# Banach algebras of symmetric functions on the polydisc

Amol Sasane  Department of Mathematics
London School of Economics
Houghton Street
London WC2A 2AE
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

Let $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ and for an integer $d\geq 1$, let
$S_{d}$ denote the symmetric group, consisting of of all permutations of the
set $\\{1,\cdots,d\\}$. A function $f:\mathbb{D}^{d}\rightarrow\mathbb{C}$ is
symmetric if $f(z_{1},\cdots,z_{d})=f(z_{\sigma(1)},\cdots,z_{\sigma(d)})$ for
all $\sigma\in S_{d}$ and all $(z_{1},\cdots,z_{d})\in\mathbb{D}^{d}$. The
polydisc algebra $A(\mathbb{D}^{d})$ is the Banach algebra of all holomorphic
functions $f$ on the polydisc $\mathbb{D}^{d}$ that can be continuously
extended to the closure of the polydisc in $\mathbb{C}^{d}$, with pointwise
operations and the supremum norm (given by
$\|f\|_{\infty}:=\sup_{\mathbf{z}\in\mathbb{D}^{d}}|f(\mathbf{z})|$). Let
$A_{\textrm{sym}}(\mathbb{D}^{d})$ be the Banach subalgebra of
$A(\mathbb{D}^{d})$ consisting of all symmetric functions in the polydisc
algebra. Algebraic-analytic properties of $A_{\textrm{sym}}(\mathbb{D}^{d})$
are investigated. In particular, the following results are shown: the corona
theorem, description of the maximal ideal space and its contractibility,
Hermiteness, projective-freeness, and non-coherence.

###### Key words and phrases:

symmetric polynomials, polydisc algebra, Wiener algebra, Banach algebras,
maximal ideal space, Hermite ring, projective free ring, coherent ring, corona
theorem, Gelfand topology

###### 2010 Mathematics Subject Classification:

Primary 32A38; Secondary 46J15, 32A65, 46J20, 54C40

##  1\. Introduction

Symmetric polynomials and the ring of symmetric functions are interesting
objects in algebra. We study an analogous object from the Banach algebra
viewpoint. The aim of this article is to study the algebraic-analytic
properties of Banach algebras of ‘symmetric functions’ (defined below) on the
polydisc.

######  Definition 1.1.

$\bullet$ Let $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$. Let
$\overline{\mathbb{D}}$ denote the closure of $\mathbb{D}$ in $\mathbb{C}$.

$\bullet$ Let $\mathbb{N}:=\\{1,2,3,\cdots\\}$. For $d\in\mathbb{N}$, let
$A(\mathbb{D}^{d})$ denote the polydisc $\bullet$ algebra of all functions
$f:\overline{\mathbb{D}}^{d}\rightarrow\mathbb{C}$ that are holomorphic in
$\mathbb{D}^{d}$ and $\bullet$ continuous on $\overline{\mathbb{D}}^{d}$.
Define
$\displaystyle\|f\|_{\infty}:=\sup_{\mathbf{z}\in\mathbb{D}^{d}}|f(\mathbf{z})|$
for $f\in A(\mathbb{D}^{d}).$

$\bullet$ Then $A(\mathbb{D}^{d})$ is a Banach algebra with pointwise
operations and the $\bullet$ supremum norm $\|\cdot\|_{\infty}$.

$\bullet$ Let $d\in\mathbb{N}$. Then $S_{d}$ denotes the symmetric group,
consisting of all $\bullet$ bijective maps
$\sigma:\\{1,\cdots,d\\}\rightarrow\\{1,\cdots,d\\}$, with composition $\circ$
taken $\bullet$ as the group operation. The group $S_{d}$ acts on
$\mathbb{C}^{d}$ as follows.

$\bullet$ If $\mathbf{z}=(z_{1},\cdots,z_{d})\in\mathbb{C}^{d}$ and $\sigma\in
S_{d}$, then we define $\sigma\;\\!\mathbf{z}\in\mathbb{C}^{d}$ by

$\sigma\;\\!\mathbf{z}:=(z_{{\scaleobj{0.75}{\sigma(1)}}},\cdots,z_{{\scaleobj{0.75}{\sigma(d)}}}).$

$\bullet$ A function $f\in A(\mathbb{D}^{d})$ is symmetric if for all
$\mathbf{z}\in\mathbb{D}^{d}$, and all $\sigma\in S_{d}$, $\bullet$ we have
$f(\mathbf{z})=f(\sigma\;\\!\mathbf{z})=:(\sigma f)(\mathbf{z})$.

$\bullet$ Define the polydisc algebra of symmetric functions by

$A_{\textrm{sym}}(\mathbb{D}^{d}):=\\{f\in A(\mathbb{D}^{d}):f\textrm{ is
symmetric}\\}.$

Then $A_{\textrm{sym}}(\mathbb{D}^{d})$ is a Banach subalgebra of
$A(\mathbb{D}^{d})$ with the same operations and the same norm. Closure under
addition, scalar multiplication, and multiplication is clear. Also, if
$(f_{n})_{n\in\mathbb{N}}$ is a Cauchy sequence in
$A_{\textrm{sym}}(\mathbb{D}^{d})$, then being Cauchy in $A(\mathbb{D}^{d})$
too, it converges to some $f\in A(\mathbb{D})$, and as uniform convergence
implies pointwise convergence,

$f(\sigma\;\\!\mathbf{z})=\lim_{n\rightarrow\infty}f_{n}(\sigma\;\\!\mathbf{z})=\lim_{n\rightarrow\infty}f_{n}(\mathbf{z})=f(\mathbf{z})\quad(\mathbf{z}\in\mathbb{D}^{d}).$

In this article, we focus on the algebraic-analytic properties of the
symmetric polydisc algebra as a concrete prototypical example. However, more
generally, for a ‘symmetric’ domain $\Omega\subset\mathbb{C}^{d}$, one could
also study the properties of appropriately defined ‘symmetric function
algebras’ such as $A_{\textrm{sym}}(\Omega)$,
$H^{\infty}_{\textrm{sym}}(\Omega)$ (symmetric subalgebra of the Hardy algebra
$H^{\infty}(\Omega)$ of bounded and holomorphic functions on $\Omega$), etc.

##  2\. Corona problem

Let $\mathbf{1}\in A_{\textrm{sym}}(\mathbb{D}^{d})$ be the constant function
taking value $1$ everywhere.

######  Theorem 2.1.

Let $n\in\mathbb{N}$ and $f_{1},\cdots,f_{n}\in A_{\textrm{\em
sym}}(\mathbb{D}^{d})$.

Then the following are equivalent:

(1) There exist $g_{1},\cdots,g_{n}\in A_{\textrm{\em sym}}(\mathbb{D}^{d})$
such that $f_{1}g_{1}+\cdots+f_{n}g_{n}=\mathbf{1}$.

(2) There exists a $\delta>0$ such that $\forall\mathbf{z}\in\mathbb{D}^{d}$,
$|f_{1}(\mathbf{z})|+\cdots+|f_{n}(\mathbf{z})|\geq\delta$.

###### Proof.

(1)$\Rightarrow$(2) follows from the triangle inequality: For all
$\mathbf{z}\in\mathbb{D}^{d}$,

$\displaystyle 0<1=|\mathbf{1}(\mathbf{z})|\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!|f_{1}(\mathbf{z})g_{1}(\mathbf{z})+\cdots+f_{n}(\mathbf{z})g_{n}(\mathbf{z})|$
$\displaystyle\leq$
$\displaystyle\\!\\!(|f_{1}(\mathbf{z})|+\cdots+|f_{n}(\mathbf{z})|)\max_{1\leq
i\leq n}\|g_{i}\|_{\infty},$

and so we may take $\delta:=\displaystyle\Big{(}\max_{1\leq i\leq
n}\|g_{i}\|_{\infty}\Big{)}^{-1}>0$.

The converse (2)$\Rightarrow$(1) follows from the corona theorem for
$A(\mathbb{D}^{d})$, and by ‘symmetrising’ the solution as follows. The corona
condition (2) gives the existence of $g_{1},\cdots,g_{n}\in A(\mathbb{D}^{n})$
such that $f_{1}g_{1}+\cdots+f_{n}g_{n}=\mathbf{1}.$ For any permutation
$\sigma\in S_{d}$, we have for all
$\mathbf{z}=(z_{1},\cdots,z_{d})\in\mathbb{D}^{d}$ that

$\displaystyle 1\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!\sum_{1\leq
i\leq
n}f_{i}(z_{{\scaleobj{0.75}{\sigma(1)}}},\cdots,z_{{\scaleobj{0.75}{\sigma(d)}}})\;\\!g_{i}(z_{{\scaleobj{0.75}{\sigma(1)}}},\cdots,z_{{\scaleobj{0.75}{\sigma(d)}}})$
$\displaystyle=$ $\displaystyle\\!\\!\\!\sum_{1\leq i\leq
n}f_{i}(z_{{\scaleobj{0.75}{1}}},\cdots,z_{{\scaleobj{0.75}{d}}})\;\\!g_{i}(z_{{\scaleobj{0.75}{\sigma(1)}}},\cdots,z_{{\scaleobj{0.75}{\sigma(d)}}}).$

Summing the above over all $\sigma\in S_{d}$, we obtain
$f_{1}\widetilde{g}_{1}+\cdots+f_{n}\widetilde{g}_{n}=\mathbf{1},$ where
$\widetilde{g}_{1},\cdots,\widetilde{g}_{n}\in
A_{\textrm{sym}}(\mathbb{D}^{d})$ are defined by

$\phantom{AAa}\widetilde{g}_{i}(\mathbf{z}):=\frac{1}{d!}\sum_{\sigma\in
S_{d}}g_{i}(z_{{\scaleobj{0.75}{\sigma(1)}}},\cdots,z_{{\scaleobj{0.75}{\sigma(d)}}}),\quad\mathbf{z}=(z_{1},\cdots,z_{d})\in\mathbb{D}^{d}.\phantom{AAa}\qed$

##  3\. Maximal ideal space of $A_{\mathrm{sym}}(\mathbb{D}^{d})$

Let $\sim$ be the relation on $\overline{\mathbb{D}}^{d}$ defined as follows:
$\mathbf{z}\sim\\!\mathbf{w}$ if there exists a permutation $\sigma\in S_{d}$
such that

$\sigma\;\\!\mathbf{z}=\mathbf{w}.$

Then $\sim$ is an equivalence relation on $\overline{\mathbb{D}}^{d}$. For
$\mathbf{z}\in\overline{\mathbb{D}}^{d}$, the orbit of $\mathbf{z}$ is

$S_{d}\;\\!\mathbf{z}:=\\{\sigma\;\\!\mathbf{z}:\sigma\in S_{d}\\},\quad$

The equivalence class of $\mathbf{z}\in\overline{\mathbb{D}}^{d}$ is denoted
by

$[\mathbf{z}]=S_{d}\;\\!\mathbf{z}.$

Let $\overline{\mathbb{D}}^{d}/\\!\sim$ denote the set of all equivalence
classes of $\overline{\mathbb{D}}^{d}$ under the equivalence relation $\sim$.
The natural projection map

$\begin{array}[]{rccc}\pi:&\overline{\mathbb{D}}^{d}&\rightarrow&\overline{\mathbb{D}}^{d}/\\!\sim\\\\[2.84544pt]
&\mathbf{z}&\mapsto&[\mathbf{z}]\end{array}$

is surjective. We use the subspace topology on $\overline{\mathbb{D}}^{d}$
induced from the usual Euclidean topology on $\mathbb{C}^{d}$. We endow the
set $\overline{\mathbb{D}}^{d}/\\!\sim$ with the quotient topology
$\tau_{\textrm{quot}}$, $\overline{\mathbb{D}}^{d}$that is, the
finest/strongest topology making the above projection map
$\pi:\overline{\mathbb{D}}^{d}\rightarrow\overline{\mathbb{D}}^{d}/\\!\sim$
continuous. As the topological space $\overline{\mathbb{D}}^{d}$ is compact,
it follows that the quotient space
$(\overline{\mathbb{D}}^{d}/\\!\sim,\tau_{\textrm{quot}})$ is compact too. For
any set $X\subset\overline{\mathbb{D}}^{d}$ and any $\sigma\in S_{d}$, define

$\sigma X:=\\{\sigma\;\\!\bm{\xi}:\bm{\xi}\in X\\}.$

######  Proposition 3.1.

$(\overline{\mathbb{D}}^{d}/\\!\sim,\tau_{\textrm{\em quot}})$ is Hausdorff.

###### Proof.

Let $[\mathbf{z}],[\mathbf{w}]\in\overline{\mathbb{D}}^{d}/\\!\sim$ (where
$\mathbf{z},\mathbf{w}\in\overline{\mathbb{D}}^{d}$) be such that
$[\mathbf{z}]\neq[\mathbf{w}]$. Then
$(S_{d}\;\\!\mathbf{z})\cap(S_{d}\;\\!\mathbf{w})=\emptyset$.
$\overline{\mathbb{D}}^{d}$ But $S_{d}\;\\!\mathbf{w}$ is a finite set
(with $d!$ elements). As $\overline{\mathbb{D}}^{d}$ with its usual Euclidean
topology is Hausdorff, it follows that there exist open sets $Z,W$ in
$\overline{\mathbb{D}}^{d}$ such that $S_{d}\;\\!\mathbf{z}\subset Z$,
$S_{d}\;\\!\mathbf{w}\subset W$, and $Z\cap W=\emptyset$. For any $\sigma\in
S_{d}$, the map
$\bm{\zeta}\mapsto\sigma\bm{\zeta}:\overline{\mathbb{D}}^{d}\rightarrow\overline{\mathbb{D}}^{d}$
is a homeomorphism, and so $\sigma Z=\\{\sigma\bm{\zeta}:\bm{\zeta}\in Z\\}$
is open in $\overline{\mathbb{D}}^{d}$. Define

$U=\bigcap_{\sigma\in S_{d}}\sigma Z\;\;\textrm{ and
}\;\;V:=\bigcap_{\sigma\in S_{d}}\sigma W.$

Then $U,V$ are open in $\overline{\mathbb{D}}^{d}$. Moreover, for all
$\sigma\in S_{d}$, $\sigma\;\\!U=U$ and $\sigma V=V$. For any $\sigma\in
S_{d}$, we have $\sigma^{-1}\mathbf{z}\in Z$, and so $\mathbf{z}\in\sigma Z$,
showing that $\mathbf{z}\in U$. Similarly, $\mathbf{w}\in V$. Define the sets

$\displaystyle\mathcal{U}\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!\\{[\bm{\zeta}]:\bm{\zeta}\in
U\\}\;\;\;\\!(\owns[\mathbf{z}]),\textrm{ and}$
$\displaystyle\mathcal{V}\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!\\{[\bm{\eta}]:\bm{\eta}\in
V\\}\;\;(\owns[\mathbf{w}]).$

We claim that $\pi^{-1}\mathcal{U}=U$. Firstly, if $\bm{\zeta}\in U$, then
$[\bm{\zeta}]\in\mathcal{U}$, that is, $\pi\bm{\zeta}\in\mathcal{U}$, and so
$\bm{\zeta}\in\pi^{-1}\mathcal{U}$, showing that
$U\subset\pi^{-1}\mathcal{U}$. Secondly, if
$\bm{\zeta}\in\pi^{-1}\mathcal{U}$, then $\pi\;\\!\bm{\zeta}\in\mathcal{U}$,
so that $[\bm{\zeta}]=\pi\;\\!\bm{\zeta}=[\bm{\eta}]$ for some $\bm{\eta}\in
U$, giving, for some $\sigma\in S_{d}$, that
$\bm{\zeta}=\sigma\bm{\eta}\in\sigma U=U$, showing that
$\pi^{-1}\mathcal{U}\subset U$ too.

Since $\pi^{-1}\mathcal{U}=U$ and $\pi^{-1}\mathcal{V}=V$ are open in
$\overline{\mathbb{D}}^{d}$, we conclude that the sets
$\mathcal{U},\mathcal{V}$ are open in
$(\overline{\mathbb{D}}^{d}/\\!\sim,\tau_{\textrm{quot}})$.

Finally, we show that $\mathcal{U}\cap\mathcal{V}=\emptyset$. Suppose not.
Then there exist $\bm{\zeta}\in U$ and $\bm{\eta}\in V$ such that
$[\bm{\zeta}]=[\bm{\eta}]$, that is, $\bm{\zeta}=\sigma\;\\!\bm{\eta}$ for
some $\sigma\in S_{d}$. But $\bm{\eta}\in U\subset Z$, and
$\sigma\;\\!\bm{\eta}\in\sigma V=V\subset W$, so that
$\bm{\zeta}=\sigma\;\\!\bm{\eta}\in Z\cap W=\emptyset$, a contradiction. ∎

The maximal ideal space of a complex unital commutative Banach algebra
$\mathcal{A}$ is denoted by $M_{\mathcal{A}}$, and it is the set of all
nontrivial complex homomorphisms $\varphi:\mathcal{A}\rightarrow\mathbb{C}$.
Let $\mathcal{L}(\mathcal{A};\mathbb{C})$ be dual space of $\mathcal{A}$, that
is, the set of all continuous linear maps
$\varphi:\mathcal{A}\rightarrow\mathbb{C}$. We equip
$\mathcal{L}(\mathcal{A};\mathbb{C})$ with the weak-$\ast$ topology (see e.g.
[12, §3.14]). Clearly
$M_{\mathcal{A}}\subset\mathcal{L}(\mathcal{A};\mathbb{C})$. The Gelfand
topology $\tau_{{\scaleobj{0.75}{\textrm{Gel}}}}$ on $M_{\mathcal{A}}$ is the
topology induced on $M_{\mathcal{A}}$ from the weak-$\ast$ topology of
$\mathcal{L}(\mathcal{A};\mathbb{C})$. As the Gelfand topology need not be
metrisable, we use the language of nets when discussing limits and convergence
in the proof of Theorem 3.4111But after proving this result, we see that the
Gelfand topology of $M_{\scaleobj{0.81}{A_{\textrm{sym}}(\mathbb{D}^{d})}}$ is
the quotient topology of
${\scaleobj{0.9}{\overline{\mathbb{D}}^{d}/\\!\sim}}$, and the latter can be
seen to be metrisable (since the space
${\scaleobj{0.9}{\overline{\mathbb{D}}^{d}}}$ is compact and metrisable, and
the quotient space ${\scaleobj{0.9}{\overline{\mathbb{D}}^{d}/\\!\sim}}$ is
Hausdorff; see [15, §23, 23K]).$\overline{\mathbb{D}}^{d}/\\!\sim$. In
particular, we will use the notion of subnets in the sense of Willard [15,
§11] (see also [10, §1.3.B]). If $(x_{i})_{i\in I}$ (where $I$ is a directed
set) is a net in a topological space $X$, then a subnet of $(x_{i})_{i\in I}$
is a net $(x_{h(j)})_{j\in J}$, where $J$ is a directed set, and
$h:J\rightarrow I$ is an increasing cofinal function, that is, it satisfies

$\bullet$ (increasing) $h(j_{1})\leq h(j_{2})$ whenever $j_{1}\leq j_{2}$, and

$\bullet$ (cofinal) for each $i\in I$, there is some $j\in J$ such that $i\leq
h(j)$.

This notion of a subnet affords us the following facts, which we shall need
later (see [15, §11, §17]):

$\bullet$ A subnet of a net convergent to $x$ also converges to $x$.

$\bullet$ A net in a compact set $K\subset X$ possesses a subnet which
converges

$\bullet$ to a point in $K$.

We will now show that
$(M_{A_{\textrm{sym}}(\mathbb{D}^{d})},\tau_{{\scaleobj{0.75}{\textrm{Gel}}}})$
can be identified as a topological space with
$(\overline{\mathbb{D}}^{d}/\\!\sim,\tau_{\textrm{quot}})$. To do this, we
will use the following:

######  Lemma 3.2.

Suppose that $\varphi,\psi\in M_{A_{\textrm{\em sym}}(\mathbb{D}^{d})}$.

Then $\varphi=\psi$ if and only if $\ker\varphi=\ker\psi$.

###### Proof.

The ‘only if’ part is trivial. ‘If’ part: Let $\ker\varphi=\ker\psi$. Suppose
$f\in A_{\textrm{sym}}(\mathbb{D}^{d})$ is such that $\varphi(f)\neq\psi(f)$.
Set $g:=f-\psi(f)\mathbf{1}\in A_{\textrm{sym}}(\mathbb{D}^{d})$. Clearly
$g\in\ker\psi$. As $\ker\varphi=\ker\psi$, $\varphi(g)=0$ too, that is,
$\varphi(f)=\psi(f)$, a contradiction. ∎

######  Lemma 3.3.

If $\tau_{1},\tau_{2}$ are topologies on a set $X$ such that

$\bullet$ $\tau_{1}\subset\tau_{2}$,

$\bullet$ $\tau_{1}$ is a Hausdorff topology, and

$\bullet$ $\tau_{2}$ is compact,

then $\tau_{1}=\tau_{2}$.

###### Proof.

This is precisely [10, Theorem 1.53] or [12, §3.8(a)]. ∎

######  Theorem 3.4.

The maximal ideal space of $A_{\textrm{\em sym}}(\mathbb{D}^{d})$ is
homeomorphic to $\overline{\mathbb{D}}^{d}/\\!\sim$.

###### Proof.

The map $\iota:\overline{\mathbb{D}}^{d}/\\!\sim\;\\!\rightarrow
M_{A_{\textrm{sym}}(\mathbb{D}^{d})}$:

For $[\mathbf{z}]\in\overline{\mathbb{D}}^{d}/\\!\sim$ (where
$\mathbf{z}\in\overline{\mathbb{D}}^{d}$), we define
$\varphi_{[\mathbf{z}]}:A_{\textrm{sym}}(\mathbb{D}^{d})\rightarrow\mathbb{C}$
to be the point evaluation at $\mathbf{z}$, that is,

$\varphi_{[\mathbf{z}]}(f)=f(\mathbf{z}),\quad f\in
A_{\textrm{sym}}(\mathbb{D}^{d}).$

Then $\varphi_{[\mathbf{z}]}$ is well-defined (since
$A_{\textrm{sym}}(\mathbb{D}^{d})$ consists of symmetric functions), and
defines a complex homomorphism. Also, $\varphi_{[\mathbf{z}]}(\mathbf{1})=1$.
We show below that the map

$\begin{array}[]{rccc}\iota:&\overline{\mathbb{D}}^{d}/\\!\sim&\rightarrow&M_{A_{\textrm{sym}}(\mathbb{D}^{d})}\\\\[2.84544pt]
&[\mathbf{z}]&\mapsto&\varphi_{[\mathbf{z}]}\end{array}$

is a homeomorphism.

The map $\iota$ is injective: Suppose that

$[\mathbf{z}]\neq[\mathbf{w}]$

for some $\mathbf{z},\mathbf{w}\in\overline{\mathbb{D}}^{d}$. Then for all
$\sigma\in S_{d}$, we have $\sigma\;\\!\mathbf{z}\neq\mathbf{w}$. Thus the
polynomials $p_{\mathbf{z}},p_{\mathbf{w}}\in\mathbb{C}[X]$ defined by

$\displaystyle p_{\mathbf{z}}(X)\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!(X-z_{1})\cdots(X-z_{d}),$ $\displaystyle
p_{\mathbf{w}}(X)\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!(X-w_{1})\cdots(X-w_{d}),$

are distinct. As the coefficients of a polynomial are symmetric polynomials of
the zeroes of the polynomial, there exists a symmetric polynomial $f$ such
that $f(\mathbf{z})\neq f(\mathbf{w})$, that is,

$\varphi_{[\mathbf{z}]}(f)\neq\varphi_{[\mathbf{w}]}(f).$

The map $\iota$ is surjective: Let $\varphi\in
M_{A_{\textrm{sym}}(\mathbb{D}^{d})}$ be such that

$\textrm{ for all
}\mathbf{z}\in\overline{\mathbb{D}}^{d},\;\varphi\neq\varphi_{[\mathbf{z}]}.$

Let $\mathfrak{m}$ be the maximal ideal in $A_{\textrm{sym}}(\mathbb{D}^{d})$
given by $\mathfrak{m}:=\ker\varphi$.

By Lemma 3.2,

$(\mathfrak{m}=)\;\ker\varphi\neq\ker\varphi_{[\mathbf{z}]}\textrm{ for all
}\mathbf{z}\in\overline{\mathbb{D}}^{d}.$

Thus for each $\mathbf{z}\in\overline{\mathbb{D}}^{d}$, there exists an
$f_{\mathbf{z}}\in A_{\textrm{sym}}(\mathbb{D}^{d})$ such that

$f_{\mathbf{z}}\in\mathfrak{m},\textrm{ but }f_{\mathbf{z}}(\mathbf{z})\neq
0.$

By the compactness of $\overline{\mathbb{D}}^{d}$, there is an
$n\in\mathbb{N}$ and $f_{1},\cdots,f_{n}\in A_{\textrm{sym}}(\mathbb{D}^{d})$
such that

$\forall\mathbf{z}\in\overline{\mathbb{D}}^{d},\;\exists
k\in\\{1,\cdots,n\\}\textrm{ such that }f_{k}(\mathbf{z})\neq 0.$

Hence there exists a $\delta>0$ such that for all
$\mathbf{z}\in\mathbb{D}^{d}$,

$|f_{1}(\mathbf{z})|+\cdots+|f_{n}(\mathbf{z})|\geq\delta.$

By the corona theorem for $A_{\textrm{sym}}(\mathbb{D}^{d})$ (Theorem 2.1
above), there exist $g_{1},\cdots,g_{n}\in A_{\textrm{sym}}(\mathbb{D}^{d})$
such that $f_{1}g_{1}+\cdots+f_{n}g_{n}=\mathbf{1}.$ But as the element
$f_{1}g_{1}+\cdots+f_{n}g_{n}$ belongs to the ideal $\mathfrak{m}$, we get
$\mathbf{1}\in\mathfrak{m}$, contradicting the maximality of $\mathfrak{m}$.

The map $\iota$ is a homeomorphism:

Via the bijection
$\iota\\!:\\!\overline{\mathbb{D}}^{d}\\!/\\!\\!\sim\;\\!\rightarrow\\!M_{A_{\textrm{sym}}(\mathbb{D}^{d})}$,
we identify $M_{A_{\textrm{sym}}(\mathbb{D}^{d})}$ and
$\overline{\mathbb{D}}^{d}\\!/\\!\sim$.

Then showing $\iota$ is a homeomorphism is tantamount to showing that the
topologies $\tau_{\textrm{quot}}$ and $\tau_{{\scaleobj{0.75}{\textrm{Gel}}}}$
coincide, in view of the above identification. To show the coincidence of
these two topologies, will use Lemma 3.3, with
$\tau_{1}=\tau_{{\scaleobj{0.75}{\textrm{Gel}}}}$ and
$\tau_{2}=\tau_{\textrm{quot}}$. We want to show that
$\tau_{{\scaleobj{0.75}{\textrm{Gel}}}}\subset\tau_{\textrm{quot}}$. It is
enough to show that if $F$ is closed in
$\tau_{{\scaleobj{0.75}{\textrm{Gel}}}}$, then it is closed in
$\tau_{\textrm{quot}}$. Suppose that $F$ is closed in
$\tau_{{\scaleobj{0.75}{\textrm{Gel}}}}$. Let $([\mathbf{z}_{i}])_{i\in I}$
(where $I$ is a directed set) be a net in $F$ that is convergent with limit
$[\mathbf{z}]$ in $(\overline{\mathbb{D}}^{d}/\\!\sim,\tau_{\textrm{quot}})$.
As $(\mathbf{z}_{i})_{i\in I}$ is a net in the compact set
$\overline{\mathbb{D}}^{d}$, it has a convergent subnet
$(\mathbf{z}_{h(j)})_{j\in J}$ (where $J$ is a directed set and
$h:J\rightarrow I$ $\overline{\mathbb{D}}^{d}_{h(j)})_{j\in
J}$ is a monotone final function), convergent with a
limit, say $\mathbf{w}$ in $\overline{\mathbb{D}}^{d}$ (in the Euclidean
topology). Then the net $([\mathbf{z}_{h(i)}])_{j\in J}$ converges to
$[\mathbf{w}]$ in $(\overline{\mathbb{D}}^{d}/\\!\sim,\tau_{\textrm{quot}})$.
But as $([\mathbf{z}_{h(i)}])_{j\in J}$ is a subnet of the convergent net
$([\mathbf{z}_{i}])_{i\in I}$ with limit $[\mathbf{z}]$ in
$(\overline{\mathbb{D}}^{d}/\\!\sim,\tau_{\textrm{quot}})$, and since
$(\overline{\mathbb{D}}^{d}/\\!\sim,\tau_{\textrm{quot}})$ is Hausdorff
(Proposition 3.1), we get

$[\mathbf{z}]=[\mathbf{w}].$

As $(\mathbf{z}_{h(j)})_{j\in J}$ converges to $\mathbf{w}$ in
$\overline{\mathbb{D}}^{d}$, we have, $\overline{\mathbb{D}}^{d}_{h(j)})_{j\in
J}$ for all $f\in A_{\textrm{sym}}(\mathbb{D}^{d})$,
that $(f(\mathbf{z}_{h(j)}))_{j\in J}$ converges to $f(\mathbf{w})$.
$\overline{\mathbb{D}}^{d}_{h(j)})_{j\in J}$ Hence
$(\varphi_{[\mathbf{z}_{h(j)}]}(f))_{j\in J}$ converges to
$\varphi_{[\mathbf{w}]}f$ for all $f\in A_{\textrm{sym}}(\mathbb{D}^{d})$.
$\overline{\mathbb{D}}^{d}_{h(j)})_{j\in J}$ Thus
$(\varphi_{[\mathbf{z}_{h(j)}]})_{j\in J}$ in $F$ converges to
$\varphi_{[\mathbf{w}]}$ in
$(M_{A_{\textrm{sym}}(\mathbb{D}^{d})},\tau_{\textrm{Gel}})$.
$\overline{\mathbb{D}}^{d}_{h(j)})_{j\in J}$ But as $F$
was closed in $\tau_{{\scaleobj{0.75}{\textrm{Gel}}}}$, we have
$[\mathbf{w}]\in F$. As $[\mathbf{z}]=[\mathbf{w}]$, we obtain
$[\mathbf{z}]\in F$. $\overline{\mathbb{D}}^{d}_{h(j)})_{j\in
J}$ Consequently, $F$ is closed in
$\tau_{\textrm{quot}}$ too. By Lemma 3.3, it follows that $\iota$ is a
homeomorphism. $\overline{\mathbb{D}}^{d}_{h(j)})_{j\in
J}$ ∎

##  4\. Contractibility of $M_{A_{\mathrm{sym}}(\mathbb{D}^{d})}$ and its
consequences

Recall that a topological space $X$ is contractible if the identity map is
null-homotopic, that is, there exists an $x_{0}\in X$ and a continuous map
$H:[0,1]\times X\rightarrow X$ such that

$\;\;\bullet$ $H(0,\cdot)=\textrm{id}_{X}$ (the identity map
$\textrm{id}:X\rightarrow X$, $X\owns x\mapsto x$) and

$\;\;\bullet$ $H(1,x)=x_{0}$ for all $x\in X$ (the constant map $X\owns
x\mapsto x_{0}$).

Here $[0,1]\times X$ is given the product topology.

######  Theorem 4.1.

$(M_{A_{\mathrm{sym}}(\mathbb{D}^{d})},\tau_{{\scaleobj{0.75}{\mathrm{Gel}}}})$
is contractible.

###### Proof.

It is enough to show that
$(\overline{\mathbb{D}}^{d}/\\!\sim,\tau_{\textrm{quot}})$ is contractible. We
show that the identity map on $\overline{\mathbb{D}}^{d}/\\!\sim$ is homotopic
to the constant map taking value $[\mathbf{0}]$ everywhere. Define
$H:[0,1]\times(\overline{\mathbb{D}}^{d}/\\!\sim)\rightarrow\overline{\mathbb{D}}^{d}/\\!\sim$
by

$H(t,[\mathbf{z}])=[(1-t)\mathbf{z}],\quad\mathbf{z}\in\overline{\mathbb{D}}^{d}.$

The map $H$ is well-defined. Indeed, if $[\mathbf{z}]=[\mathbf{w}]$ for some
$\mathbf{z},\mathbf{w}\in\overline{\mathbb{D}}^{d}$, then
$\mathbf{z}\sim\mathbf{w}$, that is, there exists a $\sigma\in S_{d}$ such
that $\sigma\;\\!\mathbf{z}=\mathbf{w}$, and so for $t\in[0,1]$,

$\sigma((1-t)\mathbf{z})=(1-t)\mathbf{w},$

showing that $(1-t)\mathbf{z}\sim((1-t)\mathbf{w})$, that is,
$[(1-t)\mathbf{z}]=[(1-t)\mathbf{w}]$.

Clearly, $H(0,[\mathbf{z}])=[\mathbf{z}]$ for all
$\mathbf{z}\in\overline{\mathbb{D}}^{d}$, and so

$H(0,\cdot)=\textrm{id}_{\overline{\mathbb{D}}^{d}/\\!\sim}.$

Moreover, $H(1,[\mathbf{z}])=[\mathbf{0}]$ for all
$\mathbf{z}\in\overline{\mathbb{D}}^{d}$, and so $H(1,\cdot)$ is the constant
map on $\overline{\mathbb{D}}^{d}/\\!\sim$, taking value $[\mathbf{0}]$
everywhere on $\overline{\mathbb{D}}^{d}/\\!\sim$.

Finally, we show the continuity of $H$. Let
$\mathbf{z}\in\overline{\mathbb{D}}^{d}$, $t\in[0,1]$, and $\mathcal{V}$ be a
neighbourhood of $[(1-t)\mathbf{z}]$. Set

$Z:=\pi^{-1}\mathcal{V}\subset\overline{\mathbb{D}}^{d}.$

Then $Z$ is an open set containing the finite set
$\\{(1-t)\sigma\mathbf{z}:\sigma\in S_{d}\\}$.

As the map

$\begin{array}[]{rccc}\Phi:&[0,1]\times\overline{\mathbb{D}}^{d}&\rightarrow&\overline{\mathbb{D}}^{d}\\\\[2.84544pt]
&\;\;(t,\bm{\zeta})&\mapsto&(1-t)\bm{\zeta}\end{array}$

is continuous, for each $\sigma\in S_{d}$, there exists an open neighbourhood
$I_{\sigma}$ of $t$ in $[0,1]$, and an open neighbourhood $W_{\sigma}$ of
$\mathbf{z}$ in $\overline{\mathbb{D}}^{d}$ such that

$\Phi(I_{\sigma}\times W_{\sigma})\subset Z.$

Set

$W:=\bigcup_{\sigma\in S_{d}}W_{\sigma},\;\;\textrm{ and
}\;\;I:=\bigcap_{\sigma\in S_{d}}I_{\sigma}.$

Then

$\Phi(I\times W)\subset Z.$

Let

$U=\bigcap_{\sigma\in S_{d}}\sigma W.$

Then $U$ is open in $\overline{\mathbb{D}}^{d}$, and for all $\sigma\in
S_{d}$, we have

$\sigma\;\\!U=U.$

Using the fact that for all $\sigma\in S_{d}$ we have $\sigma\mathbf{z}\in W$,
it can be seen that $\mathbf{z}\in U$. Define

$\mathcal{U}:=\pi U:=\\{[\bm{\zeta}]:\bm{\zeta}\in U\\}.$

Then

$\pi^{-1}\mathcal{U}=U.$

So $\mathcal{U}$ is open in $\overline{\mathbb{D}}^{d}/\\!\sim$. As
$\mathbf{z}\in U$, we have $[\mathbf{z}]\in\mathcal{U}$. Hence
$I\times\mathcal{U}$ is an open set in
$[0,1]\times(\overline{\mathbb{D}}^{d}/\\!\sim)$ containing $(t,[\mathbf{z}])$
such that $H(I\times\mathcal{U})\subset\mathcal{V}$. Consequently, $H$ is
continuous. $\overline{\mathbb{D}}^{d}$ ∎

###  4.1. Applications

A commutative unital ring $R$ is projective-free if every finitely generated
projective $R$-module is free. Recall that if $M$ is a finitely generated
$R$-module, then

$\;\;\bullet$ $M$ is called free if $M\simeq R^{k}$ for some integer $k\geq
0$;

$\;\;\bullet$ $M$ is called projective if there exists an $R$-module $N$ and
an integer $m\geq 0$ such that $M\oplus N=R^{m}$.

For $m,n\in\mathbb{N}$, $R^{m\times n}$ denotes the set of matrices with $m$
rows and $n$ columns having entries from $R$. The identity element in
$R^{k\times k}$ having diagonal elements $1$, and zeroes elsewhere will be
denoted by $I_{k}$. In terms of matrices (see [4, Proposition 2.6] or [2,
Lemma 2.2]), the ring $R$ is projective-free if and only if every idempotent
matrix $P$ is conjugate (by an invertible matrix $S$) to a diagonal matrix
with elements $1$ and $0$ on the diagonal, that is, for every $m\in\mathbb{N}$
and every $P\in R^{m\times m}$ satisfying $P^{2}=P$, there exists an $S\in
R^{m\times m}$ such that $S$ is invertible as an element of $R^{m\times m}$,
and for some $k\geq 0$,

$S^{-1}PS=\left[\begin{array}[]{cc}I_{k}&0\\\ 0&0\end{array}\right].$

In 1976, it was shown independently by Quillen and Suslin, that if
$\mathbb{F}$ is a field, then the polynomial ring
$\mathbb{F}[x_{1},\dots,x_{n}]$ is projective-free, settling Serre’s
conjecture from 1955 (see [7]). In the context of a commutative semisimple
unital complex Banach algebra $\mathcal{A}$, [3, Corollary 1.4] says that the
contractibility of the maximal ideal space $M_{\mathcal{A}}$ in the Gelfand
topology, suffices for $\mathcal{A}$ to be projective-free. Theorem 4.1 gives:

######  Corollary 4.2.

$A_{\textrm{\em sym}}(\mathbb{D}^{d})$ is a projective-free ring.

The study of Serre’s conjecture and algebraic $K$-theory naturally led to the
notion of Hermite rings. A commutative unital ring $R$ is Hermite if every
finitely generated stably free $R$-module is free. (A finitely generated
$R$-module $M$ is stably free if there exist free finitely generated
$R$-modules $F$ and $G$ such that $M\oplus F=G$.) Clearly, every projective-
free ring is Hermite. In terms of matrices, $R$ is Hermite if and only if
left-invertible tall matrices over $R$ can be completed to invertible ones
(see e.g. [7, p.VIII], [14, p.1029]):

 $\begin{array}[]{llll}\textrm{for all }k\in\mathbb{N}\textrm{ and
}K\in\mathbb{N}\textrm{ such that }k<K,\textrm{ and}\\\ \textrm{for all }f\in
R^{K\times k}\textrm{ such that there exists a }g\in R^{k\times K}\textrm{ so
that }gf=I_{k},\\\ \textrm{there exists an }f_{c}\in R^{K\times(K-k)}\textrm{
and there exists a }G\in R^{K\times K}\\\ \textrm{such that
}G\left[\begin{array}[]{c|c}f&f_{c}\end{array}\right]=I_{K}.\end{array}\\!\\!$

######  Corollary 4.3.

$A_{\textrm{\em sym}}(\mathbb{D}^{d})$ is a Hermite ring.

##  5\.
${\textrm{SL}}_{n}(A_{\mathrm{sym}}(\mathbb{D}^{d}))={\textrm{E}}_{n}(A_{\mathrm{sym}}(\mathbb{D}^{d}))$

Let $R$ be a commutative unital ring with multiplicative identity $1$ and
additive identity element $0$. Let $n\in\mathbb{N}$.

$\bullet$ Let $\textrm{GL}_{n}(R)$ denote the general linear group of all
invertible matrices $\bullet$ in $R^{\;\\!n\times n}$ (with matrix
multiplication).

$\bullet$ The special linear group ${\textrm{SL}}_{n}(R)$ denotes the subgroup
of $\textrm{GL}_{n}(R)$ of $\bullet$ all matrices $M$ whose determinant $\det
M=1$.

$\bullet$ An elementary matrix $E_{ij}(\alpha)$ over $R$ has the form
$E_{ij}=I_{n}+\alpha\;\\!\mathbf{e}_{ij}$, $\bullet$ $\bullet$ where $i\neq
j$, $\alpha\in R$, and $\mathbf{e}_{ij}$ is the $n\times n$ matrix whose entry
in the $\bullet$ $\bullet$ $i$th row and $j$th column is $1$, and all the
other entries of $\mathbf{e}_{ij}$ are zeros.

${\textrm{E}}_{n}(R)$ is the subgroup of ${\textrm{SL}}_{n}(R)$ generated by
the elementary matrices. A classical question in algebra is: For all
$n\in\mathbb{N}$, is ${\textrm{SL}}_{n}(R)={\textrm{E}}_{n}(R)$? The answer
depends on the ring $R$. For Banach algebras, the following result is known
[9, §7]:

######  Proposition 5.1.

Let $\mathcal{A}$ be a commutative unital Banach algebra, $n\in\mathbb{N},$
$M\in{\textrm{\em SL}}_{n}(\mathcal{A})$. Then the following are equivalent:

  * •

$M\in{\textrm{\em E}}_{n}(\mathcal{A})$.

  * •

$M$ is null-homotopic.

######  Definition 5.2 (Null-homotopic element of
${\textrm{SL}}_{n}(\mathcal{A})$).

Let $\mathcal{A}$ be a commutative unital Banach algebra, and
$n\in\mathbb{N}$. An element $M\in{\textrm{SL}}_{n}(\mathcal{A})$ is null-
homotopic if $M$ is homotopic to the identity matrix $I_{n}$, that is there
exists a continuous map $H:[0,1]\rightarrow{\textrm{SL}}_{n}(\mathcal{A})$
such that $H(0)=M$ and $H(1)=I_{n}$.

We also elaborate on the Banach algebra structure of $\mathcal{A}^{n\times n}$
for a Banach algebra $\mathcal{A}$. Let $(\mathcal{A},\|\cdot\|)$ be a
commutative unital Banach algebra. Then $\mathcal{A}^{n\times n}$ is a complex
algebra with the usual matrix operations. Let $\mathcal{A}^{n\times 1}$ denote
the vector space of all column vectors of size $n$ with entries from
$\mathcal{A}$ and componentwise operations. Then $\mathcal{A}^{n\times 1}$ is
a normed space with the ‘Euclidean norm’ defined by
$\displaystyle\|\mathbf{v}\|_{2}^{2}:=\|v_{1}\|^{2}+\cdots+\|v_{n}\|^{2}$ for
all $\mathbf{v}$ in $\mathcal{A}^{n\times 1}$, where $\mathbf{v}$ has
components denoted by $v_{1},\cdots,v_{n}\in\mathcal{A}$. If
$M\in\mathcal{A}^{n\times n}$, then the matrix multiplication map,

$\mathcal{A}^{n\times 1}\owns\mathbf{v}\mapsto
M\mathbf{v}\in\mathcal{A}^{n\times 1},$

is a continuous linear transformation, and we equip $\mathcal{A}^{n\times n}$
with the induced operator norm, denoted by $\|\cdot\|$ again. Then
$\mathcal{A}^{n\times n}$ with this operator norm is a unital Banach algebra.
Subsets of $\mathcal{A}^{n\times n}$ are given the induced subspace topology.
We also state the following observation which will be used later.

######  Lemma 5.3.

Let $(\mathcal{A},\|\cdot\|)$ be a commutative unital Banach algebra, and
$M=[m_{ij}]\in\mathcal{A}^{n\times n}$, where the entry in the $i^{\textrm{\em
th}}$ row and $j^{\textrm{\em th}}$ column of $M$ is denoted by $m_{ij}$,
$1\leq i,j\leq n$. Then

$\|M\|^{2}\leq\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{n}\|m_{ij}\|^{2}.$

###### Proof.

Let $\mathbf{v}\in\mathcal{A}^{n\times 1}$ have components
$v_{1},\cdots,v_{n}\in A$.

Using the Cauchy-Schwarz inequality in $\mathbb{R}^{n}$, we have

$\displaystyle\|M\mathbf{v}\|_{2}^{2}=\sum_{i=1}^{n}\Big{\|}\sum_{j=1}^{n}m_{ij}v_{j}\Big{\|}^{2}\\!\\!\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!\\!\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}\|m_{ij}v_{j}\|\Big{)}^{2}\leq\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}\|m_{ij}\|\|v_{j}\|\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!\\!\sum_{i=1}^{n}\sum_{j=1}^{n}\|m_{ij}\|^{2}\sum_{k=1}^{n}\|v_{k}\|^{2}=\sum_{i=1}^{n}\sum_{j=1}^{n}\|m_{ij}\|^{2}\|\mathbf{v}\|_{2}^{2}.$

As $\mathbf{v}\in\mathcal{A}^{n\times 1}$ was arbitrary, it follows that
$\|M\|^{2}\leq\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{n}\|m_{ij}\|^{2}$. ∎

######  Corollary 5.4.

For all $n\in\mathbb{N},$ ${\textrm{\em
SL}}_{n}(A_{\mathrm{sym}}(\mathbb{D}^{d}))={\textrm{\em
E}}_{n}(A_{\mathrm{sym}}(\mathbb{D}^{d}))$.

###### Proof.

We produce the homotopy using dilations (see [13], [5]).

We need to show
${\textrm{SL}}_{n}(A_{\textrm{sym}}(\mathbb{D}^{d}))\subset{\textrm{E}}_{n}(A_{\textrm{sym}}(\mathbb{D}^{d}))$.
Let

$M=\left[\\!\\!\begin{array}[]{ccc}f_{11}&\cdots&f_{1n}\\\
\vdots&\ddots&\vdots\\\
f_{n1}&\cdots&f_{nn}\end{array}\\!\\!\right]\in{\textrm{SL}}_{n}(A_{\textrm{sym}}(\mathbb{D}^{d})).$

For $t\in[0,1]$, define

$H(t)=M_{t}=\left[\\!\\!\begin{array}[]{ccc}f((1-t)\cdot)&\cdots&f((1-t)\cdot)\\\
\vdots&\ddots&\vdots\\\
f_{n1}((1-t)\cdot)&\cdots&f_{nn}((1-t)\cdot)\end{array}\\!\\!\right].$

Then $M_{t}\in{\textrm{SL}}_{n}(A_{\textrm{sym}}(\mathbb{D}^{d}))$ for all
$t\in[0,1]$, since $\det(M_{t})=\mathbf{1}$. The continuity of the map

$[0,1]\owns t\mapsto
M_{t}\in{\textrm{SL}}_{n}(A_{\textrm{sym}}(\mathbb{D}^{d}))$

follows from the compactness of $\overline{\mathbb{D}}^{d}$ (making elements
of $A_{\textrm{sym}}(\mathbb{D}^{d})$ uniformly continuous), and from Lemma
5.3. We note that $M_{1}=C\mathbf{1}$, where
$C\in{\textrm{SL}}_{n}(\mathbb{C})$ is the constant matrix given by

$C=\left[\\!\\!\begin{array}[]{ccc}f_{11}(\mathbf{0})&\cdots&f_{1n}(\mathbf{0})\\\
\vdots&\ddots&\vdots\\\
f_{n1}(\mathbf{0})&\cdots&f_{nn}(\mathbf{0})\end{array}\\!\\!\right].$

But ${\textrm{SL}}_{n}(\mathbb{C})$ is path-connected, and so there exists a
homotopy, say $h:[0,1]\rightarrow{\textrm{SL}}_{n}(\mathbb{C})$, taking $C$ to
the identity matrix $I_{n}\in{\textrm{SL}}_{n}(\mathbb{C})$. Combining $H$
with $h$, we obtain a homotopy
$\widetilde{H}:[0,1]\rightarrow{\textrm{SL}}_{n}(A)$ that takes $M$ to
$I_{n}\in{\textrm{SL}}_{n}(A_{\textrm{sym}}(\mathbb{D}^{d}))$:

$\widetilde{H}(t)=\left\\{\begin{array}[]{ll}H(2t)&\textrm{if
}\;t\in\Big{[}0,\displaystyle{\scaleobj{0.75}{\frac{1}{2}}}\Big{]},\\\\[5.97527pt]
h(2t-\\!1)\mathbf{1}&\textrm{if
}\;t\in\Big{[}{\scaleobj{0.75}{\displaystyle\frac{1}{2}}},1\Big{]}.\end{array}\right.$

So we have shown that $M\in{\textrm{SL}}_{n}(A)$ is null-homotopic.
Consequently, by Propposition 5.1, we have
$M\in{\textrm{E}}_{n}(A_{\textrm{sym}}(\mathbb{D}^{d}))$. ∎

##  6\. Noncoherence of $A_{\mathrm{sym}}(\mathbb{D}^{d})$

Let $R$ be a commutative ring. For $n\in\mathbb{N}$ and elements
$r_{1},\cdots,r_{n}\in R$, we denote by $\langle r_{1},\cdots,r_{n}\rangle$
the ideal in $R$ generated by $r_{1},\cdots,r_{n}$. The ring
$A_{\mathrm{sym}}(\mathbb{D}^{d})$ is not Noetherian, since the ascending
chain condition fails, as demonstrated by the chain of ideals

$\langle B_{1}(z_{1})\cdots B_{1}(z_{d})\rangle\subsetneq\langle
B_{2}(z_{1})\cdots B_{2}(z_{d})\rangle\subsetneq\langle B_{3}(z_{1})\cdots
B_{3}(z_{d})\rangle\subsetneq\cdots,$

where $B_{n}\in A(\mathbb{D})$ are the rational functions defined by

$\displaystyle
B_{n}(z):=\prod_{k=1}^{n}\frac{\alpha_{n}-z}{1-\alpha_{n}z}\;\;\textrm{ and
}\;\;\displaystyle\alpha_{n}=1-\frac{1}{n^{2}}\;\;(n\in\mathbb{N}).$

In absence of the Noetherian ‘finiteness’ property, the next best finiteness
condition in commutative algebra is presented when one has a coherent ring.

######  Definition 6.1.

A unital commutative ring $R$ is said to be coherent if

$\bullet$ the intersection of any two finitely generated ideals in $R$ is
finitely  aa$\bullet$ finitely generated, and

$\bullet$ for every $a\in R$, the annihilator $\textrm{Ann}(a):=\\{x\in
R:ax=0\\}$ is Aaa aa$\bullet$ finitely generated.

We refer the reader to the article [6] for the relevance of the property of
coherence in commutative algebra. The following, which we will need, is a
slight abstraction of [11, Observation 2.1].

######  Theorem 6.2.

Let $\mathcal{R}_{d}$ be the class of all commutative unital subrings of
$\mathbb{C}^{\mathbb{D}^{d}}:=\\{f:\mathbb{D}^{d}\rightarrow\mathbb{C}\\}$
under the usual pointwise operations.

Let $D:R_{d}\rightarrow R_{1}$ and $U:R_{1}\rightarrow R_{d}$ be ring
homomorphisms, such that

$DUg=g\textrm{ for all }g\in R_{1}.$

Then if $R_{d}\in\mathcal{R}_{d}$ is a coherent ring, then $R_{1}$ is a
coherent ring.

###### Proof.

222The proof is the same, mutatis mutandis, as that of [11, Observation 2.1]
(since it turns out that there the above abstract properties of $D$ and $U$
were crucial, rather than their concrete explicit form). So we include the
details of the short proof here for the convenience of the reader and to keep
the discussion self-contained.

Suppose that $I$ and $J$ are two finitely generated ideals in $R_{1}$, say
$I=\langle f_{1},\cdots,f_{K}\rangle$ and $J=\langle
g_{1},\cdots,g_{L}\rangle$, and let $I_{d}$ and $J_{d}$ be the ideals in
$R_{d}$ generated by the functions $Uf_{k}$ ($k=1,\cdots,K$), respectively
$Ug_{\ell}$ ($\ell=1,\cdots,L$). The coherence of $R_{d}$ implies that
$I_{d}\cap J_{d}$ is finitely generated. Let $\\{p_{1},\cdots,p_{M}\\}$ be a
set of generators for the ideal $I_{d}\cap J_{d}$ in $R_{d}$. Then
$\\{Dp_{1},\cdots,Dp_{M}\\}$ is a set of generators for $I\cap J$. Indeed, if
$f\in I\cap J$, then in particular, $f\in I=\langle
f_{1},\cdots,f_{K}\rangle$, so that there exist
$\alpha_{1},\cdots,\alpha_{K}\in R_{1}$ such that
$f=\alpha_{1}f_{1}+\cdots+\alpha_{K}f_{K}$. Hence we have
$Uf=(U\alpha_{1})(Uf_{1})+\cdots+(U\alpha_{K})(Uf_{K})\in\langle
Uf_{1},\cdots,Uf_{K}\rangle=I_{d}$. Similarly, $Uf\in J_{d}$ too. Thus $Uf\in
I_{d}\cap J_{d}=\langle p_{1},\cdots,p_{M}\rangle$, and so there exist
$\gamma_{1},\cdots,\gamma_{M}\\!\in\\!R_{d}$ such that
$Uf=\gamma_{1}p_{1}+\cdots+\gamma_{M}p_{M}$. Consequently,
$f=DUf=(D\gamma_{1})(Dp_{1})+\cdots+(D\gamma_{M})(Dp_{M})\in\langle
Dp_{1},\cdots,Dp_{M}\rangle$, and so $I\cap J\subset\langle
Dp_{1},\cdots,Dp_{M}\rangle$. Vice versa, for any $m=1,\cdots,M$, we have
$p_{m}\in I_{d}=\langle Uf_{1},\cdots,Uf_{K}\rangle$. So there exist
$\theta_{1},\cdots,\theta_{K}\in R_{d}$ such that
$p_{m}=\theta_{1}(Uf_{1})+\cdots+\theta_{K}(Uf_{K})$. Applying $D$,

$\displaystyle Dp_{m}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!(D\theta_{1})(DUf_{1})+\cdots+(D\theta_{K})(DUf_{K})$
$\displaystyle=$
$\displaystyle\\!\\!\\!(D\theta_{1})f_{1}+\cdots+(D\theta_{K})f_{K}\in\langle
f_{1},\cdots,f_{K}\rangle=I.$

Similarly, each $Dp_{m}\in J$. Hence $\\{Dp_{1},\cdots,Dp_{M}\\}\subset I\cap
J$, and so the ideal $\langle Dp_{1},\cdots,Dp_{M}\rangle\subset I\cap J$ too.
Hence $I\cap J=\langle Dp_{1},\cdots,Dp_{M}\rangle$, and so $I\cap J$ is
finitely generated.

It remains to verify the condition on the annihilators. Suppose $f\in R_{1}$.
Then $Uf\in R_{d}$. As $\textrm{Ann}(Uf)$ is finitely generated, there exist
$h_{1},\cdots,h_{r}\in R_{d}$ which generate $\textrm{Ann}(Uf)$. We claim that
$\textrm{Ann}(f)$ is generated by $Dh_{1},\cdots,Dh_{r}$. Suppose
$g\in\textrm{Ann}(f)$, that is, $fg=0$. Then $(Uf)(Ug)=U(fg)=U(0)=0$, and so
$Ug\in\textrm{Ann}(Uf)=\langle h_{1},\cdots,h_{r}\rangle$. So there exist
$\beta_{1},\cdots,\beta_{r}\in R_{d}$ such that
$Ug=\beta_{1}h_{1}+\cdots+\beta_{r}h_{r}$. Applying $D$, we obtain
$g=DUg=(D\beta_{1})(Dh_{1})+\cdots+(D\beta_{r})(Dh_{r})$ and so $g\in\langle
Dh_{1},\cdots,Dh_{r}\rangle$, showing that $\textrm{Ann}(f)\subset\langle
Dh_{1},\cdots,Dh_{r}\rangle$. Vice versa, since for $1\leq k\leq r$, we have
$h_{k}\in\textrm{Ann}(Uf)$, it follows that $(Uf)h_{k}\\!=\\!0$, which yields
$f(Dh_{k})\\!=\\!(DUf)(Dh_{k})\\!=\\!D((Uf)h_{k})\\!=\\!D0\\!=\\!0$, that is,
$Dh_{k}\in\textrm{Ann}(f)$. Thus $\langle
Dh_{1},\cdots,Dh_{r}\rangle\subset\textrm{Ann}(f)$ too. Consequently,
$\textrm{Ann}(f)$ is finitely generated, completing the proof that $R_{1}$ is
coherent. ∎

In [11, Observation 2.1], maps $D:R_{d}\rightarrow DR_{d}$ and
$U:DR_{d}\rightarrow R_{d}$ were defined as follows:

$\begin{array}[]{l}(Df)(z)=f(z,0,\cdots,0),\quad z\in\mathbb{D},\;\;f\in
R_{d},\\\
(Ug)(z_{1},\cdots,z_{d})=g(z_{1}),\quad(z_{1},\cdots,z_{d})\in\mathbb{D}^{d},\;\;g\in
DR_{d}.\end{array}$

However, these cannot be used for $R_{d}:=A_{\textrm{sym}}(\mathbb{D}^{d})$,
since the property $UDf\in R_{d}$ for all $f\in R_{d}$, demanded in [11,
Observation 2.1], fails: For example, for $d\\!=\\!2$ and with $f\in
R_{2}\\!=\\!A_{\mathrm{sym}}(\mathbb{D}^{2})$ given by $f(z,w)\\!=\\!z+w$ for
$(z,w)\in\mathbb{D}^{2}$, we get $(UDf)(z,w)\\!=\\!z$, so that $UDf\not\in
A_{\textrm{sym}}(\mathbb{D}^{2})\\!=\\!R_{2}$.

In order to apply Theorem 6.2, we instead use the following.

######  Lemma 6.3.

Let $R_{d}:=A_{\mathrm{sym}}(\mathbb{D}^{d})$, and define the maps
$D:R_{d}\rightarrow R_{1}$ and $U:R_{1}\rightarrow R_{d}$ by

$\begin{array}[]{l}(Df)(z)=f(z,\cdots,z),\quad z\in\mathbb{D},\;\;f\in
R_{d},\\\\[2.84544pt] (Ug)(z_{1},\cdots,z_{d})=\displaystyle
g\Big{(}\frac{z_{1}+\cdots+z_{d}}{d}\Big{)},\quad(z_{1},\cdots,z_{d})\in\mathbb{D}^{d},\;\;g\in
R_{1}.\end{array}$

Then $D,U$ are well-defined ring homomorphisms, and

$DUg=g\textrm{ for all }g\in R_{1}.$

###### Proof.

For $f\in R_{d}$, $Df$ is clearly holomorphic in $\mathbb{D}$, and possesses a
continuous extension to $\overline{\mathbb{D}}$. Thus $Df\in
A(\mathbb{D})=R_{1}$. Also, it is clear that $D$ is a ring homomorphism.

The map
$\mathbb{D}^{d}\owns(z_{1},\cdots,z_{d})\mapsto\frac{z_{1}+\cdots+z_{d}}{d}\in\mathbb{D}$
is continuous. So it follows that $Ug\in A(\mathbb{D}^{d})$. For $g\in R_{1}$,
$\sigma\in S_{d}$, and $(z_{1},\cdots,z_{d})\in\mathbb{D}^{d}$,

$\displaystyle(Ug)(z_{\sigma(1)},\cdots,z_{\sigma(d)})\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!g\Big{(}\frac{z_{\sigma(1)}+\cdots+z_{\sigma(d)}}{d}\Big{)}=g\Big{(}\frac{z_{1}+\cdots+z_{d}}{d}\Big{)}$
$\displaystyle=$ $\displaystyle\\!\\!\\!(Ug)(z_{1},\cdots,z_{d}),$

showing that $Ug\in A_{\mathrm{sym}}(\mathbb{D}^{d})=R_{d}$. Moreover, it is
clear that $U$ is a ring homomorphism.

Finally, we check that $DUg=g$ for all $g\in R_{1}=A(\mathbb{D})$. Let $g\in
A(\mathbb{D})$. Then for all $(z_{1},\cdots,z_{d})\in\mathbb{D}^{d}$, we have

$\Phi(z_{1},\cdots,z_{d}):=(Ug)(z_{1},\cdots,z_{d})=g\Big{(}\frac{z_{1}+\cdots+z_{d}}{d}\Big{)}.$

Hence for all $z\in\mathbb{D}$,

$(D(Ug))(z)=(D\Phi)(z)=\Phi(z,\cdots,z)=g\Big{(}\frac{z+\cdots+z}{d}\Big{)}=g(z).$

This completes the proof. ∎

As the disk algebra $R_{1}:=A(\mathbb{D})$ is not coherent (see [8], or [11]
for a simpler proof), it follows from Theorem 6.2 and Lemma 6.3 that:

######  Corollary 6.4.

$A_{\mathrm{sym}}(\mathbb{D}^{d})$ is not coherent.

##  7\. The Wiener algebra $W^{+}_{\mathrm{sym}}(\mathbb{D}^{d})$ of symmetric
functions

Analogous to the polydisc algebra of symmetric functions, one can also
consider $W^{+}_{\mathrm{sym}}(\mathbb{D}^{d})$, the analytic Wiener algebra
on the polydisc, consisting of symmetric functions. All the results of the
previous sections can be proved in the same way (mutatis mutandis) also for
this algebra. We discuss this algebra here for a different reason. A
fundamental theorem in algebra states that every symmetric polynomial can be
expressed uniquely as a polynomial in the symmetric polynomials. One may ask
if an analogue of this result carries over for the
$W^{+}_{\mathrm{sym}}(\mathbb{D}^{d})$, and we investigate this question here.
We begin with the pertinent definitions.

Let

$\mathbb{N}_{0}:=\\{0,1,2,3,\cdots\\}.$

For $d\in\mathbb{N}$, $\mathbf{n}=(n_{1},\cdots,n_{d})\in\mathbb{N}_{0}^{d}$,
and $\mathbf{z}=(z_{1},\cdots,z_{d})\in\mathbb{C}^{d}$, we set

$\displaystyle|\mathbf{n}|\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!n_{1}+\cdots+n_{d},\;\;\textrm{ and }$
$\displaystyle\mathbf{z}^{\mathbf{n}}\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!z_{1}^{n_{1}}\cdots
z_{d}^{n_{d}},\;\;z_{k}^{0}:=1\;(1\leq k\leq d).$

The classical analytic Wiener polydisc algebra $W^{+}(\mathbb{D}^{d})$ is the
Banach algebra

$W^{+}(\mathbb{D}^{d})=\Big{\\{}f:\mathbb{D}^{d}\rightarrow\mathbb{C}:f(\mathbf{z})=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{d}}c_{\mathbf{n}}\mathbf{z}^{\mathbf{n}}\;(\mathbf{z}\in\mathbb{D}^{d}),\;\sum_{\mathbf{n}\in\mathbb{N}_{0}^{d}}|c_{\mathbf{n}}|<\infty\Big{\\}},$

with pointwise operations, and the norm $\|\cdot\|_{1}$ given by

$\|f\|_{1}:=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{d}}|c_{\mathbf{n}}|,\textrm{
for
}f=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{d}}c_{\mathbf{n}}\mathbf{z}^{\mathbf{n}}\in
W^{+}(\mathbb{D}^{d}).$

We define the analytic Wiener algebra of symmetric functions,
$W^{+}_{\textrm{sym}}(\mathbb{D}^{d})$, to be the subalgebra of
$W^{+}(\mathbb{D}^{d})$ consisting of symmetric functions:

$W^{+}_{\textrm{sym}}(\mathbb{D}^{d}):=\\{f\in W^{+}(\mathbb{D}^{d}):f\textrm{
is a symmetric function}\\}.$

Then $W^{+}_{\textrm{sym}}(\mathbb{D}^{d})$ is a Banach subalgebra of
$W^{+}(\mathbb{D}^{d})$ with the same norm $\|\cdot\|_{1}$. We also have
$W^{+}_{\textrm{sym}}(\mathbb{D}^{d})\subset
A_{\textrm{sym}}(\mathbb{D}^{d})$, and for all $f\in
W^{+}_{\textrm{sym}}(\mathbb{D}^{d})$,

$\|f\|_{\infty}\leq\|f\|_{1}$

A polynomial $p\in\mathbb{C}[z_{1},\cdots,z_{d}]$ is symmetric if

$p(z_{1},\cdots,z_{d})=p(z_{{\scaleobj{0.75}{\sigma(1)}}},\cdots,z_{{\scaleobj{0.75}{\sigma(d)}}})\textrm{
for all }\sigma\in S_{d}.$

######  Proposition 7.1.

The set of symmetric polynomials is dense in
$W^{+}_{\mathrm{sym}}(\mathbb{D}^{d})$.

###### Proof.

As polynomial functions are dense in $W^{+}(\mathbb{D}^{d})$, we know that
given any $f\in W^{+}_{\textrm{sym}}(\mathbb{D}^{d})\subset
W^{+}(\mathbb{D}^{d})$, there is a sequence of polynomials
$(p_{n})_{n\in\mathbb{N}}$ that converges to $f$ in the $\|\cdot\|_{1}$-norm.
But then it follows that $(\sigma\;\\!p_{n})_{n\in\mathbb{N}}$ conerges to
$\sigma f=f$ in the $\|\cdot\|_{1}$-norm. Hence

$q_{n}:=\frac{1}{d!}\sum_{\sigma\in
S_{d}}\sigma\;\\!p_{n}\rightarrow\frac{1}{d!}\sum_{\sigma\in S_{d}}f=f$

as $n\rightarrow\infty$ in the $\|\cdot\|_{1}$-norm. As
$(q_{n})_{n\in\mathbb{N}}$ is a sequence of symmetric polynomials, the proof
is complete. ∎

In fact, we can show the same result in a more constructive manner as follows.
Let $f\in W^{+}_{\mathrm{sym}}(\mathbb{D}^{d})$ have the Taylor expansion

$f=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{d}}c_{\mathbf{n}}\mathbf{z}^{\mathbf{n}}\quad(\mathbf{z}\in\mathbb{D}^{d}).$

Then defining the polynomials

$p_{k}=\sum_{|\mathbf{n}|=k}c_{\mathbf{n}}\mathbf{z}^{\mathbf{n}}\quad(k\geq
0),$

we have that $p_{k}$ is a symmetric polynomial (shown below) of (total) degree
$k$, and

$f=\sum_{k=0}^{\infty}p_{k}.\quad\;\;\;\phantom{(k\geq 0)}$ (1)

Thus the sequence of symmetric polynomials
$(p_{0}+p_{1}+\cdots+p_{n})_{n\in\mathbb{N}}$ converges to $f$ in
$W^{+}_{\mathrm{sym}}(\mathbb{D}^{d})$. To see that $p_{k}$ is a symmetric
polynomial, firstly

$c_{\mathbf{n}}=\frac{1}{n_{1}!\cdots
n_{d}!}\frac{\partial^{|\mathbf{n}|}f\hfill}{\partial z_{1}^{n_{1}}\cdots
z_{d}^{n_{d}}\hfill}(\mathbf{0}),$

and since $f$ is symmetric, it follows that for any $\sigma\in S_{d}$,

$\frac{\partial^{|\mathbf{n}|}f\hfill}{\partial z_{1}^{n_{1}}\cdots
z_{d}^{n_{d}}\hfill}(\mathbf{0})=\frac{\partial^{|\mathbf{n}|}f\hfill}{\partial
z_{1}^{n_{{\scaleobj{0.81}{\sigma^{{\scaleobj{0.81}{-1}}}(1)}}}}\cdots
z_{d}^{n_{{\scaleobj{0.81}{\sigma^{{\scaleobj{0.81}{-1}}}(d)}}}}\hfill}(\mathbf{0}).$

Consequently,

$\displaystyle\\!\\!\\!(\sigma\;\\!p_{k})(z_{1},\cdots,z_{d})=p_{k}(z_{{\scaleobj{0.81}{\sigma(1)}}},\cdots,z_{{\scaleobj{0.81}{\sigma(d)}}})\phantom{\sum}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{|\mathbf{n}|=k}\frac{1}{n_{1}!\cdots
n_{d}!}\frac{\partial^{|\mathbf{n}|}f\hfill}{\partial z_{1}^{n_{1}}\cdots
z_{d}^{n_{d}}\hfill}(\mathbf{0})z_{{\scaleobj{0.81}{\sigma(1)}}}^{n_{1}}\cdots
z_{{\scaleobj{0.81}{\sigma(d)}}}^{n_{d}}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{|\mathbf{n}|=k}\frac{1}{{n_{{\scaleobj{0.81}{\sigma^{{\scaleobj{0.81}{-1}}}(1)}}}}!\cdots{n_{{\scaleobj{0.81}{\sigma^{{\scaleobj{0.81}{-1}}}(d)}}}}!}\frac{\partial^{|\mathbf{n}|}f\hfill}{\partial
z_{1}^{n_{{\scaleobj{0.81}{\sigma^{{\scaleobj{0.81}{-1}}}(1)}}}}\cdots
z_{d}^{n_{{\scaleobj{0.81}{\sigma^{{\scaleobj{0.81}{-1}}}(d)}}}}\hfill}(\mathbf{0})z_{1}^{n_{{\scaleobj{0.81}{\sigma^{{\scaleobj{0.81}{-1}}}(1)}}}}\cdots
z_{d}^{n_{{\scaleobj{0.81}{\sigma^{{\scaleobj{0.81}{-1}}}(d)}}}}$
$\displaystyle=$ $\displaystyle\\!\\!\\!p(z_{1},\cdots,z_{d}).$

Let $d\in\mathbb{N}$, and $n\in\mathbb{N}$ is such that $n\leq d$. An
elementary symmetric polynomial of degree $n$ in $d$ variables is a symmetric
polynomial $s_{n}$ of the form

$s_{n}=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{n}\leq d}z_{i_{1}}\cdots z_{i_{d}}.$

For example if $d=2$, and we call the two variables $z,w$, then

$\displaystyle s_{1}\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!z+w,$
$\displaystyle s_{2}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!zw;\phantom{AAAAAAAAAAA}$

if $d=3$, then

$\displaystyle s_{1}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!z_{1}+z_{2}+z_{3},$ $\displaystyle s_{2}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3},\textrm{ and}$
$\displaystyle s_{3}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!z_{1}z_{2}z_{3};$

and so on.

A fundamental result on symmetric polynomials states that every symmetric
polynomial can be expressed uniquely as a polynomial in the symmetric
polynomials (see e.g. [1, Chap.14, Thm. 3.4]):

######  Proposition 7.2.

Every symmetric polynomial $p\in\mathbb{C}[z_{1},\cdots,z_{d}]$ can be written
in a unique way as a polynomial in the elementary symmetric polynomials, that
is, there exists a unique polynomial $q\in\mathbb{C}[s_{1},\cdots,s_{d}]$ such
that
$p(z_{1},\cdots,z_{d})=q\;\\!\boldsymbol{(}s_{1}(z_{1},\cdots,z_{d}),\cdots,s_{d}(z_{1},\cdots,z_{d})\boldsymbol{)}.$

For example, if $d=2$, then

$z^{2}+w^{2}=(z+w)^{2}-2zw=s_{1}^{2}-2s_{2}.$

An immediate corollary of this result and of Proposition 7.1 is the following.

######  Corollary 7.3.

The set $\mathbb{C}[s_{1},\cdots,s_{d}]$ of polynomials in the elementary
symmetric polynomials is dense in $W^{+}_{\mathrm{sym}}(\mathbb{D}^{d})$.

###  7.1. An open question

For $f\in W^{+}_{\mathrm{sym}}(\mathbb{D}^{d})$, by using the decomposition
(1) above, and expressing each $p_{k}$ in terms of a polynomial in
$s_{1},\cdots,s_{d}$, one obtains a formal power series
$g\in\mathbb{C}\llbracket s_{1},\cdots,s_{d}\rrbracket$. One could hope that
$g$ then converges on a domain containing the set

$\\{(\zeta_{1},\cdots,\zeta_{d})\in\mathbb{C}^{d}:\exists\mathbf{z}\in\mathbb{D}^{d}\textrm{
such that
}s_{1}(\mathbf{z})=\zeta_{1},\;\cdots,s_{d}(\mathbf{z})=\zeta_{d}\\},$

so that

$f(\mathbf{z})=g(s_{1}(\mathbf{z}),\cdots,s_{d}(\mathbf{z}))\quad(\mathbf{z}\in\mathbb{D}^{d})$

could hold. Is this true?

For example, let $d=2$, and consider $f$ given by

$f=\sum_{n=1}^{\infty}\frac{1}{n^{2}2^{n}}(z^{2}+w^{2})^{n}.$

As each term in the summand $(z^{2}+w^{2})^{n}$ has total degree $2n$, we have

$\displaystyle\|(z^{2}+w^{2})^{n}\|_{1}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\binom{n}{0}+\cdots+\binom{n}{n}$ $\displaystyle=$
$\displaystyle\\!\\!\\!(1+1)^{n}=2^{n},$

and so

$\|f\|_{1}=\sum_{n=1}^{\infty}\frac{1}{n^{2}2^{n}}2^{n}=\sum_{n=1}^{\infty}\frac{1}{n^{2}}<\infty.$

Hence $f\in W^{+}_{\textrm{sym}}(\mathbb{D}^{d})$. Using

$z^{2}+w^{2}=(z+w)^{2}-2zw=s_{1}^{2}-2s_{2},$

we obtain for $(z,w)\in\mathbb{D}^{2}$ that

$\displaystyle f(z,w)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{n=1}^{\infty}\frac{1}{n^{2}2^{n}}(z^{2}+w^{2})^{n}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{n=1}^{\infty}\frac{1}{n^{2}2^{n}}((s_{1}(z,w))^{2}-2s_{2}(z,w))^{n}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{n=1}^{\infty}\sum_{k=0}^{n}\frac{1}{n^{2}2^{n}}\binom{n}{k}(-2)^{k}(s_{1}(z,w))^{2n-2k}(s_{2}(z,w))^{k}.$

One would then like to conclude that for $(z,w)\in\mathbb{D}^{2}$,

$\displaystyle f(z,w)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!g(s_{1}(z,w),s_{2}(z,w)),$

where $g\in\mathbb{C}\llbracket s_{1},s_{2}\rrbracket$ is given by

$\displaystyle g(s_{1},s_{2})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{n=1}^{\infty}\sum_{k=0}^{n}\frac{1}{n^{2}2^{n}}\binom{n}{k}(-2)^{k}s_{1}^{2n-2k}s_{2}^{k}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\frac{s_{1}^{2}-2s_{2}}{1^{2}2^{1}}+\frac{(s_{1}^{2}-2s_{2})^{2}}{2^{2}2^{2}}+\frac{(s_{1}^{2}-2s_{2})^{3}}{3^{2}2^{3}}+\cdots$
$\displaystyle=$
$\displaystyle\\!\\!\\!-s_{2}+\frac{1}{2}s_{1}^{2}+\frac{1}{4}s_{2}^{2}-\frac{1}{9}s_{2}^{3}-\frac{1}{2}s_{1}^{2}s_{2}+\cdots.$

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