2512.23733
Model: nano-banana-pro
# Solvability of the BĂ©zout Equation for Banach AlgebraâValued HâH^{\infty} Functions on the Polydisk
**Authors**: Alexander Brudnyi, Mahishanka Withanachchi
> Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada
## Abstract
In connection with the still unsolved multidimensional corona problem for algebras of bounded holomorphic functions on convex domains, we study the solvability of the BĂ©zout equation for the algebra of bounded holomorphic functions on the polydisk with values in a complex Banach algebra. Assuming local solvability of the BĂ©zout equation on a special open cover of the maximal ideal space of the algebra, we combine a dimension-induction scheme with a careful analysis of the topological structure of this space to glue local solutions into a global one. As a corollary, we obtain the solvability of the BĂ©zout equation for a broader class of subalgebras containing the slice algebra of bounded holomorphic functions, the case of the latter having been previously proved by the first author.
Key words and phrases: Corona problem; BĂ©zout equation; bounded holomorphic functions; Banach-valued holomorphic functions; maximal ideal space; Gleason parts; slice algebra; polydisk
2020 Mathematics Subject Classification: Primary 46J15, 30H50; Secondary 32A38, 46J10
## 1. Introduction
Let $A$ be a unital commutative complex Banach algebra. Its maximal ideal space $M(A)\subset A^{*}$ is the set of nonzero homomorphisms $A\to\mathbb{C}$ . Every element of $M(A)$ is a continuous linear functional on $A$ of norm at most one, so $M(A)$ lies in the closed unit ball of $A^{*}$ . It is a compact Hausdorff space in the weak- $\ast$ topology of $A^{*}$ . The Gelfand transform
$$
\hat{a}(\xi):=\xi(a),\qquad a\in A,\ \xi\in M(A),
$$
is a norm nonincreasing morphism of Banach algebras $A\to C(M(A))$ . If $A\subset C_{b}(X)$ , the algebra of bounded continuous functions on a Hausdorff space $X$ with the supremum norm, and $A$ separates points of $X$ , then the Gelfand transform is an isometric embedding. In this case, the evaluation map
$$
\iota:X\hookrightarrow M(A),\qquad\iota(x)(f)=f(x),\ f\in A,
$$
is continuous, and the complement of the closure of $\iota(X)$ in $M(A)$ is called the corona of $A$ .
The corona problem asks whether this complement is empty. Equivalently (cf. Garnett [20, Ch. V, Thm. 1.8]), it asks whether for every finite collection $f_{1},\dots,f_{k}\in A$ satisfying the corona condition
$$
\inf_{x\in X}\ \max_{1\leq j\leq k}|f_{j}(x)|>0,
$$
the corresponding BĂ©zout equation is solvable, that is, whether there exist $g_{1},\dots,g_{k}\in A$ such that
$$
f_{1}g_{1}+\cdots+f_{k}g_{k}=1\qquad\text{on }X.
$$
For $A=H^{\infty}(\mathbb{D})$ , the algebra of bounded holomorphic functions on the unit disk, the problem was raised by Kakutani in 1941. Newman [34] introduced the term corona, and showed that the problem is equivalent to an interpolation question later solved by Carleson [12]. Carleson proved that the corona is empty, equivalently that every collection of bounded holomorphic functions on $\mathbb{D}$ satisfying (1.1) admits a solution to (1.2). Hörmander [27] introduced $\bar{\partial}$ -methods together with the Koszul complex to simplify the proof, and Wolff [42] later gave the shortest argument by directly obtaining uniform estimates on the boundary circle of the solution of the relevant $\bar{\partial}$ -equation (see also Garnett [20]).
Following Carlesonâs theorem, the corona problem was settled for many classes of Riemann surfaces: finite bordered surfaces [1, 2, 15, 17, 36, 37, 38, 24, 35], certain infinitely connected domains (âroadrunnerâ domains) by Behrens [5, 6] (extended in [13, 14, 32]), and finitely sheeted coverings by Nakai [31]. It was later shown, starting with the work of Cole (see Gamelin [18]), that there exist Riemann surfaces with nontrivial corona (see also BarrettâDiller [3], LĂĄrusson [28]).
For planar domains the general corona problem is still unsolved. In this direction, Gamelin [19] has shown that the corona problem for planar domains is local in the sense that it depends only on the behavior of the domain locally about each boundary point. Moore [30] proved that the corona is empty for domains whose boundary lies on the graph of a $C^{1+\varepsilon}$ function, extending the earlier result of GarnettâJones [21] for Denjoy domains. Further extensions include Handyâs theorem for complements of certain Cantor sets [23] and NewDelmanâs result for subsets of Lipschitz graphs [33].
The problem also remains open in several variables for the ball and the polydisk. In fact, no examples are known of domains in $\mathbb{C}^{n}$ , $n\geq 2$ , whose algebra $H^{\infty}$ has a nontrivial corona.
## 2. Main Results
In this paper we study the corona problem for the Banach algebra $H^{\infty}(\mathbb{D}^{n})$ of bounded holomorphic functions on the polydisk $\mathbb{D}^{n}\subset\mathbb{C}^{n}$ . As before, $M(H^{\infty}(\mathbb{D}^{n}))$ denotes its maximal ideal space, equipped with the weak-* topology.
Given a unital commutative complex Banach algebra $(A,\lVert\cdot\rVert_{A})$ , we write $H^{\infty}(\mathbb{D}^{n},A)$ for the Banach algebra of bounded holomorphic functions from $\mathbb{D}^{n}$ into $A$ , equipped with the norm
$$
\|f\|_{\infty}=\sup_{z\in\mathbb{D}^{n}}\|f(z)\|_{A}.
$$
One of our main results is the following:
**Theorem 2.1**
*Let $A$ be a unital commutative complex Banach algebra, and let $f_{1},\dots,f_{k}\in H^{\infty}(\mathbb{D}^{n},A)$ . Suppose there exist open sets $\widehat{U}_{1},\dots,\widehat{U}_{k}\subset[M(H^{\infty}(\mathbb{D}))]^{n}$ such that
$$
[M(H^{\infty}(\mathbb{D}))]^{n}=\bigcup_{i=1}^{k}\widehat{U}_{i},\quad\text{and}\quad U_{i}:=\mathbb{D}^{n}\cap\widehat{U}_{i}.
$$
Assume that for each $i$ , there exist $g_{i1},\dots,g_{ik}\in H^{\infty}(U_{i},A)$ such that
$$
\sum_{j=1}^{k}f_{j}g_{ij}=1\quad\text{on }U_{i}.
$$
Then there exist $g_{1},\dots,g_{k}\in H^{\infty}(\mathbb{D}^{n},A)$ such that
$$
\sum_{j=1}^{k}f_{j}g_{j}=1\quad\text{on }\mathbb{D}^{n}.
$$*
Next, we study the structure of the maximal ideal space $M(H^{\infty}(\mathbb{D}^{n}))$ . To this end, we consider the natural continuous projection
$$
\pi_{n}:M(H^{\infty}(\mathbb{D}^{n}))\longrightarrow\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n},
$$
defined as the transpose of the canonical isometric embedding
$$
\widehat{\otimes}_{\varepsilon}^{n}H^{\infty}(\mathbb{D})\hookrightarrow H^{\infty}(\mathbb{D}^{n}),
$$
where $\widehat{\otimes}_{\varepsilon}^{n}H^{\infty}(\mathbb{D})$ denotes the $n$ -fold injective (or $\varepsilon$ -) tensor product of $H^{\infty}(\mathbb{D})$ , regarded as the closed subalgebra of $H^{\infty}(\mathbb{D}^{n})$ generated by bounded holomorphic functions in the individual coordinate variables. In particular, $\pi_{n}$ restricts to the identity on $\mathbb{D}^{n}\subset M(H^{\infty}(\mathbb{D}^{n}))$ .
For each point $\xi\in\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ , define the corresponding fiber
$$
F_{\xi}:=\pi_{n}^{-1}(\xi)\cap\operatorname{cl}(\mathbb{D}^{n})\subset M(H^{\infty}(\mathbb{D}^{n})),
$$
where $\operatorname{cl}(\mathbb{D}^{n})$ denotes the closure of $\mathbb{D}^{n}$ in $M(H^{\infty}(\mathbb{D}^{n}))$ .
We equip $\operatorname{cl}(\mathbb{D}^{n})$ with the topology induced from $M(H^{\infty}(\mathbb{D}^{n}))$ . Given an open subset $U\subset\operatorname{cl}(\mathbb{D}^{n})$ , we write $\mathcal{O}(U)\subset C(U)$ for the algebra of continuous functions on $U$ that are holomorphic on $U\cap\mathbb{D}^{n}$ in the usual sense.
For each fiber $F_{\xi}$ , we define the associated uniform algebra $A(F_{\xi})\subset C(F_{\xi})$ as the uniform closure of the algebras $\mathcal{O}(U)\rvert_{F_{\xi}}$ , where $U$ ranges over all open neighborhoods of $F_{\xi}$ in $\operatorname{cl}(\mathbb{D}^{n})$ .
Since $F_{\xi}$ is compact, the family of sets of the form
$$
\pi_{n}^{-1}(V)\cap\operatorname{cl}(\mathbb{D}^{n}),
$$
with $V$ ranging over open neighborhoods of $\xi\in\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ , forms a basis of neighborhoods of $F_{\xi}$ in $\operatorname{cl}(\mathbb{D}^{n})$ ; that is, every open neighborhood of $F_{\xi}$ contains a set of this form. Moreover, each such set $V$ contains a compact subset of the form $W=W_{1}\times\dots\times W_{n}$ , where, for $\xi=(\xi_{1},\dots,\xi_{n})\in\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ , each component $W_{i}\subset M(H^{\infty}(\mathbb{D}))$ is given by
$$
W_{i}:=\left\{\eta\in M(H^{\infty}(\mathbb{D})):|\hat{f}(\eta)|\leq 1\right\},
$$
for some function $f\in H^{\infty}(\mathbb{D})$ satisfying $\hat{f}(\xi_{i})=0$ . Such a choice is guaranteed by the classical result of SuĂĄrez [39, Thm. 2.4], which asserts that the algebra $H^{\infty}(\mathbb{D})$ is separating.
In particular, the set
$$
W\cap\mathbb{D}^{n}=(W_{1}\cap\mathbb{D})\times\dots\times(W_{n}\cap\mathbb{D})
$$
satisfies the hypotheses of the Runge-type approximation theorem established in [11]. That result asserts that every function in $H^{\infty}\bigl(\mathbb{D}^{n}\cap\pi_{n}^{-1}(V)\bigr)$ can be uniformly approximated on $W\cap\mathbb{D}^{n}$ by functions from $H^{\infty}(\mathbb{D}^{n})$ .
Summarizing the above, we obtain that for every open neighborhood $U\subset\operatorname{cl}(\mathbb{D}^{n})$ of $F_{\xi}$ , the algebra $\mathcal{O}(U)\rvert_{F_{\xi}}$ is contained in the uniform closure of the algebra $\mathcal{O}\bigl(M(H^{\infty}(\mathbb{D}^{n}))\bigr)\rvert_{F_{\xi}}$ , where
$$
\mathcal{O}\bigl(M(H^{\infty}(\mathbb{D}^{n}))\bigr):=\widehat{H^{\infty}(\mathbb{D}^{n})}=\left\{\hat{f}:f\in H^{\infty}(\mathbb{D}^{n})\right\}.
$$
In view of the definition of the algebra $A(F_{\xi})$ , it follows that $A(F_{\xi})$ coincides with the uniform closure of the algebra $\widehat{H^{\infty}(\mathbb{D}^{n})}\rvert_{F_{\xi}}$ . It then follows from a standard result in the theory of uniform algebras that the maximal ideal space of $A(F_{\xi})$ coincides with the $H^{\infty}$ -hull of $F_{\xi}$ , namely
$$
M(A(F_{\xi}))=\left\{\zeta\in M(H^{\infty}(\mathbb{D}^{n})):|\hat{f}(\zeta)|\leq\sup_{F_{\xi}}|\hat{f}|\ \text{for all }f\in H^{\infty}(\mathbb{D}^{n})\right\}.
$$
**Theorem 2.2**
*For each $\xi\in\left[M(H^{\infty}(\mathbb{D}))\right]^{n}$ , we have
$$
\pi_{n}^{-1}(\xi)=M(A(F_{\xi})).
$$ The corona problem for $H^{\infty}(\mathbb{D}^{n})$ is solvable (i.e., $\operatorname{cl}(\mathbb{D}^{n})=M(H^{\infty}(\mathbb{D}^{n}))$ ) if and only if
$$
M(A(F_{\xi}))=F_{\xi}\quad\text{for all }\xi\in\left[M(H^{\infty}(\mathbb{D}))\right]^{n}.
$$*
To describe certain fibers $F_{\xi}$ more explicitly, we begin by recalling classical results of K. Hoffman on the structure of the maximal ideal space $M(H^{\infty}(\mathbb{D}))$ .
The pseudohyperbolic metric on the unit disk $\mathbb{D}$ is given by
$$
\rho(z,w):=\left|\frac{z-w}{1-\bar{w}z}\right|,\quad z,w\in\mathbb{D},
$$
and admits a natural extension to $M(H^{\infty}(\mathbb{D}))$ via the formula
$$
\rho(\xi,\eta):=\sup\left\{|\hat{f}(\eta)|:f\in H^{\infty}(\mathbb{D}),\ \hat{f}(\xi)=0,\ \lVert f\rVert_{\infty}\leq 1\right\},\quad\xi,\eta\in M(H^{\infty}(\mathbb{D})).
$$
The Gleason part of a point $x\in M(H^{\infty}(\mathbb{D}))$ is
$$
\pi(\xi):=\left\{\eta\in M(H^{\infty}(\mathbb{D})):\rho(\xi,\eta)<1\right\}.
$$
These parts form a partition of $M(H^{\infty}(\mathbb{D}))$ : for any $\xi,\eta$ , either $\pi(\xi)=\pi(\eta)$ or $\pi(\xi)\cap\pi(\eta)=\emptyset$ .
A fundamental result of Hoffman [26] asserts that each Gleason part in $M(H^{\infty}(\mathbb{D}))$ is either a singleton or an analytic disk (i.e., the image of a continuous injective map $\tau\colon\mathbb{D}\to M(H^{\infty}(\mathbb{D}))$ such that $\hat{f}\circ\tau\in H^{\infty}(\mathbb{D})$ for all $f\in H^{\infty}(\mathbb{D})$ ). A point $\xi\in M(H^{\infty}(\mathbb{D}))$ lies in a nontrivial part if and only if it belongs to the closure of an interpolating sequence. The union $M_{a}$ of all such parts is open in the maximal ideal space, and its complement $M_{s}:=M(H^{\infty}(\mathbb{D}))\setminus M_{a}$ is closed and totally disconnected by a theorem of SuĂĄrez [40].
The space $[M(H^{\infty}(\mathbb{D}))]^{n}$ admits a natural stratification according to the number of coordinates lying in the open subset $M_{a}$ of nontrivial Gleason parts. Each stratum is determined by the subset of indices $j\in\{1,\dots,n\}$ for which the $j$ -th coordinate belongs to $M_{a}$ (and hence the remaining coordinates lie in $M_{s}$ ).
We now describe the structure of fibers of the map
$$
\pi_{n}:M(H^{\infty}(\mathbb{D}^{n}))\to[M(H^{\infty}(\mathbb{D}))]^{n}
$$
over the first two strata: (i) the open stratum $M_{a}^{n}$ , and (ii) the codimension-one stratum consisting of those $\xi\in[M(H^{\infty}(\mathbb{D}))]^{n}$ with exactly one coordinate in $M_{s}$ and the remaining $n-1$ in $M_{a}$ .
**Theorem 2.3**
*Let $\xi\in\bigl[M\bigl(H^{\infty}(\mathbb{D})\bigr)\bigr]^{n}$ . Then: - If $\xi\in M_{a}^{n}$ , then the fiber $\pi_{n}^{-1}(\xi)$ coincides with
$$
F_{\xi}:=\pi_{n}^{-1}(\xi)\cap\operatorname{cl}(\mathbb{D}^{n}),
$$
and $F_{\xi}$ lies in the closure of an interpolating sequence for $H^{\infty}(\mathbb{D}^{n})$ . A sequence $\{z_{k}\}\subset\mathbb{D}^{n}$ is interpolating for $H^{\infty}(\mathbb{D}^{n})$ if for every bounded sequence $\{a_{k}\}\in\ell^{\infty}$ there exists $f\in H^{\infty}(\mathbb{D}^{n})$ such that $f(z_{k})=a_{k}$ for all $k$ . In particular, $F_{\xi}$ is totally disconnected.
- If exactly one coordinate of $\xi$ lies in $M_{s}$ and the remaining $n-1$ in $M_{a}$ , then the fiber $\pi_{n}^{-1}(\xi)$ has covering dimension $\leq 2$ and decomposes as the disjoint union of a compact totally disconnected subset and a (possibly empty) open subset consisting of pairwise disjoint analytic disks. In both cases, $M\bigl(A(F_{\xi})\bigr)=F_{\xi}$ .*
In case (ii), the fiber $F_{\xi}$ is a disjoint union of Gleason parts for the algebra $H^{\infty}(\mathbb{D}\times\mathbb{N})$ of bounded holomorphic functions on the disjoint countable union of open disks. Each such part is either an analytic disk or a single point. For a detailed study of the structure of $M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ , see, e.g., [9].
Recall that the covering dimension of a normal space $X$ , denoted $\dim X$ , is defined as follows:
- $\dim X<n$ if every open cover of $X$ has a finite open refinement of order $\leq n$ (i.e., no point lies in more than $n+1$ sets of the refinement);
- $\dim X=n$ if $\dim X<n+1$ and $\dim X\nless n$ ;
- $\dim X=\infty$ if no such integer $n$ exists.
We mention that every compact Hausdorff totally disconnected space has covering dimension $0$ .
See [16, Ch. 1, §6.7].
**Remark 2.4**
*For strata of $[M(H^{\infty}(\mathbb{D}))]^{n}$ of codimension between $2$ and $n-1$ , which arise only for $n\geq 3$ , the structure of fibers $F_{\xi}$ is more subtle. Nevertheless, one can still show that $M(A(F_{\xi}))=F_{\xi}$ provided the corona theorem holds for the algebras $H^{\infty}(\mathbb{D}^{k})$ with $k=2,3,\dots,n-1$ . The main difficulty arises for the lowest stratum $M_{s}^{n}$ . In this case, the fiber $F_{\xi}$ may contain the image of an injective continuous map $\tau\colon\mathbb{D}^{n}\to\operatorname{cl}(\mathbb{D}^{n})$ such that $\hat{f}\circ\tau\in H^{\infty}(\mathbb{D}^{n})$ for all $f\in H^{\infty}(\mathbb{D}^{n})$ . The existence of such âembeddedâ $n$ -dimensional analytic structures obstructs an inductive approach to proving the corona theorem, as no reduction in dimension is possible. We will explore this phenomenon in more detail in a forthcoming paper.*
We now illustrate the scope of Theorems 2.1 and 2.2 by extending the class of subalgebras of $H^{\infty}(\mathbb{D}^{n})$ over which the corresponding BĂ©zout equations admit global solutions, and by showing that such solvability is preserved under all perturbations in $H^{\infty}(\mathbb{D}^{n})$ up to the maximal size allowed by the corona condition.
Let $S_{n}(H^{\infty}(\mathbb{D}))\subset H^{\infty}(\mathbb{D}^{n})$ denote the slice algebra, i.e. the closed subalgebra consisting of those holomorphic functions whose boundary values on
$$
\mathbb{T}^{n}:=\{z=(z_{1},\dots,z_{n})\in\mathbb{C}^{n}:|z_{i}|=1,\ \text{for all }i\}
$$
belong to
$$
\widehat{\otimes}_{\varepsilon}^{\,n}L^{\infty}(\mathbb{T})\subset L^{\infty}(\mathbb{T}^{n}).
$$
Equivalently, these are the holomorphic functions on $\mathbb{D}^{n}$ that can be uniformly approximated by bounded functions which are separately harmonic in each variable.
It is not known in general whether $S_{n}(H^{\infty}(\mathbb{D}))$ coincides with the uniform algebra generated by $H^{\infty}(\mathbb{D})$ in each coordinate, that is, whether
$$
S_{n}\bigl(H^{\infty}(\mathbb{D})\bigr)=\widehat{\otimes}_{\varepsilon}^{n}H^{\infty}(\mathbb{D}),
$$
a question related to the longstanding open problem of whether $H^{\infty}(\mathbb{D})$ has the approximation property (see Lindenstrauss [29], Question 7; see also BourgainâReinov [7] for related results).
Let $H_{1},\dots,H_{m}\in H^{\infty}(\mathbb{D}^{n})\setminus S_{n}\bigl(H^{\infty}(\mathbb{D})\bigr)$ , and set
$$
H:=(H_{1},\dots,H_{m}):\mathbb{D}^{n}\longrightarrow\mathbb{C}^{m}.
$$
We denote by
$$
A(S_{n};H):=\langle\,S_{n}\bigl(H^{\infty}(\mathbb{D})\bigr),\;H_{1},\dots,H_{m}\,\rangle\subset H^{\infty}(\mathbb{D}^{n})
$$
the uniform closure of the algebra generated by the slice algebra and the functions $H_{1},\dots,H_{m}$ .
**Theorem 2.5**
*Assume that for every point $x\in\mathbb{D}^{n}$ there exists a neighborhood $U_{x}\subset\mathbb{D}^{n}$ of $x$ and a oneâdimensional complex analytic subset $\Sigma_{x}\subset\mathbb{C}^{m}$ such that $H(U_{x})\subset\Sigma_{x}$ . Fix $\delta>0$ and suppose $g_{1},\dots,g_{k}\in A(S_{n};H)$ satisfy the corona condition
$$
\inf_{z\in\mathbb{D}^{n}}\ \max_{1\leq i\leq k}|g_{i}(z)|\ \geq\ \delta>0.
$$
Then for every $h_{1},\dots,h_{k}\in H^{\infty}(\mathbb{D}^{n})$ such that
$$
\max_{1\leq i\leq k}\,\|h_{i}\|_{\infty}<\delta,
$$
the BĂ©zout equation
$$
(g_{1}+h_{1})f_{1}+\cdots+(g_{k}+h_{k})f_{k}=1\quad\text{on }\ \mathbb{D}^{n}
$$
admits a solution $f_{1},\dots,f_{k}\in H^{\infty}(\mathbb{D}^{n})$ .*
**Remark 2.6**
*(a) The hypothesis of Theorem 2.5 on the map $H=(H_{1},\dots,H_{m})$ is equivalent to the existence of holomorphic maps
$$
h:\mathbb{D}^{n}\to\mathbb{D},\qquad\widetilde{H}:\mathbb{D}\to\mathbb{C}^{m}
$$
with $H=\widetilde{H}\circ h$ ; see Lemma 6.1. In particular, Theorem 2.5 applies whenever the components of $H$ factor through a holomorphic map from $\mathbb{D}^{n}$ to $\mathbb{D}$ . (b) If $g_{1},\dots,g_{k}\in S_{n}\bigl(H^{\infty}(\mathbb{D})\bigr)$ satisfy the corona condition
$$
\inf_{z\in\mathbb{D}^{n}}\ \max_{1\leq i\leq k}|g_{i}(z)|>0,
$$
then the BĂ©zout equation
$$
g_{1}f_{1}+\cdots+g_{k}f_{k}=1\quad\text{on }\mathbb{D}^{n}
$$
admits a solution $f_{1},\dots,f_{k}\in S_{n}\bigl(H^{\infty}(\mathbb{D})\bigr)$ . This was established in [8, Cor. 1.16] as a consequence of the more general result [8, Th. 1.12]. We note that Theorem 2.5 is significantly stronger: it applies already when $H$ consists of a single function from $H^{\infty}(\mathbb{D}^{n})\setminus S_{n}\bigl(H^{\infty}(\mathbb{D})\bigr)$ . (c) The perturbation condition in Theorem 2.5 is optimal. Indeed, assume
$$
\inf_{z\in\mathbb{D}^{n}}\ \max_{1\leq i\leq k}|g_{i}(z)|=\delta>0.
$$
Then there exist constants $c_{1},\dots,c_{k}\in\mathbb{C}$ with
$$
\max_{1\leq i\leq k}|c_{i}|=\delta
$$
such that
$$
\inf_{z\in\mathbb{D}^{n}}\ \max_{1\leq i\leq k}|g_{i}(z)-c_{i}|=0.
$$
Thus the corresponding BĂ©zout equation
$$
(g_{1}-c_{1})f_{1}+\cdots+(g_{k}-c_{k})f_{k}=1
$$
has no solution.*
## 3. Proof of Theorem 2.1
* Proof*
We argue by induction on the number of variables $n$ . For $n=1$ the statement is Theorem 8.3 of [10]. Assume the result holds for dimension $n$ , and consider $H^{\infty}(\mathbb{D}^{\,n+1},A)$ . Let $(\widehat{U}_{i})_{1\leq i\leq k}$ be an open cover of $[M(H^{\infty}(\mathbb{D}))]^{\,n+1}$ . By the product topology there exist a finite open cover $(\widehat{V}_{j})_{1\leq j\leq l}$ of $[M(H^{\infty}(\mathbb{D}))]^{n}$ and, for each $j$ , a finite open cover $(\widehat{W}_{jm})_{1\leq m\leq l_{j}}$ of $M(H^{\infty}(\mathbb{D}))$ such that the cover $(\widehat{V}_{j}\times\widehat{W}_{jm})_{j,m}$ of $[M(H^{\infty}(\mathbb{D}))]^{\,n+1}$ refines $(\widehat{U}_{i})_{1\leq i\leq k}$ . Set
$$
U_{jm}:=V_{j}\times W_{jm},\qquad V_{j}=\mathbb{D}^{n}\cap\widehat{V}_{j},\quad W_{jm}=\mathbb{D}\cap\widehat{W}_{jm}.
$$
Then $(U_{jm})_{j,m}$ is a finite cover of $\mathbb{D}^{\,n+1}$ refining $(U_{i})_{1\leq i\leq k}$ . Suppose the BĂ©zout equation
$$
\sum_{i=1}^{k}f_{i}g_{i}=1
$$
admits local solutions $(g_{1;jm},\dots,g_{k;jm})\in[H^{\infty}(U_{jm},A)]^{k}$ on the cover $(U_{jm})_{j,m}$ . For each $i\in\{1,\dots,k\}$ define slice functions on $W_{jm}$ ,
$$
g_{i;jm}^{s}(z)(z_{1},\dots,z_{n}):=g_{i;jm}(z_{1},\dots,z_{n},z),\qquad(z_{1},\dots,z_{n})\in V_{j},\ z\in W_{jm}.
$$
These belong to $H^{\infty}(W_{jm},H^{\infty}(V_{j},A))$ . Likewise, each $f_{i}$ defines
$$
f_{i;j}^{s}\in H^{\infty}(\mathbb{D},H^{\infty}(V_{j},A)).
$$
Passing to slice functions in (3.1) yields that the $H^{\infty}(V_{j},A)$ -valued BĂ©zout equation on $\mathbb{D}$ ,
$$
\sum_{i=1}^{k}f_{i;j}^{s}\,g_{i}=1,
$$
has local solutions $(g_{1;jm}^{s},\dots,g_{k;jm}^{s})\in[H^{\infty}(W_{jm},H^{\infty}(V_{j},A))]^{k}$ on the cover $(W_{jm})$ of $\mathbb{D}$ . Since $W_{jm}=\mathbb{D}\cap\widehat{W}_{jm}$ and $\mathcal{W}_{j}=(\widehat{W}_{jm})_{1\leq m\leq l_{j}}$ is an open cover of $M(H^{\infty}(\mathbb{D}))$ , applying [10, Thm. 8.3] we obtain global solutions $(h_{1;j},\dots,h_{k;j})\in[H^{\infty}(\mathbb{D},H^{\infty}(V_{j},A))]^{k}$ , $1\leq i\leq k$ , of (3.2). Define
$$
g_{i;j}(z_{1},\dots,z_{n+1}):=(h_{i;j}(z_{n+1}))(z_{1},\dots,z_{n}),\qquad(z_{1},\dots,z_{n+1})\in V_{j}\times\mathbb{D}.
$$
Then $g_{i;j}\in H^{\infty}(V_{j}\times\mathbb{D},A)$ and
$$
\sum_{i=1}^{k}f_{i}g_{i;j}=1\quad\text{on }V_{j}\times\mathbb{D}.
$$
Hence we obtain local bounded holomorphic solutions of the original BĂ©zout equation (3.1) on the cover $(V_{j}\times\mathbb{D})_{1\leq j\leq l}$ of $\mathbb{D}^{\,n+1}$ . Now view each $g_{i;j}$ as a slice function
$$
g_{i;j}^{s}\in H^{\infty}(V_{j},H^{\infty}(\mathbb{D},A)).
$$
Similarly, $f_{i}$ induce slice functions
$$
f_{i}^{s}\in H^{\infty}(\mathbb{D}^{n},H^{\infty}(\mathbb{D},A)).
$$
Then the solvability of the BĂ©zout equation (3.1) on the cover $(V_{j}\times\mathbb{D})$ transforms into the identities
$$
\sum_{i=1}^{k}f_{i}^{s}g_{i;j}^{s}=1\quad\text{in }H^{\infty}(\mathbb{D},A)\ \text{for each }V_{j},
$$
i.e. we obtain local bounded holomorphic solutions of the BĂ©zout equation on $\mathbb{D}^{n}$ ,
$$
\sum_{i=1}^{k}f_{i}^{s}h_{i}=1
$$
with values in the algebra $H^{\infty}(\mathbb{D},A)$ , over the cover $(V_{j})_{1\leq j\leq k}$ of $\mathbb{D}^{n}$ , where $V_{j}=\mathbb{D}^{n}\cap\widehat{V}_{j}$ and $\mathcal{V}=(\widehat{V}_{j})_{1\leq j\leq l}$ is a finite open cover of $[M(H^{\infty}(\mathbb{D}))]^{n}$ . By the induction hypothesis, there exists a solution $(h_{1},\dots,h_{k})\in[H^{\infty}(\mathbb{D}^{n},H^{\infty}(\mathbb{D},A))]^{k}$ of (3.3). Finally, set
$$
g_{i}(z_{1},\dots,z_{n+1}):=(h_{i}(z_{1},\dots,z_{n}))(z_{n+1}),\qquad(z_{1},\dots,z_{n+1})\in\mathbb{D}^{n+1}.
$$
Then $g_{i}\in H^{\infty}(\mathbb{D}^{n+1},A)$ and
$$
\sum_{i=1}^{k}f_{i}g_{i}=1.
$$
This completes the induction. â
## 4. Proof of Theorem 2.2
* Proof*
Recall that $A(F_{\xi})$ is the uniform closure of $\widehat{H^{\infty}(\mathbb{D}^{n})}|_{F_{\xi}}$ , hence $M\!\left(A(F_{\xi})\right)$ is exactly the $H^{\infty}$ -hull of $F_{\xi}$ . Also recall that
$$
\pi_{n}\colon M\!\left(H^{\infty}(\mathbb{D}^{n})\right)\longrightarrow\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}
$$
is the transpose of the canonical embedding
$$
\widehat{\otimes}_{\varepsilon}^{\,n}H^{\infty}(\mathbb{D})\ \hookrightarrow\ H^{\infty}(\mathbb{D}^{n}),
$$
and that the embedded algebra separates the fibers of $\pi_{n}$ . Since the $H^{\infty}$ -hull of $\pi_{n}^{-1}(\xi)$ coincides with $\pi_{n}^{-1}(\xi)$ , we have $M\!\left(A(F_{\xi})\right)\subset\pi_{n}^{-1}(\xi)$ . For the reverse inclusion, suppose $\zeta\in\pi_{n}^{-1}(\xi)\setminus M\!\left(A(F_{\xi})\right)$ . By the definition of the $H^{\infty}$ -hull, there exists $f\in H^{\infty}(\mathbb{D}^{n})$ with $\hat{f}(\zeta)=1$ and
$$
m:=\max_{F_{\xi}}|\hat{f}|<1.
$$
Set $g:=1-f$ . Then $\hat{g}(\zeta)=0$ and $|\hat{g}(x)|\geq 1-m>0$ for all $x\in F_{\xi}$ . By continuity of $\hat{g}$ and compactness of $F_{\xi}$ , there exists an open neighborhood $U\subset\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ of $\zeta$ such that
$$
|\hat{g}(y)|\ \geq\ \tfrac{1}{2}(1-m)\qquad\text{for all }y\in\pi_{n}^{-1}(U).
$$ By the product topology on $\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ , there exist functions
$$
g_{1},\dots,g_{k}\ \in\ \widehat{\otimes}_{\varepsilon}^{\,n}H^{\infty}(\mathbb{D})
$$
whose Gelfand transforms satisfy, for some $\delta>0$ and for a neighborhood $W\subset U$ of $\zeta$ with $\overline{W}\subset U$ ,
$$
\hat{g}_{i}|_{F_{\xi}}=0\ \ (1\leq i\leq k),\qquad\max_{1\leq i\leq k}|\hat{g}_{i}(y)|\ \geq\ \delta\ \ \text{for all }y\in\pi_{n}^{-1}(W).
$$
Since each $g_{i}$ extends continuously to $\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ , write $h_{i}$ for that extension. Then $\hat{g}_{i}=h_{i}\circ\pi_{n}$ , so $\hat{g}_{i}$ is constant on the fibers of $\pi_{n}$ . In particular, from (4.2) it follows that for every $x\in\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}\setminus W$ there exist an open neighborhood $W_{x}$ and an index $i=i(x)$ with
$$
|\hat{g}_{i}(y)|\ \geq\ \delta_{x}>0\qquad\text{for all }y\in\pi_{n}^{-1}(W_{x}).
$$
By compactness, there is a finite subcover $(W_{x_{j}})_{j=1}^{s}$ of $\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}\setminus W$ . Thus the tuple $(g,g_{1},\dots,g_{k})\subset H^{\infty}(\mathbb{D}^{n})$ satisfies a local corona condition on the finite open cover
$$
\bigl(W\cap\mathbb{D}^{n},\ W_{x_{1}}\cap\mathbb{D}^{n},\ \dots,\ W_{x_{s}}\cap\mathbb{D}^{n}\bigr)
$$
of $\mathbb{D}^{n}$ . Indeed, on $W\cap\mathbb{D}^{n}$ , inequality (4.1) gives a uniform lower bound for $|g|$ , and on each $W_{x_{j}}\cap\mathbb{D}^{n}$ inequality (4.3) gives a uniform lower bound for $|g_{i(x_{j})}|$ . By Theorem 2.1 (recall it asserts that local solvability of a BĂ©zout equation on a finite open cover of $\mathbb{D}^{n}$ , arising from a cover of $\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ via $\pi_{n}$ , yields a global solution in $H^{\infty}(\mathbb{D}^{n})$ ), there exist $f_{0},f_{1},\dots,f_{k}\in H^{\infty}(\mathbb{D}^{n})$ such that
$$
g\,f_{0}+g_{1}f_{1}+\cdots+g_{k}f_{k}=1\quad\text{on }\mathbb{D}^{n}.
$$ Taking Gelfand transforms and restricting to $\pi_{n}^{-1}(\xi)$ , and using $\hat{g}_{i}|_{F_{\xi}}=0$ for $1\leq i\leq k$ , we obtain
$$
\hat{f}_{0}(x)\,\hat{g}(x)=1\qquad\text{for all }x\in\pi_{n}^{-1}(\xi).
$$
But $\hat{g}(\zeta)=0$ , a contradiction. Hence $\pi_{n}^{-1}(\xi)\subset M\!\left(A(F_{\xi})\right)$ , and therefore
$$
\pi_{n}^{-1}(\xi)=M\!\left(A(F_{\xi})\right).
$$ Finally,
$$
\mathrm{cl}(\mathbb{D}^{n})=M\!\left(H^{\infty}(\mathbb{D}^{n})\right)
$$
(i.e., the corona theorem holds for $H^{\infty}(\mathbb{D}^{n})$ ) if and only if
$$
M\!\left(A(F_{\xi})\right)=F_{\xi}\qquad\text{for all }\xi\in\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n},
$$
equivalently, the $H^{\infty}$ -hull of each fiber $F_{\xi}$ coincides with $F_{\xi}$ . â
## 5. Proof of Theorem 2.3
* Proof of (i)*
Let $\xi=(\xi_{1},\dots,\xi_{n})\in M_{a}^{n}$ . Since each coordinate $\xi_{j}\in M(H^{\infty}(\mathbb{D}))$ lies in a nontrivial Gleason part, a classical result of Hoffman (see, e.g., [26, Thm. 5.5]) guarantees the existence of an interpolating sequence
$$
\zeta_{j}=\{z_{j,k}\}_{k=1}^{\infty}\subset\mathbb{D}
$$
whose closure in $M(H^{\infty}(\mathbb{D}))$ contains $\xi_{j}$ , for $j=1,\dots,n$ . Define the product sequence
$$
\zeta:=\zeta_{1}\times\cdots\times\zeta_{n}=\left\{z_{k}:=(z_{1,k},\dots,z_{n,k})\right\}_{k=1}^{\infty}\subset\mathbb{D}^{n},
$$
whose closure in the product topology on $[M(H^{\infty}(\mathbb{D}))]^{n}$ contains $\xi$ . CLAIM 1. The sequence $\zeta$ is interpolating for $H^{\infty}(\mathbb{D}^{n})$ . Indeed, for each $j$ and $k$ , let $\varphi_{j,k}\in H^{\infty}(\mathbb{D})$ be bounded holomorphic functions satisfying
$$
\varphi_{j,k}(z_{j,\ell})=\delta_{k,\ell},\qquad\sum_{k=1}^{\infty}|\varphi_{j,k}(z)|\leq C_{j},\quad\text{for all }z\in\mathbb{D},
$$
for some constants $C_{j}=C_{j}(\zeta_{j})>0$ ; see [20, Ch. VII, Thm. 2.1]. Define
$$
\psi_{k}(z):=\prod_{j=1}^{n}\varphi_{j,k}(z_{j}),\quad z=(z_{1},\dots,z_{n})\in\mathbb{D}^{n},\quad k\in\mathbb{N}.
$$
Then $\psi_{k}\in H^{\infty}(\mathbb{D}^{n})$ , and
$$
\psi_{k}(z_{\ell})=\delta_{k,\ell},\quad\forall\,k,\ell\in\mathbb{N},\qquad\sum_{k=1}^{\infty}|\psi_{k}(z)|\leq C:=\prod_{j=1}^{n}C_{j},\quad\forall\,z\in\mathbb{D}^{n}.
$$
Thus, the map
$$
L\colon\ell^{\infty}(\zeta)\to H^{\infty}(\mathbb{D}^{n}),\qquad L\big((c_{k})_{k=1}^{\infty}\big):=\sum_{k=1}^{\infty}c_{k}\psi_{k},
$$
is a bounded linear operator of norm $\leq C$ , with $L((c_{k}))|_{\zeta}=(c_{k})$ for all $(c_{k})\in\ell^{\infty}(\zeta)$ . Hence $\zeta$ is interpolating for $H^{\infty}(\mathbb{D}^{n})$ , as claimed. It follows from Claim 1 that the restriction map
$$
H^{\infty}(\mathbb{D}^{n})\to\ell^{\infty}(\zeta),\qquad f\mapsto f|_{\zeta},
$$
induces, by transposition, an embedding of maximal ideal spaces
$$
i_{\zeta}\colon M\big(\ell^{\infty}(\zeta)\big)\hookrightarrow M\big(H^{\infty}(\mathbb{D}^{n})\big).
$$
Since $\ell^{\infty}(\zeta)$ is self-adjoint, its maximal ideal space $M(\ell^{\infty}(\zeta))$ is homeomorphic to the StoneâÄech compactification $\beta(\mathbb{N})$ , and
$$
i_{\zeta}\big(M(\ell^{\infty}(\zeta))\big)=\operatorname{cl}(\zeta)\subset\operatorname{cl}(\mathbb{D}^{n})
$$
is totally disconnected. Let $B_{\zeta_{j}}\in H^{\infty}(\mathbb{D})$ be the Blaschke product vanishing precisely on $\zeta_{j}$ , and denote by $\hat{B}_{\zeta_{j}}\in C\big(M(H^{\infty}(\mathbb{D}))\big)$ its Gelfand transform. Let $p_{j}\colon[M(H^{\infty}(\mathbb{D}))]^{n}\to M(H^{\infty}(\mathbb{D}))$ be the projection onto the $j$ -th coordinate, and define
$$
\widetilde{B}_{\zeta_{j}}:=\hat{B}_{\zeta_{j}}\circ p_{j}\in C\big([M(H^{\infty}(\mathbb{D}))]^{n}\big).
$$
By the product topology, the joint zero set of the functions $\widetilde{B}_{\zeta_{j}}$ , $1\leq j\leq n$ , coincides with the closure of the interpolating sequence $\zeta$ in $[M(H^{\infty}(\mathbb{D}))]^{n}$ and hence contains $\xi$ . By definition, the pullbacks $\pi_{n}^{*}\widetilde{B}_{\zeta_{j}}\in C\big(M(H^{\infty}(\mathbb{D}^{n}))\big)$ are the Gelfand transforms of the functions $(p_{j}|_{\mathbb{D}^{n}})^{*}(B_{\zeta_{j}})\in H^{\infty}(\mathbb{D}^{n})$ , given by
$$
(p_{j}|_{\mathbb{D}^{n}})^{*}(B_{\zeta_{j}})(z_{1},\dots,z_{n})=B_{\zeta_{j}}(z_{j}),\qquad 1\leq j\leq n,
$$
and they vanish on $F_{\xi}$ . We next identify the joint zero set of these pullbacks. CLAIM 2. The joint zero set of the functions $\pi_{n}^{*}\widetilde{B}_{\zeta_{j}}$ on $\operatorname{cl}(\mathbb{D}^{n})\subset M(H^{\infty}(\mathbb{D}^{n}))$ , $1\leq j\leq n$ , is precisely $\operatorname{cl}(\zeta)$ . To prove this, suppose, to the contrary, that there exists a point $\eta\in\operatorname{cl}(\mathbb{D}^{n})\setminus\operatorname{cl}(\zeta)$ such that
$$
\pi_{n}^{*}\widetilde{B}_{\zeta_{j}}(\eta)=0\quad\text{for all }j=1,\dots,n.
$$
Let $U\subset M(H^{\infty}(\mathbb{D}^{n}))$ be an open neighborhood of $\eta$ whose closure is disjoint from $\operatorname{cl}(\zeta)$ . For each $\varepsilon\in(0,1)$ , consider the open set
$$
U_{\varepsilon}:=U\cap\bigl\{\theta\in M(H^{\infty}(\mathbb{D}^{n})):|\pi_{n}^{*}\widetilde{B}_{\zeta_{j}}(\theta)|<\varepsilon\text{ for all }1\leq j\leq n\bigr\}.
$$
Define the pseudohyperbolic metric on $\mathbb{D}^{n}$ by
$$
\rho_{n}(z,w):=\max_{1\leq j\leq n}\rho(z_{j},w_{j}),\qquad z,w\in\mathbb{D}^{n},
$$
where $\rho$ is the pseudohyperbolic metric on $\mathbb{D}$ , see (2.5). Since $\mathbb{D}^{n}$ is dense in $\operatorname{cl}(\mathbb{D}^{n})$ , we may select a point
$$
z_{\varepsilon}\in U_{\varepsilon}\cap\mathbb{D}^{n}.
$$
By a result of Hoffman (see, e.g., [20, Ch. X, Lemma 1.4]), we have
$$
\lim_{\varepsilon\to 0}\operatorname{dist}(z_{\varepsilon},\zeta)=0,
$$
where, for $z\in\mathbb{D}^{n}$ and $Z\subset\mathbb{D}^{n}$ ,
$$
\operatorname{dist}(z,Z):=\inf_{y\in Z}\rho_{n}(z,y).
$$
Let $\nu\in\operatorname{cl}(\mathbb{D}^{n})$ be a limit point of the sequence $\{z_{1/k}\}_{k\in\mathbb{N}}$ . By the SchwarzâPick lemma (see, e.g., [20, Ch. I, Lemma 1.2]) and (5.1), we obtain $\nu\in\operatorname{cl}(\zeta)$ . On the other hand, since $\nu\in\operatorname{cl}(U)$ and $\operatorname{cl}(U)\cap\operatorname{cl}(\zeta)=\varnothing$ , a contradiction, proving Claim 2. Claim 2 implies that
$$
F_{\xi}:=\pi_{n}^{-1}(\xi)\cap\operatorname{cl}(\mathbb{D}^{n})\subset\left\{\nu\in\operatorname{cl}(\mathbb{D}^{n}):\pi_{n}^{*}\widetilde{B}_{\zeta_{j}}(\nu)=0,\ 1\leq j\leq n\right\}\subset\operatorname{cl}(\zeta).
$$
Since $\zeta$ is an interpolating sequence for $H^{\infty}(\mathbb{D}^{n})$ , the restriction of the algebra $\mathcal{O}\big(M(H^{\infty}(\mathbb{D}^{n}))\big)$ to $\operatorname{cl}(\zeta)$ coincides with the full algebra $C\big(\operatorname{cl}(\zeta)\big)$ . By the TietzeâUrysohn extension theorem, the same property holds for any closed subset of $\operatorname{cl}(\zeta)$ , and in particular for the compact set $F_{\xi}\subset\operatorname{cl}(\zeta)$ . Hence $A(F_{\xi})=C(F_{\xi})$ , and therefore the maximal ideal space $M(A(F_{\xi}))$ coincides with $F_{\xi}$ . By Theorem 2.2, we conclude that
$$
\pi_{n}^{-1}(\xi)=F_{\xi}\subset\operatorname{cl}(\zeta).
$$
As $\operatorname{cl}(\zeta)$ is totally disconnected, the same is true for $F_{\xi}$ . This completes the proof of part (i) of the theorem. Proof of (ii). If exactly one coordinate of $\xi$ lies in $M_{s}$ and the remaining $n-1$ coordinates lie in $M_{a}$ , then the natural action of the symmetric group on coordinates of $\mathbb{D}^{n}$ preserves both algebras $\widehat{\otimes}_{\varepsilon}^{n}H^{\infty}(\mathbb{D})$ and $H^{\infty}(\mathbb{D}^{n})$ . Consequently, it extends to actions on their maximal ideal spaces $\bigl[M\big(H^{\infty}(\mathbb{D})\big)\bigr]^{n}$ and $M\big(H^{\infty}(\mathbb{D}^{n})\big)$ , and these actions commute with the projection
$$
\pi_{n}\colon M\big(H^{\infty}(\mathbb{D}^{n})\big)\to\bigl[M\big(H^{\infty}(\mathbb{D})\big)\bigr]^{n}.
$$
In particular, the fibers of $\pi_{n}$ corresponding to points in the same orbit are homeomorphic and have identical analytic structure. Hence, without loss of generality, we may and will assume that
$$
\xi\in M_{a}^{n-1}\times M_{s}.
$$ We first show that
$$
\pi_{n}^{-1}(\xi)=F_{\xi}:=\pi_{n}^{-1}(\xi)\cap\mathrm{cl}(\mathbb{D}^{n}),
$$
see Lemma 5.1 below. Let $\xi=(\xi^{\,n-1},\xi_{n})$ , where $\xi^{\,n-1}\in M_{a}^{\,n-1}$ and $\xi_{n}\in M_{s}$ . By Claim 1 in the proof of part (i) of the theorem, there exists an interpolating sequence $\zeta=\{z_{k}\}\subset\mathbb{D}^{\,n-1}$ for $H^{\infty}(\mathbb{D}^{\,n-1})$ , which is the Cartesian product of interpolating sequences for $H^{\infty}(\mathbb{D})$ in the coordinate disks, such that the closure of $\zeta$ in $\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n-1}$ contains the point $\xi^{\,n-1}$ . Moreover, there exist functions $\psi_{k}\in H^{\infty}(\mathbb{D}^{\,n-1})$ , $k\in\mathbb{N}$ , such that
$$
\psi_{k}(z_{i})=\delta_{k,i},\quad\forall\,k,i\in\mathbb{N},\qquad\sum_{k=1}^{\infty}\lvert\psi_{k}(z)\rvert\leq C=C(\zeta),\quad\forall\,z\in\mathbb{D}^{\,n-1}.
$$
Then for every uniformly bounded sequence of functions $\{f_{k}\}\subset H^{\infty}(\mathbb{D})$ , there exists $f\in H^{\infty}(\mathbb{D}^{n})$ such that
$$
f(z_{k},z)=f_{k}(z)\quad\forall\,z\in\mathbb{D},\ \forall\,k\in\mathbb{N}.
$$
Indeed, set
$$
f(w,z):=\sum_{k=1}^{\infty}\psi_{k}(w)\,f_{k}(z),\qquad(w,z)\in\mathbb{D}^{\,n-1}\times\mathbb{D}.
$$
Since $\sum_{k=1}^{\infty}\lvert\psi_{k}(w)\rvert\leq C$ for all $w\in\mathbb{D}^{\,n-1}$ and $\sup_{k}\lVert f_{k}\rVert_{\infty}<\infty$ , the series defining $f$ converges uniformly on compact subsets of $\mathbb{D}^{\,n-1}\times\mathbb{D}$ , and
$$
\lVert f\rVert_{\infty}\leq C\,\sup_{k}\lVert f_{k}\rVert_{\infty}.
$$
Uniform convergence on compacta shows that $f\in H^{\infty}(\mathbb{D}^{n})$ . Moreover, for each $i\in\mathbb{N}$ and every $z\in\mathbb{D}$ ,
$$
f(z_{i},z)=\sum_{k=1}^{\infty}\psi_{k}(z_{i})\,f_{k}(z)=\sum_{k=1}^{\infty}\delta_{k,i}\,f_{k}(z)=f_{i}(z),
$$
as required. Thus the restriction map
$$
R\colon H^{\infty}(\mathbb{D}^{n})\to H^{\infty}(\mathbb{D}\times\mathbb{N}),\qquad R(f)(k):=f\bigl|_{\{z_{k}\}\times\mathbb{D}},
$$
is a surjective homomorphism of Banach algebras, where $H^{\infty}(\mathbb{D}\times\mathbb{N})$ is equipped with the norm
$$
\lVert(g_{k})_{k\in\mathbb{N}}\rVert:=\sup_{k\in\mathbb{N}}\lVert g_{k}\rVert_{\infty}.
$$ In particular, the induced transpose map on maximal ideal spaces
$$
R^{*}\colon M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)\longrightarrow M\bigl(H^{\infty}(\mathbb{D}^{n})\bigr),\qquad R^{*}(\varphi):=\varphi\circ R,
$$
is a continuous embedding, and its image is $H^{\infty}$ -convex; that is, for every character $\eta\in M\bigl(H^{\infty}(\mathbb{D}^{n})\bigr)\setminus R^{*}\!\bigl(M(H^{\infty}(\mathbb{D}\times\mathbb{N}))\bigr)$ there exists $f\in H^{\infty}(\mathbb{D}^{n})$ such that
$$
\lvert\hat{f}(\eta)\rvert>\sup_{R^{*}(M(H^{\infty}(\mathbb{D}\times\mathbb{N})))}\lvert\hat{f}\rvert.
$$
**Lemma 5.1**
*We have
$$
R^{*}\bigl(M(H^{\infty}(\mathbb{D}\times\mathbb{N}))\bigr)\supset\pi_{n}^{-1}(\xi)=F_{\xi}.
$$*
* Proof*
Let us prove first that $\pi_{n}^{-1}(\xi)\subset\operatorname{im}R^{*}$ . Set
$$
I:=\ker R=\bigl\{f\in H^{\infty}(\mathbb{D}^{n}):f(z_{k},z)=0\ \forall\,k\in\mathbb{N},\ \forall z\in\mathbb{D}\bigr\}.
$$
Let $U$ be an open neighborhood of
$$
F_{\xi}:=\pi_{n}^{-1}(\xi)\cap\mathrm{cl}(\mathbb{D}^{n})
$$
in $M\bigl(H^{\infty}(\mathbb{D}^{n})\bigr)$ . As observed above, we may assume without loss of generality that
$$
U=\pi_{n}^{-1}(V)\cap\mathrm{cl}(\mathbb{D}^{n})
$$
for some open neighborhood $V$ of $\xi$ in $\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ . Since the closure of $\zeta\times\mathbb{D}$ in $\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ contains $\xi$ , there exists a subsequence $\zeta_{U}\subset\zeta$ such that $\zeta_{U}\times\mathbb{D}\subset V$ . Varying $U$ over a basis of open neighborhoods of $F_{\xi}$ , we conclude that
$$
F_{\xi}\subset\mathrm{cl}(\zeta\times\mathbb{D})\qquad\text{(recall that $\mathrm{cl}=\mathrm{cl}_{M(H^{\infty}(\mathbb{D}^{n}))}$)}.
$$ Thus, the Gelfand transform of every $f\in I$ vanishes on $F_{\xi}$ , and hence on its $H^{\infty}$ -hull. By Theorem 2.2 this hull equals $\pi_{n}^{-1}(\xi)$ . Therefore every $\eta\in\pi_{n}^{-1}(\xi)$ annihilates $I$ and thus factors through
$$
H^{\infty}(\mathbb{D}^{n})/I\simeq H^{\infty}(\mathbb{D}\times\mathbb{N}),
$$
i.e.,
$$
\eta=R^{*}(\varphi)\qquad\text{for some }\varphi\in M\!\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr).
$$
This proves $\pi_{n}^{-1}(\xi)\subset\operatorname{im}R^{*}$ . Using this fact, let us prove that $\pi_{n}^{-1}(\xi)=F_{\xi}$ . Indeed, the Corona Theorem for $H^{\infty}(\mathbb{D}\times\mathbb{N})$ (which follows from uniform bounds in the proof of Carlesonâs original theorem, see, e.g., [20]) implies that $\mathbb{D}\times\mathbb{N}$ is an open dense subset of $M(H^{\infty}(\mathbb{D}\times\mathbb{N}))$ . By construction of $R^{*}$ ,
$$
R^{*}(\mathbb{D}\times\mathbb{N})=\zeta\times\mathbb{D}.
$$
Since $R^{*}$ is continuous and injective, it follows that
$$
R^{*}\bigl(M(H^{\infty}(\mathbb{D}\times\mathbb{N}))\bigr)=\mathrm{cl}\bigl(\zeta\times\mathbb{D}\bigr)\subset\mathrm{cl}(\mathbb{D}^{n}).
$$
Hence
$$
\pi_{n}^{-1}(\xi)=\pi_{n}^{-1}(\xi)\cap R^{*}\bigl(M(H^{\infty}(\mathbb{D}\times\mathbb{N}))\bigr)\subset\pi_{n}^{-1}(\xi)\cap\mathrm{cl}(\mathbb{D}^{n})=F_{\xi}.
$$
Thus, $\pi_{n}^{-1}(\xi)=F_{\xi}$ , completing the proof of the lemma. â
By Lemma 5.1 and Theorem 2.2 we conclude that $M(A(F_{\xi}))=F_{\xi}$ . We now use the known structure of $M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ to analyze the internal geometry of $F_{\xi}$ . As shown in [9], every Gleason part in $M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ is either a singleton or an analytic disk; the set of nontrivial parts is open and dense, the set of trivial parts is totally disconnected, and
$$
\dim M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)=2.
$$
We relate this structure to the fibers of the projection $T^{*}$ introduced below. The Banach algebra homomorphism
$$
T\colon\ell^{\infty}\longrightarrow H^{\infty}(\mathbb{D}\times\mathbb{N}),\qquad T\bigl((c_{k})_{k\in\mathbb{N}}\bigr):=(g_{k})_{k\in\mathbb{N}},\quad g_{k}(z)\equiv c_{k},\ z\in\mathbb{D},
$$
is an isometric embedding. Identifying $M(\ell^{\infty})$ with the StoneâÄech compactification $\beta\mathbb{N}$ , its transpose
$$
T^{*}\colon M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)\longrightarrow\beta\mathbb{N}
$$
is surjective. For characters corresponding to points $(z,n)\in\mathbb{D}\times\mathbb{N}$ , the map $T^{*}$ assigns evaluation at the index $n$ ; in particular,
$$
T^{*}\bigl(\mathbb{D}\times\mathbb{N}\bigr)=\mathbb{N}\subset\beta\mathbb{N}.
$$ Fix $\chi\in\beta\mathbb{N}\setminus\mathbb{N}$ . If $G\subset M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ is an analytic disk with $T^{*}(G)$ containing $\chi$ , then, since $T^{*}$ is continuous and $\beta\mathbb{N}$ is totally disconnected, the connected set $T^{*}(G)$ must be a singleton. Hence $T^{*}(G)=\{\chi\}$ , so $G\subset(T^{*})^{-1}(\chi)$ . Consequently, for every $\chi\in\beta\mathbb{N}\setminus\mathbb{N}$ , the fiber $(T^{*})^{-1}(\chi)$ is a disjoint union of analytic disks and points belonging to trivial Gleason parts of $M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ (each such part is a singleton).
**Lemma 5.2**
*Let $G$ be an analytic disk in $M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ such that $R^{*}(G)$ contains a point $\eta\in F_{\xi}=\pi_{n}^{-1}(\xi)$ , where $\xi=(\xi^{\,n-1},\xi_{n})\in M_{a}^{n-1}\times M_{s}\subset\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}$ . Then
$$
R^{*}(G)\subset F_{\xi}.
$$*
* Proof*
Let
$$
p^{\,n-1}\colon\mathbb{D}^{n}\to\mathbb{D}^{\,n-1}
$$
be the projection onto the first $n-1$ coordinates. It induces the following algebra homomorphisms. First, the pullback
$$
(p^{\,n-1})^{*}\colon H^{\infty}(\mathbb{D}^{\,n-1})\longrightarrow H^{\infty}(\mathbb{D}^{n}),\qquad(p^{\,n-1})^{*}(f)=f\circ p^{\,n-1},
$$
whose transpose is the map
$$
\hat{p}^{\,n-1}\colon M\bigl(H^{\infty}(\mathbb{D}^{n})\bigr)\longrightarrow M\bigl(H^{\infty}(\mathbb{D}^{\,n-1})\bigr),\qquad\hat{p}^{\,n-1}(\varphi)=\varphi\circ(p^{\,n-1})^{*},
$$
and $\hat{p}^{\,n-1}|_{\mathbb{D}^{n}}=p^{\,n-1}$ . Second, the restriction
$$
(p^{\,n-1})^{*}\big|_{\widehat{\otimes}_{\varepsilon}^{\,n-1}H^{\infty}(\mathbb{D})}\;\colon\;\widehat{\otimes}_{\varepsilon}^{\,n-1}H^{\infty}(\mathbb{D})\longrightarrow\widehat{\otimes}_{\varepsilon}^{\,n}H^{\infty}(\mathbb{D})
$$
whose transpose
$$
\bar{p}^{\,n-1}\colon\bigl[M\bigl(H^{\infty}(\mathbb{D})\bigr)\bigr]^{n}\longrightarrow\bigl[M\bigl(H^{\infty}(\mathbb{D})\bigr)\bigr]^{\,n-1},
$$
coincides with the projection onto the first $n-1$ coordinates. The embeddings $\widehat{\otimes}_{\varepsilon}^{\,k}H^{\infty}(\mathbb{D})\hookrightarrow H^{\infty}(\mathbb{D}^{k})$ are compatible with the pullback $(p^{\,n-1})^{*}$ . Passing to transposes (see (2.2)) yields
$$
\bar{p}^{\,n-1}\circ\pi_{n}\;=\;\pi_{\,n-1}\circ\hat{p}^{\,n-1}.
$$ Let $\zeta=\{z_{k}\}\subset\mathbb{D}^{\,n-1}$ be the interpolating sequence fixed at the beginning of the proof, whose closure in $\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n-1}$ contains $\xi^{\,n-1}$ . Define the restriction homomorphism
$$
R_{\zeta}\colon H^{\infty}(\mathbb{D}^{\,n-1})\longrightarrow\ell^{\infty}(\mathbb{N}),\qquad R_{\zeta}(f)=\bigl(f(z_{k})\bigr)_{k\in\mathbb{N}}.
$$ By the definitions of $R$ , $T$ , and $(p^{\,n-1})^{*}$ , see (5.2)â(5.4), we have the commutative relation
$$
T\circ R_{\zeta}\;=\;R\circ(p^{\,n-1})^{*}.
$$
Passing to transposes and combining with (5.5) gives the commutative diagram: Since $R^{*}(G)\subset M\bigl(H^{\infty}(\mathbb{D}^{n})\bigr)$ meets the fibre $F_{\xi}=\pi_{n}^{-1}(\xi)$ and $T^{*}$ is constant on $G$ , the diagram (5.6) gives, for $\eta\in R^{*}(G)\cap F_{\xi}$ ,
$$
(\pi_{\,n-1}\circ\hat{p}^{\,n-1})(\eta)=\xi^{\,n-1}.
$$
Let $T^{*}(G)=\{\omega\}\subset\beta\mathbb{N}$ . Then the top square in (5.6) yields
$$
\hat{p}^{\,n-1}\bigl(R^{*}(G)\bigr)=\{(R_{\zeta})^{*}(\omega)\}.
$$
Applying $\pi_{\,n-1}$ and using (5.7), we obtain
$$
(\pi_{\,n-1}\circ\hat{p}^{\,n-1})\bigl(R^{*}(G)\bigr)=\{\xi^{\,n-1}\}.
$$ For the last coordinate, let
$$
p_{n}:\bigl[M(H^{\infty}(\mathbb{D}))\bigr]^{n}\longrightarrow M(H^{\infty}(\mathbb{D}))
$$
be the projection onto the $n$ -th factor. Then $(p_{n}\circ\pi_{n})\bigl(R^{*}(G)\bigr)$ is an analytic disk in $M(H^{\infty}(\mathbb{D}))$ passing through $\xi_{n}$ . Since $\xi_{n}\in M_{s}$ is a trivial Gleason part, Hoffmanâs results [26, Thm. 4.3, 5.5, 5.6] imply that this image is a singleton:
$$
(p_{n}\circ\pi_{n})\bigl(R^{*}(G)\bigr)=\{\xi_{n}\}.
$$ Combining the two displays gives
$$
\pi_{n}\bigl(R^{*}(G)\bigr)=\{(\xi^{\,n-1},\xi_{n})\}=\{\xi\},
$$
hence $R^{*}(G)\subset\pi_{n}^{-1}(\xi)=F_{\xi}$ , as claimed. â
Combining Lemmas 5.1 â 5.2 with the structure of $M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ established in [9], we obtain that $F_{\xi}$ is a disjoint union of its Gleason parts, each of which is either a single point or an analytic disk. Moreover, since $F_{\xi}$ embeds into $M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ , its covering dimension satisfies
$$
\dim F_{\xi}\leq 2.
$$ To complete the proof of part (ii), it remains to show that the subset of trivial Gleason parts in $F_{\xi}$ is nonempty. Equivalently, if the fiber $F_{\xi}$ contains a nontrivial Gleason part (an analytic disk), then it must also contain at least one trivial Gleason part.
**Lemma 5.3**
*The closure in $M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ of any nontrivial Gleason part of $H^{\infty}(\mathbb{D}\times\mathbb{N})$ contains a trivial Gleason part of this algebra.*
* Proof*
It was shown in [4, Lemma 2] that the algebra $H^{\infty}(\mathbb{D}\times\mathbb{N})$ is logmodular on $\mathbb{D}\times\mathbb{N}$ . This, together with the general result of K. Hoffman [25], implies that the Gleason parts of $H^{\infty}(\mathbb{D}\times\mathbb{N})$ are either points or analytic disks. The latter means that there exists a continuous injective map
$$
\pi:\mathbb{D}\longrightarrow M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)
$$
onto this Gleason part such that $\hat{f}\circ\pi\in H^{\infty}(\mathbb{D})$ for all $f\in H^{\infty}(\mathbb{D}\times\mathbb{N})$ . Suppose, to the contrary, that there exists a nontrivial Gleason part $G\subset M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ whose closure $\overline{G}$ consists entirely of nontrivial Gleason parts, so that $\overline{G}$ is a disjoint union of analytic disks. For $f\in H^{\infty}(\mathbb{D}\times\mathbb{N})$ , let $\mathcal{C}_{\hat{f}}$ denote the set of all analytic disks in $\overline{G}$ on which the Gelfand transform $\hat{f}$ is constant. Claim. Each $\mathcal{C}_{\hat{f}}$ is a nonempty compact subset of $\overline{G}$ . Indeed, the function $|\hat{f}|$ , being continuous on the compact set $\overline{G}$ , attains its maximum at some point $\eta$ . Let $G_{\eta}$ be the nontrivial Gleason part containing $\eta$ , and let $\pi:\mathbb{D}\to G_{\eta}$ be its analytic parametrization (as in the above definition). Then the function $|\hat{f}\circ\pi|$ is subharmonic on $\mathbb{D}$ and attains its maximum at the interior point $\pi^{-1}(\eta)$ . By the maximum modulus principle, it must be constant, and hence $\hat{f}\circ\pi$ , and therefore $\hat{f}$ , is constant on $G_{\eta}$ . Thus $G_{\eta}\subset\mathcal{C}_{\hat{f}}$ , showing that $\mathcal{C}_{\hat{f}}$ is nonempty. Next, we show that $\mathcal{C}_{\hat{f}}$ is compact. Let $\theta$ be a limit point of $\mathcal{C}_{\hat{f}}$ . By the hypothesis on $\overline{G}$ , the point $\theta$ belongs to some maximal analytic disk, i.e., to the Gleason part of $\theta$ . By [9, Thm. 2.5 (b),(d)], there exists an open neighborhood $U$ of $\theta$ in the set of nontrivial Gleason parts of $H^{\infty}(\mathbb{D}\times\mathbb{N})$ , and a homeomorphism
$$
h:S\times\mathbb{D}\longrightarrow U,
$$
where $S$ is a compact subset of the StoneâÄech compactification $\beta\mathbb{N}$ , such that each slice $h(\{\omega\}\times\mathbb{D})$ lies in a nontrivial Gleason part of $H^{\infty}(\mathbb{D}\times\mathbb{N})$ , and for every $f\in H^{\infty}(\mathbb{D}\times\mathbb{N})$ the pullback $\hat{f}\circ h(\omega,\cdot)$ belongs to $H^{\infty}(\mathbb{D})$ . Since $\theta$ is a limit point of $\mathcal{C}_{\hat{f}}$ , there exists a net $(\theta_{\alpha})\subset\mathcal{C}_{\hat{f}}\cap U$ converging to $\theta$ . If $\theta_{\alpha}$ lies in the image under $h$ of the slice corresponding to $\omega_{\alpha}$ , then $\hat{f}$ is constant on this image. Passing to the limit within $U$ shows that $\hat{f}$ is constant on the image under $h$ of the slice containing $\theta$ . By analyticity, $\hat{f}$ is constant on the entire Gleason part containing $\theta$ ; hence $\theta\in\mathcal{C}_{\hat{f}}$ . Thus $\mathcal{C}_{\hat{f}}$ is closed in $\overline{G}$ , and compactness follows since $\overline{G}$ is compact. Further, given any finite collection $f_{1},\dots,f_{k}\in H^{\infty}(\mathbb{D}\times\mathbb{N})$ , apply the maximum modulus principle to the continuous function
$$
\hat{F}:=\sum_{j=1}^{k}|\hat{f_{j}}|^{2}
$$
restricted to $\overline{G}$ . As in the case of a single function considered in the proof of the Claim, the function $|\hat{F}|$ , being continuous on the compact set $\overline{G}$ , attains its maximum at some point $\eta$ . Let $G_{\eta}$ be the nontrivial Gleason part containing $\eta$ , and let $\pi:\mathbb{D}\to G_{\eta}$ be its analytic parametrization. Then the function $|\hat{F}\circ\pi|$ is subharmonic on $\mathbb{D}$ and attains its maximum at the interior point $\pi^{-1}(\eta)$ . By the maximum modulus principle, it must be constant, and therefore $\hat{F}\circ\pi$ , being a holomorphic map from $\mathbb{D}$ into a Euclidean sphere in $\mathbb{C}^{k}$ , is constant. Hence each $\hat{f_{j}}$ is constant on $G_{\eta}$ , and consequently
$$
\mathcal{C}_{\hat{f_{1}}}\cap\cdots\cap\mathcal{C}_{\hat{f_{k}}}\neq\varnothing.
$$ Since each $\mathcal{C}_{\hat{f}}$ is compact and all finite intersections are nonempty, the finite intersection property implies that
$$
\bigcap_{f\in H^{\infty}(\mathbb{D}\times\mathbb{N})}\mathcal{C}_{\hat{f}}\neq\varnothing.
$$
Hence there exists an analytic disk in $\overline{G}$ on which $\hat{f}$ is constant for every $f\in H^{\infty}(\mathbb{D}\times\mathbb{N})$ . This yields a contradiction, since the Gelfand transforms of functions from the algebra $H^{\infty}(\mathbb{D}\times\mathbb{N})$ separate points of its maximal ideal space. Therefore $\overline{G}$ cannot consist solely of analytic disks, that is, it must contain a trivial Gleason part, as claimed. â
Since $F_{\xi}$ is contained in the homeomorphic image of $M\bigl(H^{\infty}(\mathbb{D}\times\mathbb{N})\bigr)$ under $R^{*}$ , which preserves the analytic structure, the subset of trivial Gleason parts in $F_{\xi}$ is nonempty. This completes the proof of part (ii) of Theorem 2.3. â
**Remark 5.4**
*It is not difficult to construct points $\xi$ as in part (ii) for which the corresponding fibers $F_{\xi}$ contain analytic disks. Whether this holds for all such $\xi$ remains an open question. We conjecture that every fiber of this type contains analytic disks.*
## 6. Proof of Theorem 2.5
We begin with the following auxiliary result.
**Lemma 6.1**
*Let $H=(H_{1},\dots,H_{m})\in[H^{\infty}(\mathbb{D}^{n})]^{m}$ . The following are equivalent: (1) For each $x\in\mathbb{D}^{n}$ there exist a neighborhood $U_{x}\subset\mathbb{D}^{n}$ of $x$ and a one-dimensional complex analytic subset $\Sigma_{x}\subset\mathbb{C}^{m}$ such that $H(U_{x})\subset\Sigma_{x}$ . (2) There exist holomorphic maps
$$
h:\mathbb{D}^{n}\to\mathbb{D},\qquad\widetilde{H}:\mathbb{D}\to\mathbb{C}^{m}
$$
such that $H=\widetilde{H}\circ h$ .*
* Proof*
We assume that $H$ is nonconstant; otherwise both statements are trivial. (1) $\Rightarrow$ (2). Choose radii $0<r_{1}<r_{2}<\cdots\nearrow 1$ and consider the exhaustion of $\mathbb{D}^{n}$ by open polydisks $\mathbb{D}_{r_{j}}^{n}$ of radii $r_{j}$ . By compactness of the closure $\overline{\mathbb{D}}_{r_{j}}^{\,n}$ , there exist $x_{1},\dots,x_{N}$ with
$$
\overline{\mathbb{D}}_{r_{j}}^{\,n}\subset U_{x_{1}}\cup\cdots\cup U_{x_{N}},
$$
hence
$$
H(\mathbb{D}_{r_{j}}^{n})\subset\Sigma_{j}:=\Sigma_{x_{1}}\cup\cdots\cup\Sigma_{x_{N}},
$$
a one-dimensional analytic set. Let $\nu_{j}:\Sigma_{j}^{\nu}\to\Sigma_{j}$ be the normalization ([22, Ch. 8]). By its universal property, the holomorphic map $H|_{\mathbb{D}_{r_{j}}^{n}}$ lifts to a holomorphic map
$$
H_{j}^{\nu}:\mathbb{D}_{r_{j}}^{n}\longrightarrow\Sigma_{j}^{\nu},\qquad H=\nu_{j}\circ H_{j}^{\nu}.
$$
Since $H_{j}^{\nu}$ is nonconstant and $\mathbb{D}_{r_{j}}^{n}$ is connected, the open mapping theorem implies that
$$
R_{j}:=H_{j}^{\nu}(\mathbb{D}_{r_{j}}^{n})
$$
is a connected open Riemann surface. Set
$$
h_{j}:=H_{j}^{\nu}:\mathbb{D}_{r_{j}}^{n}\to R_{j},\qquad\widetilde{H}_{j}:=\nu_{j}|_{R_{j}}:R_{j}\to\mathbb{C}^{m},
$$
so $H|_{\mathbb{D}_{r_{j}}^{n}}=\widetilde{H}_{j}\circ h_{j}$ . If $r_{i}<r_{j}$ , uniqueness of normalization factorizations yields a unique holomorphic map $\varphi_{ij}:R_{i}\to R_{j}$ such that $h_{j}=\varphi_{ij}\circ h_{i}$ on $\mathbb{D}_{r_{i}}^{n}$ . Thus $\{R_{j},\varphi_{ij}\}$ forms a direct system of Riemann surfaces. Formation of the limit surface. Define the inductive limit
$$
R:=\varinjlim R_{j}.
$$
Each $\varphi_{ij}$ is a local biholomorphism on the relevant component of $R_{i}$ , so the complex structures are compatible under pullback. Hence $R$ becomes a Hausdorff one-dimensional complex manifold. Since each $R_{j}$ is connected and the overlaps $\varphi_{ij}(R_{i})\cap R_{j}$ are nonempty open subsets, the union $R=\bigcup_{j}R_{j}$ is connected, and hence $R$ is a connected Riemann surface. Since the factorizations are compatible, the $h_{j}$ glue to a holomorphic map
$$
h_{0}:\mathbb{D}^{n}\to R,
$$
and the $\widetilde{H}_{j}$ glue to a holomorphic map
$$
\widetilde{H}_{0}:R\to\mathbb{C}^{m},
$$
yielding
$$
H=\widetilde{H}_{0}\circ h_{0}.
$$ Uniformization. As $\widetilde{H}_{0}$ is bounded and nonconstant, $R$ admits a nonconstant bounded holomorphic function, hence $R$ is hyperbolic. Thus its universal cover is $\mathbb{D}$ . Let $p:\mathbb{D}\to R$ be the covering map. Since $\mathbb{D}^{n}$ is simply connected, the covering homotopy theorem implies the existence of a holomorphic map
$$
h:\mathbb{D}^{n}\to\mathbb{D},\qquad h_{0}=p\circ h,
$$
and setting $\widetilde{H}:=\widetilde{H}_{0}\circ p$ gives $H=\widetilde{H}\circ h$ . (2) $\Rightarrow$ (1). Assume $H=\widetilde{H}\circ h$ with $h:\mathbb{D}^{n}\to\mathbb{D}$ and $\widetilde{H}:\mathbb{D}\to\mathbb{C}^{m}$ nonconstant holomorphic maps. Fix $x\in\mathbb{D}^{n}$ , set $\zeta_{0}:=h(x)$ , $y_{0}:=H(x)$ . Since $h$ is nonconstant, the open mapping theorem implies that $h(\mathbb{D}^{n})$ is an open subset of $\mathbb{D}$ . Hence we can choose an open disk $W_{0}\subset h(\mathbb{D}^{n})$ centered at $\zeta_{0}$ and set
$$
W:=W_{0}-\zeta_{0}=\{\zeta-\zeta_{0}:\zeta\in W_{0}\},
$$
so that $W$ is an open disk centered at $0$ . Define
$$
\varphi(\zeta):=\widetilde{H}(\zeta+\zeta_{0})-y_{0},\qquad\zeta\in W,
$$
so that $\varphi(0)=0$ . Write $\varphi=(\varphi_{1},\dots,\varphi_{m})$ and set
$$
k:=\min_{1\leq j\leq m}\operatorname{ord}_{0}\varphi_{j}<\infty.
$$
By permuting the coordinates in $\mathbb{C}^{m}$ , if necessary, and precomposing $\varphi$ with a biholomorphic change of variable on a smaller disk $W^{\prime}\subset W$ fixing $0$ (and then renaming $W^{\prime}$ by $W$ ), we may assume without loss of generality that
$$
\varphi_{1}(\zeta)=\zeta^{k},\qquad\varphi_{j}(\zeta)=\zeta^{k}\psi_{j}(\zeta),\quad j\geq 2,\ \ \zeta\in W,
$$
with each $\psi_{j}$ holomorphic near $0$ . Thus
$$
\varphi(\zeta)=\bigl(\zeta^{k},\,\zeta^{k}\psi_{2}(\zeta),\dots,\zeta^{k}\psi_{m}(\zeta)\bigr),\qquad\zeta\in W.
$$ Consider the projection onto the first coordinate,
$$
\pi_{1}:\mathbb{C}^{m}\to\mathbb{C},\qquad\pi_{1}(z_{1},\dots,z_{m})=z_{1}.
$$
Then for all $\zeta\in W$ ,
$$
\pi_{1}\bigl(\varphi(\zeta)\bigr)=\zeta^{k}.
$$
Hence for some $r>0$ ,
$$
\pi_{1}\circ\varphi:W\setminus\{0\}\longrightarrow\{w\in\mathbb{C}:0<|w|<r\}
$$
is a finite unbranched covering of degree $k$ . Consequently,
$$
\widetilde{H}(W_{0})=\varphi(W)+y_{0}
$$
is a one-dimensional complex analytic subset of a neighborhood of $y_{0}\in\mathbb{C}^{m}$ . Finally, let $U_{x}:=h^{-1}(W_{0})$ . Then $U_{x}$ is a neighborhood of $x$ and
$$
H(U_{x})=\widetilde{H}(h(U_{x}))=\widetilde{H}(W_{0}),
$$
establishing (1). â
Proof of Theorem 2.5. We must show that whenever $g_{1},\dots,g_{k}\in A(S_{n};H)$ , with $H$ as in the hypotheses of the theorem, satisfy the corona condition
$$
\inf_{z\in\mathbb{D}^{n}}\ \max_{1\leq i\leq k}|g_{i}(z)|\geq\delta>0,
$$
then for every $h_{1},\dots,h_{k}\in H^{\infty}(\mathbb{D}^{n})$ with $\max_{i}\lVert h_{i}\rVert_{\infty}<\delta$ , the ideal generated by
$$
F_{i}:=g_{i}+h_{i},\qquad i=1,\dots,k,
$$
is the whole algebra $H^{\infty}(\mathbb{D}^{n})$ .
Suppose, to the contrary, that the ideal generated by $F_{1},\dots,F_{k}$ is proper. Then there exists a maximal ideal $\chi\in M\bigl(H^{\infty}(\mathbb{D}^{n})\bigr)$ containing it; hence
$$
\widehat{F}_{i}(\chi)=0,\qquad i=1,\dots,k.
$$
Let
$$
\pi_{n}:M\bigl(H^{\infty}(\mathbb{D}^{n})\bigr)\longrightarrow\bigl[M\bigl(H^{\infty}(\mathbb{D})\bigr)\bigr]^{n}
$$
be the canonical projection introduced in (2.2), and set $\xi:=\pi_{n}(\chi)$ . The associated fiber is
$$
F_{\xi}:=\pi_{n}^{-1}(\xi)\cap\operatorname{cl}(\mathbb{D}^{n}).
$$
Let $A(F_{\xi})$ denote the uniform closure in $C(F_{\xi})$ of the restrictions of $H^{\infty}(\mathbb{D}^{n})$ to $F_{\xi}$ . By the first assertion of Theorem 2.2 we have
$$
\pi_{n}^{-1}(\xi)=M\bigl(A(F_{\xi})\bigr),
$$
so, in particular, $\chi\in M\bigl(A(F_{\xi})\bigr)$ . We therefore regard $\chi$ as a character of $A(F_{\xi})$ , and continue to denote it by $\chi$ .
By Lemma 6.1, there exist holomorphic maps $h:\mathbb{D}^{n}\to\mathbb{D}$ and $\widetilde{H}:\mathbb{D}\to\mathbb{C}^{m}$ such that $H=\widetilde{H}\circ h$ . Since $H$ is bounded on $\mathbb{D}^{n}$ , it follows that $\widetilde{H}$ is bounded on $\mathbb{D}$ .
The map $h$ induces a homomorphism
$$
h^{*}:H^{\infty}(\mathbb{D})\longrightarrow H^{\infty}(\mathbb{D}^{n}),\qquad\varphi\longmapsto\varphi\circ h,
$$
whose transpose determines a continuous map of maximal ideal spaces
$$
\hat{h}^{*}:M\bigl(H^{\infty}(\mathbb{D}^{n})\bigr)\longrightarrow M\bigl(H^{\infty}(\mathbb{D})\bigr).
$$
Set
$$
K:=\hat{h}^{*}(F_{\xi})\subset M\bigl(H^{\infty}(\mathbb{D})\bigr).
$$
Since the Gelfand transform of every function in $S_{n}(H^{\infty}(\mathbb{D}))$ is constant on the fibres of $\pi_{n}$ , the restriction of each $\hat{g}_{i}$ to $F_{\xi}$ is the uniform limit in $C(F_{\xi})$ of the restrictions to $F_{\xi}$ of Gelfand transforms of a sequence of holomorphic polynomials in the coordinate functions of $H$ , that is, polynomials in $H_{1},\dots,H_{m}$ . Using the factorization $H=\widetilde{H}\circ h$ , each such polynomial in $H_{1},\dots,H_{m}$ can be written as a polynomial in the coordinate functions of $\widetilde{H}$ composed with $h$ . Hence its Gelfand transform descends, via $\hat{h}^{*}$ , to the Gelfand transform of a holomorphic polynomial on $\mathbb{D}$ in the coordinates of $\widetilde{H}$ .
Therefore the restriction of each $\hat{g}_{i}$ to $F_{\xi}$ factors through $\hat{h}^{*}$ ; that is, there exist continuous functions $\widetilde{g}_{i}\in C(K)$ such that
$$
\hat{g}_{i}=\widetilde{g}_{i}\circ\hat{h}^{*}\quad\text{on }\ F_{\xi},\quad i=1,\dots,k.
$$
Moreover, each $\widetilde{g}_{i}$ is the uniform limit on $K$ of Gelfand transforms of functions from $H^{\infty}(\mathbb{D})$ .
Let $\mathcal{A}(K)$ denote the uniform closure on $K$ of the algebra of restrictions of continuous functions defined on open neighbourhoods $U$ of $K$ whose restrictions to $U\cap\mathbb{D}$ are holomorphic. By results of SuĂĄrez ([41, Th. 2.4 and Lem. 1.1] together with [39, Th. 2.4]), the maximal ideal space of $\mathcal{A}(K)$ is canonically identified with $K$ .
The restriction of $\hat{h}^{*}$ to $F_{\xi}$ induces a pullback homomorphism
$$
\Phi:\mathcal{A}(K)\longrightarrow A(F_{\xi}),\qquad f\longmapsto f\circ\hat{h}^{*}\big|_{F_{\xi}}.
$$
Indeed, if $U$ is an open neighbourhood of $K$ and $f$ is continuous on $U$ with $f|_{U\cap\mathbb{D}}$ holomorphic, then $(\hat{h}^{*})^{-1}(U)$ is an open neighbourhood of $F_{\xi}$ in $M\bigl(H^{\infty}(\mathbb{D}^{n})\bigr)$ , and $f\circ\hat{h}^{*}$ is continuous on this neighbourhood and holomorphic on its intersection with $\mathbb{D}^{n}$ (since $\hat{h}^{*}(z)=h(z)$ for $z\in\mathbb{D}^{n}$ ). Hence the restriction of $f\circ\hat{h}^{*}$ to $F_{\xi}$ lies in $A(F_{\xi})$ (cf. the definition in Section 2 after (2.3)). Since $\mathcal{A}(K)$ is the uniform closure of such functions $f$ on $K$ , it follows that $\Phi(\mathcal{A}(K))\subset A(F_{\xi})$ .
The transpose of $\Phi$ ,
$$
\Phi^{*}:M(A(F_{\xi}))\longrightarrow M(\mathcal{A}(K))\cong K,
$$
is therefore well defined and continuous. Moreover, $\Phi^{*}$ coincides with the restriction of $\hat{h}^{*}$ to $M(A(F_{\xi}))=\pi_{n}^{-1}(\xi)$ . Indeed, for any $f\in H^{\infty}(\mathbb{D})$ ,
$$
\Phi(\hat{f}|_{K})=\hat{f}\circ\hat{h}^{*}\big|_{\pi_{n}^{-1}(\xi)}=\widehat{f\circ h}\big|_{\pi_{n}^{-1}(\xi)}=\widehat{h^{*}(f)}\big|_{\pi_{n}^{-1}(\xi)},
$$
and evaluation at $\chi\in M(A(F_{\xi}))$ gives
$$
\Phi^{*}(\chi)(\hat{f}|_{K})=\chi\bigl(\widehat{h^{*}(f)}\bigr)=\hat{f}\bigl(\hat{h}^{*}(\chi)\bigr),
$$
which means precisely that $\Phi^{*}(\chi)=\hat{h}^{*}(\chi)$ . Thus we may identify $\Phi^{*}$ with the restriction of $\hat{h}^{*}$ , and we continue to denote it by
$$
\hat{h}^{*}:M(A(F_{\xi}))\longrightarrow K.
$$
In particular, for our chosen $\chi\in M(A(F_{\xi}))$ ,
$$
\eta:=\hat{h}^{*}(\chi)\in K,\qquad\hat{g}_{i}(\chi)=\widetilde{g}_{i}(\eta),\quad i=1,\dots,k.
$$
The corona condition now gives
$$
\max_{1\leq i\leq k}|\widetilde{g}_{i}(\eta)|=\max_{1\leq i\leq k}|\hat{g}_{i}(\chi)|\,\geq\,\delta.
$$
Hence for some index $i_{0}$ ,
$$
|\hat{g}_{i_{0}}(\chi)|\geq\delta.
$$
On the other hand,
$$
|\hat{h}_{i_{0}}(\chi)|\leq\|h_{i_{0}}\|_{\infty}<\delta.
$$
Therefore
$$
|\hat{F}_{i_{0}}(\chi)|=|\hat{g}_{i_{0}}(\chi)+\hat{h}_{i_{0}}(\chi)|\geq|\hat{g}_{i_{0}}(\chi)|-|\hat{h}_{i_{0}}(\chi)|>\delta-\delta=0,
$$
which contradicts $\hat{F}_{i_{0}}(\chi)=0$ .
Thus no maximal ideal of $H^{\infty}(\mathbb{D}^{n})$ can contain the ideal generated by $F_{1},\dots,F_{k}$ , and hence this ideal is all of $H^{\infty}(\mathbb{D}^{n})$ . The BĂ©zout equation for $F_{1},\dots,F_{k}$ therefore has a solution in $H^{\infty}(\mathbb{D}^{n})$ , completing the proof.
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