigcap\limits_{x \in (\tau^* \vert_{\overline{B}_{\ell_{\infty}}} )^{-1} (M) } Cl_{B_{c_0}} (f, x).
 \]
On the other hand, by Proposition \ref{prop:same_cluster_sets}
\begin{align*}
\bigcap\limits_{x \in (\tau^* \vert_{\overline{B}_{\ell_{\infty}}} )^{-1} (M) } Cl_{B_{c_0}} (f, x) &= \bigcap\limits_{x \in (\tau^* \vert_{\overline{B}_{\ell_{\infty}}} )^{-1} (M) } Cl_{\mathbb{D}_X} (\Psi (f), \tau^* (x) ) \\
&= \bigcap\limits_{w \in M} Cl_{\mathbb{D}_X} (\Psi (f), w ).
\end{align*}
This implies that $ \mathbb{D} (0, r) \subset \cap_{w \in M } Cl_{\mathbb{D}_X} (\Psi (f), w )$; hence $\Psi (f) \in \mathcal{F}_M (\mathbb{D}_X)$. Since $f$ is chosen arbitrarily, we have that
$\Psi ( \mathcal{F}_{(\tau^* \vert_{\overline{B}_{\ell_{\infty}}} )^{-1} (M)} (B_{c_0}) ) \subset \mathcal{F}_{M} (\mathbb{D}_X)$. The reverse inclusion can be obtained similarly.
\end{proof}

At this moment, we present the result which is an improvement of the result \cite[Theorem 1.4]{AC} related to $\mathcal{E}_{\rho B_{c_0}} (B_{c_0})$. Our claim is to show that it is possible to pass from $\rho B_{c_0}$ to its bidual ball $\rho B_{\ell_\infty}$ by slightly modifying its proof. For the sake of completeness, we include the proof.

\begin{prop}\label{prop:general_AC}
For $0 < \rho < 1$, the set $\mathcal{E}_{\rho \overline{B}_{\ell_\infty}} (B_{c_0}) \cup \{0\}$ contains an almost isometric copy of $\ell_1$.
\end{prop}

\begin{proof}
Choose a sequence $(r_n)_{n\in\mathbb{N}} \subset (\rho,1)$ so that $r_n \rightarrow 1$ rapidly and let $\delta = \prod_{n \in \mathbb{N}} r_n$. For each infinite subset $\Theta = \{ n_1 < n_2 < \cdots \} \subset \mathbb{N}$, define $f_{\Theta } : B_{c_0} \rightarrow \mathbb{C}$ by
	\[
	f_{\Theta} (x) = \delta^{-1} \prod_{j =1}^\infty \frac{r_j - x_{n_j}}{1-r_j x_{n_j}} \quad (x \in B_{c_0}).
	\]
	It is clear that $f_{\Theta} \in \mathcal{H}^\infty (B_{c_0})$ with $\|f \| \leq \delta^{-1}$. Now, choose a family $\{\Theta_k : k \in \mathbb{N} \}$ of pairwise disjoint infinite subsets of $\mathbb{N}$. Let $T : \ell_1 \rightarrow \mathcal{H}^\infty (B_{c_0})$ be defined as
	\[
	T (\beta)(x) = \sum_{k=1}^\infty \beta_k f_{\Theta_k} (x)
	\]
	for every $\beta = (\beta_n)_{n =1}^\infty \in \ell_1$ and $x \in B_{c_0}$. Notice that $T$ is an well defined bounded linear mapping with $\| T (\beta) \| \leq \delta^{-1} \|\beta\|_{\ell_1}$ for every $\beta \in \ell_1$.

	Let us denote $\Theta_k$ by $\{ n_1^k < n_2^k < \cdots \}$ and choose $m_j \in \mathbb{N}$ so that $m_j > \max_{1 \leq i, \ell \leq j} n_i^\ell$ for each $j \in \mathbb{N}$. For fixed $1 \leq k \leq N$ and $j > N$, let $s_{j,k} \in \mathbb{N}$ be an integer such that $n_{\ell}^k > m_j$ for every $\ell \geq s_{j,k}$.

	Suppose $\beta \neq 0$ and fix $N \in \mathbb{N}$ so that $(\beta_1, \ldots \beta_N, 0, 0, \ldots ) \neq 0$. Choose $\mu_1, \ldots, \mu_N \in \mathbb{D}$ arbitrarily and take $\lambda_{n_j^k} \in \mathbb{D}$ such that
	\[
	\mu_k = \frac{ r_j - \lambda_{n_j^k}}{1-r_j \lambda_{n_j^k}}
	\]
	for each $k = 1, \ldots, N$ and $j \in \mathbb{N}$.
	Let us fix $y = (y_n) \in \rho {\overline{B}_{\ell_\infty}}$. Given $j > N$, let us consider $z_j^N \in {\ell_\infty}$ given by
	\[
	z_j^N = \sum_{k=1}^{N} \left[ \left( 1 - \frac{|y_{n_j^k}|}{j} \right) \lambda_{n_j^k} - y_{n_j^k} \right] e_{n_j^k} + (P_{m_j} ( y) - y),
	\]
	where $P_n : \ell_\infty \rightarrow \ell_\infty$ the restriction mapping on the first $n$ coordinates for each $n \in \mathbb{N}$. Notice that $\| y+ z_j^N \| \leq \max \big\{ \max\limits_{1 \leq k \leq N} | \lambda_{n_j^k} | , \|y\| \big\} < 1$; hence $y + z_j^N \in B_{c_0}$ for each $j > N $.
	As $P_{m_j} ( y) \xrightarrow{w(\ell_\infty, \,\, \ell_1)} y$ as $j \rightarrow \infty$, we have that $z_j^N \xrightarrow{w^*} 0$; hence $y + z_j^N \xrightarrow{w^*} y$ as $j \rightarrow \infty$. Moreover, for fixed $1 \leq k \leq N$ and $j > N$,
	\begin{align}\label{eq:y+z_j^N}
	(y + z_j^N)_{n_\ell^k} =
	\begin{cases}
	y_{n_\ell^k}, &\quad \text{if } \ell = 1, \ldots, j-1;\\
	\left( 1-\frac{|y_{n_j^k}|}{j} \right) \lambda_{n_j^k}, &\quad \text{if } \ell = j; \\
	(P_{m_j}( y))_{n_{\ell}^k}, &\quad \text{if } \ell > j,
	\end{cases}
	\end{align}
	where $(y+ z_j^N)_{n_{\ell}^k}$ and $(P_{m_j}( y))_{n_{\ell}^k}$ denote the $n_\ell^k$-th coordinate of $y+z_j^N$ and $P_{m_j} ( y)$, respectively.

	Observe from \eqref{eq:y+z_j^N} that for fixed $1 \leq k \leq N$ and $j > N$,
	\begin{align*}
	f_{\theta_k} (y+ z_j^N) &= \delta^{-1} \prod_{\ell =1}^\infty \frac{ r_\ell - (y + z_j^N)_{n_\ell^k}}{1- r_\ell (y + z_j^N)_{n_\ell^k}} \\
	&= \delta^{-1} \left( \prod_{\ell =1}^{j-1} \frac{ r_\ell - y_{n_\ell^k}}{1- r_\ell y_{n_\ell^k}} \right) \left( \frac{ r_j - \left( 1 - j^{-1}{|y_{n_j^k}|} \right) \lambda_{n_j^k} }{ 1 - r_j \left( 1 - j^{-1}{|y_{n_j^k}|} \right) \lambda_{n_j^k} } \right) A_{k, j},
	\end{align*}
	where
	\[
	A_{k,j} = \prod_{\ell=j+1}^\infty \frac{r_\ell - (P_{m_j}( y))_{n_{\ell}^k} }{1 - r_\ell (P_{m_j}( y))_{n_{\ell}^k} }.
	\]
	On the other hand, if $k > N$ and $j > N$, then
	\[
	f_{\theta_k} (y + z_j^N) = f_{\theta_k} ( P_{m_j} (y))
	\]
	since $\{ n_\ell^k : \ell \in \mathbb{N} \} \cap \{ n_j^1, \ldots, n_j^N \} = \O$.
	It follows that for $j > N$,
	\begin{align}\label{eq:Psi_beta}
	T (\beta) ( y + z_j^N )
	&= \sum_{k=1}^{N} \beta_k \delta^{-1} \frac{ r_j - \left( 1 - j^{-1}{|y_{n_j^k}|} \right) \lambda_{n_j^k} }{ 1 - r_j \left( 1 - j^{-1}{|y_{n_j^k}|} \right) \lambda_{n_j^k} } \prod_{\ell =1}^{j-1} \frac{r_{\ell} - y_{n_{\ell}^k}}{1 - r_{\ell} y_{n_{\ell}^k}} A_{k,j} \\
	&\hspace{7cm}+ \sum_{k=N+1}^\infty \beta_k f_{\Theta_k} (P_{m_j} ( y)). \nonumber
	\end{align}
	By definition of $(s_{j,k})_{j \in \mathbb{N}}$, we have that

	\[
	A_{k, j } = \left( \prod_{\ell = j+1}^{s_{j,k} -1} \frac{r_\ell - y_{n_{\ell}^k} }{1 - r_\ell y_{n_{\ell}^k} } \right) \prod_{\ell = s_{j,k} }^\infty r_\ell
	\]
	for each $1 \leq k \leq N$.
	By passing to its subsequence, which we denote again by $(s_{j,k})_{j \in \mathbb{N}}$, we may assume that $(s_{j,k})_{j \in \mathbb{N}}$ is an increasing sequence for each $k \in \mathbb{N}$. Thereby, we have that $A_{k, j } \rightarrow 1$ as $j \rightarrow \infty$ for each $1 \leq k \leq N$.

	As $c_{N, j} := \sum_{k=N+1}^\infty \beta_k f_{\Theta_k} (P_{m_j} ( y))$ lies in $\| \beta\| \overline{\mathbb{D}}$ for each $j > N$, passing to a subsequence and keeping the same index, we may assume that $c_{N, j} \rightarrow c_N$ as $j \rightarrow \infty$ for some $c_N \in \| \beta \| \overline{\mathbb{D}}$.
	On the other hand,
	\begin{equation}\label{eq:delta_k}
	\left| \frac{ r_j - \left( 1 - j^{-1}{|y_{n_j^k}|} \right) \lambda_{n_j^k} }{ 1 - r_j \left( 1 - j^{-1}{|y_{n_j^k}|} \right) \lambda_{n_j^k} } - \mu_k \right| \xrightarrow[j \rightarrow \infty]{} 0.
	\end{equation}
	So, we have from \eqref{eq:Psi_beta} and \eqref{eq:delta_k} that
	\[
	T (\beta) (y + z_j^N ) \xrightarrow[j \rightarrow \infty]{} \sum_{k=1}^N \beta_k \delta^{-1} \mu_k \delta_k + c_N,
	\]
	where $\displaystyle \delta_k = \prod_{\ell=1}^\infty \frac{r_\ell - y_{n_{\ell}^k} }{ 1 - r_\ell y_{n_{\ell}^k} } \neq 0 $.

	Since $\mu_1, \ldots, \mu_k \in \mathbb{D}$ was chosen arbitrarily, this proves that the disk centered at $c_N$ of radius $\| (\beta_k \delta^{-1} \delta_k)_{k=1}^N \|_1$ is contained in the cluster set of $T (\beta)$ at $y$. For the case when $y = 0$, we have that $\delta_k = \delta$. Noting that $|c_N| \rightarrow 0$ as $N \rightarrow \infty$ as well, we conclude that $\| T ( \beta ) \| \geq \| \beta \|$. It follows that $\| \beta \| \leq \| T (\beta)\| \leq \delta^{-1} \|\beta\|$. Since $\delta$ can be taken arbitrarily close to $1$, this completes the proof.

\end{proof}

We come back to the sets just defined with respect to the algebra $\mathcal{H}^\infty (\mathbb{D}_X)$. Having in mind that an image of a strongly $\mathfrak{c}$-algebrable set under an isometric isomorphism is again strongly $\mathfrak{c}$-algebrable and that
$\tau^* (\rho \overline{B}_{\ell_{\infty}} ) = \rho \tau^* ( \overline{B}_{\ell_{\infty}} )= \rho \overline{\mathbb{D}}_X^{\sigma}$ for $0 < \rho < 1$,
we get the following result by combining Proposition \ref{prop:F_M_and_E_M}, \ref{prop:general_AC} with the above results in \cite{AC}.

\begin{theorem}
Let $X$ be a Banach space with a normalized basis. Then
\begin{enumerate}	\setlength\itemsep{0.4em}
\item $\mathcal{F}_{\overline{\mathbb{D}}_X^{\sigma}} (\mathbb{D}_X)$ is strongly $\mathfrak{c}$-algebrable,
\item for each $\rho \in (0,1)$, $\mathcal{E}_{\rho \overline{\mathbb{D}}_X^{\sigma}} (\mathbb{D}_X)$ is strongly $\mathfrak{c}$-algebrable and contains an almost isometric copy of $\ell_1$.
\end{enumerate}
\end{theorem}

\section{Gleason parts of $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$.}\label{sec:Gleason}

Before we start, let us introduce some terminology. Given an open set $U$ of a Banach space $X$, let $\mathcal{A}$ be a Banach algebra of holomorphic functions on $U$. If we consider the spectrum $\mathcal{M} (\mathcal{A})$ as a subset of $\mathcal{A}^*$, then $\mathcal{M} (\mathcal{A})$ can be naturally equipped with the metric induced from $\mathcal{A}^*$. In other words, for $\phi, \psi \in \mathcal{M} (\mathcal{A})$,
\[
\| \phi - \psi \|_{\mathcal{A}} := \sup\{ |\phi(f)- \psi(f)| : f \in \mathcal{A} \text{ with } \|f\| \leq 1 \}.
\]
From this metric, which we call \emph{Gleason metric}, we can partite the space $\mathcal{M} (\mathcal{A})$ into equivalence classes by the relation $\phi \sim \psi $ if and only if $\| \phi - \psi \|_{\mathcal{A}} < 2$ for $\phi, \psi \in \mathcal{M} (\mathcal{A})$. These equivalence classes are called the \emph{Gleason parts} in the spectrum $\mathcal{M} (\mathcal{A})$. In other words, the Gleason part containing $\phi \in \mathcal{M} (\mathcal{A})$, which we will denote by $\mathcal{GP} (\phi, \mathcal{A})$ is the set given as
$\mathcal{GP} (\phi, \mathcal{A}) = \{ \psi \in \mathcal{M} (\mathcal{A}) : \| \phi - \psi \|_{\mathcal{A}} < 2 \}.$
It is worth mentioning that for $\phi, \psi \in \mathcal{M} (\mathcal{A})$ the \emph{pseudohyperbolic distance} defined by $\rho_{\mathcal{A}} (\phi ,\psi) := \sup \{ \phi (f) : f \in \mathcal{A} \text{ with } \|f\| \leq 1, \psi (f) = 0 \}$ satisfies that
\[
\| \phi - \psi \|_{\mathcal{A}} = \frac{2 -2 \sqrt{1 - \rho_{\mathcal{A}}(\phi, \psi)^2}}{\rho_{\mathcal{A}} (\phi, \psi)}.
\]
It is well known that $\| \phi - \psi \|_{\mathcal{A}} < 2$ if and only if $\rho_{\mathcal{A}} (\phi, \psi ) < 1$ for $\phi$ and $\psi$ in $\mathcal{M}( \mathcal{A})$. For further information on Gleason parts, we refer \cite{B}.

In this section, whenever there is no possible confusion, we shall simply write $\| \phi - \psi \|$ and $\rho (\phi, \psi)$ for $\phi, \psi \in \mathcal{M} (\mathcal{A})$ omitting their subscripts.

As $\Psi$ is an isometric isomorphism from $\mathcal{H}^\infty (B_{c_0})$ onto $\mathcal{H}^\infty (\mathbb{D}_X)$, it is clear that
the isomorphism ${\Phi}$ from $\mathcal{M} ( \mathcal{H}^\infty ( \mathbb{D}_X ))$ onto $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$ is a Gleason isometry, that is,
$\| \Phi ( \phi ) - \Phi (\psi) \| = \| \phi - \psi \|$ for all $\phi$ and $\psi$ in $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$.
This completes the proof of the following result which will be frequently used in the sequel.

\begin{prop}\label{isomorphic_image_of_Gleason_parts}
Let $X$ be a Banach space with a normalized basis. Then
\begin{equation*}
\Phi( \mathcal{GP} (\phi, \mathcal{H}^\infty (\mathbb{D}_X)) ) = \mathcal{GP} (\Phi (\phi), \mathcal{H}^\infty (B_{c_0}) )
\end{equation*}
for every $\phi \in \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X ))$.
\end{prop}

Let us keep to use the notation $\delta_u$ for the point evaluation at $u \in B_{\ell_{\infty}}$ in both $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$ and $\mathcal{M} (\mathcal{A}_u (B_{c_0}))$. Notice from \cite[Theorem 2.4]{ADLM} that
\begin{equation*}
\rho_{\mathcal{A}_u (B_{c_0})}( \delta_u , \delta_ v ) = \sup_{ n \in \mathbb{N}} \left| \frac{u_n - v_n}{1 - \overline{u_n}v_n}\right|
\end{equation*}
for every $u = (u_n), v = (v_n) \in \overline{B}_{\ell_\infty}$. If we consider the canonical restriction $\Upsilon$ from $\mathcal{M} ( \mathcal{H}^\infty (B_{X}))$ to $\mathcal{M} (\mathcal{A}_u (B_{X}))$, then it is true in general that $\rho_{\mathcal{H}^\infty (B_{c_0})} (\phi, \psi) \geq \rho_{\mathcal{A}_u (B_{c_0})} (\Upsilon (\phi), \Upsilon (\psi))$ for every $\phi, \psi \in \mathcal{M} (\mathcal{H}^\infty (B_{X}))$. From this, we can observe that
\begin{equation}\label{Gleason_inclusion}
\rho_{\mathcal{H}^\infty (B_{c_0})} (\delta_u, \delta_v) \geq \rho_{\mathcal{A}_u (B_{c_0})} (\Upsilon(\delta_u), \Upsilon(\delta_v) ) = \sup_{ n \in \mathbb{N}} \left| \frac{u_n - v_n}{1 - \overline{u_n}v_n}\right|
\end{equation}
for every $u, v \in B_{\ell_\infty}$. As a matter of fact, B.J. Cole and T.W. Gamelin proved in \cite[Equation 6.1]{CG} that

\begin{equation}\label{prop:distance_point_evaluations}
\rho_{\mathcal{H}^\infty (B_{c_0})} (\delta_u, \delta_v) = \sup_{ n \in \mathbb{N}} \left| \frac{u_n - v_n}{1 - \overline{u_n}v_n}\right|.
\end{equation}

Let us give a different proof of this fact. Given $u, v \in B_{\ell_\infty}$, let us fix $r \in (0,1)$ so that $\| u\|, \|v \| < r $. Given $f \in \mathcal{H}^\infty (B_{c_0})$ with $AB(f) (u) = 0$, consider $f_r : r^{-1} B_{c_0} \rightarrow \mathbb{C}$ defined as $f_r (x) = f(r x)$ for every $x \in r^{-1} B_{c_0}$. It follows that $f_r \in \mathcal{H}^\infty (r^{-1} B_{c_0})$; hence $f_r \in \mathcal{A}_u (B_{c_0})$. By definition of the Aron-Berner extension, it is clear that $AB(f_r) (r^{-1} u) = 0$. Thus,
\begin{align*}
|AB(f) (v) | = |AB(f_r) (r^{-1} v)| \leq \rho_{\mathcal{A}_u (B_{c_0})} (\delta_{r^{-1} u}, \delta_{ r^{-1} v} ) = \sup_{n \in \mathbb{N}} \left| \frac{r^{-1} u_n - r^{-1} v_n}{1 - r^{-2} \overline{u_n} v_n } \right|.
\end{align*}
By letting $r \rightarrow 1$, we obtain that $|AB (f) (v) | \leq \sup\limits_{ n \in \mathbb{N}} \left| \frac{u_n - v_n}{1 - \overline{u_n}v_n}\right|$. As $f \in \mathcal{H}^\infty (B_{c_0})$ was arbitrary, we completes the proof.

Combining Proposition \ref{isomorphic_image_of_Gleason_parts} with Proposition \ref{prop:point_evaluation} and equation \eqref{prop:distance_point_evaluations}, we can calculate the Gleason metric between point evaluation homomorphisms induced from $\accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}}$ in $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$.

\begin{prop}\label{prop:Gleason_parts_basics}
Let $X$ be a Banach space with a normalized basis $(e_j)_{j=1}^{\infty}$. Then for each $z, w \in \accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}}$, we have that
\[
\rho (\widetilde{\delta}_z, \widetilde{\delta}_w ) = \sup_{n \in \mathbb{N}} \left| \frac{z(e_n^*) - w(e_n^*)}{1 - \overline{z(e_n^*)} w(e_n^*)}\right|.
\]
In particular, the set $\{ \widetilde{\delta}_z : z \in \accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}} \}$ is contained in $\mathcal{GP} (\delta_0, \mathcal{H}^\infty (\mathbb{D}_X))$. Moreover, for a sequence $\bm{r} = (r_n)_{n=1}^\infty$ of positive reals with $\sup\limits_{n \in \mathbb{N}} r_n < 1$, the $\sigma$-closure of
$\mathbb{D}_X (\bm{r} )$
is contained in $\mathcal{GP} (\delta_0, \mathcal{H}^\infty (\mathbb{D}_X))$.
\end{prop}

\begin{proof}
Note that if $z \in \accentset{\circ\,\,\,\,\,}{\overline{\D}_X^{w^*}}$, then $\sup\limits_{n \in \mathbb{N}} |z(e_n^*)| < 1$; hence $\rho (\widetilde{\delta}_z, \delta_0) < 1$.
It remains to prove the last statement and this can be proved similarly as \cite[Proposition 1.2]{ADLM}. Let $\phi$ in the $\sigma$-closure of $\mathbb{D}_X (\bm {r})$ be given. Let $f \in \mathcal{H}^\infty (\mathbb{D}_X)$ be such that $f(0) = 0$ and $\|f \| = 1$. Given $\varepsilon > 0$ such that $\sup\limits_{n\in\mathbb{N}} r_n + \varepsilon < 1$, there exists $x_0 \in \mathbb{D}_X (\bm{r})$ such that $| \phi (f) - \delta_{x_0} (f) | < \varepsilon$. Observe that
\[
|\phi (f) - \delta_0 (f)| \leq \varepsilon + |\delta_{x_0} (f)-\delta_0 (f)| \leq \varepsilon + \rho (\delta_0, \delta_{x_0}) < 1,
\]
since $\rho (\delta_0, \delta_{x_0}) = \sup\limits_{n \in \mathbb{N}} |e_n^* (x_0)| \leq \sup\limits_{n \in \mathbb{N}} r_n$.
\end{proof}

Moreover, we can obtain the following result analogous to \cite[Proposition 1.1]{ADLM}.

\begin{prop} Let $X$ be a Banach space with a normalized basis $(e_j)_{j=1}^{\infty}$. If $\phi \in \mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$ and $\psi \in \mathcal{M}_w (\mathcal{H}^\infty (\mathbb{D}_X))$, where $z \in \overline{\mathbb{D}}_X^{\sigma}$ and $w \in \overline{\mathbb{D}}_X^{\sigma}$ satisfy $\sup\limits_{j \in \mathbb{N}} |z(e_j^*)| = 1$ and $\sup\limits_{j \in \mathbb{N}} |w(e_j^*)| < 1 $, then $\phi$ and $\psi$ are in different Gleason parts.
\end{prop}

\begin{proof}
Note that $\Phi (\phi)$ lies in $\mathcal{M}_{(\tau^{-1})^* (z)} (\mathcal{H}^\infty (B_{c_0}))$ and $\Phi (\psi)$ in $\mathcal{M}_{(\tau^{-1})^* (w)} (\mathcal{H}^\infty (B_{c_0}))$. As $\| (\tau^{-1})^* (z) \| = 1$ and $\| (\tau^{-1})^* (w) \| < 1$, homomorphisms $\Phi(\phi)$ and $\Phi(\psi)$ lie in different Gleason parts \cite[Proposition 1.1]{ADLM}. Thus, Proposition \ref{isomorphic_image_of_Gleason_parts} completes the proof.
\end{proof}

There are results \cite[Remark 3.1, Proposition 3.2]{ADLM} which is related to the Gleason part containing a homomorphism in $\mathcal{M}(\mathcal{H}^\infty (B_{c_0}))$ lying in the fiber over the boundary point. We can translate these results to our setting as follows with the aid of Proposition \ref{prop:fiber_representation} and \ref{isomorphic_image_of_Gleason_parts}.

\begin{prop} Let $X$ be a Banach space with a normalized basis, and $z, w \in \overline{\mathbb{D}}_X^{\sigma}$ with $\sup\limits_{n \in \mathbb{N}} |z(e_n^*)| = \sup\limits_{n \in \mathbb{N}} |w(e_n^*)| = 1$.
\begin{enumerate}
\item If $\mathcal{GP} ( \delta_{ (\tau^{-1})^* (z) }, \mathcal{A}_u (B_{c_0})) \neq \mathcal{GP} ( \delta_{ (\tau^{-1})^* (w) }, \mathcal{A}_u (B_{c_0})) $, then we have
\[
\mathcal{GP} ( \phi, \mathcal{H}^\infty (\mathbb{D}_X)) \neq \mathcal{GP} ( \psi, \mathcal{H}^\infty (\mathbb{D}_X)),
\]
for any $\phi \in \mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$ and $\psi \in \mathcal{M}_w (\mathcal{H}^\infty (\mathbb{D}_X))$.
In particular, if $z \in \overline{\mathbb{D}}_X^{\sigma}$ satisfies that $|z(e_j^*)| = 1$ for every $j \in \mathbb{N}$, then
\[
\mathcal{GP} (\phi, \mathcal{H}^\infty (\mathbb{D}_X)) \subset \mathcal{M}_z ( \mathcal{H}^\infty (\mathbb{D}_X)).
\]
for any $\phi \in \mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$.
\item If $\mathcal{GP} ( \delta_{ (\tau^{-1})^* (z) }, \mathcal{A}_u (B_{c_0})) = \mathcal{GP} ( \delta_{ (\tau^{-1})^* (w) }, \mathcal{A}_u (B_{c_0})) $, then there exists $\phi \in \mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$ and $\psi \in \mathcal{M}_w (\mathcal{H}^\infty (\mathbb{D}_X))$ such that
\[
\mathcal{GP} ( \phi, \mathcal{H}^\infty (\mathbb{D}_X)) = \mathcal{GP} ( \psi, \mathcal{H}^\infty (\mathbb{D}_X)).
\]

\end{enumerate}
\end{prop}

To present the result which can be viewed as a generalization of the second item in the previous result, we will follow the arguments in \cite[Proposition 3.3]{ADLM}. Recall that a basis $\{e_j\}_{j=1}^\infty$ of a Banach space $X$ is said to be \emph{symmetric} if $\{e_{\theta(j)} \}_{j \in \mathbb{N}}$ is equivalent to $\{e_j\}$ for every permutation $\theta$ of the integers. For a moment, let $X$ be a Banach space with a symmetric basis $\{e_j\}_{j=1}^\infty$.
Given $b \in \mathbb{D}$ and $x \in \mathbb{D}_X $, let us denote by $(b,x)$ the element
\[
b e_1 + \sum_{j=1}^{\infty} e_j^*(x) e_{j+1} \in \mathbb{D}_X,
\]
which converges well in $X$ due to that $\{e_j\}_{j=1}^\infty$ is symmetric.
Recall \cite[Lemma 2.9]{AFGM} that for given $b \in \mathbb{D}$ and $u \in \overline{B}_{\ell_{\infty}}$, the mapping $R_b : \mathcal{M}_u (\mathcal{H}^\infty( B_{c_0})) \rightarrow \mathcal{M}_{(b,u_1,u_2,\ldots)} (\mathcal{H}^\infty( B_{c_0})) $ given by
\[
R_b (\varphi ) (f) = \varphi \left(x \in B_{c_0} \leadsto f(b, x_1, x_2, \ldots) \right),
\]
for every $\varphi \in \mathcal{M}_u (\mathcal{H}^\infty( B_{c_0}))$ and $f \in \mathcal{H}^\infty( B_{c_0})$,
is a homeomorphism. If we denote by $\widetilde{R_b}$ the mapping from $\mathcal{M} (\mathcal{H}^\infty( \mathbb{D}_X ))$ to $\mathcal{M} (\mathcal{H}^\infty( \mathbb{D}_X ))$ defined by
\[
\widetilde{R_b}(\phi) (f) = \phi (x \in \mathbb{D}_X \leadsto f(b,x) ),
\]
for every $\phi \in \mathcal{M} (\mathcal{H}^\infty( \mathbb{D}_X ))$ and $f \in \mathcal{H}^\infty (\mathbb{D}_X)$, then
\begin{align*}
R_b ( \Phi (\phi) ) (f) &= \Phi (\phi) \left(x \in B_{c_0} \leadsto f(b, x_1, x_2, \ldots) \right) \\
&= \phi \left( x \in \mathbb{D}_X \leadsto f (b, e_1^*(x), e_2^*(x), \ldots ) \right) \\
&= \phi (x \in \mathbb{D}_X \leadsto (f \circ \iota) (b,x)) \\
&= \Phi (\widetilde{R_b} (\phi) ) (f),
\end{align*}
for every $f \in \mathcal{H}^\infty (B_{c_0})$. This implies that $R_b \circ \Phi = \Phi \circ \widetilde{R_b}$ for every $b \in \mathbb{D}$, i.e., the following diagram commutes.
\[
	\begin{tikzcd}[row sep=large]
\mathcal{M}(\mathcal{H}^\infty (\mathbb{D}_X)) \arrow{r}{\widetilde{R_b}} \arrow{d}{\Phi} &\mathcal{M}(\mathcal{H}^\infty (\mathbb{D}_X)) \arrow{d}{\Phi} \\
\mathcal{M}(\mathcal{H}^\infty (B_{c_0})) \arrow{r}{R_b} & \mathcal{M}(\mathcal{H}^\infty (B_{c_0})).
\end{tikzcd}
	\]
Moreover, as a composition of homeomorphisms, $\widetilde{R_b}$ is a homeomorphisms between fibers $\mathcal{M}_z (\mathcal{H}^\infty( \mathbb{D}_X ))$ and $\mathcal{M}_{\tau^{*} (b, z(e_1^*), z(e_2^*), \ldots)} (\mathcal{H}^\infty( \mathbb{D}_X ))$. Using the homeomorphisms $\widetilde{R_b}$, roughly speaking, we can observe that any Gleason part containing an element outside the distinguished boundary point contains elements from \emph{disk}-many different fibers.

\begin{prop}\label{prop:Gleason_parts_disk_in_fiber}
Let $X$ be a Banach space with a normalized symmetric Schauder basis.
For each $b \in \mathbb{D}$, there exists $r_b > 0$ such that if $|c-b|< r_b$, then $\widetilde{R_b} (\phi)$ and $\widetilde{R_c} (\phi)$ are in the same Gleason part for all $\phi \in \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$.
\end{prop}

\begin{proof}
Consider $r_b >0 $ chosen so that $\| R_b (\varphi) - R_c (\varphi) \| < 2$ for all $\varphi \in \mathcal{M}(\mathcal{H}^\infty( B_{c_0}))$ whenever $|c-b| < r_b$ (see \cite[Proposition 3.3]{ADLM}). Let $\phi \in \mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$ be given, then
\begin{align*}
\| \widetilde{R_b} (\phi) - \widetilde{R_c} (\phi) \| = \| \Phi^{-1} \left[ R_b ( \Phi (\phi) ) - R_c ( \Phi (\phi) ) \right] \| = \| R_b ( \Phi (\phi) ) - R_c ( \Phi (\phi) ) \| < 2,
\end{align*}
which implies that $\widetilde{R_b} (\phi)$ and $\widetilde{R_c} (\phi)$ are in the same Gleason part.
\end{proof}

From Proposition \ref{prop:Gleason_parts_basics} and \ref{prop:Gleason_parts_disk_in_fiber}, we can particularly notice that there are homomorphisms $\varphi$ and $\psi$ in $\mathcal{M}(\mathcal{H}^\infty (\mathbb{D}_X))$ lying in different fibers but in the same Gleason part. Thus, it seems to natural to ask in the opposite direction, i.e., whether there exist homomorphisms in $\mathcal{M}(\mathcal{H}^\infty (\mathbb{D}_X))$ which belong to the same fiber but not to the same Gleason part. Combining \cite[Corollary 3.12]{ADLM} with Proposition \ref{prop:point_evaluation} and \ref{prop:fiber_representation} gives the answer that there are uncountably many homomorphisms in the same fiber but each of these lie in the different Gleason parts.

\begin{prop}
Let $X$ be a Banach space with a normalized Schauder basis. Let $z \in \tau^* (B_{\ell_\infty}) = B_{(X^\sharp)^*}$ be given. Then there exists an embedding $\xi : (\beta\mathbb{N} \setminus \mathbb{N}) \times \mathbb{D} \rightarrow \mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X))$ that is analytic on each slide $\{\theta\} \times \mathbb{D}$ and satisfies
\begin{enumerate}
	\setlength\itemsep{0.4em}
\item[(a)] $\xi (\theta, \lambda) \notin \mathcal{GP} (\delta_z, \mathcal{H}^\infty (\mathbb{D}_X))$ for each $(\theta, \lambda) \in (\beta\mathbb{N} \setminus \mathbb{N}) \times \mathbb{N}$,
\item[(b)] $\mathcal{GP} (\xi (\theta, \lambda)) \cap \mathcal{GP} (\xi(\theta', \lambda')) = \O$ for each $\theta \neq \theta' \in \beta\mathbb{N} \setminus \mathbb{N}$ and $\lambda, \lambda' \in \mathbb{D}$.
\end{enumerate}
\end{prop}

We finish this section by introducing a simple way to obtain homomorphisms in $\mathcal{M} (\mathcal{H}^\infty (\mathbb{D}_X))$ which lie in the same fiber but in the different Gleason parts. As we use a bijective biholomorphic mapping on the open ball $B_{\ell_\infty}$ in the argument, the result only covers for homomorphisms in fibers over $B_{\ell_\infty}$.

\begin{prop}
Let $u \in {B}_{\ell_\infty}$ be given, then there exist homomorphisms $\varphi$ and $\psi$ in $\mathcal{M}_u (\mathcal{H}^\infty (B_{c_0} ))$ but $\mathcal{GP} (\varphi, \mathcal{H}^\infty (B_{c_0})) \cap \mathcal{GP} (\psi, \mathcal{H}^\infty (B_{c_0})) =\emptyset$.
\end{prop}

\begin{proof}
Choose $(r_n) \subset (0,1)$ so that $r_n \rightarrow 1$ rapidly so that
\[
f(x) := \prod_{n=1}^\infty \frac{r_n -x_n}{1-r_n x_n} \quad (x = (x_n) \in B_{c_0})
\]
is a well-defined bounded holomorphic function on $B_{c_0}$. Note that $\frac{r_n -t}{1-r_n t} \rightarrow -1$ as $t \rightarrow 1$. For each $n \in \mathbb{N}$, choose $t_n \in (0,1)$ such that
\[
\left| \frac{r_n -t_n}{1-r_n t_n} + 1 \right| < \frac{1}{n}.
\]
Now, consider the family of homomorphisms $\{  \delta_{t_n e_n} : n \in \mathbb{N}\}$ in $\mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$ and let $\varphi$ be an accumulate point of this family in $\mathcal{M}(\mathcal{H}^\infty (B_{c_0}))$. As $t_n e_n \xrightarrow{w} 0$, we have that $\varphi \in \mathcal{M}_0(\mathcal{H}^\infty (B_{c_0}))$. However,
\begin{align*}
\| \varphi - \delta_0 \| \geq | \varphi(f) - f(0) | &= \lim_{n\rightarrow\infty} | f(t_n e_n) - f(0) | \\
&= \lim_{n\rightarrow\infty} \left| \left(\prod_{j \neq n} r_j \right) \frac{r_n - t_n}{1- r_n t_n} - \prod_{j =1}^\infty r_j \right| = 2 \prod_{j=1}^\infty r_j.
\end{align*}
As the infinite product $\prod_{j=1}^\infty r_j$ can be arbitrarily close to $1$, we conclude that $\| \varphi - \delta_0 \| = 2$, i.e., $\mathcal{GP} (\varphi, \mathcal{H}^\infty (B_{c_0})) \cap \mathcal{GP} (\delta_0, \mathcal{H}^\infty (B_{c_0})) =\O$. Now, let us fix a point $u \in B_{\ell_\infty}$. Then there exists a bijective biholomorphic mapping $\Lambda_{u} : B_{\ell_\infty} \rightarrow B_{\ell_\infty}$ with $\Lambda_u (u) = 0$ (see \cite[Proposition 3.9]{ADLM}).
Consider the surjective Gleason isometry $C_{\Lambda_u}^* : \mathcal{M} (\mathcal{H}^\infty (B_{c_0})) \rightarrow \mathcal{M} (\mathcal{H}^\infty (B_{c_0}))$ with inverse $\left(C_{\Lambda_u}^*\right)^{-1} = C_{\Lambda_u^{-1}}^*$ which satisfies
\[
C_{\Lambda_u}^* (\mathcal{M}_v (\mathcal{H}^\infty (B_{c_0})) ) = \mathcal{M}_{\Lambda_u (v)} (\mathcal{H}^\infty (B_{c_0}))
\]
for every $z \in B_{\ell_\infty}$ (see \cite[Theorem 3.11]{ADLM}).

It follows that $ C_{\Lambda_u^{-1}}^* (\varphi)$ and $C_{\Lambda_u^{-1}}^* (\delta_0)$ belongs to $\mathcal{M}_{\Lambda_u^{-1} (0)} (\mathcal{H}^\infty (B_{c_0})) = \mathcal{M}_{u} (\mathcal{H}^\infty (B_{c_0}))$ and satisfies that
\[
\mathcal{GP} (C_{\Lambda_u^{-1}}^* (\varphi), \mathcal{H}^\infty (B_{c_0})) \cap \mathcal{GP} (C_{\Lambda_u^{-1}}^* (\delta_0), \mathcal{H}^\infty (B_{c_0})) =\O.
\]
\end{proof}

Having in mind Proposition \ref{prop:fiber_representation} and Proposition \ref{isomorphic_image_of_Gleason_parts}, the above result establishes the following result:

\begin{prop}
Let $X$ be a Banach space with a normalized Schauder basis.
If $z \in \tau^* (B_{\ell_\infty}) = B_{(X^\sharp)^*}$, then there exist homomorphisms $\varphi$ and $\psi$ in $\mathcal{M}_z (\mathcal{H}^\infty (\mathbb{D}_X ))$ but $\mathcal{GP} (\varphi, \mathcal{H}^\infty (\mathbb{D}_X)) \cap \mathcal{GP} (\psi, \mathcal{H}^\infty (\mathbb{D}_X)) =\emptyset$.
\end{prop}

		\noindent \textbf{Acknowledgment:\ } We would like to thank Daniel Carando and Veronica Dimant for valuable comments and fruitful conversations.