# Projective freeness and Hermiteness
of complex function algebras

Alexander Brudnyi  Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta, Canada T2N 1N4<EMAIL_ADDRESS>and  Amol Sasane
Department of Mathematics
London School of Economics
Houghton Street
London WC2A 2AE
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

The paper studies projective freeness and Hermiteness of algebras of complex-
valued continuous functions on topological spaces, Stein algebras, and
commutative unital Banach algebras. New sufficient cohomology conditions on
the maximal ideal spaces of the algebras are given that guarantee the
fulfilment of these properties. The results are illustrated by nontrivial
examples. Based on the Borsuk theory of shapes, a new class $\mathscr{C}$ of
commutative unital complex Banach algebras is introduced (an analog of the
class of local rings in commutative algebra) such that the projective tensor
product with algebras in $\mathscr{C}$ preserves projective freeness and
Hermiteness. Some examples of algebras of class $\mathscr{C}$ and of other
projective free and Hermite function algebras are assembled. These include,
e.g., Douglas algebras, finitely generated algebras of symmetric functions,
Bohr-Wiener algebras, algebras of holomorphic semi-almost periodic functions,
and algebras of bounded holomorphic functions on Riemann surfaces.

###### Key words and phrases:

Projective free rings, Hermite rings, algebras of continuous functions, Stein
space, Banach algebras, maximal ideal space, homotopy equivalence, Čech
cohomology, trivial shape, topological tensor products

###### 2020 Mathematics Subject Classification:

Primary 46J10; Secondary 46M10, 13C10, 46J15.

##  1\. Introduction

In this article, we study projective freeness and Hermiteness of certain
function algebras. Recall that a unital commutative ring $R$ is said to be
projective free if every finitely generated projective $R$-module is free
(i.e., if $M$ is an $R$-module such that $M\oplus N\cong R^{n}$ for an
$R$-module $N$ and $n\in\mathbb{Z}_{+}\,(:=\mathbb{N}\cup\\{0\\})$, then
$M\cong R^{m}$ for some $m\in\mathbb{Z}_{+}$). In terms of matrices, the ring
$R$ is projective free if and only if for each $n\in\mathbb{N}$ every $n\times
n$ matrix $X\not\in\\{0_{n},I_{n}\\}$ over $R$ such that $X^{2}=X$ (i.e., an
idempotent) has a form $X=S(I_{r}\oplus 0_{n-r})S^{-1}$ for some $S\in
GL_{n}(R)$, $r\in\\{1,\dots,n-1\\}$; here $GL_{n}(R)$ denotes the group of
invertible $n\times n$ matrices over $R$ and $0_{k}$ and $I_{k}$ are zero and
identity $k\times k$ matrices; see [12, Proposition 2.6].

Quillen and Suslin (see, e.g., [13]) proved, independently, that the
polynomial ring over a projective free ring is again projective free; in
particular settling affirmatively Serre’s problem from 1955, which asked if
the polynomial ring $\mathbb{F}[x_{1},\dots,x_{n}]$ is projective free for any
field $\mathbb{F}$ (see [31]). Also, if $R$ is any projective free ring, then
the formal power series ring $R\llbracket x\rrbracket$ is again projective
free [13, Theorem 7], and so the formal power series ring
$\mathbb{F}\llbracket x_{1},\dots,x_{n}\rrbracket$ is projective free. In the
context of rings arising in analysis, it is known that for example the algebra
of complex continuous functions on a contractible topological space is
projective free [53]. According to the Grauert theorem [23] and the
Novodvorski-Taylor theory (see, e.g.,[38], [51, §7.5]), the same holds for the
algebra of holomorphic functions on a contractible reduced Stein space and a
commutative unital complex Banach algebra with a contractible maximal ideal
space.

In control theory, projective freeness of rings of stable transfer functions
plays an important role in the stabilisation problem, since if the underlying
ring of stable transfer functions is projective free, then the stabilisability
of an unstable system is equivalent to the existence of a doubly coprime
factorisation, see [39, Theorem 6.3].

The concept of a Hermite ring is a weaker notion than that of a projective
free ring. A unital commutative ring $R$ is Hermite if every finitely
generated stably free $R$-module is free. An $R$-module $M$ is stably free if
it is stably isomorphic to a free module, i.e., $M\oplus R^{k}$ is free for
some $k\in\mathbb{Z}_{+}$. Since every stably free module is projective, every
projective free ring is Hermite. In terms of matrices, $R$ is Hermite if and
only if every left-invertible $n\times k$ matrix, $k,n\in\mathbb{N}$, $k<n$,
over $R$ can be completed to an invertible one (see, e.g., [31, p.VIII], [52,
p.1029]).

In control theory, Hermiteness of the underlying ring of stable transfer
functions implies that if the transfer function of an unstable system has a
right (or left) coprime factorisation, then it has a doubly coprime
factorisation; see [54, Theorem 66].

In the article, we study successively projective freeness and Hermiteness of
algebras of complex-valued continuous functions on topological spaces (§2),
Stein algebras (§3) and commutative unital complex Banach algebras (§4). We
begin with a survey of some results describing the structure of finitely
generated projective modules over these algebras in terms of complex vector
bundles over their maximal ideal spaces using the Swan and Vaserstein theorems
[49] and [53], the Grauert Oka principle [22], [23] and the Novodvorski-Taylor
theory [38], [51]. Then, based on this description, we give new sufficient
cohomology conditions for projective freeness and Hermiteness of the
considered algebras and illustrate our results by nontrivial examples. In §5,
we introduce and study a new class $\mathscr{C}$ of commutative unital complex
Banach algebras $A$ (namely those whose maximal ideal spaces are of trivial
shape) such that for every commutative unital complex Banach algebra $B$ the
isomorphism classes of finitely generated projective modules over $B$ are in a
one-to-one correspondence with those over the projective tensor product
$A\widehat{\otimes}_{\pi}B$. In particular, the projective tensor product with
algebras in $\mathscr{C}$ preserves projective freeness and Hermiteness.
Finally, in §6 several examples of algebras of class $\mathscr{C}$ and of
projective free function algebras are assembled. These include finitely
generated algebras of symmetric functions (§6.1), Bohr-Wiener algebras (§6.2),
algebras of holomorphic semi-almost periodic functions (§6.3), and algebras of
bounded holomorphic functions on Riemann surfaces (§6.4). The Appendix
contains auxiliary results used in §6.

##  2\. Algebras of continuous functions

For a topological space $X$, we denote by $C(X)$ the algebra (with pointwise
multiplication) of complex-valued continuous functions on $X$. Also, by
$C_{b}(X)\subset C(X)$ we denote the subalgebra of bounded functions.
$C_{b}(X)$ equipped with the supremum norm, $\|f\|_{\infty}=\sup_{x\in
X}|f(x)|$, is a complex commutative unital Banach algebra.

We recall that a bundle over $X$ is of finite type if there is a finite set
$S$ of nonnegative continuous functions on X whose sum is $1$ such that the
restriction of the bundle to the set $\\{x\in X\,:\,f(x)\neq 0\\}$ is trivial
for each $f$ in $S$.

Every bundle over a compact Hausdorff space $X$ is of finite type. In turn, a
bundle over a normal space $X$ is of finite type if and only if there is a
finite open covering $\mathfrak{U}$ of $X$ such that the restriction of the
bundle to each $U\in\mathfrak{U}$ is trivial.

Let ${\textrm{\bf Gr}}(\mathbb{C}^{n})$ be the set of all subspaces of
$\mathbb{C}^{n}$ equipped with the topology of the disjoint union of the
Grassmanian manifolds ${\textrm{\bf Gr}_{m}}(\mathbb{C}^{n})$, $0\leq m\leq
n$. For each $n\in\mathbb{Z}_{+}$ there is a natural embedding ${\textrm{\bf
Gr}}(\mathbb{C}^{n})\subset{\textrm{\bf Gr}}(\mathbb{C}^{n+1})$ so that the
space ${\textrm{\bf
Gr}}(\mathbb{C}^{\infty})=\bigcup_{n\in\mathbb{Z}_{+}}\\!{\textrm{\bf
Gr}}(\mathbb{C}^{n})$, equipped with the inductive limit topology, is well-
defined.

The next result follows from Vaserstein’s extension [53, Theorem 2] of the
Swan theorem [49].

######  Proposition 2.1.

The following statements are equivalent:

  1. (1)

The algebra $C(X)$ is projective free.

  2. (2)

The algebra $C_{b}(X)$ is projective free.

  3. (3)

The space $X$ is connected and each finite type complex vector bundle over $X$
is trivial.

  4. (4)

Each continuous map $X\rightarrow{\textrm{\bf Gr}}(\mathbb{C}^{\infty})$ with
a relatively compact image is homotopic, via a homotopy with relatively
compact image, to a constant map.

In what follows, for a topological space $Y$ and an abelian group $G$, the
$n^{\textrm{th}}$ Čech cohomology group of $Y$ with values in $G$ is denoted
by $H^{n}(Y,G)$, $n\in\mathbb{Z}_{+}$.

Let $\beta X$ be the Stone-Čech compactification of $X$. Then $\beta X$ is
naturally homeomorphic to the maximal ideal space of $C_{b}(X)$, and $C(\beta
X)$ is isometrically isomorphic to $C_{b}(X)$. Proposition 2.1 implies that if
$C(X)$ is projective free, then $C(\beta X)$ is projective free, and so $\beta
X$ is connected and all finite rank complex vector bundles over $\beta X$ are
trivial.

Since isomorphism classes of rank one complex vector bundles over $\beta X$
are in one-to-one correspondence (determined by assigning to a bundle its
first Chern class) with elements of $H^{2}(\beta X,\mathbb{Z})$, projective
freeness of $C(X)$ implies that $H^{2}(\beta X,\mathbb{Z})=0$. Also, in this
case Proposition 2.1 implies that the Grothendieck group $K_{0}(C_{b}(X))$ of
the algebraic $K$-theory for $C_{b}(X)$ (isomorphic to the group $K^{0}(\beta
X)$ of the Atiyah-Hirzebruch theory) is $\mathbb{Z}$. Since the Chern
character ch determines a ring isomorphism
$K_{0}(C_{b}(X))\otimes\mathbb{Q}\rightarrow H^{\textrm{even}}(\beta
X,\mathbb{Q})$, see, e.g., [30, Theorem 3, p.16-09], we obtain the following
result.

######  Corollary 2.2.

If the algebra $C(X)$ is projective free, then $\beta X$ is connected,
$H^{2}(\beta X,\mathbb{Z})=0,$ and $H^{2n}(\beta X,\mathbb{Q})=0$ for all
$n\geq 2$.

For instance, if $X$ is a connected closed orientable manifold such that the
algebra $C(X)$ is projective free, then the corollary along with the Poincaré
duality imply that $X$ has an odd dimension $n$ and $H^{k}(X,\mathbb{Q})=0$
for all $k\not\in\\{0,n\\}$. Note that even if $H^{k}(X,\mathbb{Z})=0$ for all
$k\not\in\\{0,n\\}$, $C(X)$ is not necessarily projective free. In fact, it is
not even necessarily Hermite; see Example 2.5 below.

Let $\Omega^{n}=X\times\mathbb{C}^{n}$ denote the standard trivial rank $n$
complex vector bundle over a topological space $X$. A complex vector bundle
$E$ over $X$ is stably trivial if there exist $m,n\in\mathbb{Z}_{+}$ such that
the bundle $E\oplus\Omega^{m}$ is isomorphic to $\Omega^{n}$.

Our next result can be easily deduced from [53, Theorem 2] as well.

######  Proposition 2.3.

The following statements are equivalent:

  1. (1)

The algebra $C(X)$ is Hermite.

  2. (2)

The algebra $C_{b}(X)$ is Hermite.

  3. (3)

Each stably trivial finite type complex vector bundle over $X$ is trivial.

Propositions 2.1 and 2.3 imply the following result.

######  Corollary 2.4.

Projective freeness and Hermiteness of the algebra $C(X)$ depend only on the
homotopy type of $X$.

######  Example 2.5.

Let $S^{n}$ denote the $n$-dimensional unit sphere. Cutting the sphere $S^{n}$
into two ‘bowls’ (each homeomorphic to an $n$-dimensional ball) such that
their intersection is a ‘collar’ $C$ containing the subsphere $S^{n-1}$, it
can be seen that the isomorphism classes $\textrm{Vect}_{k}(S^{n})$ of rank
$k$ complex vector bundles on $S^{n}$ are in a one-to-one correspondence with
the elements of the $(n-1)^{\textrm{st}}$ homotopy group $\pi_{n-1}(U(k))$ of
the unitary group $U(k)$. As the homotopy group
$\pi_{4}(U(2))\cong\pi_{3}(S^{4})\cong\mathbb{Z}_{2}$, it follows that there
exists a nontrivial rank $2$ complex vector bundle $E$ over $S^{5}$. Hence,
$C(S^{5})$ is not projective free, although $H^{k}(S^{5},\mathbb{Z})=0$ for
all $k\not\in\\{0,5\\}$. Moreover, since
$K_{0}(C(S^{5}))=K^{0}(S^{5})=\mathbb{Z}$ (see, e.g., [29, Corollary 5.2 on
p.121 and p.150 of Chap. 11]), every finite rank complex vector bundle on
$S^{5}$ is stably trivial. Hence, $C(S^{5})$ is not Hermite, as $E$ is
nontrivial.

Next, we formulate an auxiliary result used in our proofs which follows
easily, e.g., from [15, Theorem 11.9, p.287], [32, Lemmas 1, 2], and [15,
Theorem 3.1, p.261].

For preliminaries on projective and injective limits, see, e.g., [15, Chap.
VIII, §2]. Recall that the projective limit of a projective system
$((X_{i})_{i\in I},(f_{ij})_{i\leq j\in I})$ of nonempty compact Hausdorff
spaces is a nonempty compact Hausdorff space in the limit topology.
Henceforth, the projective limit will be denoted by $\varprojlim X_{i}$, with
the projective system $((X_{i}),(f_{ij}))$ being understood. As usual, the
abbreviation ANR stands for absolute neighbourhood retract (for the
corresponding definition, see, e.g., [28, p.80]). Throughout the article,
$\cong$ denotes an isomorphism of objects of a category in question, and
${\begin{smallmatrix}\phantom{g}\\\ {\scaleobj{1.33}{\sim}}\\\\[-2.56064pt]
{\scaleobj{0.72}{h}}\end{smallmatrix}}$ a homotopy of maps or a homotopy
equivalence of topological spaces.

######  Lemma 2.6.

Suppose $X=\varprojlim X_{i}$ and $\pi_{i}:X\rightarrow X_{i}$ are canonical
continuous projections, where the $X_{i}$ are nonempty compact Hausdorff
spaces, and $Y$ is an ANR.

  * (1)

Given a continuous map $f:X\rightarrow Y,$ there exists an index $i$ and a
continuous map $f_{i}:X_{i}\rightarrow Y$ such that
$f_{i}\circ\pi_{i}{\begin{smallmatrix}\phantom{g}\\\
{\scaleobj{1.33}{\sim}}\\\\[-2.56064pt]
{\scaleobj{0.72}{h}}\end{smallmatrix}}f$.

  * (2)

Given a finite rank complex vector bundle $E$ over $X,$ there exists an index
$i$ and a complex vector bundle $E_{i}$ over $X_{i}$ such that the pullback
bundle $\pi_{i}^{*}(E_{i})\cong E$. Moreover, if $E$ is stably trivial, then
$E_{i}$ is stably trivial as well.

  * (3)

For each $k\in\mathbb{Z}_{+}$, the Čech cohomology group
$H^{k}(X,\mathbb{Z})=\cup_{i}\,\pi_{i}^{*}(H^{k}(X_{i},\mathbb{Z}))$.111In
other words, $H^{k}(X,\mathbb{Z})\cong\varinjlim H^{k}(X_{i},\mathbb{Z})$, the
injective limit of the related injective system of groups
$(H^{k}(X_{i},\mathbb{Z}),f_{ij}^{*})$.

The following result expresses continuity of projective freeness and
Hermiteness for algebras $C(X)$ on compact Hausdorff spaces $X$.

######  Proposition 2.7.

Suppose $X{\begin{smallmatrix}\phantom{g}\\\
{\scaleobj{1.33}{\sim}}\\\\[-2.56064pt]
{\scaleobj{0.72}{h}}\end{smallmatrix}}\varprojlim X_{i}$, where the $X_{i}$
are nonempty compact Hausdorff spaces such that either all algebras $C(X_{i})$
are projective free, or all algebras $C(X_{i})$ are Hermite. Then $C(X)$ is
projective free or Hermite, respectively.

###### Proof.

Due to Corollary 2.4, we can assume $X=\varprojlim X_{i}$. Then the required
result follows from Propositions 2.1, 2.3 (applied to the algebras $C(X_{i})$)
and Lemma 2.6. ∎

To formulate our next result, we recall some definitions.

The covering dimension of a topological space $X$, denoted by $\dim X$, is the
smallest integer $d$ such that every open covering of $X$ has an open
refinement of order at most $d+1$. If no such integer exists, then $X$ is said
to have infinite covering dimension.

Given a finite open cover $\alpha=\\{U_{1},\dots,U_{r}\\}$ of a compact
topological space $X$, its nerve is the abstract simplicial complex
$A_{\alpha}$, whose vertex set is $\alpha$ and such that the simplex
$\sigma=[U_{i_{0}},U_{i_{1}},\dots,U_{i_{n}}]\subset A_{\alpha}$ if and only
if $\cap_{j=0}^{n}U_{i_{j}}\neq\emptyset$.

We also recall that a path-connected topological space $X$ is $n$-simple if
for each $x\in X$, the fundamental group $\pi_{1}(X,x_{0})$ acts trivially on
the $n^{\textrm{th}}$ homotopy group $\pi_{n}(X,x_{0})$, see, e.g., [27, Chap.
IV, §16] for details. For instance, every path-connected topological group is
$n$-simple for all $n\in\mathbb{N}$.

The following result, generalising [7, Corollary 1.4] and [52, Theorem 5],
gives a sufficient condition for $C(X)$ to be projective free or Hermite.

######  Theorem 2.8.

If $X{\begin{smallmatrix}\phantom{g}\\\
{\scaleobj{1.33}{\sim}}\\\\[-2.56064pt]
{\scaleobj{0.72}{h}}\end{smallmatrix}}\varprojlim X_{i}$, where the
$X_{i}\,(\neq\emptyset)$ are finite-dimensional compact Hausdorff spaces such
that $H^{n}(X_{i},\mathbb{Z})=0$ for all $n\geq 5,$ then $C(X)$ is Hermite. In
addition, if all spaces $X_{i}$ are connected and satisfy
$H^{2}(X_{i},\mathbb{Z})=H^{4}(X_{i},\mathbb{Z})=0,$ then $C(X)$ is projective
free.

###### Proof.

In the proof, we use the fact that the rank of a stably trivial complex vector
bundle or a complex vector bundle over a connected space is well-defined. Due
to Proposition 2.7, it suffices to prove the result for each space $X_{i}$.

In what follows, we use the terminology from [26]. So $A_{\alpha}$ denotes the
nerve of a finite open covering $\alpha=\\{U_{1},U_{2},\dots,U_{r}\\}$ of
$X_{i}$ and $\phi_{\alpha}:X_{i}\rightarrow A_{\alpha}$ denotes a canonical
map of $\alpha$, i.e., for each point $x\in X_{i}$, $\phi_{\alpha}(x)$ is
contained in the closure of the simplex
$[U_{i_{0}},U_{i_{1}},\dots,U_{i_{n}}]$ of $A_{\alpha}$, where
$U_{i_{0}},U_{i_{1}},\dots,U_{i_{n}}$ denote the members of $\alpha$
containing $x$.

Given a continuous map $f:X_{i}\rightarrow Y$, a continuous map
$\psi_{\alpha}:A_{\alpha}\rightarrow Y$ is called a bridge if
$\psi_{\alpha}\circ\phi_{\alpha}{\begin{smallmatrix}\phantom{g}\\\
{\scaleobj{1.33}{\sim}}\\\\[-2.56064pt]
{\scaleobj{0.72}{h}}\end{smallmatrix}}f$ for each canonical map
$\phi_{\alpha}:X_{i}\rightarrow A_{\alpha}$ of the covering $\alpha$. The
family $(A_{\alpha})_{\alpha}$, together with the natural simplicial
projection maps $p_{\alpha}^{\beta}:A_{\beta}\rightarrow A_{\alpha}$, where
$\beta$ is a finite refinement of $\alpha$, forms a projective system with
limit $X_{i}$.

Let $E_{i}$ be a stably trivial complex vector bundle over $X_{i}$. Applying
Proposition 2.6 (2), we obtain for some $\alpha$, a stably trivial bundle
$E_{\alpha}$ over $A_{\alpha}$ such that $\phi_{\alpha}^{*}(E_{\alpha})\cong
E_{i}$; in particular its Chern classes $c_{k}(E_{\alpha})=0\,(\in
H^{2k}(A_{\alpha},\mathbb{Z}))$, $k\in\mathbb{N}$. If $E_{\alpha}$ is of
complex rank one, then the latter implies that $E_{i}$ is trivial. Next,
suppose $E_{\alpha}$ has complex rank $n\geq 2$. Let $A_{\alpha}^{4}$ be the
$4^{\textrm{th}}$ skeleton of $A_{\alpha}$, that is, the totality of the
simplexes of $A_{\alpha}$ with dimensions not exceeding $4$. Consider the
bundle $E^{\prime}:=E_{\alpha}|_{A_{\alpha}^{4}}$. Since the rank $n$ of
$E_{i}$ is $\geq 2$, $E^{\prime}$ is isomorphic to the Whitney sum
$E^{\prime\prime}\oplus\Omega^{n-2}$, where
$\Omega^{n-2}:=A_{\alpha}^{4}\times\mathbb{C}^{n-2},$ and $E^{\prime\prime}$
is a rank $2$ complex vector bundle over $A_{\alpha}^{4}$ (see [29, Part II,
Chap. 9, Theorem 1.2]). Since the Chern classes of $E^{\prime}$ are zeros, the
latter implies that Chern classes of $E^{\prime\prime}$ are zeros as well. In
turn, $A_{\alpha}^{4}$ is four-dimensional, and therefore the vanishing of
Chern classes of $E^{\prime\prime}$ implies that $E^{\prime\prime}$ is trivial
(it follows, e.g., from [35, Problem 14-C, p.171]). Hence, the bundle
$E_{\alpha}|_{A_{\alpha}^{4}}=E^{\prime}=E^{\prime\prime}\oplus\Omega^{n-2}$
is trivial as well. In particular, due to the construction of [29, Part I,
Theorem 3.5.5], there is a continuous map into a complex Grassmanian,
$h:A_{\alpha}\rightarrow{\textrm{\bf Gr}}_{n}(\mathbb{C}^{n+k})$, constant in
a neighbourhood of $A_{\alpha}^{4}$, such that $h^{*}(\gamma_{n,n+k})\cong
E_{\alpha}$, where $\gamma_{n,n+k}$ is the tautological bundle over
$\textrm{\bf Gr}_{n}(\mathbb{C}^{n+k})$. Let the value of $h$ on
$A_{\alpha}^{4}$ be $x$. Consider the continuous maps
$f_{0}:=h\circ\phi_{\alpha}:X_{i}\rightarrow{\textrm{\bf
Gr}}_{n}(\mathbb{C}^{n+k})$ and $f_{1}:X_{i}\rightarrow{\textrm{\bf
Gr}}_{n}(\mathbb{C}^{n+k})$ of constant value $x$. According to the definition
of $n$-homotopy222Let $X$ be a normal space, $Y$ be a connected ANR, and
either $X$ or $Y$ be compact. Continuous maps $f_{0},f_{1}:X\rightarrow Y$ are
$n$-homotopic if there exists a bridge $\alpha$ for the pair $(f_{0},f_{1})$
with bridge map $\psi_{i}:A_{\alpha}\rightarrow Y$ for $f_{i}$ ($i=0,1$) such
that $\psi_{0}=\psi_{1}$ on the $n$-dimensional skeleton $A_{\alpha}^{n}$ of
the nerve $A_{\alpha}$ of the covering $\alpha$ of $X$. in [26, §6], $f_{0}$
and $f_{1}$ are $4$-homotopic. Moreover, by the hypotheses we have
$H^{s}(X_{i},\mathbb{Z})=0$ for all $s\geq 5$ which implies that
$H^{s}(X,G)=0$ for all $s\geq 5$, for any abelian group $G$ (by the Universal
Coefficient Theorem for Čech cohomology [45, Chap. 6, §8, Theorem 10]). In
particular, $H^{s}(X,\pi_{s})=0$ for all $s\geq 5$ where $\pi_{s}$ is the
$s^{\textrm{th}}$ homotopy group of ${\textrm{\bf Gr}}_{n}(\mathbb{C}^{k+n})$.
Since the latter is a simply connected manifold, it is $r$-simple for all
$r\geq 1$. Hence, the previous facts based on [28, Theorem 8.1] imply that
$f_{0}$ and $f_{1}$ are homotopic maps. Thus $f_{0}^{*}(\gamma_{n,n+k})$ and
$f_{1}^{*}(\gamma_{n,n+k})$ are isomorphic bundles over $X_{i}$. But the
former is isomorphic to the original bundle $E_{i}$ by our construction, and
the latter is trivial. This proves that $E_{i}$ is trivial. Thus every stably
trivial complex vector bundle over $C(X_{i})$ is trivial. Hence, all
$C(X_{i})$, and therefore $C(X)$, are Hermite due to Proposition 2.7.

Now, assume in addition that $X_{i}$ is connected and
$H^{2}(X_{i},\mathbb{Z})=H^{4}(X_{i},\mathbb{Z})=0$. Let $E_{i}$ be a finite
rank complex vector bundle over $X_{i}$. Then as above, there exist $\alpha$
and a complex vector bundle $E_{\alpha}$ over $A_{\alpha}$, such that
$\phi_{\alpha}^{*}(E_{\alpha})\cong E_{i}$. Due to Lemma 2.6 (3), we can
choose the $\alpha$ such that the Chern classes $c_{i}(E_{\alpha})=0\,(\in
H^{2i}(A_{\alpha},\mathbb{Z}))$, $i=1,2$. Hence, the complex vector bundle
$E^{\prime}:=E_{\alpha}|_{A_{\alpha}^{4}}$ is trivial. So arguing as above, we
obtain that the bundle $E_{i}$ is trivial. Thus every finite rank complex
vector bundle over $C(X_{i})$ is trivial. Hence, all $C(X_{i})$, and therefore
$C(X)$, are projective free by Proposition 2.7. ∎

We deduce from the theorem the following result.

######  Theorem 2.9.

Let $X$ be a Hausdorff paracompact space of finite covering dimension such
that $H^{n}(X,\mathbb{Z})=0$ for all $n\geq 5$. Then the algebra $C(X)$ is
Hermite. If, in addition, $X$ is connected and
$H^{2}(X,\mathbb{Z})=H^{4}(X,\mathbb{Z})=0,$ then $C(X)$ is projective free.

###### Proof.

Due to the Dieudonné theorem, a Hausdorff paracompact space is normal; see,
e.g., [16, Theorem 5.1.5]. So the embedding $X\hookrightarrow\beta X$ induces
a natural isomorphism $H_{f}^{*}(X,\mathbb{Z})\cong H^{*}(\beta
X,\mathbb{Z})$; here $H_{f}^{*}(X,\mathbb{Z})$ are Čech cohomology groups of
$X$ defined with respect to finite open coverings, see [15, p. 282]. Further,
for a paracompact space $X$, the groups $H^{*}(X,\mathbb{Z})$ coincide with
the Čech cohomology groups of $X$ defined with respect to numerable open
coverings. Thus, according to [10, Corollary (6.3)], the group
$H_{f}^{n}(X,\mathbb{Z})$ coincides with the Čech cohomology group
$H^{n}(X,\mathbb{Z})$ for all $n\geq 2$. These facts and our hypotheses imply
that $H^{n}(\beta X,\mathbb{Z})=0$ for all $n\geq 5$. Moreover, since $X$ is
normal, $\dim\beta X=\dim X$, and in particular, $\beta X$ is finite-
dimensional; see, e.g., [16, Theorem 7.1.17, p.390]. Thus, applying Theorem
2.8 to $\beta X$, we obtain that the algebra $C(\beta X)$ is Hermite. Then
Proposition 2.3 implies that the algebra $C(X)$ is Hermite as well.

In the second case, the previous argument implies that $H^{2}(\beta
X,\mathbb{Z})=H^{4}(\beta X,\mathbb{Z})=0$ also. Moreover, if $X$ is
connected, then $\beta X$ is connected as well. Then, under these additional
assumptions, Theorem 2.8 implies that the algebra $C(\beta X)$ is projective
free. Therefore due to Proposition 2.1, the algebra $C(X)$ is projective free
as well, as required. ∎

######  Example 2.10.

Let $X\subset\mathbb{R}^{5}$ be a closed subset. Then $H^{n}(X,\mathbb{Z})=0$
for all $n\geq 5$. (In fact, the Čech cohomology groups of a closed subset of
$\mathbb{R}^{m}$ are isomorphic to the injective limit of Čech cohomology
groups of its open neighbourhoods; see, e.g., [45, Chap. 6, §1, Theorem 12,
§8, Corollary 8]. Also, by a result due to Whitehead [55, Theorem 3.2], an
open subset $U$ of $\mathbb{R}^{m}$ is homotopy equivalent to an $m-1$
dimensional simplicial complex $\Gamma\subset U$. As the dimension of $\Gamma$
is $m-1$, $H^{n}(U,\mathbb{Z})=H^{n}(\Gamma,\mathbb{Z})=0$ for all $n\geq
m-1$.) Thus, due to Theorem 2.9, the algebra $C(X)$ is Hermite.

In connection with Theorems 2.8, 2.9 the following question seems quite
natural:

######  Question.

Is there a Hausdorff topological space $X$ with $H^{n}(X,\mathbb{Z})\neq 0$
for some $n\geq 5$ such that the algebra $C(X)$ is Hermite?

##  3\. Stein Algebras

For basic facts about complex analytic spaces and Stein spaces we refer the
readers to the book [24].

Let $\Gamma(X,\mathcal{O}_{X})$ be the ring of global sections of the
structure sheaf $\mathcal{O}_{X}$ on a finite-dimensional complex analytic
space $(X,\mathcal{O}_{X})$. There is a natural algebra homomorphism
$\,\hat{\,}:\Gamma(X,\mathcal{O}_{X})\rightarrow C(X)$ with image
$\mathcal{O}(X)$, the ring of holomorphic functions on $X$, injective if
$(X,\mathcal{O}_{X})$ is reduced. A space $(X,\mathcal{O}_{X})$ is said to be
Stein if it is holomorphically convex (i.e., for each infinite discrete set
$D\subset X$ there exists an $f\in\mathcal{O}(X)$ which is unbounded on $D$)
and holomorphic separable (i.e., for all $x,y\in X$, $x\neq y$, there exists
an $f\in\mathcal{O}(X)$ such that $f(x)\neq f(y)$).

By the Cartan and Oka theorem, the nilradical $\mathfrak{n}(\mathcal{O}_{X})$
of $\mathcal{O}_{X}$ (i.e., the union of nilradicals of stalks
$\mathcal{O}_{x}$, $x\in X$) is a coherent sheaf of ideals on $X$ and so if
$(X,\mathcal{O}_{X})$ is Stein, then by Cartan’s Theorem B we have the
following exact sequence of global sections of sheaves

$0\rightarrow\Gamma(X,\mathfrak{n}(\mathcal{O}_{X}))\rightarrow\Gamma(X,\mathcal{O}_{X})\stackrel{{\scriptstyle
r^{*}}}{{\rightarrow}}\Gamma(X,\mathcal{O}_{{\rm red}\,X})\rightarrow 0,$
(3.1)

where $\mathcal{O}_{{\rm
red}\,X}:=\mathcal{O}_{X}/\mathfrak{n}(\mathcal{O}_{X})$ is the structure
sheaf on the reduction of $X$. It easily seen that
$\Gamma(X,\mathfrak{n}(\mathcal{O}_{X}))$ is the Jacobson radical of
$\Gamma(X,\mathcal{O}_{X})$, i.e., the intersection of all maximal ideals of
$\Gamma(X,\mathcal{O}_{X})$; see, e.g., [18, §1.4]. Moreover, the algebra
$\Gamma(X,\mathcal{O}_{X})$ is
$\Gamma(X,\mathfrak{n}(\mathcal{O}_{X}))$-complete, i.e., the natural
homomorphism from $\Gamma(X,\mathcal{O}_{X})$ to the projective limit of
quotient algebras
$\varprojlim\Gamma(X,\mathcal{O}_{X})\big{/}\Gamma(X,\mathfrak{n}(\mathcal{O}_{X}))^{N}$
is an isomorphism333Equivalently, in the topology on
$\Gamma(X,\mathcal{O}_{X})$ determined by letting the family of ideals
$\left\\{\Gamma(X,\mathfrak{n}(\mathcal{O}_{X}))^{N}\right\\}_{N\in\mathbb{N}}$
be a base of open neighbourhoods of $0$, every Cauchy sequence converges to a
unique limit., see, e.g., [24, Ch. V, §4.3].

######  Theorem 3.1.

Let $(X,\mathcal{O}_{X})$ be a finite-dimensional Stein space. The
homomorphism $\,\hat{\,}:\Gamma(X,\mathcal{O}_{X})\rightarrow C(X)$ induces a
bijection between isomorphism classes of finitely generated projective
$\Gamma(X,\mathcal{O}_{X})$ and $C(X)$ modules.

###### Proof.

Since the algebra $\Gamma(X,\mathcal{O}_{X})$ is
$\Gamma(X,\mathfrak{n}(\mathcal{O}_{X}))$-complete, the correspondence

$P\cong\Gamma(X,\mathcal{O}_{X})\otimes_{\Gamma(X,\mathcal{O}_{X})}\\!P\stackrel{{\scriptstyle
r^{*}\otimes{\rm id}_{P}}}{{\xrightarrow{\hskip
19.91692pt}}}\Gamma(X,\mathcal{O}_{{\rm
red}\,X})\otimes_{\Gamma(X,\mathcal{O}_{X})}\\!P$ (3.2)

determines a bijection between isomorphism classes of finitely generated
projective $\Gamma(X,\mathcal{O}_{X})$ modules and finitely generated
projective $\Gamma(X,\mathcal{O}_{{\rm red}\,X})$ modules, see, e.g., [50,
Theorem 2.26].

For the reduced Stein space $(X,\mathcal{O}_{{\rm red}\,X})$, the algebra
$\Gamma(X,\mathcal{O}_{{\rm red}\,X})$ can be naturally identified with
$\mathcal{O}(X)$. Then it follows from [19, Sätze 6.7, 6.8] (see also [37,
Theorem 2.1]) that there is a bijection between isomorphism classes of
finitely generated projective $\mathcal{O}(X)$ modules and isomorphism classes
of holomorphic vector bundles over $X$ of bounded rank.444I.e., the complex
ranks of restrictions of such bundles to connected components of $X$ are
uniformly bounded from above. Moreover, according to the Grauert theorem (see
[22], [23] and [11]) the inclusion of sheaves $i:\mathcal{O}_{{\rm
red}\,X}\hookrightarrow C_{X}$ (the sheaf of germs of continuous functions on
$X$) induces a bijection between isomorphism classes of holomorphic and
continuous complex vector bundles over $X$ of bounded rank. Next, since $X$ is
a Hausdorff paracompact of finite covering dimension (by the definition of a
finite-dimensional complex analytic space), each continuous complex vector
bundle over $X$ is of bounded rank if and only it is of finite type (see,
e.g., [29, Ch. 3, Proposition 5.4]) and, hence, by Swan’s theorem (see [53,
Theorem 2]), there is a bijection between isomorphism classes of continuous
complex vector bundles over $X$ of bounded rank and of finitely generated
projective $C(X)$ modules. This implies that the correspondence

$P\cong\mathcal{O}(X)\otimes_{\mathcal{O}(X)}\\!P\stackrel{{\scriptstyle
i\otimes{\rm id}_{P}}}{{\xrightarrow{\hskip
19.91692pt}}}C(X)\otimes_{\mathcal{O}(X)}\\!P$ (3.3)

determines a bijection between isomorphism classes of finitely generated
projective $\mathcal{O}(X)$ modules and finitely generated projective $C(X)$
modules.

The composition of the bijections in (3.2) and (3.3) gives the required
statement: the correspondence

$P\cong\Gamma(X,\mathcal{O}_{X})\otimes_{\Gamma(X,\mathcal{O}_{X})}\\!P\stackrel{{\scriptstyle\,\hat{\,}\,\otimes{\rm
id}_{P}}}{{\xrightarrow{\hskip
19.91692pt}}}C(X)\otimes_{{\Gamma(X,\mathcal{O}_{X})}}\\!P$ (3.4)

determines a bijection between isomorphism classes of finitely generated
projective $\Gamma(X,\mathcal{O}_{X})$ modules and finitely generated
projective $C(X)$ modules. ∎

Theorems 3.1 and 2.9 imply the following:

######  Theorem 3.2.

The algebra $\Gamma(X,\mathcal{O}_{X})$ is projective free or Hermite if and
only if the algebra $C(X)$ is projective free or Hermite.
In particular, if $H^{n}(X,\mathbb{Z})=0$ for all $n\geq 5$, then
$\Gamma(X,\mathcal{O}_{X})$ is Hermite, and if, in addition, $X$ is connected
and $H^{2}(X,\mathbb{Z})=H^{4}(X,\mathbb{Z})=0$, then it is projective free.

######  Example 3.3.

(1) According to [25], a reduced Stein space $X$ of (complex) dimension $k$ is
homotopy equivalent to a $k$-dimensional $CW$ complex. Hence,
$H^{n}(X,\mathbb{Z})=0$ for all $n>k$. Thus, due to Theorem 3.2 if $X$ is of
dimension $\leq 4$, then $\mathcal{O}(X)$ is Hermite, and if $X$ is one-
dimensional and connected, then $\mathcal{O}(X)$ is projective free.

(2) Let $U$ be an open subset of a Stein manifold $X$. Equipped with the
topology of uniform convergence on compact subsets of $U$, the algebra
$\mathcal{O}(U)$ becomes a complex Fréchet space. Each nonzero homomorphism
$\mathcal{O}(U)\to\mathbb{C}$ is an element of the dual space
$\mathcal{O}(U)^{*}$ (see, e.g., [24, Chap. 5, §7.1]). The space of such
homomorphisms equipped with the weak-$*$ topology of $\mathcal{O}(U)^{*}$ is
denoted by $M(\mathcal{O}(U))$. If $f\in\mathcal{O}(U)$, then $\hat{f}\in
C(M(\mathcal{O}(U)))$ is defined by $\hat{f}(\alpha)=\alpha(f)$ for each
$\alpha\in M(\mathcal{O}(U))$.

Since $X$ is Stein, $M(\mathcal{O}(X))=X$ (see, e.g., [24, Chap. 5, §7]), so
we have the natural restriction map $\pi_{U}:M(\mathcal{O}(U))\rightarrow X$
given by $\pi_{U}(\alpha)(f)=\alpha(f|_{U})$. Rossi [40] has shown that
$M(\mathcal{O}(U))$ admits the structure of a Stein manifold in such a way
that: (i) the map $U\rightarrow M(\mathcal{O}(U))$ sending $z\in U$ to the
evaluation homomorphism at $z$ is a biholomorphism of $U$ with an open subset
of $M(\mathcal{O}(U))$ (we will regard $U$ as an open subset of
$M(\mathcal{O}(U))$); (ii) if $f\in\mathcal{O}(U)$, then $\hat{f}$ is the
unique holomorphic extension of $f$ to $M(\mathcal{O}(U))$ (so that
$\mathcal{O}(M(\mathcal{O}(U)))\cong\mathcal{O}(U))$; (iii) $\pi_{U}$ is
locally a biholomorphism. This and Theorem 3.2 imply that

$\bullet$ if $X$ is of dimension $\leq 4$, then the algebra $\mathcal{O}(U)$
is Hermite.

Assume, in addition, that the set $U$ is holomorphically contractible (e.g.,
$X=\mathbb{C}^{k}$ and $U\subset X$ is a star-shaped domain), then

$\bullet$ the algebra $\mathcal{O}(U)$ is projective free.

Indeed, let the holomorphic contraction be given by a continuous map
$H:U\times[0,1]\to U$, so that $H(\cdot,1)={\rm id}_{U}$, $H(\cdot,0)=z_{o}\in
U$, and $H(\cdot,t):U\to U$ is holomorphic for all $t\in[0,1]$. Then $H$
determines the map $H^{*}$ from $[0,1]$ to the set of homomorphisms
$\mathcal{O}(U)\to\mathcal{O}(U)$ given by $(H^{*}(t))(f)=f(H(\cdot,t))$. The
transpose of each $H^{*}(t)$ induces a holomorphic map
$\hat{H}(\cdot,t):M(\mathcal{O}(U))\to M(\mathcal{O}(U))$ such that
$\hat{H}(\cdot,t)|_{U}=H(\cdot,t)$. Let us show that
$\hat{H}:M(\mathcal{O}(U))\times[0,1]\to M(\mathcal{O}(U))$ is continuous. To
this end, let $\\{(z_{n},t_{n})\\}_{n\in\mathbb{N}}\subset
M(\mathcal{O}(U))\times[0,1]$ be a sequence converging to $(z,t)\in
M(\mathcal{O}(U))\times[0,1]$. For each $f\in\mathcal{O}(U)$,

${\scaleobj{0.97}{\begin{array}[]{l}\displaystyle\varlimsup\limits_{n\to\infty}|(\hat{H}(z_{n},t_{n})-\hat{H}(z,t))(f)|\\\\[8.5359pt]
\leq\\!\varlimsup\limits_{n\to\infty}|(\hat{H}(z_{n},t_{n})-\hat{H}(z_{n},t))(f)|+\varlimsup\limits_{n\to\infty}|(\hat{H}(z_{n},t)-\hat{H}(z,t))(f)|\\\\[8.5359pt]
=\\!\varlimsup\limits_{n\to\infty}|z_{n}(f(H(\cdot,t_{n}))\\!-\\!f(H(\cdot,t)))|\\!+\\!\varlimsup\limits_{n\to\infty}|(z_{n}\\!-\\!z)(f(H(\cdot,t)))|\\!=:\\!I\\!+\\!II.\end{array}}}$

By the definition of convergence in the weak-∗ topology, the limit $II$ equals
$0$. Similarly, by continuity of $H$, the sequence of functions
$\\{f(H(\cdot,t_{n}))\\}_{n\in\mathbb{N}}\subset\mathcal{O}(U)$ converges
uniformly on compact subsets of $U$ to the function
$f(H(\cdot,t))\in\mathcal{O}(U)$. Since $z_{n}\in\mathcal{O}(U)^{*}$, the
latter implies that the limit $\,I$ equals $0$ as well. Hence,
$\lim_{n\to\infty}\hat{H}(z_{n},t_{n})=\hat{H}(w,t)$ in the topology of
$M(\mathcal{O}(U))$, as required.

Thus, $\hat{H}:M(\mathcal{O}(U))\times[0,1]\to M(\mathcal{O}(U))$ is a
homotopy between $\hat{H}(\cdot,1)={\rm id}_{M(\mathcal{O}(U))}$ and
$\hat{H}(\cdot,0)=z_{o}$, i.e., the Stein manifold $M(\mathcal{O}(U))$ is
holomorphically contractible. From here and Theorem 3.2, it follows that the
algebra $\mathcal{O}(U)\cong\mathcal{O}(M(\mathcal{O}(U)))$ is projective
free.

##  4\. Commutative Unital Complex Banach Algebras

Recall that for a commutative unital complex Banach algebra $A$, the maximal
ideal space $M(A)\subset A^{\ast}$ is the set of nonzero homomorphisms
$A\\!\rightarrow\\!\mathbb{C}\;$ 555Every such homomorphism is continuous,
see, e.g., [33, §23(A), Theorem]. endowed with the Gelfand topology, the
weak-$\ast$ topology of $A^{\ast}$. It is a compact Hausdorff space contained
in the unit sphere of $A^{\ast}$. The Gelfand transform
$\,\hat{\,}:A\rightarrow C(M(A))$, defined by $\hat{a}(\varphi):=\varphi(a)$
for $a\in A$ and $\varphi\in M(A)$, is a nonincreasing-norm morphism of Banach
algebras.

######  Theorem 4.1.

Let $A$ be a commutative unital complex Banach algebra. Then:

  * (1)

$A$ is projective free if and only if $C(M(A))$ is projective free.

  * (2)

$A$ is Hermite if and only if $C(M(A))$ is Hermite.

In particular, if $M(A){\begin{smallmatrix}\phantom{g}\\\
{\scaleobj{1.33}{\sim}}\\\\[-2.56064pt]
{\scaleobj{0.72}{h}}\end{smallmatrix}}\varprojlim X_{i}$, where the $X_{i}$
are finite-dimensional compact Hausdorff spaces such that
$H^{n}(X_{i},\mathbb{Z})=0$ for all $n\geq 5,$ then $A$ is Hermite. If, in
addition, each space $X_{i}$ is connected and
$H^{2}(X_{i},\mathbb{Z})=H^{4}(X_{i},\mathbb{Z})=0,$ then $A$ is projective
free.

Parts (1) and (2) of the theorem follow from a one-to-one correspondence
(determined via the Gelfand transform) between the isomorphism classes of
finitely generated projective $A$ modules and the isomorphism classes of
complex vector bundles over $M(A)$ (see [38], and also [51, §7.5, Theorem on
p.199]) along with the Swan theorem [49, Theorem 2]. In turn, the last
statement follows from Theorem 2.8.

Let $\bar{A}$ be the uniform closure in $C(M(A))$ of the image under the
Gelfand transform of algebra $A$. It is known that $M(\bar{A})=M(A)$, see,
e.g., [41, Proposition 3]. Then we obtain from Theorem 4.1:

######  Corollary 4.2.

$A$ is projective free or Hermite if and only if $\bar{A}$ is projective free
or Hermite.

######  Example 4.3.

(1) Let $\textrm{L}^{1}[0,1]$ be the Banach space of complex-valued Lebesgue
integrable functions on $[0,1]$ with the norm
$\|f\|_{1}:=\int_{0}^{1}|f(t)|dt$. The space $\textrm{L}^{1}[0,1]$ equipped
with the multiplication given by truncated convolution $(f\ast
g)(t):=\int_{0}^{t}f(\tau)g(t-\tau)d\tau$ becomes a complex commutative Banach
algebra $V$ called the Volterra algebra. The algebra $V$ is non-unital without
maximal ideals, see, e.g., [36, Example 9.82]. Let $V_{1}$ denote the algebra
of pairs $(f,c)$, where $f\in V$ and $c\in\mathbb{C}$ with addition and
multiplication given by

$(f,c)+(f^{\prime},c^{\prime}):=(f+f^{\prime},c+c^{\prime}),\quad(f,c)\cdot(f^{\prime},c^{\prime}):=(f\ast
f^{\prime}+c\cdot f^{\prime}+c^{\prime}\cdot f,c\cdot c^{\prime}).$

We equip $V_{1}$ with the norm $\|(f,c)\|:=\|f\|_{1}+|c|$. Then $V_{1}$
becomes a commutative unital complex Banach algebra. Since $V$ is without
maximal ideals, $V\times\\{0\\}$ is the only maximal ideal of $V_{1}$. Thus
due to Theorem 4.1 the algebra $V_{1}$ is projective free.

(2) Let $A$ be a commutative unital complex Banach algebra such that the
algebra $\bar{A}\subset C(M(A))$ is generated by $k$ elements (this is true,
e.g., if $A$ itself is generated by $k$ elements). Then $A$ is Hermite if
$k\leq 5$, and projective free if $k\leq 2$ and $A$ does not contain
nontrivial idempotent elements. Indeed, in this case $M(A)$ is homeomorphic to
a polynomially convex subset of $\mathbb{C}^{k}$ (see, e.g., [20, Chap. III,
Theorem 1.4]). Recall that a compact set $K\subset\mathbb{C}^{k}$ is
polynomially convex if for every $\mathbf{z}\not\in K$ there is a polynomial
$p\in\mathbb{C}[z_{1},\dots,z_{k}]$ such that
$|p(\mathbf{z})|>\sup_{\mathbf{w}\in K}|p(\mathbf{w})|$. It is known, see,
e.g., [47, Corollary 2.3.6], that if $K\subset\mathbb{C}^{k}$ is a compact
polynomially convex set, then $H^{n}(K,\mathbb{Z})=0$ for all $n\geq k$. This
and Theorem 4.1 imply that $A$ is Hermite if $k\leq 5$. If $k\leq 2$ and $A$
does not contain nontrivial idempotent elements, then due to the Shilov
idempotent theorem (see, e.g., [20, Chap. III, Corollary 6.5]) $M(A)$ is
connected. Hence, in this case Theorem 4.1 implies that $A$ is projective
free.

(3) Let $L^{\infty}(S)$ be the Banach algebra of essentially bounded
measurable complex-valued functions on a measure space $S$, with pointwise
operations and the supremum norm. Then the maximal ideal space
$M(L^{\infty}(S))$ is totally disconnected (see, e.g., [20, Chap. I, Lemma
9.1]) and, hence, ${\rm dim}\,M(L^{\infty}(S))=0$. Thus, Theorem 4.1 implies
that the algebra $L^{\infty}(S)$ is Hermite.

(4) Let $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$ and
$\mathbb{T}:=\\{z\in\mathbb{C}:|z|=1\\}$. With pointwise operations and the
supremum norm, $\textrm{L}^{\infty}$ denotes the Banach algebra of essentially
bounded Lebesgue measurable functions on $\mathbb{T}$, and $H^{\infty}$ the
Banach algebra of all bounded holomorphic functions in $\mathbb{D}$. Via
identification with boundary values, $H^{\infty}$ is a uniformly closed
subalgebra of $\textrm{L}^{\infty}$. According to the Chang-Marshall theorem,
see, e.g., [21, Chap. IX, §3], any uniformly closed subalgebra $A$ between
$H^{\infty}$ and $\textrm{L}^{\infty}$ is a Douglas algebra generated by
$H^{\infty}$ and a family $\mathscr{B}_{A}\subset\overline{H^{\infty}}$ of
functions conjugate to some inner functions of $H^{\infty}$ (written
$A=[H^{\infty},\mathscr{B}_{A}]$). If $H^{\infty}\subsetneq A$, then the
maximal ideal space $M(A)$ is a closed subset of
$M(H^{\infty})\setminus\mathbb{D}$ of the form (see, e.g., [21, Chap. IX,
Theorem 1.3]):

$M(A)=\bigcap_{\bar{u}\in\mathscr{B}_{A}}\\{x\in
M(H^{\infty})\,:\,|x(u)|=1\\}.$

According to the results of Suárez [48], for each closed set $K\subset
M(H^{\infty})$ $\dim K\leq 2$ and $H^{2}(K,\mathbb{Z})=0$. This and Theorem
4.1 imply that $A$ is Hermite and it is projective free if $M(A)$ is
connected. Next, due to the Shilov idempotent theorem, $M(A)$ is connected if
and only if $A$ does not contain nontrivial idempotents in
$\textrm{L}^{\infty}$. For instance, $M(A)$ is connected if $A$ is one of the
algebras: $H^{\infty}$, $H^{\infty}+C$, where $C:=C(\mathbb{T})$, or
$B_{1}=[H^{\infty},C_{1}]$ (the closed subalgebra generated by $H^{\infty}$
and the Banach algebra $C_{1}$ of all complex-valued functions on $\mathbb{T}$
which are continuous except possibly at $z=1$ but which have one-sided limits
at $z=1$; for details, see e.g., [43] or [21, Chap. IX, Exercise 7]). Thus, in
these special cases, $A$ is projective free.

##  5\. The class $\mathscr{C}$

A topological space $X$ is said to be of trivial shape if every continuous map
from $X$ to an ANR is homotopic to a constant map; see, e.g., [34, p.248]. A
space of trivial shape generalises the notion of a contractible space, and, in
particular, if a space of trivial shape is homotopy equivalent to an ANR, then
it is contractible. If $X$ is a compact Hausdorff space of trivial shape, then
it is connected and Čech cohomology groups $H^{k}(X,\mathbb{Z})=0$ for all
$k\geq 1$.

We say that a commutative unital complex Banach algebra $A$ belongs to the
class $\mathscr{C}$ if $M(A)$ is a space of trivial shape. In this section, we
study some properties of class $\mathscr{C}$.

Let $B,C$ be unital closed subalgebras of a commutative unital complex Banach
algebra $\mathfrak{A}$, and let
$B\widehat{\otimes}_{\mathfrak{A}}C\subset\mathfrak{A}$ be the closure of the
subalgebra $\langle B,C\rangle$ generated by $B$ and $C$. Assume that $B,C$
and $\mathfrak{A}$ satisfy the condition:

> There exists a constant $c$ such that for all $\xi\in M(B)$ and every
> $n\in\mathbb{N}$ and $b_{i}\in B$, $c_{i}\in C$, $1\leq i\leq n$,
>
> $\left\|\sum_{k=1}^{n}\xi(b_{k})c_{k}\right\|_{C}\leq
> c\left\|\sum_{k=1}^{n}b_{k}c_{k}\right\|_{\mathfrak{A}}.$ (5.1)

Condition (5.1) is valid, e.g., if $\mathfrak{A}$ is isomorphic to (a) the
projective tensor product $B\widehat{\otimes}_{\pi}C$; (b) the injective
tensor product $B\widehat{\otimes}_{\varepsilon}C$ where either $B$ or $C$ is
a uniform algebra, see, e.g., [14, §1.3] for the references.

Let $i_{B}$ denote the embedding $B\hookrightarrow
B\;\\!\widehat{\otimes}_{\mathfrak{A}}C$.

######  Theorem 5.1.

If $C\in\mathscr{C}$, then the correspondence

$P\cong B\otimes_{B}\\!P\stackrel{{\scriptstyle i_{B}\otimes{\rm
id}_{P}}}{{\xrightarrow{\hskip
19.91692pt}}}(B\;\\!\widehat{\otimes}_{\mathfrak{A}}C)\otimes_{B}\\!P$ (5.2)

determines a bijection between isomorphism classes of finitely generated
projective $B$ modules and finitely generated projective
$B\;\\!\widehat{\otimes}_{\mathfrak{A}}C$ modules.

######  Remark 5.2.

The result shows that the class $\mathscr{C}$ is an analog of the class of
local rings (i.e., those with unique maximal ideals) in commutative algebra
(see, e.g., [1] for the corresponding definitions and results). Indeed, if $S$
is a local commutative ring with the maximal ideal $\mathfrak{m}$ and $R$ is a
commutative ring, then for the completion in the Krull topology
$\widehat{R\otimes S}_{I}:=\varprojlim(R\otimes S)/I^{n}$ with respect to the
ideal $I=R\otimes\mathfrak{m}$, we obtain an analog of Theorem 5.1, i.e., the
correspondence $R\mapsto\widehat{R\otimes S}_{I}$ induces a bijection between
isomorphism classes of finitely generated projective $R$ modules and finitely
generated projective $\widehat{R\otimes S}_{I}$ modules. Note that the
canonical map $\pi$ from $R\otimes S$ to $\widehat{R\otimes S}_{I}$ is
injective on $R\otimes 1_{S}\,(\cong R)$. If, in addition, $S$ is Noetherian,
then by the Krull intersection theorem,
$\cap_{n=1}^{\infty}\mathfrak{m}^{n}=0$ and, hence, $\pi$ is injective on
$1_{R}\times S\,(\cong S)$ (here $1_{S}$ and $1_{R}$ are units in $S$ and
$R$). Moreover, the subalgebra generated by $\pi(R\otimes 1_{S})$ and
$\pi(1_{R}\otimes S)$ is dense in $\widehat{R\otimes S}_{I}$. Thus, in this
case the completion $\widehat{R\otimes S}_{I}$ is an analog of
$B\widehat{\otimes}_{\mathfrak{A}}C$ in Theorem 5.1.

To prove Theorem 5.1, first, we prove the following general result.

######  Lemma 5.3.

Under the condition (5.1), $M(B\;\\!\widehat{\otimes}_{\mathfrak{A}}C)$ is
homeomorphic to $M(B)\\!\times\\!M(C)$.

###### Proof.

Condition (5.1) implies that for $\xi\in M(B)$ the map $\langle
B,C\rangle\rightarrow C$,

$\left(\sum_{k=1}^{n}b_{k}c_{k}\right)\mapsto\sum_{k=1}^{n}\xi(b_{k})c_{k}\in
C,$ (5.3)

extends by continuity to a bounded multiplicative projection

$P_{\xi}:B\widehat{\otimes}_{\mathfrak{A}}C\rightarrow C.$

In particular, $\eta\circ P_{\xi}\in
M(B\;\\!\widehat{\otimes}_{\mathfrak{A}}C)$ for each $\eta\in M(C)$. It is
easily seen that the map

$F:M(B)\\!\times\\!M(C)\rightarrow
M(B\;\\!\widehat{\otimes}_{\mathfrak{A}}C),\quad F(\xi,\eta):=\eta\circ
P_{\xi},$

is continuous. Also, $F$ is injective, as if $\eta_{1}\circ
P_{\xi_{1}}=\eta_{2}\circ P_{\xi_{2}}$ for some $(\xi_{i},\eta_{i})\in
M(B)\\!\times\\!M(C)$, $i=1,2$, then for all $b\in B$, $c\in C$,

$\xi_{1}(b)\eta_{1}(c)=\xi_{2}(b)\eta_{2}(c).$ (5.4)

Applying (5.4) with $c=1$ (the unit of $\mathfrak{A}$) and then again with
$b=1$, we get $\xi_{1}=\xi_{2}$ and $\eta_{1}=\eta_{2}$, as required.

Further, the map $F$ is surjective, as if $\varphi\in
M(B\;\\!\widehat{\otimes}_{\mathfrak{A}}C)$, then clearly
$\xi:=\varphi|_{B}\in M(B)$ and $\eta:=\varphi|_{C}\in M(C)$ and due to (5.3)

$\begin{split}F(\xi,\eta)\left(\sum_{k=1}^{n}b_{k}c_{k}\right)&=(\eta\circ
P_{\xi})\left(\sum_{k=1}^{n}b_{k}c_{k}\right)=\eta\left(\sum_{k=1}^{n}\varphi(b_{k})c_{k}\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
&=\sum_{k=1}^{n}\varphi(b_{k})\varphi(c_{k})=\varphi\left(\sum_{k=1}^{n}b_{k}c_{k}\right),\end{split}$

i.e., $F(\xi,\eta)=\varphi$.

This completes the proof of the lemma. ∎

###### Proof of Theorem 5.1.

Due to Lemma 5.3 without loss of generality we will identify
$M(B\;\\!\widehat{\otimes}_{\mathfrak{A}}C)$ with $M(B)\\!\times\\!M(C)$. Then
the transpose of $i_{B}$ restricted to
$M(B\;\\!\widehat{\otimes}_{\mathfrak{A}}C)$ is the map
$p_{B}:M(B)\\!\times\\!M(C)\rightarrow M(B)$, $p_{B}(x,y)=x$ for all $(x,y)\in
M(B)\\!\times\\!M(C)$.

According to the Novodvorski-Taylor theorem ([38], [51, §7.5]) and the Swan
theorem [49], to prove the result we must show that the pullback by $p_{B}$
determines a bijection between isomorphism classes of complex vector bundles
over $M(B)$ and $M(B)\times M(C)$. In turn, it suffices to prove the same for
complex vector bundles of constant rank over clopen subsets $U\subset M(B)$
and $U\times M(C)\subset M(B)\\!\times\\!M(C)$.

To this end, we present $U$ as $\varprojlim U_{i}$, where all $U_{i}$ are
finite-dimensional compact simplicial complexes (cf. the argument of the proof
of Theorem 2.8). Then $U\times M(C)=\varprojlim(U_{i}\times M(C))$. Moreover,
if $\pi_{i}:U\rightarrow U_{i}$ are canonical projections for the first limit,
then $\tilde{\pi}_{i}:=(\pi_{i},{\rm id}_{M(C)}):U\times M(C)\rightarrow
U_{i}\times M(C)$ are canonical projections for the second one.

Suppose $E$ is a complex vector bundle of rank $n$ over $U\times M(C)$. Due to
Lemma 2.6 (2) there is an index $i$ and a complex vector bundle $E_{i}$ over
$U_{i}\times M(C)$ such that the pullback bundle
$\tilde{\pi}_{i}^{*}(E_{i})\cong E$. Then there is a map
$h\\!\in\\!C(U_{i},{\textrm{\bf Gr}}_{n}(\mathbb{C}^{m}))\;$666For topological
spaces $X,Y$ we denote by $C(X,Y)$ the set of continuous maps from $X$ to $Y$.
from $U_{i}$ into a complex Grassmanian such that $h^{*}(\gamma_{n,m})\cong
E_{i}$, where $\gamma_{n,m}$ is the tautological bundle over $\textrm{\bf
Gr}_{n}(\mathbb{C}^{m})$, see, e.g., [29, Part I, Theorem 3.5.5]. Consider the
map $H:M(C)\rightarrow C(U_{i},\textrm{\bf Gr}_{n}(\mathbb{C}^{m}))$,

$H(y)(x):=h(x,y),\quad y\in M(C),\quad x\in U_{i}.$

Since $U_{i}$ is an ANR (as it is a compact simplicial complex; see, e.g.,
[28, Chap. III, Corollary 8.4]), and also $\textrm{\bf
Gr}_{n}(\mathbb{C}^{m})$ is an ANR (as it is a compact complex manifold [28,
Chap. III, Corollary 8.3]), the space $C(U_{i},\textrm{\bf
Gr}_{n}(\mathbb{C}^{m}))$ of continuous maps from $U_{i}$ to $\textrm{\bf
Gr}_{n}(\mathbb{C}^{m})$ equipped with the topology of uniform convergence is
an ANR as well; see, e.g. [28, Chapter VI, Theorem 2.4]. Moreover, since every
continuous map between compact metrisable spaces is uniformly continuous, the
map $H$ is continuous. Thus by the definition of a space of trivial shape, $H$
is homotopic to a constant map $M(C)\rightarrow C(U_{i},\textrm{\bf
Gr}_{n}(\mathbb{C}^{m}))$. This homotopy gives rise to a homotopy between $h$
and a map $h_{o}:U_{i}\times M(C)\rightarrow\textrm{\bf
Gr}_{n}(\mathbb{C}^{m})$, $h_{o}(x,y):=h(x,o)$ for all $(x,y)\in U_{i}\times
M(C)$, where $o\in M(C)$ is a fixed point. In particular, we have the
following isomorphisms of bundles

$E_{i}\cong h^{*}(\gamma_{n,m})\cong h_{o}^{*}(\gamma_{n,m})\cong
p_{i}^{*}(E_{i}),$ (5.5)

where $p_{i}:U_{i}\times M(C)\rightarrow U_{i}\times\\{o\\}$ is defined by the
formula $p_{i}(x,y)=(x,o)$ for all $x\in U_{i}$. Applying to (5.3) the
pullback map $\tilde{\pi}_{i}^{*}$ we obtain that the bundle $E$ is isomorphic
to the pullback by the natural projection $U\times M(C)\rightarrow
U\times\\{o\\}$ of the restriction of $E$ to $U\times\\{o\\}$. Since $p_{B}$
maps $U_{i}\times\\{o\\}$ homeomorphically onto $U$, this shows that $E$ is
isomorphic to a bundle pulled back by $p_{B}$ from $U$. Thus the map
$p_{B}^{*}$ determines a surjection between isomorphism classes of complex
vector bundles over $M(B)$ and $M(B)\times M(C)$. Clearly, it determines an
injection between these sets as well, since if $E_{1},E_{2}$ are bundles over
$M(B)$ such that the pullback bundles $p_{B}^{*}(E_{1})$ and
$p_{B}^{*}(E_{2})$ are isomorphic, then their restrictions to
$M(B)\times\\{o\\}$ are isomorphic and so $E_{1}$ and $E_{2}$ must be
isomorphic.

The proof of the theorem is complete. ∎

Theorem 5.1 leads straightforwardly to the following:

######  Corollary 5.4.

Let $B,C\subset\mathfrak{A}$ be Banach algebras satisfying conditions of
Theorem 5.1. If $C\in\mathscr{C}$ and $B$ is Hermite, then
$B\;\\!\widehat{\otimes}_{\mathfrak{A}}C$ is Hermite. If $C\in\mathscr{C}$ and
$B$ is projective free, then $B\;\\!\widehat{\otimes}_{\mathfrak{A}}C$ is
projective free.

Let $B,C$ be commutative unital complex Banach algebras and $\alpha$ be a norm
on the algebraic tensor product $B\otimes C$. Let
$\mathfrak{A}:=B\widehat{\otimes}_{\alpha}C$ be the completion of $B\otimes C$
with respect to $\alpha$. Suppose that $\mathfrak{A}$ is a Banach algebra.
Identifying $B$ and $C$ with subalgebras $B\otimes 1_{C}$ and $1_{B}\otimes C$
of $\mathfrak{A}$ (here $1_{B}$ and $1_{C}$ are units of $B$ and $C$), we
assume that the triple $B,C,\mathfrak{A}$ satisfies condition (5.1). Then we
have:

######  Proposition 5.5.

If $B,C\in\mathscr{C}$, then $\mathfrak{A}\in\mathscr{C}$.

###### Proof.

According to Lemma 5.3, $M(\mathfrak{A})$ is homeomorphic to the direct
product $M(B)\times M(C)$. Let $f:M(B)\times M(C)\rightarrow X$ be a
continuous map to an ANR $X$. Since by the hypotheses $M(C)$ is of trivial
shape, repeating the arguments of the proof of Theorem 5.1, we obtain that $f$
is homotopic to the continuous map $f_{o}:M(B)\times M(C)\rightarrow X$, where
$f_{o}(x,y):=f(x,o)$ for all $(x,y)\in M(B)\times M(C)$; here $o\in M(C)$ is a
fixed point. In turn, since $M(B)$ is of trivial shape, the restriction
$f_{o}|_{M(B)}$ is homotopic to a constant map. This implies that $f_{o}$
(and, hence, $f$) is homotopic to a constant map. Thus, $M(B)\times M(C)$ is
of trivial shape, as required. ∎

##  6\. Examples

In Sections 6.1 and 6.2 we present some examples of algebras of class
$\mathscr{C}$. We restrict ourselves to the case of semisimple algebras only.
The choice of examples of Sections 6.3 and 6.4 is based on the research
interests of the authors.

###  6.1. Banach algebras of symmetric functions

Recall that the polynomial convex hull of a bounded set
$K\subset\mathbb{C}^{n}$ is the minimal polynomially convex set
$\widehat{K}\Subset\mathbb{C}^{n}$ containing $K$, i.e.,

$\widehat{K}:=\bigl{\\{}\mathbf{z}\in\mathbb{C}^{n}:|p(\mathbf{z})|\leq\sup_{\mathbf{w}\in
K}|p(\mathbf{w})|\textrm{ for all
}p\in\mathbb{C}[z_{1},\dots,z_{n}]\bigr{\\}}.$

If $A$ is a finitely generated semisimple commutative unital complex Banach
algebra with generators $f_{1},\dots,f_{n}$, then the map

$F(x):=(x(f_{1}),\dots,x(f_{n})),\quad x\in M(A),$ (6.1)

is an embedding with image a polynomially convex subset of $\mathbb{C}^{n}$
(see, e.g., [20, Chap. III, Theorem 1.4]). If we identify $M(A)$ with
$F(M(A))$, then $A$ becomes a (not necessarily closed) subalgebra of
$C(F(M(A)))$ such that the restriction of polynomial algebra
$\mathbb{C}[z_{1},\dots,z_{n}]$ to $F(M(A))$ is dense in $A$. In what follows,
we consider a more general situation of a unital complex Banach algebra
$A\subset C_{b}(K)$ on a subset $K\Subset\mathbb{C}^{n}$ such that
$\mathbb{C}[z_{1},\dots,z_{n}]|_{K}$ is dense in $A$. Then $M(A)$ is naturally
identified with $\widehat{K}$.

Let $G\subset GL_{n}(\mathbb{C})$ be a finite group of order $|G|$. We say
that the Banach algebra $A\subset C(K)$ is $G$-invariant if $G(K)=K$ and $A$
is invariant with respect to the action of $G$ on $C(K)$: $g^{*}(f)=f\circ g$
for all $f\in C(K)$, $g\in G$. (Since $\mathbb{C}[z_{1},\dots,z_{n}]$ is
invariant with respect to the action of $G$, this implies that
$G(\widehat{K})=\widehat{K}$ as well.) In this case, each $g^{*}:A\to A$,
$g\in G$, is an automorphism of $A$, and since $A$ is semisimple, each $g^{*}$
is continuous (see, e.g., [33, §24B, Theorem]).

For a $G$-invariant algebra $A$, the subalgebra $A_{G}\subset A$ of elements
invariant with respect to the action of $G$ on $A$ (i.e., such $f\in A$ that
$g^{*}(f)=f$ for all $g\in G$) is said to be $G$-symmetric.

Let $\mathcal{P}_{G}\subset\mathbb{C}[z_{1},\dots,z_{n}]$ be the subalgebra of
polynomials invariant with respect to the action of $G$. If $f\in A_{G}$ and
$(p_{j})_{j\in\mathbb{N}}\subset\mathbb{C}[z_{1},\dots,z_{n}]$ is a sequence
such that $f=\lim_{j\to\infty}p_{j}|_{K}$, then all
$\tilde{p}_{j}:=\frac{1}{|G|}\sum_{g\in G}g^{*}(p_{j})\in\mathcal{P}_{G}$ and
$f=\lim_{j\to\infty}\tilde{p}_{j}|_{K}$ (because all $g^{*}:A\to A$ are
continuous). Hence, $\mathcal{P}_{G}|_{K}$ is a dense subalgebra of $A_{G}$.

By the Hilbert basis theorem, see, e.g., [1, Theorem 7.5], there exist
homogeneous polynomials $h_{1},\dots,h_{m}\in\mathbb{C}[z_{1},\dots,z_{n}]$,
$m\geq n$, invariant with respect to the action of $G$ which generate
$\mathcal{P}_{G}$.777For instance, if $G=S_{n}$ acts by permutation of
coordinates, then we can choose $h_{i}$ to be the elementary symmetric
polynomial of degree $i$, $1\leq i\leq n$. Hence, $A_{G}$ is generated by
elements $h_{1}|_{K},\dots,h_{m}|_{K}$. In particular, the continuous map
$F_{G}:M(A_{G})\to\mathbb{C}^{m}$,

$F_{G}(x):=(x(h_{1}|_{K}),\dots,x(h_{m}|_{K})),\quad x\in M(A_{G}),$ (6.2)

embeds $M(A_{G})$ into $\mathbb{C}^{m}$ as a polynomially convex subset. On
the other hand, by Lemma 7.1 we obtain that $M(A_{G})$ can be identified with
the quotient space $\widehat{K}/G$. In this identification, if
$\pi:\widehat{K}\to\widehat{K}/G$ is the quotient map, then $A_{G}$ is
isomorphic to a subalgebra $\tilde{A}_{G}\subset C(\widehat{K}/G)$ such that
$\pi^{*}(f)|_{K}\in A_{G}$ for all $f\in\tilde{A}_{G}$. Let
$\tilde{h}_{i}\in\tilde{A}_{G}$ be such that
$\pi^{*}(\tilde{h}_{i})=h_{i}|_{\widehat{K}}$, $1\leq i\leq m$. Then the map
$F_{G}$ becomes

$F_{G}(x):=(\tilde{h}_{1}(x)),\dots,\tilde{h}_{m}(x)),\quad
x\in\widehat{K}/G.$

This implies that the map
$H:=(h_{1},\dots,h_{m}):\mathbb{C}^{n}\to\mathbb{C}^{m}$ maps $\widehat{K}$
onto $F_{G}(\widehat{K}/G)$.

######  Remark 6.1.

The image $H(\mathbb{C}^{n})$ is an $n$-dimensional complex algebraic
subvariety of $\mathbb{C}^{m}$. For each polynomially convex set
$S\subset\mathbb{C}^{n}$ invariant with respect to the action of $G$ on
$\mathbb{C}^{n}$, the uniform algebra $P(S)$ (defined as the closure in $C(S)$
of the algebra $\mathbb{C}[z_{1},\dots,z_{n}]|_{S}$) is $n$-generated,
$G$-invariant, and $M(P(S))=S$. Hence, the map $H$ determines a one-to-one
correspondence $S\mapsto H(S)$ between $G$-invariant polynomially convex
subsets of $\mathbb{C}^{n}$ and polynomially convex subsets of
$\mathbb{C}^{m}$ lying in $H(\mathbb{C}^{n})$.

######  Proposition 6.2.

Suppose $A\subset C(K)$ is $G$-invariant for some finite group $G\subset
GL_{n}(\mathbb{C})$ and $K\Subset\mathbb{C}^{n}$ star-shaped with respect to
the origin. Then $A_{G}\in\mathscr{C}$. In particular, $A_{G}$ is projective
free.

###### Proof.

Since $K$ is star-shaped with respect to the origin and all $h_{i}$ are
homogeneous polynomials, we have $t\cdot\mathbf{z}\in K$ for all
$\mathbf{z}\in K$, $t\in[0,1]$ and

$H(t\cdot\mathbf{z})=\bigl{(}t^{{\rm deg}\,h_{1}}\cdot
h_{1}(\mathbf{z}),\dots,t^{{\rm deg}\,h_{m}}\cdot h_{m}(\mathbf{z})\bigr{)}.$

We set for $\mathbf{w}=(w_{1},\dots,w_{m})\in\mathbb{C}^{m}$ and $t\in[0,1]$,

$D(\mathbf{w},t):=\bigl{(}t^{{\rm deg}\,h_{1}}\cdot w_{1},\dots,t^{{\rm
deg}\,h_{m}}\cdot w_{m}\bigr{)}.$

Then $D:\mathbb{C}^{m}\times[0,1]\to\mathbb{C}^{m}$ is continuous, maps
$F_{G}(\widehat{K}/G)\times[0,1]$ into $F_{G}(\widehat{K}/G)$ and is such that
$D(\cdot,1)={\rm id}_{F_{G}(\widehat{K}/G)}$ and $D(\cdot,0)={\bf 0}$. Hence,
$F_{G}(\widehat{K}/G)$ is contractible and therefore $A_{G}\in\mathscr{C}$, as
required. ∎

Let us consider several explicit examples of nonuniform algebras $A$.

(1) Let $\omega:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a nondecreasing concave
function, not identically zero, and such that $\omega(0)=0$. We set

$d(z_{1},z_{2}):=\omega(\|z_{1}-z_{2}\|),\quad z_{1},z_{2}\in\mathbb{C}^{n},$

where $\|\cdot\|$ is the Euclidean norm on
$\mathbb{C}^{n}\,(\cong\mathbb{R}^{2n})$.

Then $d$ is a metric on $\mathbb{C}^{n}$ compatible with its topology.

For a bounded set $K\subset\mathbb{C}^{n}$, let $\textrm{Lip}_{d}(K)\subset
C(K)$ be the Banach algebra of complex-valued Lipschitz functions on $K$ with
respect to $d$, equipped with norm

$\|f\|_{\textrm{Lip}}:=\sup_{K}|f|+\sup_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)}.$

Then $A$ is the completion in $\textrm{Lip}_{d}(K)$ of the algebra
$\mathbb{C}[z_{1},\dots,z_{n}]|_{K}$.

(2) Let $C^{p}(K)\subset C(K)$ be the restriction to $K$ of the algebra
$C^{p}(\mathbb{C}^{n})$ of bounded complex-valued functions $f$ on
$\mathbb{C}^{n}$ having bounded continuous partial derivatives up to order $p$
with the norm the sum of supremum norms of $f$ and of all its partial
derivatives. Equipped with the quotient norm, $C^{p}(K)$ becomes a unital
commutative Banach algebra. Then $A$ is the completion in $C^{p}(K)$ of the
algebra $\mathbb{C}[z_{1},\dots,z_{n}]|_{K}$.

If $G\subset GL_{n}(\mathbb{C})$ is a finite group and $K$ is invariant with
respect to the action of $G$ on $\mathbb{C}^{n}$, then it is easy to check
that algebras $\textrm{Lip}_{d}(K)$ and $C^{p}(K)$ are $G$-invariant. Since
$\mathbb{C}[z_{1},\dots,z_{n}]|_{K}$ is invariant with respect to the
corresponding action of $G$, this implies that the algebras $A_{G}$ of
$G$-symmetric $\textrm{Lip}_{d}(K)$ and $C^{p}(K)$ functions are $G$-invariant
as well. If, in addition $K$ is star-shaped with respect to the origin, then
by Proposition 6.2, these algebras are projective free.

###  6.2. Bohr-Wiener algebras

Let $G$ be a connected compact abelian group and let $\Gamma$ be its
(multiplicative) character group. Thus $\Gamma$ consists of continuous
homomorphisms of $G$ into the group $\mathbb{T}$ of unimodular complex numbers
and separates points of $G$. As $G$ is connected, $\Gamma$ can be made into a
linearly ordered group (see e.g. [42, 8.1.8]). Let $\preccurlyeq$ be a fixed
linear order such that $(\Gamma,\preccurlyeq)$ is an ordered group. We
henceforth write $\Gamma$ additively and denote its identity element by $0$.

Standard widely used examples of $\Gamma$ are $\mathbb{Z}^{k}$ and
$\mathbb{R}^{k}$ ($k\in\mathbb{Z}_{+}$) with a lexicographic ordering; here we
use usual addition in $\mathbb{Z}$ and in $\mathbb{R}$, and the discrete
topology in both cases.

For a nonempty set $I$, we denote by $\ell^{1}(I)$ the complex Banach space of
complex-valued sequences $\mathbf{a}=(a_{i})_{i\in I}$ with pointwise
operations and the norm

$\|\mathbf{a}\|_{1}:=\sum\limits_{i\in I}|a_{i}|.$

If $\emptyset\neq J\subset I$, then we view $\ell^{1}(J)$ as a subset of
$\ell^{1}(I)$.

If $I=\Gamma$, then $\ell^{1}(\Gamma)$ is a commutative unital complex Banach
algebra with multiplication given by convolution:

$(\mathbf{a}\ast\mathbf{b})_{j}=\sum\limits_{k\in\Gamma}a_{k}b_{j-k},\quad\mathbf{a}=(a_{j})_{j\in\Gamma},\;\mathbf{b}=(b_{j})_{j\in\Gamma}\in\ell^{1}(\Gamma).$

The algebra $\ell^{1}(\Gamma)$ is semisimple and its maximal ideal space is
$G$. The Gelfand transform $\,\hat{\,}\,:\ell^{1}(\Gamma)\to C(G)$ is given by
the formula

$\hat{\mathbf{a}}(g):=\sum\limits_{j\in\Gamma}\,a_{j}\mathbf{e}_{j}(g),\quad
g\in G,\quad\mathbf{a}=(a_{j})_{j\in\Gamma}\in\ell^{1}(\Gamma).$ (6.3)

Here $\mathbf{e}_{j}(g):=\langle j,g\rangle\in\mathbb{T}$ is the action of
$j\in\Gamma$ on $g\in G$.

The function $\hat{\mathbf{a}}$ is called the symbol of $\mathbf{a}$ with
Bohr-Fourier coefficients $a_{j}$ and with the Bohr-Fourier spectrum
$\\{j\in\Gamma:a_{j}\neq 0\\}$. The image of $\ell^{1}(\Gamma)$ under
$\,\hat{\,}\,$ is denoted by $W(G)$. For a subsemigroup $\Sigma$ of $\Gamma$,
we denote by $W(G)_{\Sigma}$ the algebra of symbols of elements in
$\ell^{1}(\Sigma)$. We let $C(G)_{\Sigma}$ be the closure of $W(G)_{\Sigma}$
in $C(G)$ (so that $C(G)_{\Gamma}=C(G)$). The notions of Bohr-Fourier
coefficients and spectrum are extended from functions in $W(G)$ to $C(G)$ by
continuity. The Bohr-Fourier spectrum of an element of $C(G)$ is at most
countable, and $C(G)_{\Sigma}$ coincides with the set of functions having the
Bohr-Fourier spectra in $\Sigma$.

Assume that $\Sigma\subset\Gamma$ is pointed, i.e., such that

$0\in\Sigma,\quad{\rm and}\quad\Sigma\cap(-\Sigma)=\\{0\\}.$ (6.4)

(E.g., if $\Gamma_{+}=\\{j\in\Gamma:0\preccurlyeq j\\}$,
$\Gamma_{-}=\\{j\in\Gamma:j\preccurlyeq 0\\}$, then $\Gamma_{+}$ and
$\Gamma_{-}$ are pointed subsemigroups.)

######  Theorem ([8, Theorem 1.2]).

If $\Sigma$ is a pointed subsemigroup, then the algebras $W(G)_{\Sigma}$ and
$C(G)_{\Sigma}$ belong to the class $\mathscr{C}$.

Next, for a commutative unital complex Banach algebra $B$ with norm
$\|\cdot\|_{B}$, we let $\ell^{1}(\Sigma,B)$ denote the complex Banach space
of $B$-valued sequences $\mathbf{b}=(b_{j})_{j\in\Sigma}$ with pointwise
operations and the norm

$\|\mathbf{b}\|_{1,B}:=\sum\limits_{j\in\Sigma}\|b_{j}\|_{B}.$

Then the projective tensor product $B\widehat{\otimes}_{\pi}W(G)_{\Sigma}$
consists of $B$-valued continuous functions on $G$

$\widetilde{\mathbf{b}}(g):=\sum\limits_{j\in\Sigma}\,b_{j}\mathbf{e}_{j}(g),\quad
g\in G,\quad\mathbf{b}=(b_{j})_{j\in\Sigma}\in\ell^{1}(\Sigma,B),$

with norm $\|\widetilde{\mathbf{b}}\|:=\|\mathbf{b}\|_{1,B}$.

In turn, the injective tensor product
$B\widehat{\otimes}_{\varepsilon}C(G)_{\Sigma}$ is the closure of the
commutative algebra $B\widehat{\otimes}_{\pi}W(G)_{\Sigma}$ in $C(G,B)$
equipped with norm $\|f\|:=\max_{g\in G}\|f(g)\|_{B}$.

Now Theorem 5.1 implies the following:

######  Corollary.

If $\Sigma$ is a pointed subsemigroup, then the correspondences

$\begin{split}P\cong&B\otimes_{B}\\!P\stackrel{{\scriptstyle({\rm
id}_{B}\otimes 1)\otimes{\rm id}_{P}}}{{\xrightarrow{\hskip
42.67912pt}}}(B\widehat{\otimes}_{\pi}W(G)_{\Sigma})\otimes_{B}\\!P\\\
P\cong&B\otimes_{B}\\!P\stackrel{{\scriptstyle({\rm id}_{B}\otimes
1)\otimes{\rm id}_{P}}}{{\xrightarrow{\hskip
42.67912pt}}}(B\widehat{\otimes}_{\varepsilon}C(G)_{\Sigma})\otimes_{B}\\!P\end{split}$

determine bijections between isomorphism classes of finitely generated
projective $B$ modules and finitely generated projective
$B\widehat{\otimes}_{\pi}W(G)_{\Sigma}$ and
$B\widehat{\otimes}_{\varepsilon}C(G)_{\Sigma}$ modules, respectively.

In particular, if $B$ is projective free $($or Hermite$)$, then the algebras
$B\widehat{\otimes}_{\pi}W(G)_{\Sigma}$ and
$B\widehat{\otimes}_{\varepsilon}C(G)_{\Sigma}$ are projective free
$($respectively, Hermite$)$.

###  6.3. Algebras of holomorphic semi-almost periodic functions

Recall that $f\in C_{b}(\mathbb{R})$ is almost periodic if the family
$\\{S_{\tau}f:\tau\in\mathbb{R}\\}$ of its translates, where
$(S_{\tau}f)(x):=f(x+\tau)$ ($x\in\mathbb{R}$), is relatively compact in
$C_{b}(\mathbb{R})$. Let $AP(\mathbb{R})$ be the Banach algebra of almost
periodic functions endowed with the supremum norm.

Let $\mathbb{T}$ be the boundary of the unit disc $\mathbb{D}$, with the
counterclockwise orientation. For $s:=e^{it}$, $t\in[0,2\pi)$, let
$\gamma^{k}_{s}(\delta):=\\{se^{ikx}:0\leq x<\delta<2\pi\\}$,
$k\in\\{-1,1\\}$, be two open arcs having $s$ as the right and the left
endpoints (with respect to the orientation), respectively.

A function $f\in\textrm{L}^{\infty}(:=\textrm{L}^{\infty}(\mathbb{T}))$ is
semi-almost periodic if for any $s\in\mathbb{T}$, and any $\varepsilon>0$
there exist a number $\delta=\delta(s,\varepsilon)\in(0,\pi)$ and functions
$f_{k}:\gamma^{k}_{s}(\delta)\rightarrow\mathbb{C}$, $k\in\\{-1,1\\}$, such
that functions

$\widetilde{f}_{k}(x):=f_{k}(se^{ik\delta e^{x}}),\quad-\infty<x<0,\quad
k\in\\{-1,1\\},$

are restrictions of some almost periodic functions from $AP(\mathbb{R})$, and

$\sup\limits_{z\in\gamma^{k}_{s}(\delta)}|f(z)-f_{k}(z)|<\varepsilon,\quad
k\in\\{-1,1\\}.$

By $SAP$ we denote the Banach algebra of semi-almost periodic functions on
$\mathbb{T}$ endowed with the supremum norm. The algebra $SAP$ contains as a
special case an algebra introduced by Sarason [44] in connection with some
problems in the theory of Toeplitz operators.

It is easy to see that the set of points of discontinuity of a function in
$SAP$ is at most countable. For a closed subset $S$ of $\mathbb{T}$, we denote
by $SAP(S)$ the Banach algebra of semi-almost periodic functions on
$\mathbb{T}$ that are continuous on $\mathbb{T}\setminus S$.

Next, for a semi-almost periodic function $f\in SAP$, $k\in\\{-1,1\\}$, and a
point $s\in S$, the left ($k=-1$) and the right ($k=1$) mean values of $f$
over $s$ are given by the formulas

$M^{k}_{s}(f):=\lim\limits_{n\rightarrow\infty}\frac{1}{b_{n}-a_{n}}\int_{a_{n}}^{b_{n}}f(se^{ike^{t}})dt,$
(6.5)

where $(a_{n})_{n\in\mathbb{N}}$ and $(b_{n})_{n\in\mathbb{N}}$ are arbitrary
sequences of real numbers converging to $-\infty$ such that
$\lim\limits_{n\rightarrow\infty}(b_{n}-a_{n})=+\infty$.

The Bohr-Fourier coefficients of $f$ over $s$ can be then defined by the
formulas

$a^{k}_{\lambda}(f,s):=M_{s}^{k}(fe^{-i\lambda\log^{k}_{s}}),$ (6.6)

where

$\log^{k}_{s}(se^{ikx}):=\log x,\quad 0<x<2\pi,\quad k\in\\{-1,1\\}.$

The spectrum of $f$ over $s$ is

$\textrm{spec}^{k}_{s}(f):=\\{\lambda\in\mathbb{R}:a_{\lambda}^{k}(f,s)\neq
0\\}.$ (6.7)

Let $\Sigma:S\times\\{-1,1\\}\rightarrow 2^{\mathbb{R}}$ be a set-valued map
which associates with each $s\in S$, $k\in\\{-1,1\\}$, a unital semi-group
$\Sigma(s,k)\subset\mathbb{R}$. By $SAP_{\Sigma}(S)\subset SAP(S)$, we denote
the Banach algebra of semi-almost periodic functions $f$ with
$\textrm{spec}^{k}_{s}(f)\subset\Sigma(s,k)$ for all $s\in S$,
$k\in\\{-1,1\\}$.

Let $H^{\infty}$ be the Banach algebra of bounded holomorphic functions in
$\mathbb{D}$ with the supremum norm. Then $SAP_{\Sigma}(S)\cap H^{\infty}$ is
called the algebra of holomorphic semi-almost period functions with spectrum
$\Sigma$.

If $f\in SAP_{\Sigma}(S)\cap H^{\infty}$, then

$\textrm{spec}^{-1}_{s}(f)=\textrm{spec}^{1}_{s}(f)=:\textrm{spec}_{s}(f)$
(6.8)

and, moreover,

$a^{-1}_{\lambda}(f,s)=e^{\lambda\pi}a^{1}_{\lambda}(f,s)\quad{\rm for\
each}\;\;\lambda\in\textrm{spec}_{s}(f),$ (6.9)

see [9, Proposition 3.3]. Thus, $SAP_{\Sigma}(S)\cap
H^{\infty}=SAP_{\Sigma_{{\rm sym}}}(S)\cap H^{\infty}$, where $\Sigma_{{\rm
sym}}(s):=\Sigma_{{\rm sym}}(s,\pm 1)=\Sigma(s,-1)\cap\Sigma(s,1)$ for all
$s\in S$.

Let $A_{\Sigma}^{S}$ be the closed subalgebra of $H^{\infty}$ generated by the
disk-algebra $A(\mathbb{D})$ and the functions of the form $ge^{\lambda h}$,
where ${\rm Re}(h)|_{\mathbb{T}}$ is the characteristic function of the closed
arc going in the counterclockwise direction from the initial point at $s$ to
the endpoint at $-s$ such that $s\in S$, $\frac{\lambda}{\pi}\in\Sigma_{{\rm
sym}}(s)$ and $g(z):=z+s$, $z\in\mathbb{D}$ (in particular, $ge^{\lambda h}$
has discontinuity at $s$ only). It is shown in [9, Corollary 3.7] that

$SAP_{\Sigma}(S)\cap H^{\infty}=A_{\Sigma}^{S}.$

Let $b^{S}_{\Sigma}$ denote the maximal ideal space of the algebra
$SAP_{\Sigma}(S)\cap H^{\infty}$. The structure of $b^{S}_{\Sigma}$ is
described in [9, Theorem 3.13]. Specifically, the transpose of the embedding
$A(\mathbb{D})\hookrightarrow SAP_{\Sigma}(S)\cap H^{\infty}$ determines a
continuous epimorphism $a_{\Sigma}^{S}:\bar{\mathbb{D}}\setminus
S\to\bar{\mathbb{D}}$ one-to-one over $\bar{\mathbb{D}}\setminus S$ and such
that for each $s\in S$ the fibre of $a_{\Sigma}^{S}$ over $s$ is homeomorphic
to the maximal ideal space $b_{\Sigma_{\rm sym}(s)}(T)$ of the algebra
$APH_{\Sigma_{\rm sym}(s)}(T)$ of holomorphic almost periodic functions on the
closed strip $T=\\{z\in\mathbb{C}\,:\,{\rm Im}(z)\in[0,\pi]\\}$ with the
spectrum in $\Sigma_{\rm sym}(s)$.

Now, we have (cf. [8, Theorem 1.2]):

######  Theorem ([9, Theorem 3.19]).

  * (1)

The map of cohomology groups induced by embeddings
$(a_{\Sigma}^{S})^{-1}(s)\hookrightarrow b_{\Sigma}^{S}$, $s\in S$, produces
an isomorphism

$H^{k}(b_{\Sigma}^{S},\mathbb{Z})\cong\bigoplus_{s\in S}H^{k}(b_{\Sigma_{\rm
sym}(s)}(T),\mathbb{Z}),\quad k\geq 1.$

  * (2)

Suppose that each $\Sigma_{\rm sym}(s)$ is a subset of $\mathbb{R}_{+}$ or
$\mathbb{R}_{-}$. Then $H^{k}(b_{\Sigma}^{S},\mathbb{Z})=0$ for all $k\geq 1$
and the algebra $SAP_{\Sigma}(S)\cap H^{\infty}$ is projective free.

###  6.4. Algebras of bounded holomorphic functions

Let $X$ be a connected Riemann surface such that the Banach algebra
$H^{\infty}(X)$ of bounded holomorphic functions on $X$ separates the points
of $X$. Since $X$ is homotopy equivalent to a one-dimensional $CW$-complex,
Theorem 2.8 and Proposition 2.1 imply that the algebras $C(X)$ and $C_{b}(X)$
are projective free. The transpose of the isometric embedding
$H^{\infty}(X)\hookrightarrow C_{b}(X)$ induces a continuous map
$p:\beta(X)\to M(H^{\infty}(X))$ of the maximal ideal spaces, with image ${\rm
cl}\,X$, the closure of $X$ (identified with the set of evaluations at points
of $X$) in $M(H^{\infty}(X))$. Since $\dim X=2$ and $H^{2}(X,\mathbb{Z})=0$,
$\dim\beta X=2$ and $H^{2}(\beta X,\mathbb{Z})=0$ as well (see the proof of
Theorem 2.9 for the references). Moreover, the map $p$ is identical on $X$.
These make the following conjecture plausible (cf. the Question in [6]):

######  Conjecture 6.3.

The covering dimension of ${\rm cl}\,X$ is $2$ and the Čech cohomology group
$H^{2}({\rm cl}\,X,\mathbb{Z})=0$.

If the conjecture is true, it implies the validity of the following:

######  Conjecture 6.4.

If ${\rm cl}\,X\\!=\\!M(H^{\infty}(X))\;$888I.e., that the corona theorem is
valid for $H^{\infty}(X)$., then $H^{\infty}(X)$ is a projective free algebra.

Let us describe some classes of Riemann surfaces $X$ for which one of the
conjectures is valid.

(1) Let $R$ be an unbranched covering of a bordered Riemann surface and $X$ be
a domain in $R$ such that inclusion $i:X\hookrightarrow R$ induces a
monomorphism $i_{*}:\pi_{1}(X)\to\pi_{1}(R)$ of fundamental groups. The corona
theorem, ${\rm cl}\,X=M(H^{\infty}(X))$, was proved in [4, Corollary 1.6] and
projective freeness of $H^{\infty}(X)$ was established in [7, Theorem 1.5]
using an analog of the classical Lax-Halmos theorem proved in [5, Theorem
1.7]. Thus Conjecture 6.4 is valid for such $X$.

In the special case $X=R$, it was proved in [6, Theorem 1.3] that Conjecture
6.3 is valid as well.

######  Remark 6.5.

Conjecture 6.4 is false if $X$ has a nonempty corona, i.e., if
$M(H^{\infty}(X))\setminus{\rm cl}\,X\neq\emptyset$. Indeed, using [3, Theorem
1.2], given an integer $n\geq 2$, one can construct an unbranched covering of
a compact Riemann surface $X$ for which $H^{\infty}(X)$ separates points,
$\dim M(H^{\infty}(X))\geq n$, and $H^{2}(M(H^{\infty}(X)),\mathbb{Z})\neq 0$.
The latter implies that $H^{\infty}(X)$ is not projective free.

(2) A wide class of planar domains, the so-called $\mathscr{B}$-domains, was
introduced and studied by Behrens [2]. A set $X\subset\mathbb{C}$ is called a
$\mathscr{B}$-domain if it is obtained from a domain $Y\subset\mathbb{C}$ by
deleting a (possibly finite) hyperbolically-rare sequence of closed discs
$(\Delta_{n})$ contained in $Y$ with centres $\alpha_{n}$, i.e., such that
there are disjoint closed discs $D_{n}$ with centres $\alpha_{n}$ satisfying
$\Delta_{n}\Subset D_{n}\subset Y$ and
$\sum\frac{\textrm{rad}\;\\!\Delta_{n}}{\textrm{rad}\;\\!D_{n}}<\infty$. It
was shown in [6, Theorem 1.1] that if $X$ is obtained from a domain
$Y\subset\mathbb{C}$ by deleting a (possibly finite) hyperbolically-rare
sequence of closed discs such that (i) ${\rm cl}\,Y=M(H^{\infty}(Y))$ and (ii)
Conjecture 6.3 is valid for $Y$, then ${\rm cl}\,X=M(H^{\infty}(X))$ and
Conjecture 6.3 is valid for $X$ as well. (In particular, algebras
$H^{\infty}(Y)$ and $H^{\infty}(X)$ are projective free.)

The class of domains $Y$ satisfying (i), (ii) includes planar unbranched
coverings of bordered Riemann surfaces (see part (1)) and domains obtained
from them by deleting compact subsets of analytic capacity zero (e.g., totally
disconnected compact subsets). Starting from such a $Y$ one can construct a
descending chain $Y:=Y_{0}\supset Y_{1}\supset Y_{2}\supset\cdots\supset
Y_{n}$, $n\in\mathbb{N}$, of $\mathscr{B}$-domains, such that each $Y_{i}$ is
defined by deleting a hyperbolically-rare sequence of closed discs and then a
compact subset of analytic capacity zero from $Y_{i-1}$. Then all $Y_{i}$
satisfy assumptions (i), (ii), and, in particular, all algebras
$H^{\infty}(Y_{i})$ are projective free.

Recall that every bordered Riemann surface $S$ is a domain in a compact
Riemann surface $\widetilde{S}$ such that $\widetilde{S}\setminus S$ is the
disjoint union of finitely many discs with analytic boundaries (see [46]).
Here are some examples of planar unbranched coverings of $S$:

(a) It is known (see, e.g., [17, Chap. X]) that each $\widetilde{S}$ is the
quotient of a planar domain $\Omega$ by the discrete action of a Schottky
group $G$ (the free group with $g$ generators, where $g$ is the genus of
$\widetilde{S}$) by Möbius transformations. The corresponding quotient map
$r:\Omega\rightarrow\widetilde{S}$ determines the regular covering of
$\widetilde{S}$ with the deck transformation group $G$. Then
$Y:=r^{-1}(S)\subset\Omega$ is a regular covering of $S$ satisfying conditions
(i),(ii). By definition, $Y$ is the complement in $\Omega$ of the finite
disjoint union of $G$-orbits of compact simply connected domains with analytic
boundaries biholomorphic by $r$ to the connected components of
$\widetilde{S}\setminus S$.

(b) Consider the universal covering
$r_{u}:\widetilde{S}_{u}\rightarrow\widetilde{S}$ of $\widetilde{S}$ (where
$\widetilde{S}_{u}=\mathbb{D}$ if $g\geq 2$, $\widetilde{S}_{u}=\mathbb{C}$ if
$g=1$, and $\widetilde{S}_{u}=\mathbb{C}\mathbb{P}$ if $g=0$), then
$Y:=r_{u}^{-1}(S)\subset\widetilde{S}_{u}$ satisfies conditions (i),(ii) as
well. Here $Y$ is the complement in $\widetilde{S}_{u}$ of the finite disjoint
union of orbits under the action by Möbius transformations of the fundamental
group $\pi_{1}(\widetilde{S})$ of $\widetilde{S}$ of compact simply connected
domains with analytic boundaries biholomorphic by $r_{u}$ to the connected
components of $\widetilde{S}\setminus S$.

Finally, let us mention that if a connected Riemann surface $X$ is such that
$H^{\infty}(X)$ separates its points, ${\rm cl}\,X=M(H^{\infty}(X))$,
$\dim{\rm cl}\,X=2$ and $H^{2}({\rm cl}\,X,\mathbb{Z})=0$, then by the Künneth
formula (see, e.g., [45, Chap. 6, Exercise E]) for $M(H^{\infty}(X))^{i}$,
$2\leq i\leq 4$, the Čech cohomology groups
$H^{n}(M(H^{\infty}(X))^{i},\mathbb{Z})=0$ for all $n\geq 5$ and $\dim
M(H^{\infty}(X))^{i}=2i$. Hence, due to Theorem 4.1, for all $2\leq i\leq 4$,
the uniform algebra $(H^{\infty}(X))^{\widehat{\otimes}_{\varepsilon}i}$ (the
$i^{\textrm{th}}$ injective tensor power of $H^{\infty}(X)$) is Hermite.

##  7\. Appendix

Let $A$ be a commutative unital complex Banach algebra and $G$ be a finite
subgroup of automorphisms of $A$. By $A_{G}\subset A$ we denote the Banach
subalgebra of elements invariant with respect to the action of $G$, i.e.,

$A_{G}:=\\{a\in A\,:\,g(a)=a\quad\forall g\in G\\}.$

There is a natural projection $P_{G}:A\to A_{G}$ given by the formula

$P_{G}(a):=\frac{1}{|G|}\sum_{g\in G}g(a)\;\;\textrm{ for all }a\in A.$ (7.1)

Here $|G|$ is the cardinality of the set $G$.

For each $g\in G$, the transpose of the map $a\mapsto g(a):A\rightarrow A$
induces a homeomorphism $g^{*}$ of the maximal ideal space $M(A)$ of $A$. Thus
we obtain an action of $G$ on $M(A)$. Let $M(A)/G$ be the quotient space by
this action and $\pi:M(A)\rightarrow M(A)/G$ be the quotient map. We equip
$M(A)/G$ with the smallest topology in which the map $\pi$ is continuous. Then
$M(A)/G$ becomes a compact Hausdorff space homeomorphic to the maximal ideal
space of the subalgebra $C(M(A))_{G}\subset C(M(A))$ of continuous functions
invariant with respect to the action of $G$ on $M(A)$. We have:

######  Lemma 7.1.

$M(A)/G$ is homeomorphic to the maximal ideal space of the algebra $A_{G}$.

###### Proof.

Let $\\{g^{*}(x)\,:\,g\in G\\}\subset M(A)$ be the equivalence class
representing $\pi(x)\in M(A)/G$ for $x\in M(A)$. Since for each $a\in A_{G}$,

$(g^{*}(x))(a)=x(g(a))=x(a)$

(as elements of $A_{G}$ are invariant with respect to the action of $G$ on
$A$), the functional $\varphi_{z}$, $z\in M(A)/G$,

$\varphi_{z}(a):=x(a),\quad x\in\pi^{-1}(z),\quad\forall a\in A_{G},$

is well-defined and is an element of $M(A_{G})$. Thus, we have a map
$\Phi:M(A)/G\to M(A_{G})$, $\Phi(z):=\varphi_{z}$. Let us show that $\Phi$ is
continuous. Indeed, by the definition of the Gelfand topology on $M(A_{G})$,
it suffices to show that for an open set $U_{a}:=\\{y\in
M(A_{G})\,:\,|y(a)|<1\\}\subset M(A_{G})$ with $a\in A_{G}$, the set
$\Phi^{-1}(U_{a})$ is open in $M(A)/G$. In turn, by the definition of topology
on $M(A)/G$, the previous set is open if and only if its preimage under $\pi$
is open in $M(A)$. We have

$\begin{split}\pi^{-1}(\Phi^{-1}(U_{a}))=&\;\\!\\{x\in
M(A)\,:\,|\varphi_{\pi(x)}(a)|<1\\}\\\ =&\;\\!\\{x\in
M(A)\,:\,|x(a)|<1\\}.\end{split}$

The latter is an open subset of $M(A)$ by the definition of the Gelfand
topology.

Next, let us show that the map $\Phi$ is injective.

Let $z_{1},z_{2}$ be distinct points of $M(A)/G$. Since every finite subset of
$M(A)$ is interpolating for $A$, there is $a\in A$ such that

$x(a)=0\;\;{\rm for\ all}\;\;x\in\pi^{-1}(z_{1}),\quad{\rm and}\quad
y(a)=1\;\;{\rm for\ all}\;\;y\in\pi^{-1}(z_{2}).$

Then, for $P_{G}(a)\in A_{G}$ (see (7.1)) we obtain

$\varphi_{z_{1}}(P_{G}(a))=0,\quad{\rm and}\quad\varphi_{z_{2}}(P_{G}(a))=1,$

i.e., $\Phi(z_{1})\neq\Phi(z_{2})$. Therefore $\Phi$ is injective and, hence,
embeds $M(A)/G$ as a compact subset of $M(A_{G})$.

Finally, let us show that the map $\Phi$ is onto. If, on the contrary, there
is $y\in M(A_{G})\setminus\Phi(M(A)/G)$, then by the definition of the Gelfand
topology on $M(A_{G})$, there exist $a_{1},\dots,a_{k}\in A_{G}$ such that

$y(a_{1})=\cdots=y(a_{k})=0\quad{\rm and}\quad\max_{1\leq i\leq k}\min_{z\in
M(A)/G}|\varphi_{z}(a_{i})|\geq\delta>0.$ (7.2)

This implies that

$\max_{1\leq i\leq k}\min_{x\in M(A)}|x(a_{i})|\geq\delta>0.$

Equivalently, the family $a_{1},\dots,a_{n}$ does not belong to a maximal
ideal of $A$, i.e., there exist $b_{1},\dots,b_{n}\in A$ such that

$b_{1}a_{1}+\cdots+b_{n}a_{n}=1.$ (7.3)

We set

$\tilde{b}_{i}:=P_{G}(b_{i}),\quad 1\leq i\leq k.$

Then each $\widetilde{b}_{i}\in A_{G}$, and since $1$ and all $a_{i}\in
A_{G}$, equation (7.3) implies that

$\tilde{b}_{1}a_{1}+\cdots+\tilde{b}_{n}a_{n}=1.$ (7.4)

Applying $y$ to (7.4), we get due to (7.2),

$1=y(1)=y(\tilde{b}_{1})y(a_{1})+\cdots+y(\tilde{b}_{n})y(a_{n})=0,$

a contradiction which shows that $\Phi$ is a surjection. Therefore
$\Phi:M(A)/G\to M(A_{G})$ is a homeomorphism, as required. ∎

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