Strict Wick–type deformation quantization on Riemann surfaces:
Rigidity and Obstructions

Daniela Kraus, Oliver Roth, Sebastian Schleißinger and Stefan Waldmann

###### Abstract

Let $X$ be a hyperbolic Riemann surface. We study a convergent Wick–type star
product $\star_{X}$ on $X$ which is induced by the canonical convergent star
product $\star_{{\mathbb{D}}}$ on the unit disk ${\mathbb{D}}$ via
Uniformization Theory. While by construction, the resulting Fréchet algebras
$(\mathcal{A}(X),\star_{X})$ are strongly isomorphic for conformally
equivalent Riemann surfaces, our work exhibits additional severe topological
obstructions. In particular, we show that the Fréchet algebra
$(\mathcal{A}(X),\star_{X})$ degenerates if and only if the connectivity of
$X$ is at least $3$, and $(\mathcal{A}(X),\star_{X})$ is noncommutative if and
only if $X$ is simply connected. We also explicitly determine the algebra
$\mathcal{A}_{X}$ and the star product $\star_{X}$ for the intermediate case
of doubly connected Riemann surfaces $X$. As a perhaps surprinsing result, we
deduce that two such Fréchet algebras are strongly isomorphic if and only if
either both Riemann surfaces are conformally equivalent to an (not neccesarily
the same) annulus or both are conformally equivalent to a punctured disk.

##  1 Introduction

00footnotetext: 2020 Mathematics Subject Classification. Primary 30F45, 30F35,
53D55; Secondary 53A5500footnotetext: Key words. Riemann surfaces, Fuchsian
groups, convergent star product

For a complex manifold $\mathbb{M}$ we denote by $\mathcal{H}(\mathbb{M})$ the
set of all complex–valued holomorphic functions on $\mathbb{M}$. Equipped with
the standard compact–open topology, $\mathcal{H}(\mathbb{M})$ becomes a
Fréchet space. In this note, we are interested in the particular case
$\mathbb{M}=\Omega$, where

$\Omega:=\hat{{\mathbb{C}}}^{2}\setminus\left(\left\\{(z,w)\in{\mathbb{C}}^{2}\,:\,zw=1\right\\}\cup\\{(0,\infty)\\}\cup\\{(\infty,0)\\}\right)\,,$

and $\hat{{\mathbb{C}}}:={\mathbb{C}}\cup\\{\infty\\}$ denotes the Riemann
sphere. Our interest in the Fréchet space $\mathcal{H}(\Omega)$ and its
function–theoretic properties comes from the insight that
$\mathcal{H}(\Omega)$ arises naturally in problems related to strict
deformation quantization of the open unit disk
${\mathbb{D}}=\\{z\in{\mathbb{C}}\,:\,|z|<1\\}$, see [4, 5] and in particular
[3, 21]. In fact, the main result111In [21] the more general situation of the
unit ball in ${\mathbb{C}}^{n}$ is considered. We restrict ourselves to the
case $n=1$ since in this paper we are primarly interested in star products on
Riemann surfaces. of [21] shows that the largest Fréchet algebra
$(\mathcal{A}({\mathbb{D}}),\star_{{\mathbb{D}}})$ of all real–analytic
functions on ${\mathbb{D}}$ generated via phase space reduction from the
polynomials in $z$ and $\overline{z}$ such that the canonical formal star
product $\star_{{\mathbb{D}}}$ of Wick–type (see [5]) is continuous can solely
be described in terms of $\mathcal{H}(\Omega)$ and its natural compact–open
topology:

######  Theorem 1.1 ([21])

$\mathcal{A}({\mathbb{D}})=\big{\\{}f:{\mathbb{D}}\to{\mathbb{C}}\,|\,f(z)=F(z,\overline{z})\text{
for all }z\in{\mathbb{D}}\text{ for some }F\in\mathcal{H}(\Omega)\big{\\}}\,.$

Moreover, the algebra $(\mathcal{A}({\mathbb{D}}),\star_{{\mathbb{D}}})$
endowed with the topology inherited from the topology of locally uniform
convergence in $\mathcal{H}(\Omega)$ is a Fréchet algebra.

We note that for each $f\in\mathcal{A}({\mathbb{D}})$ there is a unique
$F\in\mathcal{H}(\Omega)$ such that $f(z)=F(z,\overline{z})$ for all
$z\in{\mathbb{D}}$. This is a consequence of the identity principle, see e.g.
[25, p. 18].

The point of departure of this paper is the conformal invariance of the
Fréchet algebra $(\mathcal{A}({\mathbb{D}}),\star_{{\mathbb{D}}})$, that is,
its invariance w.r.t. precomposition with the elements of the group
$\mathop{{\rm Aut}}({\mathbb{D}})$ of conformal automorphisms (biholomorphic
selfmaps) of ${\mathbb{D}}$. For later reference, we state this crucial
property explicitly:

######  Proposition 1.2 (Conformal invariance of the Fréchet algebra
$(\mathcal{A}({\mathbb{D}}),\star_{{\mathbb{D}}})$)

Let $f,g\in\mathcal{A}({\mathbb{D}})$ and $\varphi\in\mathop{{\rm
Aut}}({\mathbb{D}})$. Then $f\circ\varphi\in A({\mathbb{D}})$ and
$(f\circ\varphi)\star_{{\mathbb{D}}}(g\circ\varphi)=(f\star_{{\mathbb{D}}}g)\circ\varphi$.

We point out that the star product $\star_{{\mathbb{D}}}$ depends on a complex
deformation parameter $\hbar$, which plays the role of Planck’s constant, and
can take any value in the deformation domain

$\mathscr{D}:={\mathbb{C}}\setminus\\{0,-1,-1/2,-1/3,\ldots\\}\,.$

Moreover, $\star_{{\mathbb{D}}}$ depends holomorphically on
$\hbar\in\mathscr{D}$, see [21, Theorem 4.1]222In [21] the deformation domain
is ${\mathbb{C}}\setminus\\{0,-1/2,-1/4,-1/6,\ldots\\}$ which is just a simple
scaled version of $\mathscr{D}$.. In most of our results, we hold
$\hbar\in\mathscr{D}$ fixed and simply write $\star_{{\mathbb{D}}}$. In
exceptional cases, when dependence on $\hbar$ is essential, we use the
notation $\star_{{\mathbb{D}},\hbar}$ instead.

As suggested in [21] the conformal invariance of
$(\mathcal{A}({\mathbb{D}}),\star_{{\mathbb{D}}})$ makes it possible to
transfer the star product $\star_{{\mathbb{D}}}$ from the unit disk
${\mathbb{D}}$ to any other Riemann surface $X$ which is covered by the unit
disk, that is, to any Riemann surface of hyperbolic type. We emphasize that,
with one exception, in the following we will merely use the conformal
invariance of the star product, but not how it is explicitly defined. We
therefore refrain for the moment from discussing any explicit form of the star
product. The exception occurs for doubly connected Riemann surfaces. In this
case we will make use of an explicit formula for $\star_{{\mathbb{D}}}$ which
has recently been obtained in [10] and [26]. See the discussion in Section 6
for details.

######  Definition (Convergent Wick–star product on hyperbolic Riemann
surfaces)

Let $X$ be a hyperbolic Riemann surface and $\pi:{\mathbb{D}}\to X$ a
universal covering projection. Then

$\mathcal{A}(X):=\left\\{f:X\to{\mathbb{C}}\,|\,f\circ\pi\in\mathcal{A}({\mathbb{D}})\right\\}\,$

and for any $f,g\in\mathcal{A}(X)$ we define $f\star_{X}g\in\mathcal{A}(X)$ by

$\left(f\star_{X}g\right)\circ\pi:=(f\circ\pi)\star_{{\mathbb{D}}}(g\circ\pi)\,.$

Moreover, we equip $\mathcal{A}(X)$ with the topology inherited from the
topology of the Fréchet algebra
$(\mathcal{A}({\mathbb{D}}),\star_{{\mathbb{D}}})$. In particular, a sequence
$(f_{n})$ in $\mathcal{A}(X)$ converges in the $\mathcal{A}(X)$–topology to
$f\in\mathcal{A}(X)$ if and only if
$(f_{n}\circ\pi)\subseteq\mathcal{A}({\mathbb{D}})$ converges to
$f\circ\pi\in\mathcal{A}({\mathbb{D}})$ in the
$\mathcal{A}({\mathbb{D}})$–topology.

###### Remark 1.3

By conformal invariance of $(\mathcal{A}({\mathbb{D}}),\star_{{\mathbb{D}}})$,
the actual choice of the universal covering $\pi:{\mathbb{D}}\to X$ is
immaterial, and the set $\mathcal{A}(X)$ as well as the product $\star_{X}$
are well–defined objects depending only on the Riemann surface $X$. Moreover,
convergence of a sequence $(f_{n})\subseteq\mathcal{A}(X)$ in the
$\mathcal{A}(X)$–topology does not depend on the choice of $\pi$ either. In
addition, the star product
$\star_{X}:\mathcal{A}(X)\times\mathcal{A}(X)\to\mathcal{A}(X)$ is continuous
in the $\mathcal{A}(X)$–topology and $(\mathcal{A}(X),\star_{X})$ is a Fréchet
algebra. These facts are not difficult to prove, but for convenience we will
supply the details in Section 2. Since the star product $\star_{{\mathbb{D}}}$
depends on the choice of the deformation parameter $\hbar$, the star product
$\star_{X}$ on $X$ also depends on $\hbar$, and we write $\star_{\hbar,X}$
whenever the dependence on $\hbar$ is relevant. However, the set
$\mathcal{A}(X)$ is independent of $\hbar$.

The main purpose of this note is to describe the Fréchet algebras
$(\mathcal{A}(X),\star_{X})$ in an explicit way. The following preliminary
observation shows that $(\mathcal{A}(X),\star_{X})$ depends on the conformal
type of $X$. Recall that two Riemann surfaces $X$ and $Y$ are called
conformally equivalent, if there is a conformal (biholomorphic) map $\Psi:Y\to
X$.

######  Proposition 1.4 (Conformal invariance of the Fréchet algebra
$(\mathcal{A}(X),\star_{X})$)

Let $X$ and $Y$ be hyperbolic Riemann surfaces. Then any conformal map
$\Psi:Y\to X$ induces a Fréchet algebra isomorphism
$\Psi^{*}:(\mathcal{A}(X),\star_{X,\hbar})\to(\mathcal{A}(Y),\star_{Y,\hbar})$
defined by

$\Psi^{*}(f)(z)=f(\Psi(z))\,,\qquad f\in\mathcal{A}(X),\,z\in Y\,.$

The isomorphism $\Psi^{*}$ in Proposition 1.4 does not depend on the
deformation parameter $\hbar$, so $\mathcal{A}(X)$ and $\mathcal{A}(Y)$ are
strongly isomorphic in the following sense.

######  Definition (Strong isomorphy)

Let $X$ and $Y$ be Riemann surfaces. We call $\mathcal{A}(X)$ and
$\mathcal{A}(Y)$ strongly isomorphic if there is a map
$T:\mathcal{A}(X)\to\mathcal{A}(Y)$ such that $T$ is a Fréchet algebra
isomorphism between $(\mathcal{A}(X),\star_{X,\hbar})$ and
$(\mathcal{A}(Y),\star_{Y,\hbar})$ for every $\hbar\in\mathscr{D}$. In this
case, we write $\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}(Y)$.

We can now state the main result of this paper.

###### Theorem 1.5

Let $X$ be a hyperbolic Riemann surface. Then the following hold.

  * (a)

$X$ is conformally equivalent to ${\mathbb{D}}$ if and only if

$\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}({\mathbb{D}})\,.$

  * (b)

$X$ is conformally equivalent to some annulus
$A_{R}=\\{z\in{\mathbb{C}}\,:\,1/R<|z|<R\\}$, $R>1$, if and only if
$\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}(A_{R})$. In this case,

$\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}(A_{R})\quad\text{ for every
}R>1\,.$

  * (c)

$X$ is conformally equivalent to the punctured disk
${\mathbb{D}}^{*}:={\mathbb{D}}\setminus\\{0\\}$ if and only

$\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}({\mathbb{D}}^{*})\,.$

  * (d)

In all other cases, $\mathcal{A}(X)$ consists only of constant functions.

The proof of Theorem 1.5 depends in an essential way on the description of
$\mathcal{A}({\mathbb{D}})$ in terms of $\mathcal{H}(\Omega)$ and classical
uniformization theory, and thereby on the complex analytic properties of the
Riemann surface $X$. However, the structure of $(\mathcal{A}(X),\star_{X})$ as
a Fréchet algebra depends only on the topological type of $X$, except when $X$
is doubly connected. This is the content of the following result, which
follows from Theorem 1.5 and the Uniformization Theorem.

###### Corollary 1.6

Let $X$ be a hyperbolic Riemann surface. Then the following hold.

  * (a)

$X$ is simply connected if and only if
$\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}({\mathbb{D}})$.

  * (b)

$X$ is doubly connected and conformally equivalent to some annulus $A_{R}$,
$R>1$, if and only if $\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}(A_{R})$.
In this case, $\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}(A_{R})$ for every
$R>1$.

  * (c)

$X$ is doubly connected and conformally equivalent to the punctured disk
${\mathbb{D}}^{*}$ if and only if
$\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}({\mathbb{D}}^{*})$.

  * (d)

$X$ is neither simply nor doubly connected if and only if $\mathcal{A}(X)$
consists only of constant functions.

In particular, if $X$ is a compact Riemann surface of genus greater or equal
than two, then $\mathcal{A}(X)$ degenerates and consists only of constant
functions. This might be viewed as a very strict topological rigidity
phenomenon. We also see that the Fréchet algebras
$(\mathcal{A}(A_{R}),\star_{A_{R}})$ with $R>1$ are all strongly isomorphic,
even though $A_{R}$ and $A_{R^{\prime}}$ are not conformally equivalent if
$R\not=R^{\prime}$. The following result gives an explicit description of
$\mathcal{A}(A_{R})$ and the star product $\star_{A_{R},\hbar}$. In order to
properly formulate this result, we introduce for each $R>0$ the auxiliary
function

$f_{R}:A_{R}\to{\mathbb{C}}\,,\qquad f_{R}(z):=-i\tan\left(\frac{\pi}{2\log
R}\log|z|\right)\,.$ (1.1)

The choice of the prefactor $-i$ in (1.1) facilitates some of the
computations. We also denote

$c_{n}(\hbar):=\begin{cases}\,\displaystyle\frac{\hbar^{n}}{\prod\limits_{j=0}^{n-1}\left(1+j\hbar\right)}&\text{
for }n\geq 1\,,\\\ \,1&\text{ for }n=0\,.\end{cases}$ (1.2)

######  Theorem 1.7 (The Fréchet algebra $(\mathcal{A}(A_{R}),\star_{A_{R}})$
for annuli $A_{R}=\\{z\in{\mathbb{C}}\,:\,1/R<|z|<R\\}$)

Let $R>1$. Then the mapping

$T_{R}:\mathcal{H}({\mathbb{C}})\to\mathcal{A}(A_{R})\,,\qquad
T_{R}(g):=g\circ f_{R}\,,$ (1.3)

is a continuous linear bijection between the Fréchet spaces
$\mathcal{H}({\mathbb{C}})$ and $\mathcal{A}(A_{R})$. In particular,

$\mathcal{A}(A_{R})=\big{\\{}g\circ
f_{R}\,:\,g\in\mathcal{H}({\mathbb{C}})\big{\\}}\,.$ (1.4)

Moreover, let $g,\tilde{g}\in\mathcal{H}({\mathbb{C}})$, and $f:=g\circ
f_{R},\tilde{g}\circ f_{R}\in\mathcal{A}(A_{R})$. Then

$f\star_{A_{R}}\tilde{f}=\sum\limits_{n=0}^{\infty}\frac{c_{n}(\hbar)}{n!}\left(f_{R}^{2}-1\right)^{n}\left(g^{(n)}\circ
f_{R}\right)\cdot\left(\tilde{g}^{(n)}\circ f_{R}\right)\,.$ (1.5)

In particular, the star product $\star_{A_{R}}$ is commutative,

$f\star_{A_{R}}\tilde{f}=\tilde{f}\star_{A_{R}}f\,.$

Furthermore, for fixed $\hbar\in\mathscr{D}$, the series (1.5) converges in
the topology of $\mathcal{A}(A_{R})$.

As a corollary we deduce that the Fréchet algebras
$(\mathcal{A}(A_{R}),\star_{A_{R}})$ and
$(\mathcal{A}(A_{R^{\prime}}),\star_{A_{R^{\prime}}})$ are strongly isomorphic
for every choice of $R>1$ and $R^{\prime}>1$:

###### Corollary 1.8

Let $R,R^{\prime}>1$. Then the mapping

$\Psi_{R^{\prime},R}:\mathcal{A}(A_{R^{\prime}})\to\mathcal{A}(A_{R})\,,\qquad\Psi_{R^{\prime},R}:=T_{R}\circ
T^{-1}_{R^{\prime}}$

is a Fréchet algebra isomorphism between
$(\mathcal{A}(A_{R^{\prime}}),\star_{A_{R^{\prime}},\hbar})$ and
$(\mathcal{A}(A_{R}),\star_{A_{R},\hbar})$ for every $\hbar\in\mathscr{D}$.

The next result gives an explicit description of
$\mathcal{A}({\mathbb{D}}^{*})$ and the star product
$\star_{{\mathbb{D}}^{*},\hbar}$ for the punctured unit disk
${\mathbb{D}}^{*}={\mathbb{D}}\setminus\\{0\\}$. We first introduce the
auxiliary function

$f_{0}:{\mathbb{D}}^{*}\to{\mathbb{C}}\,,\qquad
f_{0}(z):=-\frac{1}{\log|z|}\,.$ (1.6)

######  Theorem 1.9 (Wick star product for the punctured disk
${\mathbb{D}}^{*}={\mathbb{D}}\setminus\\{0\\}$)

The mapping

$T_{0}:\mathcal{H}({\mathbb{C}})\to\mathcal{A}({\mathbb{D}}^{*})\,,\qquad
T_{0}(g):=g\circ f_{0}$

is a continuous linear bijection between the Fréchet spaces
$\mathcal{H}({\mathbb{C}})$ and $\mathcal{A}({\mathbb{D}}^{*})$. In
particular,

$\mathcal{A}({\mathbb{D}}^{*})=\big{\\{}g\circ
f_{0}\,:\,g\in\mathcal{H}({\mathbb{C}})\big{\\}}\,.$ (1.7)

Moreover, let $g,\tilde{g}\in\mathcal{H}({\mathbb{C}})$, and $f:=g\circ
f_{0},\tilde{g}\circ f_{0}\in\mathcal{A}({\mathbb{D}}^{*})$. Then

$f\star_{{\mathbb{D}}^{*}}\tilde{f}=\sum\limits_{n=0}^{\infty}\frac{c_{n}(\hbar)}{n!}\,f_{0}^{2}\cdot\left(g^{(n)}\circ
f_{0}\right)\cdot\left(\tilde{g}^{(n)}\circ f_{0}\right)\,.$ (1.8)

In particular, the star product $\star_{{\mathbb{D}}^{*}}$ is commutative,

$f\star_{{\mathbb{D}}^{*}}\tilde{f}=\tilde{f}\star_{{\mathbb{D}}^{*}}f\,.$

Furthermore, for fixed $\hbar\in\mathscr{D}$, the series (1.8) converges in
the topology of $\mathcal{A}({\mathbb{D}}^{*})$.

###### Remark 1.10

The series in (1.5) as well as in (1.8) is a factorial series in
$\hbar\in\mathscr{D}$ and converges locally uniformly in $\mathscr{D}$.

The following result was a bit surprising for us.

###### Theorem 1.11

Let $R>1$. Then the commutative Fréchet algebras
$(\mathcal{A}(A_{R}),\star_{A_{R}})$ and
$(\mathcal{A}({\mathbb{D}}^{*}),\star_{{\mathbb{D}}^{*}})$ are not strongly
isomorphic.

The proof of Theorem 1.11 is based on the explicit formulas (1.5) for
$\star_{\hbar,A_{R}}$ and (1.8) for $\star_{\hbar,{\mathbb{D}}^{*}}$, but also
on an asymptotic series for $\star_{\hbar,{\mathbb{D}}}$ which has recently
been established in [10, Theorem 6.9].

###### Remark 1.12

As it has been observed in [21, Section 4.5], the algebra
$\mathcal{A}({\mathbb{D}})$ possesses a “$\mathbb{Z}_{2}$–symmetry”: if
$f\in\mathcal{A}({\mathbb{D}}),f(z)=g(z,\overline{z})$ with
$g\in\mathcal{H}(\Omega)$, then $g(1/z,1/w)$ also belongs to
$\mathcal{H}(\Omega)$. This implies that $f$ can be extended in a natural way
to $\hat{{\mathbb{C}}}\setminus\overline{{\mathbb{D}}}$ and that
${\mathbb{D}}\ni z\mapsto f(1/z)$ also belongs to $\mathcal{A}({\mathbb{D}})$.
This symmetry is carried over to $\mathcal{A}({\mathbb{D}}^{*})$ and
$\mathcal{A}(A_{R})$:

  * (a)

If $f=g\circ f_{0}$ is in $\mathcal{A}({\mathbb{D}}^{*})$, then $f$ can be
extended to ${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$ and
$\tilde{f}(z):=f(1/z)$ again belongs to $\mathcal{A}({\mathbb{D}}^{*})$.

  * (b)

If $f=g\circ f_{R}$ is in $\mathcal{A}(A_{R})$, then $f$ extends naturally to
$\\{z\in\hat{{\mathbb{C}}}\,:\,|z|\not=R^{2k+1},k\in\mathbb{Z}\cup\\{\pm\infty\\}\\}$.

Thus, in all three cases, $f$ extends to the sphere $\hat{{\mathbb{C}}}$ minus
a set of circles (including radius zero).

We can factor out the $\mathbb{Z}_{2}$–symmetry and consider the set

$\mathcal{A}^{\prime}({\mathbb{D}})=\left\\{g(z,\overline{z})\,:\,g\in\mathcal{H}(\Omega),g(z,w)=g\left(\frac{1}{z},\frac{1}{w}\right)\right\\}\,.$

Then $\mathcal{A}^{\prime}({\mathbb{D}})$ is a
$\star_{{\mathbb{D}}}$-invariant subset of $\mathcal{A}({\mathbb{D}})$.
Factoring out the symmetry $z\mapsto 1/z$ for $\mathcal{A}({\mathbb{D}}^{*})$
simply gives the $\star_{{\mathbb{D}}^{*}}$-invariant set

$\mathcal{A}^{\prime}({\mathbb{D}}^{*})=\left\\{g\circ
f_{0}^{2}\,:\,g\in\mathcal{H}({\mathbb{C}})\right\\}.$

Theorem 1.5, Theorem 1.7 and Theorem 1.9 imply the following characterizations
of simply connected hyperbolic Riemann surfaces in terms of properties of
$\mathcal{A}(X)$.

###### Corollary 1.13

Let $X$ be a hyperbolic Riemann surface. Then the following are equivalent.

  * (a)

$X$ is simply connected.

  * (b)

$\mathcal{A}(X)$ separates points.

  * (c)

The Fréchet algebra $(\mathcal{A}(X),\star_{X})$ is noncommutative.

######  Remark 1.14 (Parameter space of $\mathcal{A}(X)$)

Combining Theorem 1.5 and Corollary 1.6, and noting that $\mathcal{H}(\Omega)$
is isomorphic (as a vector space) to the space $\mathcal{H}({\mathbb{C}}^{2})$
of entire functions on ${\mathbb{C}}^{2}$, see [21] as well as [11, Section
4], it is tempting to think of the Fréchet space $\mathcal{A}(X)$ as being
“parametrized” by the Fréchet space $\mathcal{H}({\mathbb{C}}^{k})$ with $k$
chosen in such a way that $X$ is $\max\\{3-k,0\\}$–connected:

Connectivity of $X$ |  Parameter space of $\mathcal{A}(X)$ |  Star product $\star_{X}$
---|---|---
$1$ | $\mathcal{H}({\mathbb{C}}^{2})$ | noncommutative
$2$ | $\mathcal{H}({\mathbb{C}}^{1})$ | commutative
$\geq 3$ |  $\mathcal{H}({\mathbb{C}}^{0})\simeq{\mathbb{C}}$ | trivial
Table 1: Topology of $X$ vs. ”Dimension” of the Parameter space

The plan of the present paper is as follows. In a preliminary Section 2 we
prove the basic properties of the star poduct stated in Proposition 1.2,
Remark 1.3 and Proposition 1.4. These properties will be needed throughout the
paper. In Section 3 we discuss and prove a number of topological and complex
analytic properties of the manifold $\Omega$. It turns out to be more
convenient to pass from $\Omega$ to the biholomorphically equivalent manifold

$G:=\hat{{\mathbb{C}}}^{2}\setminus\\{(z,z)\,:z\in\hat{{\mathbb{C}}}\\}\,.$

The set $G$ is a specific example of a configuration space and is also called
second ordered configuration space of the sphere. Such spaces arise in various
problems in algebraic topology, knot theory and mathematical physics, see e.g.
[7]. Section 4 is partly based on Section 3 and is concerned with holomorphic
automorphic functions on $G$. We obtain explicit representations of all
functions in $\mathcal{H}(G)$ which are invariant w.r.t. to certain discrete
subgroups of automorphisms of $G$. More precisely, those subgroups are induced
by fixed–point free Fuchsian groups $\Gamma$ of automorphisms of the upper
half–plane ${\mathbb{H}}$, which by classical Riemann surface theory represent
the category of all hyperbolic Riemann surfaces. The results of Section 4 are
among the main technical contributions of this paper; their proofs are given
in Section 5. Section 6 is devoted to the exceptional case of doubly connected
Riemann surfaces, and we prove Theorem 1.7 and Corollary 1.8 as well as
Theorem 1.9 and Theorem 1.11. In Section 7 we finally conclude with the proof
of Theorem 1.5 and Corollary 1.13.

##  2 Basic properties of the star products

In this section we establish the basic properties of the continuous Wick star
product on the unit disk and hyperbolic Riemann surfaces. In particular, we
prove Proposition 1.2, Remark 1.3 and Proposition 1.4.

Let us first make clear that for each $f\in\mathcal{A}({\mathbb{D}})$ there is
a unique $F\in\mathcal{H}(\Omega)$ such that $f(z)=F(z,\overline{z})$ for all
$z\in{\mathbb{D}}$. In fact, let $f\in\mathcal{A}({\mathbb{D}})$ and suppose
$F,\tilde{F}\in\mathcal{H}(\Omega)$ such that
$F(z,\overline{z})=f(z)=\tilde{F}(z,\overline{z})$ for all $z\in{\mathbb{D}}$.
Hence a well–known variant of the identity principle ([25, p. 18]) implies
$F=\tilde{F}$, first on ${\mathbb{D}}\times{\mathbb{D}}$ and then by
connectedness on the entire manifold $\Omega$.

###### Proof of Proposition 1.2.

This follows from the construction of the algebra
$(\mathcal{A}({\mathbb{D}}),\star_{{\mathbb{D}}})$ in [21]. In fact, the
conformal invariance of the set $\mathcal{A}({\mathbb{D}})$ also follows
directly from Theorem 1.1. In order to see this let
$f\in\mathcal{A}({\mathbb{D}})$ and $\varphi\in\mathop{{\rm
Aut}}({\mathbb{D}})$. By definition, there is $F\in\mathcal{H}(\Omega)$ such
that $f(z)=F(z,\overline{z})$ for all $z\in{\mathbb{D}}$. Since $\varphi$ is a
Möbius transformation which maps $\hat{{\mathbb{C}}}$ bijectively onto
$\hat{{\mathbb{C}}}$, we immediately see that

$T_{\varphi}(z,w):=\left(\varphi(z),\frac{1}{\varphi(1/w)}\right)$

defines a biholomorphic selfmap of $\Omega$. Now, in view of the symmetry
property $\overline{\varphi(z)}=1/\varphi(1/\overline{z})$, we get

$(f\circ\varphi)(z)=F\left(\varphi(z),\overline{\varphi(z)}\right)=F(\varphi(z),1/\varphi(1/\overline{z}))=(F\circ
T_{\varphi})(z,\overline{z})$

for all $z\in{\mathbb{D}}$. Since $F\circ T_{\varphi}\in H(\Omega)$, we deduce
$f\circ\varphi\in\mathcal{A}({\mathbb{D}})$. ∎

We can now easily see that the definition of the algebra
$(\mathcal{A}(X),\star_{X})$ does not depend on the choice of the universal
covering map $\pi:{\mathbb{D}}\to X$ and that $(\mathcal{A}(X),\star_{X})$ is
a Fréchet algebra w.r.t. the topology inherited from
$(\mathcal{A},\star_{X})$.

###### Proof of Remark 1.3.

Let $X$ be a hyperbolic Riemann surface and $\pi,\tilde{\pi}:{\mathbb{D}}\to
X$ universal covering maps. Then $\pi=\tilde{\pi}\circ\varphi$ for some
$\varphi\in\mathop{{\rm Aut}}({\mathbb{D}})$. Proposition 1.2 shows that if
$f:X\to{\mathbb{C}}$ is a function, then
$f\circ\pi\in\mathcal{A}({\mathbb{D}})$ implies
$f\circ\tilde{\pi}\in\mathcal{A}({\mathbb{D}})$. Therefore, the set
$\mathcal{A}(X)=\\{f:X\to{\mathbb{C}}\,|\,f\circ\pi\in\mathcal{A}({\mathbb{D}})\\}$
does not depend on the choice of $\pi$. Now let $\Gamma$ denote the covering
group of $X$ w.r.t. the covering $\pi:{\mathbb{D}}\to X$, so
$\pi\circ\gamma=\pi$ for any $\gamma\in\Gamma$. Let $p\in X$ and
$z_{0},z_{1}\in{\mathbb{D}}$ such that $\pi(z_{0})=\pi(z_{1})=p$. Since
$\Gamma$ acts transitively on the fiber $\pi^{-1}(\\{p\\})$ there is an
$\gamma\in\Gamma$ such that $\gamma(z_{0})=z_{1}$. This implies

$\displaystyle\left[(f\circ\pi)\star_{{\mathbb{D}}}(g\circ\pi)\right](z_{1})$
$\displaystyle=$
$\displaystyle\left[\left((f\circ\pi)\star_{{\mathbb{D}}}(g\circ\pi)\right)\circ\gamma\right](z_{0})=\left[(f\circ\pi\circ\gamma)\star_{{\mathbb{D}}}(g\circ\pi\circ\gamma)\right](z_{0})$

by Proposition 1.2. In view of $\pi\circ\gamma=\pi$, we see that
$(f\star_{X}g)(p)$ is well–defined. If $\tilde{\pi}:{\mathbb{D}}\to X$ is
another universal covering, then $\pi=\tilde{\pi}\circ\varphi$ for some
$\varphi\in\mathop{{\rm Aut}}({\mathbb{D}})$ and Proposition 1.2 implies

$\left[(f\circ\pi)\star_{{\mathbb{D}}}(g\circ\pi)\right]=\left[(f\circ\tilde{\pi}\circ\varphi)\star_{{\mathbb{D}}}(g\circ\tilde{\pi}\circ\varphi)\right]=\left[(f\circ\tilde{\pi})\star_{{\mathbb{D}}}(g\circ\tilde{\pi})\right]\circ\varphi.$

Hence $f\star_{X}g:X\to{\mathbb{C}}$ does not depend on the choice of $\pi$.

We next show that convergence in $\mathcal{A}(X)$ does not depend on the
choice of $\pi$ either. Let $(f_{n})\subseteq\mathcal{A}(X)$ and
$f\in\mathcal{A}(X)$ such that $f_{n}\circ\pi\to f\circ\pi$ in the
$\mathcal{A}({\mathbb{D}})$–topology. This means that there are functions
$F_{n},F\in\mathcal{H}(\Omega)$ such that
$(f_{n}\circ\pi)(z)=F_{n}(z,\overline{z})$ and
$(f\circ\pi)(z)=F(z,\overline{z})$ for all $z\in{\mathbb{D}}$ and $F_{n}\to F$
locally uniformly in $\Omega$. If $\tilde{\pi}:{\mathbb{D}}\to X$ is another
universal covering, then $\tilde{\pi}=\pi\circ\gamma$ for some
$\gamma\in\mathop{{\rm Aut}}({\mathbb{D}})$. As in the proof of Proposition
1.2,

$T_{\gamma}(z,w):=\left(\gamma(z),\frac{1}{\gamma(1/w)}\right)$

is a biholomorphic selfmap of $\Omega$, and $F_{n}\circ T_{\gamma}$ converges
locally uniformly in $\Omega$ to $F\circ T_{\gamma}$. By definition, this
means $f_{n}\circ\tilde{\pi}\to f\circ\tilde{\pi}$ in the
$\mathcal{A}({\mathbb{D}})$–topology.

Finally, let us convince ourselves that $\mathcal{A}(X)$ is a Fréchet space
such that $\star_{X}:\mathcal{A}(X)\times\mathcal{A}(X)\to\mathcal{A}(X)$ is
continuous. Let $(f_{n})$ be a Cauchy sequence in $\mathcal{A}(X)$. This means
that there are $F_{n}\in\mathcal{H}(\Omega)$ such that
$g_{n}(z):=(f_{n}\circ\pi)(z)=F_{n}(z,\overline{z})$ on ${\mathbb{D}}$ and
$(F_{n})$ is Cauchy in $\mathcal{H}(\Omega)$. By completeness of
$\mathcal{H}(\Omega)$ there is $F\in\mathcal{H}(\Omega)$ such that $F_{n}\to
F$ locally uniformly in $\Omega$. Then $g(z):=F(z,\overline{z})$ belongs to
$\mathcal{A}({\mathbb{D}})$ by definition, and $g_{n}\to g$ locally uniformly
in ${\mathbb{D}}$. Now fix a point $x_{0}\in X$, and choose
$z_{0}\in{\mathbb{D}}$ such that $\pi(z_{0})=x_{0}$. It is easy to see that
$g(z_{0})$ only depends on $x_{0}$, but not on $z_{0}$. Hence, we can define
$f(x_{0}):=g(z_{0})$ and get a function $f:X\to{\mathbb{C}}$ such that
$f\circ\pi=g$ and $f_{n}\to f$ in the $\mathcal{A}(X)$–topology. Thus
$\mathcal{A}(X)$ is a Fréchet space. The continuity of
$\star_{X}:\mathcal{A}(X)\times\mathcal{A}(X)\to\mathcal{A}(X)$ follows at
once from the definitions. ∎

###### Proof of Proposition 1.4.

Let $X$ and $Y$ be hyperbolic Riemann surfaces, $\pi:{\mathbb{D}}\to Y$ a
universal covering of $Y$ and $\Psi:Y\to X$ a conformal map. Then
$\Psi\circ\pi:{\mathbb{D}}\to X$ is a universal covering of $X$. Hence we have
for $\Psi^{*}:\mathcal{A}(X)\to\mathcal{A}(Y)$, $\Psi^{*}(f)=f\circ\Psi$ that

$\Psi^{*}(\mathcal{A}(X))=\\{\Psi^{*}(f):X\to{\mathbb{C}}\,|\,f\circ\Psi\circ\pi\in\mathcal{A}({\mathbb{D}})\\}=\\{g:Y\to{\mathbb{C}}\,:g\circ\pi\in
A\\}=\mathcal{A}(Y)\,.$

Moreover,

$\displaystyle\Psi^{*}(f\star_{X}g)\circ\pi$ $\displaystyle=$
$\displaystyle(f\star_{X}g)\circ\Psi\circ\pi=(f\circ\Psi\circ\pi)\star_{{\mathbb{D}}}(g\circ\Psi\circ\pi)$
$\displaystyle=$
$\displaystyle\left(\Psi^{*}(f)\circ\pi\right)\star_{{\mathbb{D}}}\left(\Psi^{*}(g)\circ\pi\right)=\left(\Psi^{*}(f)\star_{Y}\Psi^{*}(g)\right)\circ\pi\,,$

so $\Psi^{*}:(\mathcal{A}(X),\star_{X})\to(\mathcal{A}(Y),\star_{Y})$ is an
algebra isomorphism. Finally, it is to see that
$\Psi^{*}:(\mathcal{A}(X),\star_{X})\to(\mathcal{A}(Y),\star_{Y})$ is
continuous. ∎

##  3 Complex structure of the second ordered configuration space of the
sphere

For some purposes of this paper, it is somewhat more convenient to work on the
upper half–plane ${\mathbb{H}}$ instead of the unit disk ${\mathbb{D}}$. Since
every conformal map $T$ from ${\mathbb{D}}$ onto ${\mathbb{H}}$ is actually a
Möbius transformation which maps $\hat{{\mathbb{C}}}$ biholomorphically onto
itself, we are in a position to replace the complex manifold $\Omega$ by its
image under the map

$(z,w)\mapsto\Psi(z,w):=\left(T(z),T(1/w)\right)\,.$

Note that $\Psi$ maps $\Omega$ biholomorphically onto

$G:=\Psi(\Omega)=\hat{{\mathbb{C}}}^{2}\setminus\\{(z,z)\,:\,z\in\hat{{\mathbb{C}}}\\}\,,$

and by conformal invariance we have

$\mathcal{A}({\mathbb{H}})=\big{\\{}f:{\mathbb{H}}\to{\mathbb{C}}\,|\,f(z)=F(z,\overline{z})\text{
for all }z\in{\mathbb{H}}\text{ for some }F\in\mathcal{H}(G)\big{\\}}\,.$

The set $G$ is also called the second configuration space of the sphere
$\hat{{\mathbb{C}}}$ and is denoted by $\text{Conf}_{2}(\hat{{\mathbb{C}}})$,
see [7]. Its fundamental group $\pi_{1}(G)$ is the $2$–strand pure braid group
of $\hat{{\mathbb{C}}}$.

###### Theorem 3.1

$G$ is a connected simply connected complex manifold. In addition, $G$

  * (a)

is a Stein manifold;

  * (b)

is not biholomorphic to ${\mathbb{C}}^{2}$ but contains an open subset which
is biholomorphic to ${\mathbb{C}}^{2}$;

  * (c)

is diffeomorphic to the real tangent bundle $T_{\mathbb{R}}S^{2}$ of the
2-sphere $S^{2}\subset{\mathbb{R}}^{3}$;

  * (d)

possesses the density property from Andersén-Lempert theory.

###### Remark 3.2

With regard to Theorem 3.1 (b), we note that one can directly show that the
open subsets $G_{+}:=G\setminus\\{(z,0)\,:z\in\hat{{\mathbb{C}}}\\}$ and
$G_{-}:=G\setminus\\{(0,w)\,:w\in\hat{{\mathbb{C}}}\\}$ of $G$ are
biholomorphically equivalent to ${\mathbb{C}}^{2}$. This is proved in [11,
Proposition 4.3] for $\Omega$ instead of $G$. In the original $\Omega$–model
the set $G_{+}$ corresponds to
$\Omega_{+}=\Omega\setminus\\{(z,\infty)\,:\,z\in\hat{{\mathbb{C}}}\\}$ and
$G_{-}$ to
$\Omega_{-}:=\Omega\setminus\\{(\infty,w)\,:\,w\in\hat{{\mathbb{C}}}\\}$. The
sets $\Omega_{+}$ and $\Omega_{-}$ play a crucial role for the Fréchet space
structure of $\mathcal{H}(\Omega)$ (see [11]), for the study of invariant
differential operators on $\mathcal{H}(\Omega)$ (see [10]) and for the
spectral theory of the invariant Laplacian on $\Omega$ (see [12, 23]).

###### Remark 3.3

In complex geometry, the manifold $G$ often appears in the biholomorphically
equivalent form of the complex two–sphere

$S^{2}_{\mathbb{C}}:=\\{(z_{1},z_{2},z_{3})\in{\mathbb{C}}^{3}\,|\,z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1\\}\,,$

and thus as a complex homogeneous surface (see the classification [22]), or as
an entire Grauert tube (see [28]). A coordinate change shows that $G$ is also
biholomorphic to

$D:=\left\\{(a,b,c)\in{\mathbb{C}}^{3}\,:\,b^{2}-4ac=1\right\\}\,,$ (3.1)

which is known as a Danielewski surface when written in the form
$\\{(a,b,c)\in{\mathbb{C}}^{3}\,:\,ac=(b^{2}-1)/4\\}$. (A Danielewski surface
has the form $\\{(x,y,z)\in{\mathbb{C}}^{3}\,:\,x^{n}y=p(z)\\}$ for a
polynomial $p$ with simple roots.)

In fact, the mapping

$f:G\to
D\,,\quad(z,w)\mapsto\left(\frac{1}{z-w},\frac{z+w}{z-w},\frac{zw}{z-w}\right).$

is biholomorphic ([22, Section 10.1]. The change of coordinates

$z_{1}:=ia+ic,\quad z_{2}:=b,\quad z_{3}:=-a+c.$ (3.2)

yields the homogeneous quadric $S_{\mathbb{C}}^{2}$.

The characterization of $G$ in Theorem 3.1 (c) suggests that our quantization
is related to a quanitzation of the 2-sphere $S^{2}$, where $S^{2}$ is
regarded as the configuration space and $T_{\mathbb{R}}S^{2}$ as the phase
space. Indeed, certain holomorphic functions on $S^{2}_{\mathbb{C}}$ appear in
this context in [8] (with corrections [9, 24]), see [8, Theorem 5] for the
space of holomorphic functions. This space is conformally invariant in the
sense of [8, Lemma 3].
Our construction is related to the 2-sphere insofar as the symmetry of
$\mathcal{A}({\mathbb{D}})$ with respect to the transform $z\mapsto 1/z$ (see
Remark 1.12) shows that we are actually considering functions that are defined
on two discs at the same time, the unit disc ${\mathbb{D}}$ and the disc
$\hat{{\mathbb{C}}}\setminus\overline{{\mathbb{D}}}$, whose union gives the
2-sphere $\hat{{\mathbb{C}}}$ (minus $\partial{\mathbb{D}}$).

###### Remark 3.4

Theorem 3.1 (d) implies that $\mathop{{\rm Aut}}(G)$ is infinite-dimensional.
This can also be seen directly by verifying that the following mappings belong
to $\mathop{{\rm Aut}}(G)$:

$(z,w)\mapsto\left(\alpha z+g\left(\frac{1}{z-w}\right),\alpha
w+g\left(\frac{1}{z-w}\right)\right),$

where $\alpha\in{\mathbb{C}}^{*}$ and $g:{\mathbb{C}}\to{\mathbb{C}}$ is an
entire mapping. Furthermore, $\mathop{{\rm Aut}}(G)$ also contains the
mappings

$(z,w)\mapsto(w,z),\quad(z,w)\mapsto\left(\varphi(z),\varphi(w)\right),\quad\varphi\in\mathop{{\rm
Aut}}(\hat{{\mathbb{C}}}).$ (3.3)

In [11, Theorem 5.1] it is shown that these automorphisms are the only ones in
the infinite–dimensional group $\mathop{{\rm Aut}}(G)$ which commute with the
action of the natural Laplacian on $G$. Even though we do not need this
result, it partly explains why we only consider automorphisms on $G$ of the
form (3.3).

###### Remark 3.5

The complex (compact Kähler) manifold
$\hat{{\mathbb{C}}}\times\hat{{\mathbb{C}}}$ can be embedded into
${\mathbb{C}}P^{3}$ via Segre’s embedding
$\varphi([z_{0}:z_{1}],[w_{0}:w_{1}])=[z_{0}w_{0}:z_{0}w_{1}:z_{1}w_{0}:z_{1}w_{1}]$.
We have
$\varphi(\hat{{\mathbb{C}}}\times\hat{{\mathbb{C}}})=\\{[z_{0}:z_{1}:z_{2}:z_{3}]\,|\,z_{0}z_{3}-z_{1}z_{2}=0\\}$
and thus we obtain yet another way to represent $G$ as

$\varphi(G)=\\{[z_{0}:z_{1}:z_{2}:z_{3}]\,|\,z_{0}z_{3}-z_{1}z_{2}=0,z_{1}\not=z_{2}\\}.$

In this context we can consider the group of all rational automorphisms of
$G$, which turns out to be generated by all automorphisms from Remark 3.4 with
$g$ being a polynomial, see [14, Theorem 4].

###### Remark 3.6

An automorphism $\tau$ of a complex manifold is called a “generalized
translation” if for any holomorphically convex compact $K\subset G$ there
exists $m\in{\mathbb{N}}$ with $\tau^{m}(K)\cap K=\emptyset$ and
$\tau^{m}(K)\cup K$ is holomorphically convex. A Stein manifold with density
property which has a generalized translation possesses universal automorphisms
in the sense of [1, Theorem 3.6]. Our manifold $G$ has these three properties.
A generalized translation (for certain Danielewski surfaces) is given in [1,
Example 4.3]. On $G$, this simply corresponds to
$\tau(z,w)=(z+\alpha,w+\alpha),$ $\alpha\in{\mathbb{C}}^{*}$.

###### Remark 3.7

Consider the set of all $g\in\mathcal{A}({\mathbb{D}})$ with
$g(z)=g(\overline{z})$, which corresponds to all $F\in\mathcal{H}(\Omega)$
with $F(z,w)=F(w,z)$. As $M:=\Omega/((z,w)\sim(w,z))$ is again a complex
manifold, the latter set simply corresponds to $\mathcal{H}(M)$. With [22,
Section 10.2] and the change of coordinates from (3.2) wee see that $M$ is
biholomorphic to

$\\{[z_{1}:z_{2}:z_{3}]\in{\mathbb{C}}P^{2}\,:\,z_{1}^{2}+z_{2}^{2}+z_{3}^{2}\not=0\\}.$

###### Proof of Theorem 3.1.

$G$ is an open subset of $\hat{{\mathbb{C}}}^{2}$ and pathconnected and hence
connected. It is known that $G$ is homotopy equivalent to
$\hat{{\mathbb{C}}}$, see [7, Example 2.4 (1)]. Thus $\pi_{1}(G)$ is trivial
and $G$ is simply connected.

(a) Using functions $f:\Omega\to{\mathbb{C}}$ of the form
$f(z,w)=g(1/(\gamma(z)-\gamma(w))$ with $\gamma$ as a Moebius transform and
$g\in\mathcal{H}({\mathbb{C}})$, it is straightforward to check that $G$ is
Stein (see [15, Def. 2.2.1]). This also follows from Remark 3.3 since closed
submanifolds of ${\mathbb{C}}^{n}$ are always Stein.

(b) Since $G$ is homotopy equivalent to $\hat{{\mathbb{C}}}$, it is not
biholomorphic to ${\mathbb{C}}^{2}$. It is is known that there exist
Fatou–Bieberbach domains in ${\mathbb{C}}^{2}$ avoiding a complex line, see
[15, Example 4.3.9], so $G\cap{\mathbb{C}}^{2}$ contains a Fatou–Bieberbach
domain (i.e. a proper subset of ${\mathbb{C}}^{2}$ which is biholomorphically
equivalent to ${\mathbb{C}}^{2}$). This statement also follows more generally
from (a), (d), and [29, Corollary 4.1].

(c) The diffeomorphic equivalence between the real tangent bundle of a real
analytic manifold and its complexification is a more general fact in the sense
of [6, p.104]. An explicit diffeomorphism for our case can be found in [27,
p.410].

(d) Remark 3.3 shows that $G$ can be represented as
$\\{(a,b,c)\in{\mathbb{C}}^{3}\,:\,ac=(b^{2}-1)/4\\}$. The density property
now follows from the main theorem in [16] with $p(x)=(x^{2}-1)/4$. ∎

##  4 Möbius–invariant holomorphic functions on the second configuration space
of the sphere

In order to describe the algebras $\mathcal{A}(X)$ for hyperbolic Riemann
surfaces $X$ which are not simply connected, we are going to use some few
basic facts about Fuchsian groups, see e.g. [13, 30, 17]. Recall that a
Fuchsian group is a discrete Möbius group $\Gamma$ acting on the upper
half–plane ${\mathbb{H}}$. In particular, every Fuchsian group is a subgroup
of the group $\mathop{{\rm Aut}}({\mathbb{H}})$ of all conformal automorphisms
of ${\mathbb{H}}$ and the quotient ${\mathbb{H}}/\Gamma$ is a hyperbolic
Riemann surface. In fact, every hyperbolic Riemann surface arises in this way.
If $X$ is a hyperbolic Riemann surface and $\pi:{\mathbb{H}}\to X$ is a
universal covering map, then its group $\Gamma$ of covering transformations is
a fixed point free Fuchsian group, that it, every
$\gamma\in\Gamma\setminus\\{id\\}$ has no fixed points in ${\mathbb{H}}$. In
this case, $X$ is conformally equivalent to ${\mathbb{H}}/\Gamma$, and every
function $g:X\to{\mathbb{C}}$ can be identified with the $\Gamma$–invariant
function $f:=g\circ\pi:{\mathbb{H}}\to{\mathbb{C}}$ and vice versa.

We conclude that in order to find $\mathcal{A}({\mathbb{H}}/\Gamma)$ we only
need to identify the $\Gamma$–invariant functions in
$\mathcal{A}({\mathbb{H}})$. Now every $\gamma\in\mathop{{\rm
Aut}}({\mathbb{H}})$ induces in a natural way an element $\hat{\gamma}$ of the
group $\mathop{{\rm Aut}}(G)$ of biholomorphic self–maps of $G$ via

$\hat{\gamma}(z,w)=(\gamma(z),\gamma(w))\,.$

###### Lemma 4.1

Let $f\in\mathcal{A}({\mathbb{H}})$ and $F\in\mathcal{H}(G)$ such that
$f(z)=F(z,\overline{z})$. Suppose that $\gamma\in\mathop{{\rm
Aut}}({\mathbb{H}})$. Then $f$ is $\gamma$–invariant if and only if $F$ is
$\hat{\gamma}$–invariant.

###### Proof.

This is immediate from

$(f\circ\gamma)(z)=F\left(\gamma(z),\overline{\gamma(z)}\right)=F(\gamma(z),\gamma(\overline{z}))=(F\circ\hat{\gamma})(z,\overline{z})$

which is based on the crucial property that every $\gamma\in\mathop{{\rm
Aut}}({\mathbb{H}})$ not only maps ${\mathbb{H}}$ onto ${\mathbb{H}}$, but is
symmetric w.r.t. the real axis, that is,
$\overline{\gamma(z)}=\gamma(\overline{z})$. ∎

We are therefore led to the following problem.

###### Problem 4.2

Let $\Gamma$ be a Fuchsian group acting on the upper half–plane
${\mathbb{H}}$. Find the set of all $F\in\mathcal{H}(G)$ which are invariant
under the subgroup

$\hat{\Gamma}:=\\{\hat{\gamma}\,:\,\gamma\in\Gamma\\}$

of $\mathop{{\rm Aut}}(G)$.

It turns out that the so–called limit set of the Fuchsian group $\Gamma$ plays
the crucial role here. The limit set of $\Gamma$, denoted by
$\Lambda(\Gamma)$, is the set of accumulation points of the $\Gamma$–orbits of
any point $z\in{\mathbb{H}}$. Since the action of $\Gamma$ is properly
discontinuous,
$\Lambda(\Gamma)\subseteq\partial{\mathbb{H}}:={\mathbb{R}}\cup\\{\infty\\}$.
A Fuchsian group $G$ is said to be elementary if $\Lambda(\Gamma)$ consists of
exactly zero, one or two elements. Otherwise, it is called non–elementary. It
is well–known (see e.g. [13, Corollary 3.4.6]) that the limit set of any
non–elementary Fuchsian group is either $\partial{\mathbb{H}}$ or a Cantor
set.

###### Theorem 4.3

Let $\Gamma$ be a non–elementary Fuchsian group. Then every
$\hat{\Gamma}$–invariant holomorphic function on $G$ is constant.

###### Corollary 4.4

Suppose that $X$ is a hyperbolic Riemann surface with a non–elementary
covering group. Then $\mathcal{A}(X)$ consists only of constant functions.

Corollary 4.4 applies in particular to all compact Riemann surfaces of higher
genus. We wish to point out that this strong rigidity result only depends on
the function–theoretic description of $\mathcal{A}({\mathbb{D}})$ in terms of
$\mathcal{H}(\Omega)$ and conformal invariance.

We now turn to the remaining cases, when $X$ is not simply connected and the
Fuchsian group $\Gamma$ is elementary. Since we are only interested in
hyperbolic Riemann surfaces, we may restrict ourselves to the case that
$\Gamma$ is a covering group, so it acts freely on ${\mathbb{H}}$. Recalling
that two fixed–point free Fuchsian groups $\Gamma$ and $\Gamma^{\prime}$ are
conjugated if and only if the corresponding Riemann surfaces
${\mathbb{D}}/\Gamma$ and ${\mathbb{D}}/\Gamma^{\prime}$ are conformally
equivalent, the well–known classification of elementary Fuchsian groups (see
[13, Proposition 3.1.2]) implies that there are only two cases to be
considered. Either $\Gamma$ is conjugate to a group generated by exactly one
parabolic element (one fixed point on $\partial{\mathbb{H}}$) or by exactly
one hyperbolic element (two fixed points on $\partial{\mathbb{H}}$), that is,
${\mathbb{D}}/\Gamma$ is either the punctured unit disk or a proper annulus.

We first consider the case of a proper annulus
$A_{R}=\\{z\in{\mathbb{C}}\,:\,1/R<|z|<R\\}$ for some $R>1$. Here
$A_{R}={\mathbb{H}}/\Gamma$, where $\Gamma$ is the cyclic group generated by
$z\mapsto c\cdot z$ with $\log c=\pi^{2}/\log R>0$.

######  Theorem 4.5 (Annulus)

Let $\Gamma=\\{z\mapsto c^{n}z\,:\,n\in\mathbb{Z}\\}$ for some $c>1$. Then
$F\in\mathcal{H}(G)$ is $\hat{\Gamma}$–invariant if and only if

$F(z,w)=g\left(\frac{w}{z-w}\right)$

for some $g\in\mathcal{H}({\mathbb{C}})$.

On the Danielewski surface $D$ (see (3.1)) the functions described in Theorem
4.5 are exactly all holomorphic functions $(a,b,c)\mapsto F(a,b,c)$ which
depend only on $b$.

We finally consider the case of the punctured unit disk
${\mathbb{D}}^{*}:={\mathbb{D}}\setminus\\{0\\}={\mathbb{H}}/\Gamma$, where
$\Gamma$ is the cyclic group generated by right–translation $z\mapsto z+1$ and
associated universal covering $z\mapsto e^{2\pi iz}$.

###### Theorem 4.6

Let $\Gamma:=\\{z\mapsto z+n\,:\,n\in\mathbb{Z}\\}$. Then $F\in\mathcal{H}(G)$
is $\hat{\Gamma}$–invariant if and only if

$F(z,w)=g\left(\frac{1}{z-w}\right)$

for some $g\in\mathcal{H}({\mathbb{C}})$.

On the Danielewski surface $D$ (see (3.1)) the functions described in Theorem
4.6 are exactly all holomorphic functions $(a,b,c)\mapsto F(a,b,c)$ which
depend only on $a$.

######  Remark 4.7 (Elementary Fuchsian groups that do not act freely on
${\mathbb{H}}$)

For completeness we describe all $\hat{\Gamma}$–invariant functions in
$\mathcal{H}(G)$ for the case that $\Gamma$ is an elementary Fuchsian group
which does not act freely on ${\mathbb{H}}$. There are basically two cases:

Case 1: $\Gamma$ is a Fuchsian group generated by an elliptic element. On
${\mathbb{D}}$, such a group is conjugated to

$\Gamma^{\prime}=\\{z\mapsto\alpha^{k}\cdot
z\,:\,k\in\mathbb{Z}\\},\quad\alpha=\exp(2\pi i/N),\,N\geq 2.$

We see that the quotient ${\mathbb{D}}/\Gamma^{\prime}$ is isomorphic to
${\mathbb{D}}$. (The function $[z]\in{\mathbb{D}}/\Gamma^{\prime}\mapsto
z^{N}$ is a chart.) The corresponding invariant functions on $\Omega$ satisfy
$F(\alpha z,\overline{\alpha}w)=F(z,w)$. Each of the functions

$f_{p,q}(z,w):=\frac{z^{p}w^{q}}{(1-zw)^{\max\\{p,q\\}}}\,,\quad
p,q\in{\mathbb{N}}_{0}\,,$

belongs to $\mathcal{H}(\Omega)$ and conversely, $\mathcal{H}(\Omega)$ is the
closure of the span of all $f_{p,q}$, [21, Theorem 3.16]. We conclude that the
space of all invariant $F\in\mathcal{H}(\Omega)$ is the closure of the span of
those $f_{p,q}$ with $(p-q)/N\in\mathbb{Z}$.

The set of all functions ${\mathbb{D}}\ni z\mapsto F(z,\overline{z})$,
$F\in\mathcal{H}(\Omega)$ $\hat{\Gamma}$–invariant, can be represented on
${\mathbb{D}}$ by using the chart $z^{N}$. It consists of all functions

$g:{\mathbb{D}}\to{\mathbb{C}}\quad\text{with}\quad
g(z^{N})=F(z,\overline{z}),\quad F\in\mathcal{H}(\Omega).$

Note that on ${\mathbb{D}}/\Gamma^{\prime}$ this set contains non-
differentiable functions (non-differentiable at $[0]$). For example, let $N=2$
and consider the function $g(z)=\frac{|z|}{1-|z|}$ on ${\mathbb{D}}$. Then
$g(z^{2})=f_{1,1}(z,\overline{z})$ with $f_{1,1}\in\mathcal{H}(\Omega)$.

Case 2: The Fuchsian group is conjugated to the group generated by $z\mapsto
cz$ for some $c>1$, and $z\mapsto-1/z$ on ${\mathbb{H}}$. This group is the
direct product of the groups generated by $z\mapsto cz$ and $z\mapsto-1/z$.
Thus we can take the quotient of ${\mathbb{H}}/(z\sim
cz)=A_{R}=\\{z\in{\mathbb{C}}\,:\,1/R<|z|<R\\}$ with respect to the involution
$J(z)=1/z$. The set of all functions $g\in\mathcal{A}(A_{R})$ which are
invariant under $z\mapsto 1/z$ can also be written as

$\left\\{z\mapsto g\left(\left(\tan\left(\frac{\pi}{2\log
R}\log|z|\right)\right)^{2}\right)\,:\,g\in\mathcal{H}({\mathbb{C}})\right\\}.$

This follows immediately from Theorem 1.7.

##  5 Möbius–invariant holomorphic functions on the second configuration space
of the sphere – Proof of Theorem 4.3 and Theorem 4.5

For the proof of Theorem 4.3 we recall that $\gamma\in\mathop{{\rm
Aut}}({\mathbb{H}})$ is said to be hyperbolic if it has two distinct
fixed–points on $\partial{\mathbb{H}}$, one repulsive and one attractive. If
$\Gamma$ is a non–elementary Fuchsian group, then its limit set
$\Lambda(\Gamma)$ has the following two well–known properties

  * (P1)

$\Lambda(\Gamma)$ is the closure of the set of fixed points of the hyperbolic
elements in $\Gamma$ ([17, Thm. 3.4.4]);

  * (P2)

$\Lambda(\Gamma)$ has no isolated points ([17, Thm. 3.4.6]).

The idea of the proof of Theorem 4.3 is as follows. In a first step, one
proves that for each fixed point $w_{0}$ of any hyperbolic element in $\Gamma$
the holomorphic functions $F(w_{0},\cdot)$ and $F(\cdot,w_{0})$ are constant.
In a second step, which is based on (P1), it is then shown that $F$ is
constant on $(\Lambda(\Gamma)\times\Lambda(\Gamma))\cap G$. From this and
using (P2), one finally can prove that $F$ is constant on $G$.

###### Proof of Theorem 4.3.

Let $F\in\mathcal{H}(G)$ be $\hat{\Gamma}$-invariant.

(i) Let $\gamma$ be a hyperbolic automorphism in $\Gamma$ with fixed point
$w_{0}\in\partial{\mathbb{H}}={\mathbb{R}}\cup\\{\infty\\}$. We show that
$F(\cdot,w_{0})\in{\mathbb{H}}(\hat{{\mathbb{C}}}\setminus\\{w_{0}\\})$ and
$F(w_{0},\cdot)\in{\mathbb{H}}(\hat{{\mathbb{C}}}\setminus\\{w_{0}\\})$ are
constant.

Since $\gamma$ is hyperbolic, it has a second fixed point
$z_{0}\in\partial{\mathbb{H}}\setminus\\{w_{0}\\}$. We may assume that
$z_{0}\in{\mathbb{R}}$, so
$\gamma^{\prime}(z_{0})\not\in\partial{\mathbb{D}}$. The function
$g:=F(\cdot,w_{0})\in\mathcal{H}(\hat{{\mathbb{C}}}\setminus\\{w_{0}\\})$ is
$\gamma$–invariant, since

$(g\circ\gamma)(z)=F(\gamma(z),w_{0})=F(\gamma(z),\gamma(w_{0}))=F(z,w_{0})=g(z)\,.$
(5.1)

In particular, $g$ has a fixed point at $z_{0}$, and taking the derivative in
(5.1) at $z_{0}$ leads to
$g^{\prime}(z_{0})\cdot\gamma^{\prime}(z_{0})=g^{\prime}(z_{0})$. Since
$\gamma^{\prime}(z_{0})\not=1$, this implies $g^{\prime}(z_{0})=0$. Taking the
second derivative in (5.1) at $z_{0}$ leads to $g^{\prime\prime}(z_{0})=0$ and
continuing this way, we easily see that $g$ is constant. In a similar way, one
can show that
$F(w_{0},\cdot)\in\mathcal{H}(\hat{{\mathbb{C}}}\setminus\\{w_{0}\\})$ is
constant.

(ii) Fix two distinct fixed points $p,q\in\partial{\mathbb{H}}$ of some
hyperbolic automorphism in $\Gamma$ and let $C:=F(p,q)$. Now, let
$r,s\in\partial{\mathbb{H}}\setminus\\{p,q\\}$ be two distinct fixed points of
another hyperbolic automorphism in $\Gamma$. By (i),
$F(p,\hat{{\mathbb{C}}}\setminus\\{p\\})=\\{C\\}$, so $F(p,s)=C$. Again by
(i), we get $F(\hat{{\mathbb{C}}}\setminus\\{s\\},s)=C$, so $F(r,s)=C$. Since
$\Gamma$ is non–elementary, the fixed points of all hyperbolic elements in
$\Gamma$ are dense in $\Lambda(\Gamma)$. This implies that $F$ is constant
$=C$ on $(\Lambda(\Gamma)\times\Lambda(\Gamma))\cap G$ as follows. Let
$(p,q)\in\left(\Lambda(\Gamma)\times\Lambda(\Gamma)\right)\cap\Omega$. In
particular, $p\not=q$. Then there are sequences $(p_{n})$,
$(q_{n})\subseteq\partial{\mathbb{H}}$, $(T_{n})\subseteq\Gamma$ such that
each $T_{n}$ is hyperbolic and $T_{n}(p_{n})=p_{n}\to p$ as well as
$T_{n}(q_{n})=q_{n}\to q$ as $n\to\infty$. Hence $q_{n}\not=p_{n}$ for $n$
sufficiently large, so $F(p_{n},\cdot)\equiv C$ on
$\hat{{\mathbb{C}}}\setminus\\{p_{n}\\}$ and thus
$F(p,q)=\lim_{n\to\infty}F(p_{n},q_{n})=C$.

(iii) Since $\Lambda(\Gamma)$ contains no isolated points, it is clear that
any $h\in\mathcal{H}(G)$ which is constant on
$(\Lambda(\Gamma)\times\Lambda(\Gamma))\cap G$ has the property that
${\partial h/\partial z}={\partial h/\partial w}=0$ on
$(\Lambda(\Gamma)\times\Lambda(\Gamma))\cap G$. In particular, this implies
that for any $(p,q)\in(\Lambda(\Gamma)\times\Lambda(\Gamma))\cap G$ all
partial derivatives of $F$ at the point $(p,q)$ vanish. Hence $F$ is constant.
∎

###### Proof of Theorem 4.5.

Since

$\hat{\Gamma}=\\{(z,w)\mapsto(c^{n}z,c^{n}w)\,:\,n\in\mathbb{Z}\\}\,,$

it is clear that for every entire function $g:{\mathbb{C}}\to{\mathbb{C}}$ the
holomorphic function

$F:G\to{\mathbb{C}}\,,\qquad F(z,w):=g\left(\frac{w}{z-w}\right)\,,$

is $\hat{\Gamma}$–invariant. In order to prove the only–if part of Theorem
4.5, let $F\in\mathcal{H}(G)$ be $\hat{\Gamma}$–invariant. We define

$H\in\mathcal{H}\left({\mathbb{C}}^{2}\setminus\left(\\{(1,v)\,:\,v\in{\mathbb{C}}\\}\cup\\{(u,0)\,:\,u\in{\mathbb{C}}\\}\right)\right)\,,\qquad
H(u,v):=F(v/u,v)\,.$

For fixed $u\in{\mathbb{C}}\setminus\\{1\\}$, we have

$H(u,cv)=F(cv/u,cv)=F(v/u,v)=H(u,v)\,,\qquad
v\in{\mathbb{C}}\setminus\\{0\\}\,.$

Comparing the coefficients in the Laurent expansions about the point $v=0$ on
both sides, we see that $H(u,\cdot)$ is constant $=H(u,1)$ on
${\mathbb{C}}\setminus\\{0\\}$. This implies that

$F(z,w)=H(w/z,w)=H(w/z,1)\qquad\text{ for all }(z,w)\in G\text{ s.t.
}z\not=0\,.$

We now set

$g(s):=H\left(\frac{s}{s-1},1\right)\,\qquad
s\in{\mathbb{C}}\setminus\\{1\\}\,.$

Then $g\in\mathcal{H}({\mathbb{C}}\setminus\\{1\\})$ has a holomorphic
extension to ${\mathbb{C}}$, since

$g(s)=F\left(1-\frac{1}{s},1\right)\,,\quad
s\in{\mathbb{C}}\setminus\\{1\\}\,,$

and $F$ is holomorphic on $G$. This shows that

$F(z,w)=g\left(\frac{1}{1-z/w}\right)$

for all $(z,w)\in G$. ∎

###### Proof of Theorem 4.6 (a).

Since

$\hat{\Gamma}=\\{(z,w)\mapsto(z+n,w+n)\,:\,n\in\mathbb{Z}\\}\,,$

it is clear that for every entire function $g:{\mathbb{C}}\to{\mathbb{C}}$ the
function

$F:G\to{\mathbb{C}}\,,\qquad F(z,w):=g\left(\frac{1}{z-w}\right)\,,$

is $\hat{\Gamma}$–invariant. In order to prove the only–if part of Theorem
4.6, let $F\in\mathcal{H}(G)$ be a $\hat{\Gamma}$–invariant function. We
proceed in two steps.

(i) We first note that for any $k=0,1,2\ldots$ the entire function

$\frac{\partial^{k}F}{\partial w^{k}}(\cdot,\infty)\,$

is constant. In fact,

$\frac{\partial^{k}F}{\partial w^{k}}\in\mathcal{H}(\Omega)$

is $\hat{\Gamma}$–invariant, so that for fixed $z,w\in{\mathbb{C}}$, we have

$\displaystyle\frac{\partial^{k}F}{\partial w^{k}}(z,\infty)$ $\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}\frac{\partial^{k}F}{\partial
w^{k}}(z,w+n)=\lim\limits_{n\to\infty}\frac{\partial^{k}F}{\partial
w^{k}}(z-n+n,w+n)$ $\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}\frac{\partial^{k}F}{\partial
w^{k}}(z-n,w)=\frac{\partial^{k}F}{\partial w^{k}}(\infty,w)\,.$

This proves (i).

(ii) Define

$H\in\mathcal{H}({\mathbb{C}}\times\hat{{\mathbb{C}}}\setminus\\{0\\})\,,\qquad
H(u,v):=F(u,u-v)\,.$

Since $F$ is $\hat{\Gamma}$–invariant, we see that for every fixed
$v\in\hat{{\mathbb{C}}}\setminus\\{0\\}$ the function $u\mapsto H(u,v)$ is a
$1$–periodic entire function and therefore can be written as

$H(u,v)=\sum\limits_{n=-\infty}^{\infty}a_{n}(v)e^{2\pi inu}$

with “Fourier coefficients”
$a_{n}\in\mathcal{H}(\hat{{\mathbb{C}}}\setminus\\{0\\})$ for $n=0,\pm 1,\pm
2,\ldots$. Now (i) implies that for each $k=0,\pm 1,\pm 2,\ldots$ the
$1$–periodic entire function

$\frac{\partial^{k}H}{\partial v^{k}}(\cdot,\infty)$

is constant and hence

$\frac{d^{k}a_{n}}{dv^{k}}(\infty)=0\quad\text{ for all }n=\pm 1,\pm
2,\ldots\,.$

For any $n=\pm 1,\pm 2,\ldots$ the holomorphic function
$a_{n}:\hat{{\mathbb{C}}}\setminus\\{0\\}\to{\mathbb{C}}$ therefore vanishes
identically and we conclude

$H(u,v)=a_{0}(v)\,,\qquad(u,v)\in{\mathbb{C}}\times\hat{{\mathbb{C}}}\setminus\\{0\\}\,.$

This shows that

$F(z,w)=H(z,z-w)=a_{0}(z-w)\text{ for all }(z,w)\in G\,$

with a function $a_{0}\in\mathcal{H}(\hat{{\mathbb{C}}}\setminus\\{0\\})$. The
proof of Theorem 4.6 is complete. ∎

##  6 The convergent Wick–star product on doubly connected Riemann surfaces

In this section we prove Theorem 1.7 and Corollary 1.8 as well as Theorem 1.9
and Theorem 1.11. The main tools will be (i) an explicit formula for the star
product $\star_{\hbar,{\mathbb{D}}}$ on the unit disk ${\mathbb{D}}$ and (ii)
an asymptotic expansion for it as $\hbar\to 0$. Both results have recently
been established in [10] and rely essentially on formula (5.24) in [26]. The
explicit formula for $\star_{{\mathbb{D}}}$ is based on invariant derivatives
of Peschl–Minda type, which are defined as follows. For $z\in{\mathbb{D}}$
denote

$T_{z}(u):=\frac{z+u}{1+\overline{z}u}\in\mathop{{\rm Aut}}({\mathbb{D}})\,,$

and by

$\partial:=\frac{1}{2}\left(\frac{\partial}{\partial
x}-i\frac{\partial}{\partial y}\right)$

the standard (Wirtinger) differential operator. Then for every
$C^{\infty}$–function $f:{\mathbb{D}}\to{\mathbb{C}}$ the Peschl–Minda
derivatives of $f$ of order $n$ at the point $z\in D$ are defined by

$D^{n}f(z):=\partial^{n}(f\circ
T_{z})(0)\,,\qquad\overline{D}^{n}f(z):=\overline{D^{n}(\overline{f})(z)}\,,$

see e.g. [10, 18]. Corollary 5.1 in [10] provides the following explicit
formula for $D^{n}f$ in terms of the Wirtinger derivative $\partial$:

$D^{n+1}f(z)=\left(1-|z|^{2}\right)\partial^{n+1}\left[\left(1-|z|^{2}\right)^{n}f(z)\right]$
(6.1)

Unlike $\partial^{n+1}=\partial^{n}\partial$ one has $D^{n+1}\not=D^{n}D^{1}$
in general. The basic result we need is now

$(f\star_{{\mathbb{D}},\hbar}\tilde{f})(z)=\sum\limits_{n=0}^{\infty}\frac{c_{n}(\hbar)}{n!}\,D^{n}\tilde{f}(z)\cdot\overline{D}^{n}f(z)\,$
(6.2)

for every $f,\tilde{f}\in\mathcal{A}({\mathbb{D}})$ and every
$\hbar\in\mathscr{D}$, and with $c_{n}(\hbar)$ defined as in (1.2). In fact,
for each fixed $z\in{\mathbb{D}}$ the series (6.2) converges absolutely and
locally uniformly w.r.t. $\hbar\in\mathscr{D}$, so
$\hbar\mapsto(f\star_{{\mathbb{D}},\hbar}\tilde{f})(z)$ is holomorphic on
$\mathscr{D}$. Moreover, for fixed $\hbar\in\mathscr{D}$, the series (6.1)
converges in the topology of $\mathcal{A}({\mathbb{D}})$. See [10, Lemma 6.5].

###### Proof of Theorem 1.7.

The proof is divided into several steps.

(i) We first prove that the linear mapping
$T_{R}:\mathcal{H}({\mathbb{C}})\to\mathcal{A}(A_{R})$, $T_{R}(g)=g\circ
f_{R}$, is bijective and establish (1.4). For this purpose we work on the
upper half plane ${\mathbb{H}}$. We note that

$\pi:{\mathbb{H}}\to A_{R}\,,\qquad\pi(z)=\exp\left(\frac{2i\log
R}{\pi}\log\left(\frac{z}{i}\right)\right)\,,$

is a universal covering of $A_{R}$ and $\Gamma=\\{z\mapsto
c^{n}z\,:\,n\in\mathbb{Z}\\}$ is the associated group of deck transformations,
where

$\log c=\frac{\pi^{2}}{\log R}\,,$

and $\log$ denotes the principal branch of the logarithm. Let
$f:A_{R}\to{\mathbb{C}}$ be a function. Then $f$ belongs to
$\mathcal{A}(A_{R})$ if and only if the $\Gamma$–invariant function
$f\circ\pi$ belongs to $\mathcal{A}({\mathbb{H}})$. By Lemma 4.1 this is
equivalent to $(f\circ\pi)(z)=F(z,\overline{z})$ for all $z\in{\mathbb{H}}$
for some uniquely determined $\hat{\Gamma}$–invariant function
$F\in\mathcal{H}(G)$. Theorem 4.5 therefore shows that
$f\in\mathcal{A}(A_{R})$ if and only if

$(f\circ\pi)(z)=\tilde{g}\left(\frac{\overline{z}}{z-\overline{z}}\right)\,,\qquad
z\in{\mathbb{H}}\,,$

for some uniquely determined function $\tilde{g}\in\mathcal{H}({\mathbb{C}})$.
In fact, the proof of Theorem 4.5 reveals that $\tilde{g}(s)=F(1-1/s,1)$ for
all $s\in{\mathbb{C}}\setminus\\{1\\}$. Also note that

$\frac{\overline{z}}{z-\overline{z}}=-\frac{i}{2}\tan\left(\frac{\pi}{2\log
R}\log|\pi(z)|\right)-\frac{1}{2}=\frac{1}{2}f_{R}(\pi(z))-\frac{1}{2}\,,\qquad
z\in{\mathbb{H}}\,,$

with $f_{R}$ as defined in (1.1). Hence, if $f\in\mathcal{A}(A_{R})$, then
$f=g\circ f_{R}$ for the entire function $g(z):=\tilde{g}(z/2-1/2)$, and
conversely, $f:=g\circ f_{R}$ belongs to $\mathcal{A}(A_{R})$ for every
$g\in\mathcal{H}({\mathbb{C}})$. This proves (1.4), and also that the mapping
$T_{R}:\mathcal{H}({\mathbb{C}})\to\mathcal{A}(A_{R})$, $T_{R}(g):=g\circ
f_{R}$, is a linear bijection.

(ii) As a next step, we prove that
$T_{R}^{-1}:\mathcal{A}(A_{R})\to\mathcal{H}({\mathbb{C}})$ is continuous.
Suppose $(f_{n})\subseteq\mathcal{A}(A_{R})$ converges in $\mathcal{A}(A_{R})$
to $f\in\mathcal{A}(A_{R})$. Then $(f_{n}\circ\pi)(z)=F_{n}(z,\overline{z})$
and $(f\circ\pi)(z)=F(z,\overline{z})$ for uniquely determined
$\hat{\Gamma}$–invariant functions $F_{n},F\in\mathcal{H}(G)$ and $f_{n}\to f$
in $\mathcal{A}(A_{R})$ is the same as $F_{n}\to F$ in $\mathcal{H}(G)$ by
definition. As shown in (i), there are entire functions $\tilde{g}_{n}$,
$\tilde{g}$ such that $\tilde{g}_{n}(s)=F_{n}(1-1/s,1)$ and
$\tilde{g}(s)=F(1-1/s,1)$ for all $s\in{\mathbb{C}}\setminus\\{1\\}$. It
follows that $\tilde{g}_{n}\to\tilde{g}$ locally uniformly on
${\mathbb{C}}\setminus\\{1\\}$, which implies locally uniform convergence on
${\mathbb{C}}$ by the maximum principle. Hence
$g_{n}(z):=\tilde{g}_{n}(z/2-1/2)$ and $g(z):=\tilde{g}(z/2-1/2)$ are entire
functions such that $g_{n}\to g$ in $\mathcal{H}({\mathbb{C}})$ and
$f_{n}=g_{n}\circ f_{R}=T_{R}(g_{n})$ as well as $f=g\circ f_{R}=T_{R}(g)$.
This means that $T_{R}^{-1}(f_{n})=g_{n}\to g=T_{R}^{-1}(f)$ in
$\mathcal{H}({\mathbb{C}})$, so
$T_{R}^{-1}:\mathcal{A}(A_{R})\to\mathcal{H}({\mathbb{C}})$ is continuous, as
required. By the bounded inverse theorem for Fréchet spaces,
$T_{R}:\mathcal{H}({\mathbb{C}})\to\mathcal{A}(A_{R})$ is continuous as well.

(iii) In order to prove the explicit formula (1.5) for the star product
$\star_{A_{R},\hbar}$ for the annulus $A_{R}$ we work on the unit disk
${\mathbb{D}}$ and note that

$\pi_{R}:{\mathbb{D}}\to
A_{R}\,,\qquad\pi_{R}(z):=\exp\left(\frac{2i}{\pi}\log\frac{1+z}{1-z}\right)$

is a universal covering of $A_{R}$. Consider the auxiliary function

$p:{\mathbb{D}}\to{\mathbb{C}}\,,\qquad
p(z):=\frac{z-\overline{z}}{1-|z|^{2}}\,,$

and recall the definition of the function $f_{R}:A_{R}\to{\mathbb{C}}$ in
(1.1). Then

$f_{R}\circ\pi_{R}=p\,.$

A straightforward proof by induction and making use of (6.1) reveals that

$\displaystyle D^{n}(g\circ p)(z)$ $\displaystyle=$
$\displaystyle\left(\frac{1-\overline{z}^{2}}{1-|z|^{2}}\right)^{n}\,\left(g^{(n)}\circ
p\right)(z)$ $\displaystyle\overline{D}^{n}(g\circ p)(z)$ $\displaystyle=$
$\displaystyle(-1)^{n}\left(\frac{1-z^{2}}{1-|z|^{2}}\right)^{n}\,\left(g^{(n)}\circ
p\right)(z)\,,$

and hence

$\overline{D}^{n}(g\circ p)D^{n}(\tilde{g}\circ
p)=\left(p^{2}-1\right)^{n}\left(g^{(n)}\circ
p\right)\left(\tilde{g}^{(n)}\circ p\right)$

for all $g,\tilde{g}\in\mathcal{H}({\mathbb{C}})$. Inserting this formula with
$f=g\circ f_{R}$ and $\tilde{f}=\tilde{g}\circ f_{R}$ into (6.2) establishes
(1.5). That for fixed $\hbar\in\mathscr{D}$, the series (1.5) converges in the
topology of $\mathcal{A}(A_{R})$ is a direct consequence of the fact that the
series (6.2) converges in the topology of $\mathcal{A}({\mathbb{D}})$, see
[10, Lemma 6.5]. Alternatively, it is not difficult to show directly that the
series

$\sum\limits_{n=0}^{\infty}\frac{c_{n}(\hbar)}{n!}\left(w^{2}-1\right)^{n}g^{(n)}(w)\tilde{g}^{(n)}(w)$

converges locally uniformly w.r.t. $w\in{\mathbb{C}}$ for every fixed
$\hbar\in\mathscr{D}$ by making appeal to the Cauchy estimates for the entire
functions $g$ and $\tilde{g}$. ∎

###### Proof of Corollary 1.8.

By Theorem 1.7, the mapping $\Psi_{R^{\prime},R}=T_{R}\circ
T_{R^{\prime}}^{-1}:\mathcal{A}(A_{R^{\prime}})\to\mathcal{A}(A_{R})$ is a
continuous linear bijection. It remains to prove that $\Psi_{R^{\prime},R}$ is
compatible with the algebra structures of $(\mathcal{A}(A_{R}),\star_{A_{R}})$
and $(\mathcal{A}(A_{R^{\prime}}),\star_{A_{R^{\prime}}})$. First note that
the definition of $\Psi_{R^{\prime},R}$ implies

$\Psi_{R^{\prime},R}(h\circ f_{R^{\prime}})=h\circ f_{R}\qquad\text{ for all
}h\in\mathcal{H}({\mathbb{C}})\,.$ (6.3)

Let $f,\tilde{f}\in\mathcal{A}(A_{R^{\prime}})$ and
$g,\tilde{g}\in\mathcal{H}({\mathbb{C}})$ such that $f=g\circ
f_{R^{\prime}}=T_{R^{\prime}}(g)$ and $\tilde{f}=\tilde{g}\circ
f_{R^{\prime}}=T_{R^{\prime}}(\tilde{g})$. Applying (6.3) for the entire
function $h(w):=(w^{2}-1)^{n}g^{(n)}(w)\tilde{g}^{(n)}(w)$ shows that

$\Psi_{R^{\prime},R}\left[\left(f_{R^{\prime}}^{2}-1\right)^{n}\left(g^{(n)}\circ
f_{R^{\prime}}\right)\left(\tilde{g}^{(n)}\circ
f_{R^{\prime}}\right)\right]=\left(f_{R}^{2}-1\right)^{n}\left(g^{(n)}\circ
f_{R}\right)\left(\tilde{g}^{(n)}\circ f_{R}\right)\,.$ (6.4)

Since the series (1.5) converges for fixed $\hbar\in\mathscr{D}$ in the
topology of $\mathcal{A}(A_{R})$ and
$\Psi_{R^{\prime},R}:\mathcal{A}(A_{R^{\prime}})\to\mathcal{A}(A_{R})$ is
continuous, we deduce

$\displaystyle\Psi_{R^{\prime},R}(f)\star_{A_{R}}\Psi_{R,^{\prime},R}(\tilde{f})$
$\displaystyle=$ $\displaystyle\Psi_{R^{\prime},R}(g\circ
f_{R^{\prime}})\star_{A_{R}}\Psi_{R^{\prime},R}(\tilde{g}\circ
f_{R^{\prime}})$ $\displaystyle\overset{(\ref{eq:Psi2})}{=}$
$\displaystyle(g\circ f_{R})\star_{A_{R}}(\tilde{g}\circ f_{R})$
$\displaystyle\overset{(\ref{eq:FormulaStarAnnulus})}{=}$
$\displaystyle\sum\limits_{n=0}^{\infty}c_{n}(\hbar)\left(f_{R}^{2}-1\right)^{n}\left(g^{(n)}\circ
f_{R}\right)\left(\tilde{g}^{(n)}\circ f_{R}\right)$
$\displaystyle\overset{(\ref{eq:Psi2})}{=}$
$\displaystyle\sum\limits_{n=0}^{\infty}c_{n}(\hbar)\Psi_{R^{\prime},R}\left[\left(f_{R^{\prime}}^{2}-1\right)^{n}\left(g^{(n)}\circ
f_{R^{\prime}}\right)\left(\tilde{g}^{(n)}\circ f_{R^{\prime}}\right)\right]$
$\displaystyle=$
$\displaystyle\Psi_{R^{\prime},R}\left[\sum\limits_{n=0}^{\infty}c_{n}(\hbar)\left(f_{R^{\prime}}^{2}-1\right)^{n}\left(g^{(n)}\circ
f_{R^{\prime}}\right)\left(\tilde{g}^{(n)}\circ f_{R^{\prime}}\right)\right]$
$\displaystyle\overset{(\ref{eq:FormulaStarAnnulus})}{=}$
$\displaystyle\Psi_{R^{\prime},R}\left(f\star_{A_{R}^{\prime}}\tilde{f}\right)\,.$

∎

###### Proof of Theorem 1.9.

The proof is very similar to the proof of Theorem 1.7, so we only indicate the
main steps. We first prove (1.7). To this end, we note that
$\pi:{\mathbb{H}}\to{\mathbb{D}}^{*}$, $\pi(z)=e^{2\pi iz}$, is a universal
covering of ${\mathbb{D}}^{*}$ and $\Gamma=\\{z\mapsto
z+n\,:\,n\in\mathbb{Z}\\}$ is the associated group of deck transformations.
Hence a function $f:{\mathbb{D}}^{*}\to{\mathbb{C}}$ belongs to
$\mathcal{A}({\mathbb{D}}^{*})$ if and only if the $\Gamma$–invariant function
$f\circ\pi$ belongs to $\mathcal{A}({\mathbb{H}})$. By Lemma 4.1 this is
equivalent to $(f\circ\pi)(z)=F(z,\overline{z})$ for some
$\hat{\Gamma}$–invariant function $F\in\mathcal{H}(G)$. Theorem 4.6 therefore
shows that $f\in\mathcal{A}({\mathbb{D}}^{*})$ if and only if

$f(e^{2\pi
iz})=(f\circ\pi)(z)=\tilde{g}\left(\frac{1}{z-\overline{z}}\right)\,,\qquad
z\in{\mathbb{H}}\,,$

for some entire function $\tilde{g}:{\mathbb{C}}\to{\mathbb{C}}$ or

$f(w)=g\left(\frac{1}{\log|w|}\right)=g(f_{0}(w))\,,\qquad
w\in{\mathbb{D}}^{*}\,,$

for the entire function $g(z):=\tilde{g}(-\pi z/i)$. Conversely, $f:=g\circ
f_{0}$ belongs to $\mathcal{A}({\mathbb{D}}^{*})$ for every
$g\in\mathcal{H}({\mathbb{C}})$, proving (1.7).

Finally, we prove the explicit formula (1.8) for the punctured disk
${\mathbb{D}}^{*}$, and for this purpose we work on the unit disk
${\mathbb{D}}$ and employ the universal covering map
$\pi_{0}:{\mathbb{D}}\to{\mathbb{D}}^{*}$ given by

$\pi_{0}(z):=\exp\left(-\frac{1+z}{1-z}\right)\,.$

Consider the auxiliary function

$q:{\mathbb{D}}\to{\mathbb{C}}\,,\qquad q(z):=\frac{|1-z|^{2}}{1-|z|^{2}}$

and the function $f_{0}:{\mathbb{D}}^{*}\to{\mathbb{C}}$ defined in (1.6).
Then

$\pi_{0}\circ f_{0}=q\,,$

and it easily follows inductively from (6.1) that

$\displaystyle D^{n}(g\circ q)(z)$ $\displaystyle=$
$\displaystyle(-1)^{n}\frac{(1-\overline{z})^{2n}}{(1-|z|^{2})^{n}}\,\left(g^{(n)}\circ
q\right)(z)$ $\displaystyle\overline{D}^{n}(g\circ q)(z)$ $\displaystyle=$
$\displaystyle(-1)^{n}\frac{(1-z)^{2n}}{(1-|z|^{2})^{n}}\,\left(g^{(n)}\circ
q\right)(z)\,$

and hence

$\overline{D}^{n}(g\circ q)D^{n}(\tilde{g}\circ q)=q^{2n}\left(g^{(n)}\circ
q\right)\left(\tilde{g}^{(n)}\circ q\right)$ (6.5)

for all $g,\tilde{g}\in\mathcal{H}({\mathbb{C}})$. Plugging this expression
into (6.2) results in (1.8) ∎

###### Proof of Theorem 1.11.

The proof is by contradiction. Assume there is a strong Fréchet algebra
isomorphism
$\Psi:(\mathcal{A}(A_{R}),\star_{A_{R}})\to(\mathcal{A}({\mathbb{D}}^{*}),\star_{{\mathbb{D}}^{*}})$.

(i) Consider $f_{R}\in\mathcal{A}(A_{R})$ as defined by (1.1). Then (1.5) for
$g(z)=\tilde{g}(z)=z$ implies

$f_{R}\star_{A_{R}}f_{R}=f_{R}^{2}+\left(f_{R}^{2}-1\right)\hbar\,$

and hence

$\Psi(f_{R}\star_{A_{R}}f_{R})=\Psi\left(f_{R}^{2}+\hbar\left(f_{R}^{2}-1\right)\right)=\Psi(f_{R}^{2})+\left(\Psi(f_{R}^{2})-1\right)\hbar\,.$

Here, we have used that $\Psi$ is a linear map and $\Psi(1)=1$, with $1$
denoting the function with constant value $1$.

(ii) By Theorem 1.9, there is a unique $g\in\mathcal{H}({\mathbb{C}})$ such
that

$\pi_{0}\circ\Psi(f_{R})=g\circ q\,.$

Here, $\pi_{0}$ and $q$ have the same meaning as in the proof of Theorem 1.9.
It follows from the definition of the star product
$\star_{\hbar,{\mathbb{D}}^{*}}$ that

$\left(\Psi(f_{R})\star_{\hbar,{\mathbb{D}}^{*}}\Psi(f_{R})\right)\circ\pi_{0}=(g\circ
q)\star_{\hbar,{\mathbb{D}}}(g\circ q)\,.$

We now make use of the asymptotic expansion for $\star_{\hbar,{\mathbb{D}}}$
as $\hbar\to 0$ established in [10, Theorem 6.9], which asserts

$\displaystyle(g\circ q)\star_{\hbar,{\mathbb{D}}}(g\circ q)$ $\displaystyle=$
$\displaystyle(g\circ q)^{2}+\hbar\overline{D}^{1}(g\circ q)D^{1}(g\circ
q)+\frac{\hbar^{2}}{2}\overline{D}^{2}(g\circ q)D^{2}(g\circ q)+O(\hbar^{3})$
$\displaystyle\overset{(\ref{eq:Bi})}{=}$
$\displaystyle\Psi(f_{R})^{2}\circ\pi_{0}+\hbar q^{2}(g^{\prime}\circ
q)^{2}+\frac{\hbar^{2}}{2}q^{4}(g^{\prime\prime}\circ q)^{2}+O(\hbar^{3})\,.$

(iii) Combing (i) and (ii), we see that under the assumption that
$\Psi(f_{R}\star_{\hbar,A_{R}}f_{R})=\Psi(f_{R})\star_{\hbar,{\mathbb{D}}^{*}}\Psi(f_{R})$
for all $\hbar\in\mathscr{D}$,

$\Psi(f_{R}^{2})\circ\pi_{0}+\left(\Psi(f_{R}^{2})\circ\pi_{0}-1\right)\hbar=\Psi(f_{R})^{2}\circ\pi_{0}+\hbar
q^{2}(g^{\prime}\circ q)^{2}+\frac{\hbar^{2}}{2}q^{4}(g^{\prime\prime}\circ
q)^{2}+O(\hbar^{3})\,.$

Comparing equal powers of $\hbar$, we get

  * (I)

$\hbar^{2}$: $\frac{1}{2}q^{4}(g^{\prime\prime}\circ q)^{2}=0$, so
$g^{\prime\prime}=0$, and hence $g(z)=\alpha z+\beta$ for some
$\alpha,\beta\in{\mathbb{C}}$.

  * (II)

$\hbar^{1}$: $\Psi(f_{R}^{2})\circ\pi_{0}-1=q^{2}\alpha^{2}$.

  * (III)

$\hbar^{0}$: $\Psi(f_{R}^{2})\circ\pi_{0}=\Psi(f_{R})^{2}\circ\pi_{0}$ and
hence $\Psi(f_{R}^{2})\circ\pi_{0}=(g\circ
q)^{2}=(\alpha^{2}q^{2}+2\alpha\beta q+\beta^{2})$.

Combining (II) and (III) gives $\alpha=0$ and $\beta=\pm 1$. Therefore,
$\Psi(f_{R})=\pm 1=\pm\Psi(1)=\Psi(\pm 1)$, and thus $f_{R}=\pm 1$, a
contradiction. ∎

##  7 Proof of Theorem 1.5, Corollary 1.6 and Corollary 1.13

###### Proof of Theorem 1.5.

The “only if” parts of (a), (b) and (c) follow from Proposition 1.4. In order
to prove the “if part” of (a) let
$\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}({\mathbb{D}})$. In particular,
$\mathcal{A}(X)$ contains non–constant functions and $\star_{X}$ is
noncommutative. Therefore Corollary 4.4 implies that the covering group of $X$
is elementary, and Theorem 1.7 and Theorem 1.9 imply that $X$ is not
conformally equivalent to a proper annulus or the punctured disk. Hence $X$ is
simply connected. Now assume
$\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}(A_{R})$ for some $R>1$. Then,
by Corollary 1.8,
$\mathcal{A}(X)\mathop{{\rm\simeq}}\mathcal{A}(A_{R^{\prime}})$ for every
$R^{\prime}>1$. Moreover, $(\mathcal{A}(X),\star_{X})$ is not strongly
isomorphic to $(\mathcal{A}({\mathbb{D}}^{*}),\star_{{\mathbb{D}}^{*}})$ by
Theorem 1.11 and hence $X$ is not conformally equivalent to ${\mathbb{D}}^{*}$
by Proposition 1.4. $X$ cannot be conformally equivalent to ${\mathbb{D}}$
since otherwise $(\mathcal{A}(X),\star_{X})$ would be noncommutative,
contradicting Theorem 1.7. We conclude that $X$ has to be conformally
equivalent to some proper annulus $A_{\rho}$, $\rho>1$. This proves the “if
part” of (b). The “if part” of (c) follows in a similar way. To prove (d) note
that if $X$ is neither conformally equivalent to ${\mathbb{D}}$ nor to some
annulus or punctured disk, then the covering group of $X$ is non–elementary,
so $\mathcal{A}(X)$ consist only of constant functions by Corollary 4.4. ∎

###### Proof of Corollary 1.13.

Note that $\mathcal{A}(A_{R})$ and $\mathcal{A}({\mathbb{D}}^{*})$ do not
separate points since they consists only of radially symmetric functions by
Theorem 1.7 and Theorem 1.9. On the other hand, $\mathcal{A}({\mathbb{D}})$
separates points, since $\mathcal{H}(\Omega)$ separates points as $\Omega$ is
a Stein manifold, see Theorem 3.1. This proves that (a) and (b) are
equivalent. The equivalence of (a) and (c) follows directly from Theorem 1.5,
Theorem 1.7 and Theorem 1.9. ∎

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Daniela Kraus, Oliver Roth, Sebsatian Schleißinger, Stefan Waldmann
Department of Mathematics
University of Würzburg
Emil Fischer Strasse 40
97074 Würzburg, Germany

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