# Hilbert Spaces of Entire Functions and
Toeplitz Quantization of Euclidean Planes

Micho D urd evich  Universidad Nacional Autónoma de México, Instituto de
Matemáticas, Area de la Investigacion Científica, Circuito Exterior, Ciudad
Universitaria, CP 04510, Mexico City, Mexico<EMAIL_ADDRESS>and  Stephen
Bruce Sontz  Centro de Investigación en Matemáticas, A.C., (CIMAT), Jalisco
S/N, Mineral de Valenciana, CP 36023, Guanajuato, Mexico<EMAIL_ADDRESS>
###### Abstract.

The theory of Toeplitz quantization presented in our previous paper is
extended and further developed to include diverse and interesting non-
commutative realizations of the classical Euclidean plane. This is done using
Hilbert spaces of entire functions, where polynomials in one complex variable
form a dense subspace. The complex coordinate naturally acts as an unbounded
multiplication operator generating, together with its adjoint, a highly non-
commutative *-algebra of operators. The Toeplitz operators are then
geometrically constructed as special elements from this algebra; they are
associated to the symbols from another quadratic non-commutative algebra,
which is interpretable as polynomials over a plane to be quantized. Such a
conceptual framework promotes interesting non-trivial conditions on the
initial scalar product. These are analyzed in detail. Various illustrative
examples are computed.

###### Contents

  1. 1 Introduction
  2. 2 Quantization Via Hilbert Spaces of Entire Functions
  3. 3 Some Algebraic Aspects of Toeplitz Quantization
  4. 4 Concluding Thoughts
  5. A Positivity and Stieltjes Moment Condition
  6. B On Hilbert Spaces of Holomorphic Functions

##  1\. Introduction

This paper is a continuation of our previous work [10], where we have provided
a detailed analysis of geometric, algebraic and analytic aspects for a
coherent states quantization of the Manin $q$-plane via Toeplitz operators. A
general method for such Toeplitz quantizations has been introduced and
extensively studied by the second author in [19, 20].

Here we would like to further expand on these principal dimensions by
exploring their deeper interrelations and establishing a link with the Hilbert
spaces of entire functions [3, 6]. On one hand this allows the use of a rich
array of techniques from complex analysis, and on the other it provides an
elegant framework for defining quantum versions of classical fundamental
geometrical objects—such as points, coordinates, observables and
transformations.

Our context fits well into the principal quantum non-commutative geometry
foundations as developed by Connes [8] and Woronowicz [22] with a variety of
interesting non-commutative algebras, representable by operators that appear
naturally in a Hilbert space. And moreover, our approach is particularly in
resonance with the pioneering formulation of quantum geometry by Prugovečki
[15], where the concept of point is quantized by morphing it into a special
wave function. These ‘wave functions of points’ can then be interpreted as
coherent states, and in our framework they are an integral part of the Hilbert
space geometry, which directly emerges from the reproducing kernel.

The reproducing kernel is a primary and defining structural part of Hilbert
spaces of holomorphic functions defined over an open domain
$\Omega\subset\mathbb{C}$. This is a map
$K\colon\Omega\times\Omega\rightarrow\mathbb{C}$, antiholomorphic in the first
coordinate and holomorphic in the second and exhibiting an appropriate matrix
complete positivity condition. Such a structure gives us the possibility of
speaking of ‘quantized’ points, these being the vectors of the Hilbert space
representing the points of $\Omega$. This association is obtained from the
kernel map by $\Omega\ni w\longmapsto K(\bar{w},*)\in\mathcal{H}$. By taking
scalar products these point wave functions evaluate in the points $w$ the
functions of $\mathcal{H}$. And by normalizing them we obtain the
corresponding coherent states. We shall here almost exclusively deal with
entire functions, which correspond to taking the domain $\Omega=\mathbb{C}$.

Let us outline the contents of the paper. In the next section in a series of
steps we establish our principal geometric and algebraic setup. We begin with
a detailed analysis of scalar products in the space $\mathbb{C}[z]$ of complex
polynomials in a complex variable $z$. Every such scalar product
$\langle|\rangle$ leads to a Hilbert space $\mathcal{H}$, namely the
completion of $\mathbb{C}[z]$ with respect to $\langle|\rangle$. So
$\mathbb{C}[z]$ is dense in $\mathcal{H}$. We would like to capture in terms
of the operators on $\mathcal{H}$ some basic idea of a plane, be it classical
or quantum. This naturally leads to interpreting the complex coordinate $z$ as
an appropriate multiplication operator $Z$ in $\mathcal{H}$, so that on
polynomials $\phi$ it acts as $Z\colon\phi(z)\mapsto z\phi(z)$. We would also
like to be able to consider the operator representation of the conjugate
coordinate $\bar{z}$ as an adjoint operator $Z^{*}$ so that an interesting
operator calculus involving $Z$ and $Z^{*}$ can be defined, reflecting the
geometrical idea of an underlying plane-like space. An important technical
problem here is to find the most effective context for defining $Z$ and
$Z^{*}$, including their domains. From consideration of this problem a number
of non-trivial conditions for the initial scalar product on $\mathbb{C}[z]$
naturally emerge. We shall call them Harmony Properties, and we shall number
them from $0$ to $3$.

Harmony Zero treats the very basic structure at the level of polynomials,
while Harmony One provides a particularly elegant and geometrically natural
answer by postulating that all the elements of the completion $\mathcal{H}$ of
$\mathbb{C}[z]$ are interpretable as entire functions. The definition of the
operator $Z$ then can be extended from the polynomials to its natural domain,
in the same way as the complex coordinate multiplication operator
$Z\colon\psi(z)\mapsto z\psi(z)$ with dense domain $\mathrm{D}(Z)$ consisting
of all $\psi(z)\in\mathcal{H}$ for which $z\psi(z)\in\mathcal{H}$. We shall
establish a variety of important properties for the operator $Z$ and its
adjoint $Z^{*}$. In particular $Z$ is closed and the point wave functions
associated to classical points $w\in\mathbb{C}$ are the unique (modulo non-
zero scalar multiples) eigenvectors for $Z^{*}$ with the eigenvalues
$\bar{w}$. This implies that the spectra of $Z$ and $Z^{*}$ coincide with the
whole $\mathbb{C}$; so they are unbounded operators.

However, this harmony property still does not ensure the existence of an
algebraically effective operational setting involving the operators $Z$ and
$Z^{*}$ so that we can meaningfully compose them in arbitrary combinations.

Our next harmony property addresses this, by postulating the continuity of
$Z^{*}$ in the normal topology of $\mathcal{H}$ (namely, the topology of
uniform convergence on compact sets) and thereby putting $Z$ and $Z^{*}$ in a
kind of balanced relationship. As we shall explain, this provides a common
acting space $\mathcal{W}$ for them, consisting of special vectors emerging
from the very geometry of $\mathcal{H}$. The point wave functions are always
in $\mathcal{W}$, and for polynomials to be included, we shall need the
topological half of Harmony Zero.

In such a framework the operators $Z$ and $Z^{*}$ generate a fundamental
*-algebra $\mathcal{A}$, in terms of which the underlying quantum space is
brought to life. We shall prove that $\mathcal{A}$ is quite non-commutative,
as it is always with a trivial centralizer in a pretty large algebra
$\mathrm{O}(\mathbb{C})$ of normally continuous linear transformations of
entire functions. We shall also prove that, under certain general additional
assumptions on interchangeability of order of the product of $Z^{*}$ and $Z$,
the underlying quantum space is always ‘pointless’. In other words there will
be no characters on $\mathcal{A}$.

We shall then proceed with interpreting the constructed framework as a natural
destination for a Toeplitz quantization of appropriate quadratic algebras
$\mathbb{C}[z,\bar{z},Q]$. These algebras are generated by coordinates $z$ and
$\bar{z}$, and equipped with a quadratic flip-over type relation $Q$ between
these coordinates. This relation can be trivial, in which case we are dealing
with the classical complex plane $\mathbb{C}$, twisted commutative as in the
case of the Heisenberg-Weyl algebra or the Manin $q$-plane, or a more
elaborate quadratic relation telling us how to exchange $z$ and $\bar{z}$.

The main construction here is that of an extended space $\mathcal{R}$ equipped
with a non-degenerate, but not necessarily positive definite form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$,
such that the common action space $\mathcal{W}$ for the operators of
$\mathcal{A}$ together with its scalar product $\langle|\rangle$ can be
singled out via a symmetric idempotent $\Pi$ acting on $\mathcal{R}$, and such
that the algebra $\mathbb{C}[z,\bar{z},Q]$ acts symmetrically on $\mathcal{R}$
with $\mathcal{W}$ being the cyclic $Z$-invariant subspace. The Toeplitz
operator with the symbol $f\in\mathbb{C}[z,\bar{z},Q]$ is then defined as $\Pi
f\Pi$ interpreted as an element of $\mathcal{A}$.

It is worth mentioning that the principal algebraic steps in a sense mirror
those of the Stinespring construction [21] for completely positive maps
between C*-algebras. As we shall see in detail, however, we are in the context
of unbounded operators and a key positivity condition is not always fulfilled.
On the other hand when
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
is positive – and this will be our final harmony condition – we can close
$\mathcal{R}$ in a Hilbert space $\mathcal{J}$ and the relationship between
$\mathbb{C}[z,\bar{z},Q]$ and $\mathcal{A}$ established by the Toeplitz
quantization turns out to have a particular richness.

We then proceed with some more detailed calculations of a number of concrete
examples by analyzing the coherent states, their resolution of the identity
and the Toeplitz quantization interpretation. We next analyze four principal
types for the algebras $\mathbb{C}[z,\bar{z},Q]$, and we also discuss their
possible realizations in Hilbert spaces of analytic (but not necessarily
entire) functions. As we shall see, the Heisenberg-Weyl algebra and its
$q$-variations are the only types allowing a realization with entire
functions. On the other hand the Manin $q$-plane is naturally realizable in a
space of Laurent series, whose common domain of definition is
$\mathbb{C}-\\{0\\}$. Some of these spaces have the unit disk or the half
plane as their common domain of definition for the holomorphic functions, and
these provide a natural setting for quantizing the classical hyperbolic plane
[9]. In all these calculations basic identities involving hypergeometric
series and $q$-special functions (such as the triple product identity, the
$q$-binomial theorem, the second Euler identity, $q$-versions of exponential,
logarithm and Gamma functions) naturally appear. The classical treaties [2]
and [13] provide an excellent in-depth exposition of these identities.

In Section 3 we focus on some general algebraic aspects of Toeplitz
quantization. We explain how the construction of the extended algebra and
module with the projector and the quadratic form can be performed over a large
class of algebras $\mathcal{C}$ equipped with a flip-over operator
$\sigma\colon\mathcal{C}\otimes\bar{\mathcal{C}}\rightarrow\bar{\mathcal{C}}\otimes\mathcal{C}$,
where $\bar{\mathcal{C}}$ is the conjugate algebra of $\mathcal{C}$. This
includes as a special case our main protagonist—the polynomials
$\mathbb{C}[z]$—and also multi-dimensional versions with several complex
variables.

Finally, in Section 4 some concluding remarks are made. The paper ends with
two extensions exhibiting perhaps some interest on their own.

In the first appendix, we provide a simple geometrical interpretation for the
positivity of the canonical quadratic form and of the Stieltjes moment
condition. And in the second appendix some formulas and illustrative examples
of Hilbert spaces of holomorphic functions are collected, including the
classical structures such as Segal-Bargmann spaces and their $q$-variations,
de Branges spaces including the Paley-Wiener space, and the Bergman spaces of
square integrable holomorphic functions over bounded domains. For all these
spaces a similar quantization scheme for their underlying classical domain can
be established.

We use standard notation such as $\mathbb{N}$ for the set of non-negative
integers and $\mathbb{C}$ for the complex plane.

##  2\. Quantization Via Hilbert Spaces of Entire Functions

### Polynomials Generating a Hilbert Space

The central object for our considerations will be the polynomial algebra
$\mathbb{C}[z]$ in one complex variable $z$. It is a complex infinite
dimensional vector space, in which the monomials
$\\{z^{n}\,|\,n\in\mathbb{N}\\}$ form a natural linear basis. For every
$n\in\mathbb{N}$ the polynomials of degree $\leq n$ form an
$(n+1)$-dimensional subspace, denoted by $\mathbb{C}_{n}[z]$.

The polynomials $\mathbb{C}[z]$ are included as a subalgebra in the larger
algebra of all entire functions, denoted by $\mathrm{H}(\mathbb{C})$. In the
normal topology we have

(2.1) $\overline{\mathbb{C}[z]}=\mathrm{H}(\mathbb{C}),$

where $\overline{A}$ denotes the topological closure of $A$.

Non-constant polynomials can also be understood as entire functions
$\phi\colon\mathbb{C}\rightarrow\mathbb{C}$ preserving the complex infinity
point $\infty$. Under composition polynomials form a non-commutative unital
semi-group. An algebraic counterpart to this geometrical view is the
interpretation of polynomials as unital algebra homomorphisms
$\phi\colon\mathbb{C}[z]\rightarrow\mathbb{C}[z]$. This linear action of
$\phi$ on $\mathbb{C}[z]$, which is called substitution, is specified on basis
elements by $\mathbb{C}[z]\ni z^{n}\longmapsto\phi(z)^{n}\in\mathbb{C}[z]$.

Of particular interest are those polynomials whose corresponding homomorphism
is invertible, which we call the invertible polynomials. They are precisely
the linear transformations $z\longmapsto az+b$ where $a,b\in\mathbb{C}$ with
$a\neq 0$. They provide all the orientation preserving symmetries of the
Euclidean plane $\mathbb{C}$. If $a=1$ we have translations given by $b$ (with
no fixed point if $b\neq 0$) and amplitwists around the unique fixed point
$b/(1-a)$ if $a\neq 1$ with rotation factor given by the phase of $a$ and
similarity factor given by $|a|$. (We continue assuming $a\neq 0$.) It is well
known that all the automorphisms of the algebra $\mathbb{C}[z]$ are of this
form, and this is the same as all the holomorphic automorphisms of the complex
plane $\mathbb{C}$.

There is a canonical anti-linear involutive algebra automorphism $J$ of
$\mathbb{C}[z]$ defined by $J(z)=z$ and which just conjugates the coefficients
of the polynomials.

The multiplication in $\mathbb{C}[z]$ can be interpreted as the left regular
representation, where $\mathbb{C}[z]$ acts by linear operators in
$\mathbb{C}[z]$. The constant polynomial $1\in\mathbb{C}[z]$ is both cyclic
and separating for this representation. Viewed in terms of the left regular
representation, $\mathbb{C}[z]$ is a maximal commutative subalgebra of linear
operators in $\mathbb{C}[z]$: any linear operator
$l:\mathbb{C}[z]\to\mathbb{C}[z]$ commuting with left multiplication by $z$
(and hence by induction and linearity with left multiplication by any
polynomial) is left multiplication by $l(1)\in\mathbb{C}[z]$.

Among other important operators acting in $\mathbb{C}[z]$ is the complex
differentiation $\partial/\partial z$, and hence all differential operators
whose coefficients are polynomial in $z$. All these operators—multiplication,
substitution, differentiation—are continuous (which will always mean in this
context with respect to the normal topology on $\mathbb{C}[z]$ induced by its
inclusion in $\mathrm{H}(\mathbb{C})$), and they naturally extend to the whole
$\mathrm{H}(\mathbb{C})$. We shall denote by $\mathrm{o}(\mathbb{C})$ the
algebra of all normal continuous linear transformations of $\mathbb{C}[z]$ to
itself:

$\mathrm{o}(\mathbb{C}):=\\{\tau:\mathbb{C}[z]\to\mathbb{C}[z]\,\big{|}\,\tau\mathrm{~{}is~{}linear~{}and~{}normal~{}continuous}\\}.$

Every $\tau\in\mathrm{o}(\mathbb{C})$ is uniquely extendible to a normal
continuous linear transformation
$\tau\colon\mathrm{H}(\mathbb{C})\rightarrow\mathrm{H}(\mathbb{C})$. We can
say equivalently that $\mathrm{o}(\mathbb{C})$ consists of continuous linear
transformations of $\mathrm{H}(\mathbb{C})$, which preserve $\mathbb{C}[z]$.
So $\mathrm{o}(\mathbb{C})$ is a subalgebra of the algebra of all continuous
linear transformations of $\mathrm{H}(\mathbb{C})$, which will be denoted by
$\mathrm{O}(\mathbb{C})$:

$\mathrm{O}(\mathbb{C}):=\\{\sigma:\mathrm{H}(\mathbb{C})\to\mathrm{H}(\mathbb{C})\,\big{|}\,\sigma\mathrm{~{}is~{}linear~{}and~{}normal~{}continuous}\\}.$

The algebra $\mathrm{H}(\mathbb{C})$ acts on itself via its left regular
representation in terms of which $\mathrm{H}(\mathbb{C})$ is a maximal
commutative subalgebra of $\mathrm{O}(\mathbb{C})$.

The dual space $\mathrm{H}(\mathbb{C})^{*}$ consists by definition of all
normally continuous complex linear functionals
$f:\mathrm{H}(\mathbb{C})\rightarrow\mathbb{C}$. This is the same as saying
normally continuous functionals on $\mathbb{C}[z]$, since each of the latter
uniquely extends by normal continuity to $\mathrm{H}(\mathbb{C})$. The
elements $f\in\mathrm{H}(\mathbb{C})^{*}$ are in one-to-one correspondence
with the complex sequences $f_{n}=f(z^{n})$ satisfying the following geometric
boundedness property: there exists a constant $\Lambda>1$ such that for all
$n\in\mathbb{N}$ we have

(2.2) $|f_{n}|\leq\Lambda^{n}.$

In addition to this very basic structure, we shall assume that $\mathbb{C}[z]$
is equipped with a positive definite scalar product $\langle|\rangle$, which
is anti-linear in its first entry and linear in the second. It is completely
specified by the infinite matrix

(2.3) $S_{\infty}=(s_{nm})\quad\mathrm{where}\quad s_{nm}=\langle
z^{n}|z^{m}\rangle\quad\mathrm{for~{}}n,m\in\mathbb{N}.$

This matrix is required to be be hermitian and moreover to be strictly
positive in the sense that the partial matrices

(2.4) $S_{n}=\begin{pmatrix}s_{00}&\cdots&s_{0n}\\\ \vdots&\ddots&\vdots\\\
s_{n0}&\cdots&s_{nn}\end{pmatrix}$

determining the scalar product in the subspaces $\mathbb{C}_{n}[z]$ are
strictly positive matrices for every $n\in\mathbb{N}$. For a hermitian matrix
this will be the case if and only if all the numbers

(2.5) $d_{n}=\det(S_{n})$

are strictly positive real numbers. In particular, $S_{n}$ is invertible and
$S_{n}^{-1}$ is strictly positive, too.

######  Remark 2.1.

All our constructions will be independent of rescalings of the scalar product.
So we can always normalize $\langle|\rangle$ by fixing a value for $s_{00}>0$,
for example by postulating $s_{00}=1$.

The following is a straightforward result.

######  Proposition 2.1.

The involution $J$ is an isometry if and only if all the coefficients $s_{nm}$
are real. ∎

By using the Gram-Schmidt procedure to orthonormalize the naturally ordered
sequence of monomials $z^{n}$, we obtain an orthonormal sequence of
polynomials $\phi_{n}(z)$ such that the degree of $\phi_{n}(z)$ is $n$ with
its highest order coefficient being positive. Also
$\phi_{n+1}(z)\bot\mathbb{C}_{n}[z]$. The space $\mathbb{C}_{n}[z]$ is spanned
by $\phi_{0}(z),\dots,\phi_{n}(z)$, and

(2.6) $B:=\Big{\\{}\phi_{n}(z)~{}\Big{|}~{}n\geq 0\Big{\\}}$

is an orthonormal vector space basis in the whole space $\mathbb{C}[z]$.

Explicitly, these canonical orthonormal polynomials are given by

(2.7)
$\phi_{n}(z)=\frac{1}{\sqrt{d_{n}d_{n-1}}}\left|\begin{matrix}s_{00}&s_{01}&\cdots&s_{0n}\\\
\vdots&\vdots&\ddots&\vdots\\\ \,s_{n-10}&s_{n-11}&\cdots&s_{n-1n}\,\\\
1&z&\cdots&z^{n}\end{matrix}\right|$

with the extra definition $d_{-1}:=1$.

######  Remark 2.2.

When appropriate we shall also write simply $|n\rangle=\phi_{n}(z)$.

The following polynomial identity holds:

(2.8)
$\sum_{k=0}^{n}\overline{\phi_{n}(w)}\phi_{n}(z)=\sum_{i,j=0}^{n}z^{i}[S_{n}^{-1}]_{ij}\bar{w}^{j}.$

The correspondence between the scalar products $\langle|\rangle$ on
$\mathbb{C}[z]$ and systems $\phi_{n}(z)$ is in fact one-to-one. If a system
$\phi_{n}(z)$ of polynomials satisfying $\deg\phi_{n}(z)=n$ and with positive
highest-order coefficients is given, then there exists a unique scalar product
$\langle|\rangle$ on $\mathbb{C}[z]$ orthonormalizing the monomials $z^{n}$
into $\phi_{n}(z)$.

Let us check how the scalar product matrix $S_{\infty}$ transforms under
holomorphic symmetries of $\mathbb{C}$. Again these are precisely the linear
maps $z\longmapsto az+b$ with $a,b\in\mathbb{C}$ and $a\neq 0$. We shall
consider separately rotations, scalings and translations.

First, under the rotations $z\longmapsto uz$ around $0$ by a unitary complex
number $u$ the coefficients $s_{nm}$ transform as

(2.9) $s_{nm}\rightsquigarrow u^{m-n}s_{nm}.$

In particular the scalar product will be invariant under all of these
transformations if and only if $s_{nm}=0$ for all $n\neq m$. Second, the
scaling $z\longmapsto rz$ with $r>0$ gives

(2.10) $s_{nm}\rightsquigarrow s_{nm}r^{n+m}.$

Finally, the transformation rule for translations $z\longmapsto z+b$ is

(2.11)
$s_{nm}\rightsquigarrow\sum_{k=0}^{n}\sum_{l=0}^{m}\binom{n}{k}\binom{m}{l}\bar{b}^{n-k}b^{m-l}s_{kl}.$

The transformations (2.9), (2.10) and (2.11) are easy to verify. From this it
is also easy to see that every transformation which is not a rotation around
some point of $\mathbb{C}$ affects non-trivially the scalar product.

######  Remark 2.3.

If the scalar product is invariant under the rotations (2.9)
($\Leftrightarrow$ the matrix $S_{\infty}$ is diagonal), we simplify the
notation by defining a sequence

(2.12) $s_{n}:=s_{nn}\quad\mathrm{for~{}}n\in\mathbb{N}.$

The canonical orthonormal polynomials are then simply

(2.13)
$\phi_{n}(z)=\frac{z^{n}}{\sqrt{\smash[b]{s_{n}}}}\quad\mathrm{for~{}}n\in\mathbb{N}.$

In our previous paper [10], we only dealt with such scalar products.

Let $\mathcal{H}$ be the Hilbert space obtained by completing the incomplete
pre-Hilbert space $\mathbb{C}[z]$ relative to $\langle|\rangle$. Then the
subspace $\mathbb{C}[z]$ is dense in $\mathcal{H}$, and the set $B$ in (2.6)
is an orthonormal basis for $\mathcal{H}$. Of course, $\mathcal{H}$ is not
unique, but it is unique up to a unique isometric isomorphism which is the
identity on $\mathbb{C}[z]$. One possible explicit construction of
$\mathcal{H}$ is as the set of all formal infinite series
$\sum_{n\in\mathbb{N}}a_{n}\phi_{n}(z)$ with coefficients $a_{n}\in\mathbb{C}$
satisfying $\sum_{n\in\mathbb{N}}|a_{n}|^{2}<\infty$. In this setting the
dense subspace $\mathbb{C}[z]$ is identified as the set of all formal sums for
which only finitely many coefficients are non-zero. This clearly shows that
$\mathbb{C}[z]$ is not equal to $\mathcal{H}$. Other realizations of
$\mathcal{H}$ will be considered later.

There exist four important, mutually subtly related, conditions for the scalar
product $\langle|\rangle$ and consequently for the resulting Hilbert space
$\mathcal{H}$. These conditions put this simple framework into special harmony
with operator theory and complex analysis, thereby establishing an elegant
context for constructing quantum models of the complex Euclidean plane
$\mathbb{C}$. We shall call them Harmony Properties and assign them numbers
$0$, $1$, $2$ and $3$.

Harmony Zero is a dual thing, namely a symbiosis of both an algebraic and a
topological condition. It provides the simplest, most straightforward
*-algebraic structure emerging from the space of polynomials and their scalar
product $\langle|\rangle$, and at the same time it maintains an elementary
topological resonance with complex analysis.

But first we shall say that an operator $S:\mathbb{C}[z]\to\mathbb{C}[z]$ is
formally adjointable with respect to $\langle|\rangle$ if there exists (a
necessarily unique) operator $T:\mathbb{C}[z]\to\mathbb{C}[z]$ satisfying
$\langle\phi|S\psi\rangle=\langle T\phi|\psi\rangle$ for all
$\phi,\psi\in\mathbb{C}[z]$, in which case we say that $T$ is the formal
adjoint of $S$. Also, for $\phi,\psi\in\mathbb{C}[z]$ we recall the Dirac
notation $\langle\phi|$ for $\psi\rightarrow\langle\phi|\psi\rangle$, a linear
functional on $\mathbb{C}[z]$, which is continuous in the $\langle|\rangle$
(or norm) topology on $\mathbb{C}[z]$.

######  Definition 2.4.

We shall say that Harmony Zero (or simply H0) holds if:

  * •

The multiplication operator by $z$ is formally adjointable in $\mathbb{C}[z]$
with respect to the inner product $\langle|\rangle$.

  * •

For every $\phi\in\mathbb{C}[z]$ the linear functional $\langle\phi|$ is
continuous in the normal topology of $\mathbb{C}[z]$.

We shall refer to the first condition as Algebraic H0 and to the second
condition as Topological H0. The reader is advised to carefully note the
distinction between the norm topology and the normal topology. In the context
of Algebraic H0 we let $z$ denote the operator of multiplication
$z:\mathbb{C}[z]\to\mathbb{C}[z]$ and denote its formal adjoint by
$z^{*}:\mathbb{C}[z]\to\mathbb{C}[z]$. The operator $z^{*}$ should not be
confused with the conjugate complex variable $\bar{z}$.

Here are some useful reformulations of these properties. As for Analytical H0
let us consider the infinite matrix $z_{nm}=\langle
n|z|m\rangle:=\langle\phi_{n}(z)|z\phi_{m}(z)\rangle$. (More Dirac notation.)
This represents the multiplication operator by $z$ in the basis $B$. By the
construction of $B$, we have that if $n-m\geq 2$, then $z_{nm}=0$. So each
column of this matrix has only finitely many non-zero entries. Moreover,
$z_{nm}>0$ if $n-m=1$. The formal adjointability of $z$ is equivalent to the
statement that every row of this matrix has only finitely many non-zero
entries. If so, then the formal adjoint operator
$z^{*}:\mathbb{C}[z]\to\mathbb{C}[z]$ to the multiplication operator by $z$ in
$\mathbb{C}[z]$ is represented in $B$ by the corresponding adjoint matrix.

As for Topological H0 let us observe first that the normal continuity of all
the dual vectors $\langle\phi|$ is equivalent to the continuity of only all
the basis vectors, that is of $\langle n|=\langle\phi_{n}(z)|$ for all
$n\in\mathbb{N}$. In accordance with (2.2) this property can be expressed as
saying that for every $m\in\mathbb{N}$ there exists a real number
$\Lambda_{m}$ such that we have the system of inequalities

(2.14)
$|s_{mn}|\leq(\Lambda_{m})^{n}\quad\mathrm{for~{}all~{}}n\in\mathbb{N}.$

In particular this is the case if the rows ($\Leftrightarrow$ the columns) of
$S_{\infty}$ possess only finitely many non-zero entries.

######  Remark 2.5.

The two components of H0, the algebraic and the topological, do not entangle
strongly within this basic polynomial context. Their mutual correlations are
manifested through the higher harmony levels.

######  Remark 2.6.

Harmony Zero always holds for diagonal scalar products. Indeed, in this case a
quick calculation shows that

(2.15)
$z|n\rangle=\left(\frac{s_{n+1}}{s_{n}}\right)^{1/2}\\!\\!\\!|n+1\rangle\quad\quad\forall
n\geq 0.$

This is a kind of creation operator with the only non-zero matrix entries
being those immediately below the main diagonal. It is formally adjointable in
$\mathbb{C}[z]$ and its formal adjoint, the annihilation operator, is given by

(2.16)
$z^{*}|n\rangle=\left(\dfrac{s_{n}}{s_{n-1}}\right)^{1/2}\\!\\!\\!|n-1\rangle\quad\quad\forall
n\geq 1,$

together with $z^{*}|0\rangle=0$. As for the topological part of H0 that
follows from the above observation on non-zero entries of $S_{\infty}$ and
comparing with (2.14). Or it follows explicitly from the formula

(2.17) $\langle
n|\phi(z)\rangle=\frac{s_{n}^{1/2}}{n!}\frac{\partial^{n}}{\partial
z^{n}}\phi(z)\Bigm{|}_{z=0},$

which tells us that all the $\langle n|$ are indeed continuous in the normal
topology of $\mathbb{C}[z]$.

The algebraic part of H0 ensures the existence of a basic *-algebra
$\mathcal{A}$.

######  Definition 2.7.

$\mathcal{A}$ is defined to be the algebra of linear operators acting on
$\mathbb{C}[z]$ generated by the operators $z$ and $z^{*}$. Specifically, the
elements of $\mathcal{A}$ are finite linear combinations of monomials in $z$
and $z^{*}$, which are by definition all of the finite products whose factors
are either $z$ or $z^{*}$ in all possible orders.

Clearly, $\mathcal{A}$ is a *-algebra, with its *-structure being the formal
adjoint operation.

The existence of the formal adjoint implies that every operator in
$\mathcal{A}$, whose domain $\mathbb{C}[z]$ is dense in $\mathcal{H}$, is
closable in $\mathcal{H}$. (See Theorem VIII.1 in [14].) Moreover,

(2.18) $a\subset\overline{a}\qquad\mathrm{and}\qquad\overline{a}^{*}\supset
a^{*}\qquad\mathrm{for~{}every~{}}a\in\mathcal{A}.$

In the second formula the ∗ in $\overline{a}^{*}$ means the Hilbert space
adjoint operator in $\mathcal{H}$ of the densely defined operator
$\overline{a}$. Also, note that the closure $\overline{a}$ is always strictly
greater than $a$.

Depending on $\langle|\rangle$ the *-algebra $\mathcal{A}$ can be very simple
or arbitrarily complicated. The simplest situation is when $z=z^{*}$, in which
case the operator of multiplication by $z$ with dense domain $\mathbb{C}[z]$
is symmetric as a densely defined operator acting in $\mathcal{H}$. This
defines the classical context of orthogonal polynomials over $\mathbb{R}$. In
this case the scalar product can always be represented in the form

(2.19)
$\langle\varphi|\psi\rangle=\int_{\\!-\infty}^{\infty}\overline{\varphi(t)}\psi(t)\,d\mu(t)\qquad\mathrm{for~{}all~{}}\varphi,\psi\in\mathbb{C}[z]$

with respect to a finite measure $\mu$ on $\mathbb{R}$. Indeed, if $z$ is
symmetric, then all the numbers $s_{nm}=\langle z^{n}|z^{m}\rangle=\langle
0|z^{n+m}|0\rangle$ are real. This implies that $J$ is isometric, and in
particular it extends to an anti-linear isometry of the whole $\mathcal{H}$.
So we have an anti-linear isometry $J$ preserving $z$. This symmetry situation
implies that $z$ is extendible to a self-adjoint operator in $\mathcal{H}$, as
$J$ exchanges $\ker(\lambda-z^{*})$ and $\ker(\bar{\lambda}-z^{*})$ for every
non-real $\lambda\in\mathbb{C}$, and so both defect indices of $z$ are the
same. Let us now take any self-adjoint extension $\widetilde{z}$ of $z$ and
consider its spectral measure $E$. This is a projector-valued measure, that is
its values $E(\Lambda)$ on Borel subsets $\Lambda$ of $\mathbb{R}$ are
orthogonal projectors in $\mathcal{H}$ so that

$\widetilde{z}=\int_{\mathbb{R}}t\,dE(t).$

The desired standard measure $\mu$ reproducing the scalar product
$\langle|\rangle$ is defined by

$\mu(\Lambda)=\langle 0|E(\Lambda)|0\rangle$

for Borel subsets $\Lambda$ of $\mathbb{R}$.

######  Remark 2.8.

Here the topological part of H0 is equivalent to the compactness of the
support of the measure $\mu$.

######  Proposition 2.2.

This classical context of orthogonal polynomials is equivalent, modulo linear
transformations of the complex variable $z$, to the commutativity of the
*-algebra $\mathcal{A}$.

###### Proof.

If $z^{*}$ commutes with $z$, then it belongs to $\mathbb{C}[z]$ in the sense
that $z^{*}=\phi(z)$, where $\phi$ is a polynomial of degree at least $1$.
Here $z$ is viewed in the same way as $z^{*}$, namely as an operator in
$\mathbb{C}[z]$. But then

$z=z^{**}=\phi(z)^{*}=(J\phi)(z^{*})=(J\phi)[\phi(z)].$

So the composition of $J\phi$ and $\phi$ is the identity map on $\mathbb{C}$.
This is possible only if

(2.20)
$\phi(z)=az+b\quad\mathrm{for~{}}a,b\in\mathbb{C}\quad\mathrm{satisfying}\quad|a|=1,\quad
a\bar{b}+b=0.$

Let $\sqrt{a}$ denote one fixed value for the square root of $a$. Using the
conditions on $a,b$ in (2.20) and $z^{*}=\phi(z)=az+b$, one then readily
calculates that

$\Big{(}\sqrt{a}z+\frac{1}{2}\frac{b}{\sqrt{a}}\Big{)}^{*}=\sqrt{a}z+\frac{1}{2}\frac{b}{\sqrt{a}}.$

Therefore the substitution $z\rightsquigarrow\sqrt{a}z+b/(2\sqrt{a})$ does the
trick. ∎

We can extend (2.19) in a sense to our non-commutative context by introducing
a canonical ‘integration functional’ on $\mathcal{A}$ defined by

(2.21) $\int F:=\langle 0|F|0\rangle=\langle\phi_{0}|F\phi_{0}\rangle=\langle
0|F\rangle\quad\forall F\in\mathcal{A},$

where $\phi_{0}=|0\rangle=1$ is the first element in the basis (2.6). In
particular we can write

(2.22)
$\langle\rho|\phi\rangle=\int\\!\rho^{*}\phi\quad\mathrm{for~{}}\rho,\phi\in\mathbb{C}[z],$

where on the right side the polynomials $\rho,\phi$ are interpreted as
elements of $\mathcal{A}$.

With the *-algebra $\mathcal{A}$ we have the very basic algebraic structure
for crafting the idea of a quantized Euclidean plane. We would like to
interpret the operators $z$ and $z^{*}$ as the quantized complex coordinate
and its conjugate. But the structure still lacks certain geometrical and
analytical contents, which are fundamental for a truly quantum interpretation.
For instance, we would expect both $z$ and $z^{*}$ to be unbounded operators
with their spectra being equal to the whole plane $\mathbb{C}$. And we also
want to be able to construct coherent states associated to every complex
number. And this is not always the case within the framework of Harmony Zero.

Our next harmony property provides these additional geometrical and analytical
dimensions by linking the theory to Hilbert spaces of entire functions.

######  Definition 2.9.

We shall say that Harmony One (or simply H1) holds (for $\mathcal{H}$ or for
$\langle|\rangle$ on $\mathbb{C}[z]$) if $\mathcal{H}$ can be realized as a
Hilbert space of entire functions on $\mathbb{C}$.

######  Remark 2.10.

Properties H0 and H1 are logically independent. However, in diverse
interesting examples they will happily work together.

Harmony One is a symbiosis of two more elementary conditions, which are
related to continuity and injectivity. The algebra of entire functions
$\mathrm{H}(\mathbb{C})$ possesses its natural normal topology of uniform
convergence on compact sets. All polynomials are entire functions. Firstly, we
require that the inclusion map
$\mathbb{C}[z]\rightarrowtail\mathrm{H}(\mathbb{C})$ be continuous, where
$\mathbb{C}[z]$ is considered equipped with the $\langle|\rangle$-induced
topology. If so, the inclusion extends to a continuous map
$\mathcal{H}\rightsquigarrow\mathrm{H}(\mathbb{C})$. Secondly, we require that
this extended map be injective. In such a way all the elements of
$\mathcal{H}$ are viewable as entire functions.

If, on the other hand, we only have the continuity but not injectivity, the
situation allows an elegant geometrical ‘renormalization’. In this case there
is a non-trivial kernel $\mathcal{K}$ of
$\mathcal{H}\rightsquigarrow\mathrm{H}(\mathbb{C})$, which is a closed
subspace of $\mathcal{H}$ transversal to $\mathbb{C}[z]$. If we orthogonally
project $\mathbb{C}[z]$ onto $\mathcal{K}^{\bot}$, the space $\mathbb{C}[z]$
will be preserved as an everywhere dense projection in $\mathcal{K}^{\bot}$.
This projection can be understood as a change of the initial scalar product in
$\mathbb{C}[z]$, so that $\langle|\rangle$ is replaced by the scalar product
induced from $\mathcal{K}^{\bot}$. The restriction of
$\mathcal{H}\rightsquigarrow\mathrm{H}(\mathbb{C})$ to $\mathcal{K}^{\bot}$ is
by construction injective. Hence $\mathcal{K}^{\bot}$ or equivalently
$\mathbb{C}[z]$ equipped with the new scalar product, satisfies Harmony One.

It is instructive to do this construction ‘backwards’ and classify all the
scalar products on $\mathbb{C}[z]$ for which the continuity property holds.

######  Proposition 2.3.

There is a canonical one-to-one correspondence between the scalar products
$\langle|\rangle$ on $\mathbb{C}[z]$ for which the above continuity property
holds and triplets $(\mathcal{K},\langle|\rangle^{\\!\sim},F)$, where
$\mathcal{K}$ is a separable Hilbert space, $\langle|\rangle^{{\\!\sim}}$ a
scalar product on $\mathbb{C}[z]$ satisfying Harmony One, and
$F\colon\mathbb{C}[z]\rightarrow\mathcal{K}$ a completely discontinuous linear
map in the sense that $\mathrm{D}(F^{*})=\\{0\\}$. The space $\mathcal{H}$ is
realized as $\mathcal{H}^{\sim}\oplus\mathcal{K}$, where $\mathcal{H}^{\sim}$
is the Hilbert space completion of $\mathbb{C}[z]$ relative to
$\langle|\rangle^{\\!\sim}$. In terms of this identification, the
$\langle|\rangle$-isometric inclusion of $\mathbb{C}[z]$ into $\mathcal{H}$ is
$p(z)\longmapsto p(z)\oplus F[p(z)]$.∎

######  Remark 2.11.

The map $F$ has the wildest possible behavior for a linear map: its
composition with any non-zero continuous functional on $\mathcal{K}$ is
discontinuous. This wildness ensures that the graph of $F$, which realizes the
embedding of $\mathbb{C}[z]$ into $\mathcal{H}$, is everywhere dense in
$\mathcal{H}$.

To see why this is so, note that $\Gamma(F^{*})=[U(\Gamma(F))]^{\perp}$, where
$\Gamma(F)$ is the graph of $F$ in the inner product space
$\mathbb{C}[z]\oplus\mathcal{K}$ and
$U:\mathbb{C}[z]\oplus\mathcal{K}\to\mathcal{K}\oplus\mathbb{C}[z]$ is the
inner product preserving map $(f,g)\mapsto(-g,f)$. Similarly, $\Gamma(F^{*})$
is the graph of $F^{*}$. Therefore $\Gamma(F)$ dense implies that
$\Gamma(F^{*})=0$ and thus $\mathrm{D}(F^{*})=\\{0\\}$. But
$\mathrm{D}(F^{*})=\\{\kappa\in\mathcal{K}\,|\,\phi\mapsto\langle\kappa|F\phi\rangle\,\mathrm{is~{}a~{}continuous~{}map~{}}\mathbb{C}[z]\to\mathbb{C}\\}$.
Moreover, $F$ composed with a continuous functional on the Hilbert space
$\mathcal{K}$ has exactly the form $\phi\mapsto\langle\kappa|F\phi\rangle$ for
some $\kappa\in\mathcal{K}$. So $\mathrm{D}(F^{*})=\\{0\\}$ implies that this
composition is continuous only for the zero functional.

Interestingly, such an extreme discontinuity naturally emerges in the study of
an important continuity property. If this continuity is granted, then the
injectivity property holds for $\langle|\rangle$ if and only if
$\mathcal{K}=\\{0\\}$ which is itself equivalent to
$\langle|\rangle=\langle|\rangle^{\\!\sim}$.

######  Remark 2.12.

On the other hand under some appropriate additional symmetry conditions on the
scalar product, the injectivity of the extended map
$\mathcal{H}\rightsquigarrow\mathrm{H}(\mathbb{C})$ automatically holds. As we
shall see, this includes our primary scenario when the monomials $z^{n}$ are
mutually orthogonal.

If Harmony One holds, then for all $z\in\mathbb{C}$ the linear functional
$l_{z}:\mathcal{H}\rightarrow\mathbb{C}$ defined for all $f\in\mathcal{H}$ by
$l_{z}(f):=f(z)$ (called evaluation at $z$) is continuous with respect to the
norm topology of $\mathcal{H}$. To see that this is so, note that uniform
convergence on compact sets implies convergence on the compact, singleton set
$\\{z\\}$, that is, point-wise convergence. And that tells us that
$l_{z}:\mathrm{H}(\mathbb{C})\to\mathbb{C}$, defined by the same formula as
above, is continuous in the normal topology. But the inclusion map
$\mathcal{H}\rightsquigarrow\mathrm{H}(\mathbb{C})$ is continuous by Harmony
One. So the composition of these two maps is continuous, as claimed. This in
turn implies that $\mathcal{H}$ is a reproducing kernel Hilbert space. So in
accordance with the general theory of Hilbert spaces of entire functions (for
example, see [6]) there exists a unique reproducing kernel
$K\colon\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ of $\mathcal{H}$
given by the series

(2.23) $K(\bar{w},z)=\sum_{n=0}^{\infty}\overline{\phi_{n}(w)}\phi_{n}(z),$

which converges absolutely and uniformly on compact sets of
$\mathbb{C}\times\mathbb{C}$. The following are the characteristic properties
of $K$:

  * •

For every $z\in\mathbb{C}$ and $f\in\mathcal{H}$ we have the Reproducing
Property:

$f(z)=\int_{\mathbb{C}}K(\bar{w},z)f(w)dw$

  * •

For each $w\in\mathbb{C}$ the function $\mathbb{C}\ni z\mapsto K(\bar{w},z)$
is an element in $\mathcal{H}$.

It follows in this case from (2.8) that the sequence of inverse matrices
$S_{n}^{-1}$, viewed naturally in $\mathrm{M}_{\infty}(\mathbb{C})$, converges
entry-by-entry as $n\to\infty$ to an infinite hermitian matrix
$S_{\infty}^{-}$ so that

(2.24)
$K(\bar{w},z)=\sum_{i,j=0}^{\infty}z^{i}[S_{\infty}^{-}]_{ij}\bar{w}^{j},$

a series which is normally convergent on $\mathbb{C}\times\mathbb{C}$.

######  Remark 2.13.

An additional explanation for this important inversion formula is, perhaps, in
order here. We know that the partial sums, the left hand side of (2.8),
converge normally on $\mathbb{C}\times\mathbb{C}$ to the reproducing kernel
$K(\bar{w},z)$. On the other hand, the reproducing kernel is expandable into a
double power series in $\bar{w}$ and $z$, normally convergent on the whole
$\mathbb{C}\times\mathbb{C}$. In particular, the coefficients of the partial
sums, the right-hand side of (2.8), converge as $n\to\infty$ to the
coefficients of the expansion of the reproducing kernel, and (2.24) indeed
holds.

For each $w\in\mathbb{C}$ its point wave function is defined for all
$z\in\mathbb{C}$ by

$w(z):=K(\bar{w},z)\quad\mathrm{or~{}more~{}simply~{}by}\quad[w]:=K(\bar{w},\cdot).$

We have that $[w]=w\in\mathcal{H}$ by the second characteristic property of
$K$. (That both $w\in\mathbb{C}$ and $w\in\mathcal{H}$ is an abuse of notation
which will be be resolved by context.) Also by the reproducing property of $K$
we obtain

(2.25) $\langle[w]|\psi\rangle=\langle w(z)|\psi(z)\rangle=\psi(w)$

for every $\psi=\psi(z)\in\mathcal{H}$ and $w\in\mathbb{C}$. In particular,
for the point wave functions associated to $v,w\in\mathbb{C}$ we obtain

(2.26) $\langle[v]|[w]\rangle=\langle
v(z)|w(z)\rangle=K(\bar{w},v)\quad\mathrm{and}\quad\|\,[w]\,\|^{2}=\|w(z)\|^{2}=K(\bar{w},w).$

We claim that a point wave function $[w]$ can not be identically equal to
zero. In fact, by (2.25) the set of dormant points defined as
$\\{w\in\mathbb{C}~{}|~{}[w]=0\\}$ is equal to

$\\{w\in\mathbb{C}~{}|~{}\psi(w)=0\quad\forall\psi\in\mathcal{H}\\}.$

But $\mathcal{H}$ contains all the elements in $\mathbb{C}[z]$, that is, all
polynomials. And there is no complex number that is the common zero of all
polynomials. So, in this setting there are no dormant points and consequently
$[w]\neq 0$ for all $w\in\mathbb{C}$. However, in other more general
reproducing kernel Hilbert spaces there are dormant points.

The following is a standard result, which goes much beyond $[w]\neq 0$.

######  Lemma 2.1.

The set of vectors $\\{~{}[w]~{}|~{}w\in\mathbb{C}\\}$ in $\mathcal{H}$ is
linearly independent.

###### Proof.

A set of vectors is linearly independent if and only if all of its finite
subsets are linearly independent. So let $w_{1},\dots,w_{n}$ be $n$ distinct
points in $\mathbb{C}$. We have to show that the vectors
$[w_{1}],\dots,[w_{n}]$ are linearly independent. So suppose that
$\sum_{j=1}^{n}\lambda_{j}[w_{j}]=0$ for some $\lambda_{j}\in\mathbb{C}$. We
have to prove that $\lambda_{j}=0$ for all $j$. Now for all
$\psi\in\mathcal{H}$ we have

(2.27)
$0=\langle\sum_{j=1}^{n}\lambda_{j}[w_{j}]|\psi\rangle=\sum_{j=1}^{n}\overline{\lambda_{j}}\langle[w_{j}]|\psi\rangle=\sum_{j=1}^{n}\overline{\lambda_{j}}\psi(w_{j}).$

Since the points $w_{1},\dots,w_{n}$ are distinct, for each $1\leq i\leq n$
there exists a polynomial $p_{i}(z)\in\mathbb{C}[z]\subset\mathcal{H}$ such
that $p_{i}(w_{j})=\delta_{ij}$, the Kronecker delta, for each $1\leq j\leq
n$. (These are the basis Lagrange polynomials for the points
$w_{1},\dots,w_{n}$.) Taking $\psi=p_{i}$ in (2.27) shows that $\lambda_{i}=0$
for every $1\leq i\leq n$ ∎

Let us consider the linear span (finite linear combinations) of all the point
wave functions:

(2.28)
$\mathcal{L}:=\Bigl{\\{}\sideset{}{{}^{*}}{\sum}_{w\in\mathbb{C}}c_{w}w(z)\Bigr{\\}}.$

This is a fundamental object directly emerging from the reproducing kernel.
From (2.25) we conclude that there is no non-zero vector in $\mathcal{H}$ that
is orthogonal to all of the point wave functions. Consequently, the vector
subspace $\mathcal{L}$ is dense in $\mathcal{H}$ in the norm topology and
hence also dense in $\mathcal{H}$ in the normal topology. Moreover, because of
(2.26) both the scalar product and the norm restricted to $\mathcal{L}$ are
completely determined by the reproducing kernel.

The reproducing kernel formula is a key to expressing Harmony One in terms of
the canonical orthonormal basis.

######  Proposition 2.4.

Harmony One holds if and only if the series

(2.29) $\sum_{n=0}^{\infty}|\phi_{n}(z)|^{2}$

is normally convergent on $\mathbb{C}$, and in addition the orthonormal
polynomials $\phi_{n}(z)$ are $\infty$-linearly independent in the sense that
all infinite linear combinations

(2.30)
$\sum_{n=0}^{\infty}c_{n}\phi_{n}(z)\quad\mathrm{with}\quad\sum_{n=0}^{\infty}|c_{n}|^{2}<+\infty,$

understood as entire functions, uniquely determine their coefficients $c_{n}$.

###### Proof.

The $\Rightarrow$ part is clear as in this case (2.29) normally converges to
$K(\bar{z},z)$, and the functions (2.30) are faithful representations of the
vectors of $\mathcal{H}$.

The $\Leftarrow$ part is a little subtler. If (2.29) is normally convergent on
$\mathbb{C}$, then we can define the function $K(\bar{w},z)$ by the series
(2.23) for which it is easy to see that it converges normally on
$\mathbb{C}\times\mathbb{C}$ in this case. Moreover, this function satisfies
by construction the primary matrix positivity condition (as discussed in
Appendix B). So it is the reproducing kernel function for a Hilbert space
$\mathcal{H}^{\sim}$ of entire functions. Let $\mathcal{L}^{\sim}$ be its
dense linear subspace of finite linear combinations of point wave functions.
The formula

$\mathcal{L}^{\sim}\ni
w(z)\rightsquigarrow\sum_{n=0}^{\infty}\overline{\phi_{n}(w)}\phi_{n}\in\mathcal{H}$

for all $w\in\mathbb{C}$ defines an isometric embedding of
$\mathcal{L}^{\sim}$ into $\mathcal{H}\leftrightarrow\ell^{2}(\mathbb{N})$.
This isometric embedding extends to an isometric embedding of
$\mathcal{H}^{\sim}$ into $\mathcal{H}$. Using this embedding, let us
calculate the orthocomplement of $\mathcal{H}^{\sim}$ in $\mathcal{H}$, namely

$\mathcal{H}^{\sim\bot}=\mathcal{L}^{\sim\bot}=\bigcap_{w\in\mathbb{C}}[w(z)]^{\bot}.$

Taking into account that

$\langle
w(z)|\sum_{n=0}^{\infty}c_{n}\phi_{n}\rangle=\sum_{n=0}^{\infty}c_{n}\phi_{n}(w)$

we see that this orthocomplement consists precisely of infinite square
summable decompositions of $0$ into $\infty$-linear combinations of the
functions $\phi_{n}(z)$ as in (2.30). The orthocomplement will be trivial
(equivalently $\mathcal{H}=\mathcal{H}^{\sim}$) if and only if (2.30)
faithfully represents the vectors of
$\mathcal{H}\leftrightarrow\ell^{2}(\mathbb{N})$. ∎

######  Remark 2.14.

In fact the normal convergence of the series (2.29) is equivalent to the
continuity part of Harmony One, while the faithfulness of (2.30) is equivalent
to the injectivity part.

The point wave functions give us corresponding coherent states, since they can
be normalized. By using some more Dirac notation, there is a natural embedding

(2.31) $\mathbb{C}\ni w\rightsquigarrow\frac{|w(z)\rangle\langle
w(z)|}{\langle
w(z)|w(z)\rangle}=\frac{|[w]\rangle\langle[w]|}{||\,[w]\,||^{2}}\in\mathrm{CP}(\mathcal{H})$

into the complex projective space $\mathrm{CP}(\mathcal{H})$, which is
naturally identified as the space of pure states of $\mathcal{H}$. In the
context given by $\mathcal{H}$ the coherent states
$|[w]\rangle\langle[w]|/||\,[w]\,||^{2}$ can be interpreted as the quantum
counterparts of classical points in the plane $\mathbb{C}$. In the literature
the corresponding unit vectors $[w]/||\,[w]\,||\in\mathcal{H}$ are also called
coherent states.

######  Proposition 2.5.

The induced metric on $\mathbb{C}$ from $\mathrm{CP}(\mathcal{H})$ via the
above map is given by

(2.32)
$\mathrm{d}s^{2}=\mathrm{d}\bar{w}\,\mathrm{d}w\Bigl{\\{}\frac{\partial^{2}}{\partial\bar{w}\,\partial
w}\,\log K(\bar{w},w)\Bigr{\\}}.$

###### Proof.

The complex projective space, viewed as the set of rank 1 projectors as above,
is a subset of the space of Hilbert-Schmidt operators acting in $\mathcal{H}$,
which itself is a Hilbert space relative to the scalar product defined by

(2.33) $\langle X|Y\rangle:=\frac{1}{2}\mathrm{Tr}(X^{*}Y).$

By definition, the metric on $\mathrm{CP}(\mathcal{H})$ is the corresponding
induced metric. And in the coordinates $w,\bar{w}$ we obtain

$\frac{1}{2}\mathrm{Tr}\Bigl{\\{}\Bigl{(}\frac{|w+\mathrm{d}w\rangle\langle
w+\mathrm{d}w|}{\langle
w+\mathrm{d}w|w+\mathrm{d}w\rangle}-\frac{|w\rangle\langle w|}{\langle
w|w\rangle}\Bigr{)}^{2}\Bigr{\\}}\sim\\\
\mathrm{d}s^{2}=\Bigl{\\{}K(\bar{w},w)\frac{\partial^{2}}{\partial\bar{w}\,\partial
w}K(\bar{w},w)-\frac{\partial}{\partial\bar{w}}K(\bar{w},w)\,\frac{\partial}{\partial
w}K(\bar{w},w)\Bigr{\\}}\frac{\mathrm{d}\bar{w}\>\mathrm{d}w}{K(\bar{w},w)^{2}}$

where we have disregarded the higher order terms in $\mathrm{d}w$ and
$\mathrm{d}\bar{w}$. The formula (2.32) follows elementarily. ∎

######  Remark 2.15.

The above formula is valid for all reproducing kernel Hilbert spaces of
analytic functions over arbitrary domains $\Omega$ in $\mathbb{C}$ as long as
the point vectors $w(z)$ are all non-zero. In other words, no point
$w\in\Omega$ is a common zero for all the elements of $\mathcal{H}$. This is a
wise thing to assume always, since in the contrary case we can simply remove
the discrete set of dormant points from $\Omega$. However, as we already
showed, there are no dormant points in the setting of this paper.

######  Remark 2.16.

The factor $1/2$ in the definition (2.33) is justified by the simple resulting
expression for the induced metric in $\mathbb{C}$. Another justification is
that in this case for unit vectors $\varphi,\psi$ the square of the distance
between the one-dimensional projectors $|\psi\rangle\langle\psi|$ and
$|\varphi\rangle\langle\varphi|$ has a direct physical meaning: the value is
$1-|\langle\varphi|\psi\rangle|^{2}$, which is the quantum mechanical
probability of not transitioning from one of the states $\varphi,\psi$ to the
other. Intuitively, this probability of non-transition (or more accurately,
its square root) is measuring how far apart the states determined by
$\varphi,\psi$ are from each other.

######  Proposition 2.6.

Assume that Harmony One holds. Then every transformation
$\tau\in\mathrm{O}(\mathbb{C})$ naturally induces a closed linear operator,
denoted by $\widehat{\tau}$, acting in $\mathcal{H}$ on the not necessarily
dense domain defined by

(2.34)
$\mathrm{D}(\widehat{\tau}):=\Bigl{\\{}\psi(z)\in\mathcal{H}\Bigm{|}\tau[\psi(z)]\in\mathcal{H}\Bigr{\\}}$

on which $\widehat{\tau}$ acts by
$\widehat{\tau}\colon\psi(z)\mapsto\tau[\psi(z)]$.

######  Remark 2.17.

The operator $\widehat{\tau}$ is the maximal restriction of $\tau$ within
$\mathcal{H}$. Its domain can be trivial, consisting of $\\{0\\}$ only. The
most interesting situations occur when its domain is dense in $\mathcal{H}$ so
that the adjoint operator $\widehat{\tau}^{*}$ is defined in $\mathcal{H}$.
Then, because of the density of $\mathcal{H}$ in $\mathrm{H}(\mathbb{C})$ in
the normal topology, $\tau$ is completely fixed by $\widehat{\tau}$. This
includes the maximal compatibility $\tau(\mathcal{H})\subseteq\mathcal{H}$, in
other words $\mathrm{D}(\widehat{\tau})=\mathcal{H}$, which by the closed
graph theorem, implies the boundedness of $\widehat{\tau}$.

###### Proof.

The closeness follows from the continuity of $\tau$ in the normal topology.
The graph $\mathrm{G}(\tau)$ of $\tau$ is closed in
$\mathrm{H}(\mathbb{C})\oplus\mathrm{H}(\mathbb{C})$. The natural inclusion
map

$\mathcal{H}\oplus\mathcal{H}\rightarrowtail\mathrm{H}(\mathbb{C})\oplus\mathrm{H}(\mathbb{C})$

is continuous. So the inverse image of $\mathrm{G}(\tau)$, which is precisely
the graph of $\widehat{\tau}$, is closed in $\mathcal{H}\oplus\mathcal{H}$. ∎

In examples many important operators $\tau$ will come from
$\mathrm{o}(\mathbb{C})$. In particular, we see that such operators include
all the multiplication operators by polynomials in $z$ (i.e., by the elements
in $\mathbb{C}[z]$, which includes multiplication by $z$), the derivative
operator $\partial/\partial z$ and all linear differential operators with
polynomial coefficients in $z$ and also the substitution operators by
polynomials $\phi(z)$.

In what follows we use the notation of the previous proposition to define the
symbol $Z:=\widehat{z}$, that is, the closed operator of multiplication by the
coordinate $z$ acting in a domain of $\mathcal{H}$ defined by

(2.35)
$\mathrm{D}(Z):=\Bigl{\\{}\psi(z)\in\mathcal{H}\Bigm{|}z\psi(z)\in\mathcal{H}\Bigr{\\}}.$

So $Z$ acts by $Z\colon\psi(z)\mapsto z\psi(z)$. The operator $Z$ is
intrinsically related to the point wave functions $w(z)$ as we shall now
explain.

Let us observe for all $w\in\mathbb{C}$ we have that

(2.36)
$w(z)^{\bot}=\Bigl{\\{}\psi(z)\in\mathcal{H}\Bigm{|}\psi(w)=0\Bigr{\\}}=\overline{(z-w)\mathbb{C}[z]},$

where the superscript line on the right denotes the closure in the norm
topology of $\mathcal{H}$. Indeed, the first equality follows from the
property (2.25) of the point wave functions, and the second equality means
that the functions from $\mathcal{H}$ vanishing at $w$ can always be
arbitrarily well approximated in $\mathcal{H}$ by polynomials with the same
property.

######  Proposition 2.7.

For every $w\in\mathbb{C}$ its associated point function $[w]=w(z)$ belongs to
the domain of the adjoint operator $Z^{*}$. In particular, $Z^{*}$ is densely
defined. Moreover, $Z^{*}[w]=\bar{w}\,[w]$ and

(2.37) $\ker(\bar{w}-Z^{*})=\mathbb{C}[w].$

The operators $Z$ and $Z^{*}$ are unbounded. Every complex number $w$ is an
eigenvalue of the annihilation operator $Z^{*}$ of multiplicity one and with
associated eigenvector being the normalized point wave function (equivalently,
coherent state) $[\bar{w}]/||\,[\bar{w}]\,||$.

###### Proof.

Take $w\in\mathbb{C}$. From the principal kernel equation (2.25) applied twice
we find

$\langle[w]|Z\psi\rangle=w\psi(w)=w\langle[w]|\psi\rangle=\langle\bar{w}[w]|\psi\rangle$

for every $\psi\in\mathrm{D}(Z)$ and every $w\in\mathbb{C}$. Therefore $[w]$
belongs to the domain of $Z^{*}$ and $Z^{*}[w]=\bar{w}\,[w].$ In particular
$\mathrm{D}(Z^{*})\supseteq\mathcal{L}$ (cp. (2.28)), and so $Z^{*}$ is
densely defined in $\mathcal{H}$. Since $[w]\neq 0$, it follows that $\bar{w}$
is an eigenvalue of $Z^{*}$ for every $w\in\mathbb{C}$. So the spectrum of
$Z^{*}$ is $\mathbb{C}$. So $Z^{*}$ is an unbounded operator, which in turn
implies that $Z$ is also an unbounded operator. In particular
$\mathrm{D}(Z)\neq\mathcal{H}$ and $\mathrm{D}(Z^{*})\neq\mathcal{H}$.

Having shown that every point in $\mathbb{C}$ is an eigenvalue of $Z^{*}$, we
now identify its multiplicity. Using a standard identity for the adjoint
operator we have

$\ker(\bar{w}-Z^{*})^{\bot}=\overline{\mathrm{im}(w-Z)}=[w]^{\bot}.$

where the second identity follows from (2.36). We conclude that
$\ker(\bar{w}-Z^{*})=\mathbb{C}[w]$, which shows (2.37). Now $\mathbb{C}[w]$
is a one dimensional subspace because $[w]\neq 0$. And this shows that each
eigenvalue of $Z^{*}$ has multiplicity one. ∎

######  Remark 2.18.

There is a certain complementarity between the two basic spaces
$\mathbb{C}[z]$ and $\mathcal{L}$: The space $\mathbb{C}[z]$ is $Z$-invariant
and the space $\mathcal{L}$ is $Z^{*}$-invariant. In a variety of interesting
examples we encounter one of the following two special configurations:

(2.38) $\displaystyle\mathbb{C}[z]$ $\displaystyle\cap\mathcal{L}=\mathbb{C}$
(2.39) $\displaystyle\mathbb{C}[z]$ $\displaystyle\subseteq\mathcal{L}.$

In general, the finite dimensional spaces $\mathbb{C}_{n}[z]$ can only contain
a finite number of the linearly independent point wave functions $w(z)$. (Cp.
Lemma 2.1.) Hence, at most countably many of them will be in $\mathbb{C}[z]$.

Since $Z^{*}$ is densely defined and $Z$ is closed, $Z^{**}=\bar{Z}=Z$. So $Z$
and $Z^{*}$ are adjoints of each other.

######  Proposition 2.8.

The subspace $\mathcal{L}$ of $\mathcal{H}$ is a core for the operator
$Z^{*}$. In other words $Z^{*}$ is the closure of its restriction
$Z^{*}|\mathcal{L}\colon\mathcal{L}\rightarrow\mathcal{L}$, that is
$\overline{Z^{*}|\mathcal{L}}=Z^{*}$.

###### Proof.

Let us compute the orthocomplement of the graph
$\mathrm{G}(Z^{*}|\mathcal{L})$ in the graph $\mathrm{G}(Z^{*})$. It consists
of the pairs $(\psi,Z^{*}\psi)$ for some $\psi\in\mathrm{D}(Z^{*})$ which are
orthogonal to all the pairs $([w],\bar{w}[w])$ where $w\in\mathbb{C}$. In
other words $\langle[w]|\psi\rangle+\langle\bar{w}[w]|Z^{*}\psi\rangle=0$. But
this means $\psi(w)+w(Z^{*}\psi)(w)=0$. We now claim under this hypothesis on
$\psi$ that $Z^{*}\psi\in\mathrm{D}(Z)$. This is because $\mathbb{C}\ni
w\rightsquigarrow wZ^{*}\psi(w)=-\psi(w)$ is the function
$-\psi\in\mathcal{H}$. It then follows from the definition of $Z$ that
$(ZZ^{*}\psi)(w)=wZ^{*}\psi(w)$ holds for all $w\in\mathbb{C}$ for this
particular vector $\psi$.

Next, this says that $\psi(w)+(ZZ^{*}\psi)(w)=0$ for all $w\in\mathbb{C}$, and
therefore $\psi$ satisfies $\psi+ZZ^{*}\psi=0$. By functional analysis
$1+ZZ^{*}$ is invertible with its inverse defined on the whole $\mathcal{H}$
and bounded. It follows that $\psi=0$, and so the orthocomplement is the zero
subspace. Thus $\mathrm{G}(Z^{*}|\mathcal{L})$ is dense in
$\mathrm{G}(Z^{*})$. ∎

######  Remark 2.19.

Another way to see this is by considering the adjoint of $Z^{*}|\mathcal{L}$.
For $\psi$ in the domain of this adjoint, that is
$\psi\in\mathrm{D}((Z^{*}|\mathcal{L})^{*})$, we have that the function

$\mathbb{C}\ni w\rightsquigarrow w\psi(w)=\langle
Z^{*}[w]|\psi\rangle=\langle[w]|(Z^{*}|{\mathcal{L}})^{*}\psi\rangle=((Z^{*}|{\mathcal{L}})^{*}\psi)(w)$

is in $\mathcal{H}$, which shows us that in fact $\psi\in\mathrm{D}(Z)$ and
$Z\supset(Z^{*}|{\mathcal{L}})^{*}$. And therefore
$Z^{*}\subset(Z^{*}|{\mathcal{L}})^{**}=\overline{Z^{*}|{\mathcal{L}}}$. The
opposite inclusion is trivial, and so $Z^{*}=\overline{Z^{*}|{\mathcal{L}}}$.
Interestingly, a similar statement does not in general hold for the operator
$Z$. In general $Z$ will be strictly greater than the closure of its
restriction on $\mathbb{C}[z]$.

The above proposition extends to all the powers of $Z^{*}|{L}$, as

$(Z^{*}|{\mathcal{L}})^{n*}=Z_{n},$

where $Z_{n}\colon\psi(z)\rightarrow z^{n}\psi(z)$ is the operator associated
to the multiplication by $z^{n}$. In general, this operator will be a non-
trivial extension of $Z^{n}$ although in many important examples
$Z_{n}=Z^{n}$. We refer to [5] for an elegant and self-contained geometrically
oriented exposition of the theory of unbounded operators in a Hilbert space.

As we have already mentioned, the polynomials $\phi\in\mathbb{C}[z]$ can be
viewed as unital homomorphisms of $\mathbb{C}[z]$ into itself given for
$p\in\mathbb{C}[z]$ by the ‘substitution map’ $p\mapsto p\circ\phi$, where
$\circ$ denotes composition of functions. Since every such a map is continuous
in the normal topology, by Proposition 2.6 we can associate to it a closed
operator $\widehat{\phi}$, which is obviously densely defined, by
$\psi\mapsto\psi\circ\phi$ for all $\psi\in\mathcal{H}$ satisfying
$\psi\circ\phi\in\mathcal{H}$.

######  Proposition 2.9.

Suppose that $\phi\in\mathbb{C}[z]$. Then the domain of the adjoint operator
$\widehat{\phi}^{*}$ contains the space $\mathcal{L}$. We have for every
$w\in\mathbb{C}$ that

(2.40) $\widehat{\phi}^{*}([w])=[\phi(w)].$

In particular, the subspace $\mathcal{L}$ is $\widehat{\phi}^{*}$-invariant.

###### Proof.

Applying the kernel equation (2.25) we find for every
$\psi\in\mathrm{D}(\widehat{\phi})$ that

$\langle[w]|\widehat{\phi}\psi\rangle=\langle[w]|\psi\circ\phi\rangle=\psi(\phi(w))=\langle[\phi(w)]|\psi\rangle.$

Thus $[w]\in\mathrm{D}(\widehat{\phi}^{*})$ and (2.40) holds. ∎

######  Remark 2.20.

The above argument is extendible without essential change from polynomials to
entire functions, understood as unital homomorphisms of
$\mathrm{H}(\mathbb{C})$ into itself, continuous in the normal topology. The
only subtlety is that we have to explicitly postulate the density of
$\mathrm{D}(\widehat{\phi})$ in $\mathcal{H}$.

The point wave functions are special case of an important class of vectors in
$\mathcal{H}$, which correspond to the linear functionals on $\mathcal{H}$
continuous in the normal topology.

######  Definition 2.21.

A vector $\xi\in\mathcal{H}$ is called normal if its dual vector, the linear
functional $\langle\xi|\colon\psi\mapsto\langle\xi|\psi\rangle$, is continuous
in the normal topology of $\mathcal{H}$.

We shall write $\mathcal{W}$ for the set of all normal vectors of
$\mathcal{H}$. Clearly, $\mathcal{W}$ is a linear subspace of $\mathcal{H}$
and $\mathcal{L}\subseteq\mathcal{W}$. So $\mathcal{W}$ is dense in
$\mathcal{H}$.

######  Remark 2.22.

In the framework of Harmony One, the topological part of Harmony Zero can be
rephrased as the inclusion $\mathbb{C}[z]\subset\mathcal{W}$, that is, the
polynomials are normal vectors.

######  Lemma 2.2.

If polynomials are normal, then the infinite matrix $S_{\infty}$ is invertible
and

(2.41) $S_{\infty}^{-}=S_{\infty}^{-1}.$

In particular, the extended scalar product
$\langle|\rangle\colon\mathbb{C}[z]\times\mathrm{H}(\mathbb{C})\rightarrow\mathbb{C}$
is non-degenerate.

###### Proof.

From the expansion (2.24) we find for all $k\in\mathbb{N}$ and
$w\in\mathbb{C}$ that

$\langle z^{k}|w(z)\rangle=\langle z^{k}|\sum_{i,j\geq
0}z^{i}[S_{\infty}^{-}]_{ij}\bar{w}^{j}\rangle=\sum_{i,j\geq
0}[S_{\infty}]_{ki}[S_{\infty}^{-}]_{ij}\bar{w}^{j})=\bar{w}^{k},$

where the second equality follows from the hypothesis that polynomials are
normal. And thus $S_{\infty}^{-}S_{\infty}=1_{\infty}$. Because of the
hermicity of $S_{\infty}$ and $S_{\infty}^{-}$ we also have
$S_{\infty}S_{\infty}^{-}=1_{\infty}$. ∎

######  Proposition 2.10.

Let a transformation $\tau\in\mathrm{O}(\mathbb{C})$ be such that
$\overline{\mathrm{D}(\widehat{\tau})}=\mathcal{H}$, where $\widehat{\tau}$ is
defined in Proposition 2.6. Then
$\mathcal{W}\subseteq\mathrm{D}(\widehat{\tau}^{*})$ and, in particular,
$\widehat{\tau}^{*}$ is densely defined. Moreover,
$\widehat{\tau}^{*}(\mathcal{W})\subseteq\mathcal{W}$.

###### Proof.

For every $\xi\in\mathcal{W}$ the map
$\mathrm{D}(\widehat{\tau})\ni\psi\longmapsto\langle\xi|\widehat{\tau}\psi\rangle\in\mathbb{C}$,
being the composition of the inclusion $\mathrm{D}(\widehat{\tau})$ into
$\mathrm{H}(\mathbb{C})$, the normally continuous tranformation $\tau$ and
normally continuous scalar product with $\xi$, is normally continuous and in
particular $\langle|\rangle$-continuous. Therefore such vectors $\xi$ are in
the domain of the adjoint operator for $\widehat{\tau}$. The same initial map
now viewed as
$\mathrm{D}(\widehat{\tau})\ni\psi\longmapsto\langle\widehat{\tau}^{*}\xi|\psi\rangle\in\mathbb{C}$
for being normally continuous, and since $\mathrm{D}(\widehat{\tau})$ is dense
in $\mathcal{H}$, extends by continuity to a normally continuous functional on
the whole $\mathcal{H}$, given by the same formula. It follows that
$\widehat{\tau}^{*}\xi\in\mathcal{W}$. ∎

In particular we see that $\mathcal{W}\subseteq\mathrm{D}(Z^{*})$. So if
Topological H0 holds, then $\mathbb{C}[z]$ is also in the domain of $Z^{*}$.
And if in addition Algebraic H0 holds (so that we have full H0), then
$Z^{*}\supset z^{*}$, where $z$ denotes the operator of multiplication by $z$
acting on $\mathbb{C}[z]$

######  Proposition 2.11.

If a transformation $\tau\in\mathrm{O}(\mathbb{C})$ satisfies
$\overline{\mathrm{D}(\widehat{\tau})}=\mathcal{H}$ (as in the previous
proposition) and $\widehat{\tau}^{*}$ is also continuous in the normal
topology of $\mathcal{H}$, then
$\mathcal{W}\subseteq\mathrm{D}(\widehat{\tau})$ and
$\widehat{\tau}(\mathcal{W})\subseteq\mathcal{W}$.

###### Proof.

Since $\widehat{\tau}^{*}$ is normally continuous, it extends by continuity to
a transformation $\varrho\in\mathrm{O}(\mathbb{C})$. Clearly
$\widehat{\varrho}\supseteq\widehat{\tau}^{*}$ and so
$\widehat{\tau}=\widehat{\tau}^{**}\supseteq\widehat{\varrho}^{*}$. In
particular,
$\mathrm{D}(\widehat{\varrho}^{*})\subseteq\mathrm{D}(\widehat{\tau})$. And by
the previous proposition
$\mathcal{W}\subseteq\mathrm{D}(\widehat{\varrho}^{*})$. Combining these two
inclusions gives $\mathcal{W}\subseteq\mathrm{D}(\widehat{\tau})$. Since
$\mathcal{W}$ is $\widehat{\varrho}^{*}$ invariant (again by the previous
proposition), $\mathcal{W}$ is also invariant for its extensions such as
$\widehat{\tau}$. ∎

### A Minimal Effective Operational Setting

Our next harmony condition quite naturally emerges when looking for the most
effective setting for constructions involving the fundamental unbounded
operators $Z$ and $Z^{*}$.

######  Definition 2.23.

We shall say that Harmony Two (or simply H2) holds if the operator $Z^{*}$ is
continuous in the normal topology of $\mathcal{H}$.

Before going further, some important observations are due. It is clear that H2
requires H1 as it builds on top of it. However, H2 is logically independent of
H0.

The operator $Z$ comes from a map in $\mathrm{o}(\mathbb{C})$, and hence it is
in $\mathrm{O}(\mathbb{C})$. However, in general, the adjoint operator $Z^{*}$
will not be continuous in the normal topology. Therefore, there is an inherent
contextual asymmetry between the two operators $Z$ and $Z^{*}$. This symmetry
is imposed by this new harmony property. In a sense, Harmony Two is a ‘grown
up’ version of Algebraic H0, which provides the simplest common setting for a
strictly polynomial version of symmetry between the operators $Z$ and $Z^{*}$.

For $Z^{*}$ to be normally continuous it is sufficient for the restriction
$Z^{*}|\mathcal{L}$ to be normally continuous, since $\mathcal{L}$ is dense in
the normal topology in $\mathcal{H}$ and so also in $\mathrm{D}(Z^{*})$. If
$Z^{*}$ is normally continuous, then it uniquely extends by continuity to
$\mathrm{H}(\mathbb{C})$, and so it is interpretable as a transformation of
$\mathrm{O}(\mathbb{C})$. We shall use the inverted symbol  $z$ to denote this
transformation
$\reflectbox{$z$}\colon\mathrm{H}(\mathbb{C})\rightarrow\mathrm{H}(\mathbb{C})$.

Therefore, if H2 holds, there is a naturally emerging subalgebra $\mathcal{A}$
of $\mathrm{O}(\mathbb{C})$ defined as the set of all (in general, non-
commutating) polynomials in $z$ and  $z$ .

######  Proposition 2.12.

In the framework of Harmony Two the space $\mathcal{W}$ is both $Z$ and
$Z^{*}$-invariant. Hence, it is invariant for all the operators in
$\mathcal{A}$. The algebra $\mathcal{A}$ is faithfully realized in
$\mathcal{W}$ and is equipped with a natural *-structure induced by the formal
adjoint operation on $\mathcal{W}$.

###### Proof.

A direct application of Proposition 2.11 implies that $\mathcal{W}$ is
contained in the domains of $Z$ and of $Z^{*}$, and it is a common invariant
subspace of these operators. When restricted to $\mathcal{W}$ the mutual
adjointness of $Z$ and $Z^{*}$ becomes the formal adjointness of their
restrictions. The entire algebra $\mathcal{A}$ is representable by
restrictions in $\mathcal{W}$. Since $\mathcal{W}$ is norm dense in
$\mathcal{H}$ it is also normally dense in $\mathrm{H}(\mathbb{C})$, which
means that the representation of $\mathcal{A}$ in $\mathcal{W}$ is faithful. ∎

######  Remark 2.24.

The same argument extends to all operators $\tau\in\mathrm{O}(\mathbb{C})$
having densely defined $\widehat{\tau}$ whose adjoint $\widehat{\tau}^{*}$ is
normally continuous.

The algebra $\mathcal{A}$ is highly non-commutative. One manifestation of this
would be the absence of characters ($\Leftrightarrow$ one-dimensional
*-representations) of $\mathcal{A}$. Quantum mechanically this means that
$\mathcal{A}$, viewed as an algebra of observables, does not admit dispersion-
free states. And geometrically, the interpretation is that the ‘underlying’
quantum Euclidean plane, whose ‘functions’ are the elements of $\mathcal{A}$,
possesses no classical points.

Any finite linear combination of the monomials $z^{k}\reflectbox{$z$}^{l}$ is
called a Wick polynomial. The word ‘polynomial’ is to be understood as a
polynomial expression of an element in $\mathcal{A}$ and not as an element in
an abstract polynomial algebra. As we shall see two such distinct expressions
can be equal.

######  Proposition 2.13.

In the framework of Harmony Two, if $z$ and  $z$ satisfy that
$\reflectbox{$z$}z$ is expressible as a Wick polynomial, then the algebra
$\mathcal{A}$ has no characters.

###### Proof.

Let us assume to the contrary, that
$\varkappa\colon\mathcal{A}\rightarrow\mathbb{C}$ is a character and define
$w:=\varkappa(z)\in\mathbb{C}$. (Recall that here $z\in\mathcal{A}$ is the
operator of multiplication by $z$ acting on $\mathrm{H}(\mathbb{C})$.) Because
$\varkappa$ is a *-representation, we have
$\varkappa(z^{k}\reflectbox{$z$}^{l})=w^{k}\bar{w}^{l}$. Since property H2
holds, by Proposition 2.12 the point wave function $[w]$ belongs to the domain
of all the polynomials in $Z$ and $Z^{*}$. Moreover, for all
$k,l\in\mathbb{N}$ we have

$\langle[w]|Z^{k}Z^{*l}[w]\rangle=\langle
Z^{*k}[w]|Z^{*l}[w]\rangle=w^{k}\bar{w}^{l}\|[w]\|^{2}=\varkappa(z^{k}\reflectbox{$z$}^{l})\|[w]\|^{2}.$

Write $\reflectbox{$z$}z=p(z,\reflectbox{$z$})$, where $p$ is a Wick
polynomial. By using the previous equation and linearity to get the second
equality below, we then obtain

$\langle[w]|Z^{*}Z[w]\rangle=\langle[w]|p(Z,Z^{*})[w]\rangle=\varkappa(p(z,\reflectbox{$z$}))\|[w]\|^{2}=\varkappa(\reflectbox{$z$}z)\|[w]\|^{2}=|w|^{2}\|[w]\|^{2}.$

The last equality holds because $\varkappa$ is a *-representation. Note that
$[w]$ is in the domain of the polynomial $Z-w$. And hence we see that

$\displaystyle\|(z-w)[w]\|^{2}$
$\displaystyle=\langle(Z-w)[w]|(Z-w)[w]\rangle$
$\displaystyle=\langle[w]|Z^{*}Z[w]\rangle-\langle Z[w]|w[w]\rangle-\langle
w[w]|Z[w]\rangle+\langle w[w]|w[w]\rangle$
$\displaystyle=|w|^{2}\|[w]\|^{2}-|w|^{2}\|[w]\|^{2}-|w|^{2}\|[w]\|^{2}+|w|^{2}\|[w]\|^{2}=0.$

where in the two middle terms on the second line we used
$Z^{*}[w]=\bar{w}[w]$. In other words $(z-w)[w]=0$ in $\mathcal{H}$, which
contradicts the fact that the holomorphic function $[w]$ is non-zero. ∎

######  Remark 2.25.

The ‘flipping’ condition given by a Wuck polynomial for the product
$\reflectbox{$z$}z$ is essential. The phenomenon is illustrated, in the last
part of this section by an example where $\mathcal{A}$ is an extension of the
Manin $q$-plane by the infinite matrix algebra
$\mathrm{M}_{\infty}(\mathbb{C})$.

######  Definition 2.26.

We shall say that Harmony Three (or simply H3) holds if both Topological H0
and H2 hold.

Harmony Three holds if and only if $1\in\mathcal{W}$, in the framework of
Harmony Two.

######  Proposition 2.14.

In the framework of Harmony Three, the only eigenvectors of  $z$ in
$\mathrm{H}(\mathbb{C})$ are those of $Z^{*}$ in $\mathcal{H}$ or, in other
words, the point wave functions $w(z)$ modulo scalar multiples.

###### Proof.

Let us assume that $\Psi(z)\in\mathrm{H}(\mathbb{C})$ satisfies
$\reflectbox{$z$}\Psi(z)=\bar{w}\Psi(z)$ for some $w\in\mathbb{C}$. This means
that $\langle(z-w)\phi(z)|\Psi(z)\rangle=0$ for every
$\phi(z)\in\mathbb{C}[z]$. Here we have extended the scalar product by normal
continuity to $\mathbb{C}[z]\times\mathrm{H}(\mathbb{C})$. The extended
$\langle|\rangle$ is non-degenerate. This leaves only one degree of freedom
for $\Psi(z)$. ∎

######  Proposition 2.15.

In the framework of Harmony Three, the commutant of $\mathcal{A}$ in
$\mathrm{O}(\mathbb{C})$ is trivial. In particular, the center of the algebra
$\mathcal{A}$ is trivial: $\mathrm{Z}(\mathcal{A})=\mathbb{C}$.

###### Proof.

If something from $\mathrm{O}(\mathbb{C})$ commutes with $z$ then it must be
of a multiplicative form $\psi(z)\longmapsto\Theta(z)\psi(z)$, where
$\Theta(z)\in\mathrm{H}(\mathbb{C})$. If $\Theta(z)$ in addition commutes with
$z$ then $\reflectbox{$z$}[\Theta(z)w(z)]=\bar{w}\Theta(z)w(z)$. Proposition
2.14 implies that $\Theta(z)w(z)$ must be proportional to $w(z)$ which is
possible only if $\Theta(z)$ is a constant function. ∎

### Projection Construction

The following list recapitulates the symbols used for the diverse operators
related intrinsically to the complex coordinate $z$.

$z$ |  The multiplication operator $\psi(z)\longmapsto z\psi(z)$ acting in $\mathbb{C}[z]$, $\mathcal{W}$ or $\mathrm{H}(\mathbb{C})$.
---|---
$Z$ |  Induced $z$-multiplication in $\mathcal{H}$ with $\mathrm{D}(Z)=\bigl{\\{}\psi(z)\in\mathcal{H}\bigm{|}z\psi(z)\in\mathcal{H}\bigr{\\}}$.
$Z^{*}$ |  The adjoint operator for the operator $Z$ in $\mathcal{H}$.
$z$ |  The continuous extension of $Z^{*}$ to the whole $\mathrm{H}(\mathbb{C})$.
$z^{*}$ |  The formal adjoint of the multiplication operator $z$ in $\mathbb{C}[z]$ and $\mathcal{W}$.

These symbols are all related to $\mathcal{H}$ and $\mathcal{A}$. Following
[10], we shall now interpret the algebra $\mathcal{A}$ as an algebra of
Toeplitz operators for the Toeplitz quantization of another, a priori
unrelated, algebra, which will be assumed to be a quadratic polynomial
*-algebra generated by $z$ and $\bar{z}$. So $z$ is a true multi-personality
creature here!

The basic example is the standard commutative *-algebra
$\mathbb{C}[z,\bar{z}]$ of polynomials in $z$ and $\bar{z}$. Its only defining
relation is simply $z\bar{z}=\bar{z}z$, and its *-structure is the anti-linear
map which exchanges $z$ and $\bar{z}$. The product between
$\mathbb{C}[\bar{z}]$ and $\mathbb{C}[z]$ induces a natural decomposition

(2.42)
$\mathbb{C}[z,\bar{z}]\leftrightarrow\mathbb{C}[\bar{z}]\otimes\mathbb{C}[z]\qquad\bar{z}^{n}z^{m}\leftrightarrow\bar{z}^{n}\otimes
z^{m}.$

A variety of interesting non-commutative variations of this emerge, if we fix
a hermitian $2\times 2$ matrix

$Q=\begin{pmatrix}q_{00}&q_{01}\\\ q_{10}&q_{11}\end{pmatrix}$

so that $\overline{q_{10}}=q_{01}$ and $q_{00},q_{11}\in\mathbb{R}$, and
define $\mathbb{C}[z,\bar{z},Q]$ to be the *-algebra generated by $z$ and
$\bar{z}$ together with the following relation:

(2.43) $z\bar{z}=\bar{z}z+(1\,\,\bar{z})\begin{pmatrix}q_{00}&q_{01}\\\
q_{10}&q_{11}\end{pmatrix}\begin{pmatrix}1\\\ z\end{pmatrix}$

where also $q_{11}\neq-1$. The *-structure, as in the commutative case, is
anti-linear and exchanges $z$ and $\bar{z}$. We can think of $Q$ as being a
quadratic form on $\mathbb{C}^{2}$. We recover the commutative case if $Q=0$.

These non-commutative polynomials keep valid the basic decomposition (2.42)
or, in other words for every $Q$ we have an isomorphism of vector spaces, but
not of algebras

(2.44)
$\mathbb{C}[z,\bar{z},Q]\leftrightarrow\mathbb{C}[\bar{z}]\otimes\mathbb{C}[z]\qquad\bar{z}^{n}z^{m}\leftrightarrow\bar{z}^{n}\otimes
z^{m}$

by the composition-diamond lemma. (See [18].) In terms of this identification
$1\otimes 1$ is the unit element of the algebra, and $\mathbb{C}[z]$ and
$\mathbb{C}[\bar{z}]$ are mutually conjugate unital subalgebras of
$\mathbb{C}[z,\bar{z},Q]$. The basis elements $\bar{z}^{n}z^{m}$ of
$\mathbb{C}[z,\bar{z},Q]$ are called anti-Wick monomials. Consequently, every
element in $\mathbb{C}[z,\bar{z},Q]$ has a unique expression as a anti-Wick
polynomial, that is as a finite linear combination of the anti-Wick monomials.

######  Remark 2.27.

The coordinates $\bar{z}$ and $z$ appear symmetrically at this point of the
theory, since the condition $q_{11}\neq-1$ ensures the presence of non-zero
coefficients of both $z\bar{z}$ and $\bar{z}z$ in the defining relation
(2.43). Then the composition-diamond lemma implies that there is also an
‘opposite’ vector space isomorphism, but not an algebra isomorphism,
$\mathbb{C}[z,\bar{z},Q]\leftrightarrow\mathbb{C}[z]\otimes\mathbb{C}[\bar{z}]$
given by $z^{n}\bar{z}^{m}\leftrightarrow z^{n}\otimes\bar{z}^{m}$. The basis
elements $z^{n}\bar{z}^{m}$ of $\mathbb{C}[z,\bar{z},Q]$ are called Wick
monomials. Symmetrically, every element in $\mathbb{C}[z,\bar{z},Q]$ is
uniquely expressible as a Wick polynomial.

It follows that the product of $\mathbb{C}[z,\bar{z},Q]$ is encoded in the
‘flipping’ rules for monomials $z^{n}$ and $\bar{z}^{m}$ which can be written
as finite sums

(2.45) $z^{n}\bar{z}^{m}=\sum_{k,l\geq
0}\prescript{n}{k}{\lozenge}^{m}_{l}\,\bar{z}^{l}z^{k}$

for an array with 4 indices $\prescript{n}{k}{\lozenge}^{m}_{l}$ of complex
numbers which can only have non-zero values for $0\leq k\leq n$ and $0\leq
l\leq m$, since the relation (2.43) has no terms of order greater than $2$ in
$z$ and $\bar{z}$. Polynomial expressions whose terms are all monomials of the
form $\bar{z}^{l}z^{k}$ as in (2.45) are said to be reduced in abstract ring
theory and in anti-Wick order in quantum theory.

This array must obey a number of consistency conditions in order to represent
a unital associative *-algebra structure on $\mathbb{C}[z,\bar{z},Q]$. At
first, by conjugating the above formula we arrive at

(2.46)
$\overline{\prescript{n}{k}{\lozenge}^{m}_{l}}=\prescript{m}{l}{\lozenge}^{n}_{k}.$

Then, the requirement for the existence of a unit translates into

(2.47)
$\prescript{n}{k}{\lozenge}^{0}_{l}=\delta_{nk}\delta_{0l}\qquad\prescript{0}{k}{\lozenge}^{m}_{l}=\delta_{0k}\delta_{ml}.$

More generally, $z^{n}$ and $\bar{z}^{m}$ commute if and only if

(2.48) $\prescript{n}{k}{\lozenge}^{m}_{l}=\delta_{nk}\delta_{ml}$

for all $k,l\geq 0$. Finally, the following charming and easy to remember
convolution identities hold, where $\times$ denotes the product of complex
numbers:

(2.49) $\prescript{u+v}{k}{\lozenge}^{m}_{l}=\sum_{\begin{subarray}{c}s\geq
0\\\
a+b=k\end{subarray}}\prescript{u}{a}{\lozenge}^{s}_{l}\times\prescript{v}{b}{\lozenge}^{m}_{s}\qquad\qquad\prescript{n}{k}{\lozenge}^{u+v}_{l}=\sum_{\begin{subarray}{c}s\geq
0\\\
a+b=l\end{subarray}}\prescript{n}{s}{\lozenge}^{u}_{a}\times\prescript{s}{k}{\lozenge}^{v}_{b}.$

They reflect the fact that both $\mathbb{C}[z]$ and $\mathbb{C}[\bar{z}]$ are
unital subalgebras of $\mathbb{C}[z,\bar{z}]$. In particular, by inductively
applying these identities, we see that the array
$\prescript{n}{k}{\lozenge}^{m}_{l}$ is completely determined by $Q$ because

(2.50)
$\prescript{1}{0}{\lozenge}^{1}_{0}=q_{00}\qquad\prescript{1}{0}{\lozenge}^{1}_{1}=q_{01}\qquad\prescript{1}{1}{\lozenge}^{1}_{0}=q_{10}\qquad\prescript{1}{1}{\lozenge}^{1}_{1}=1+q_{11}$

which is the same as the basic generating relation (2.43).

In order to clarify different roles of the same space, we shall write
$\mathcal{Q}$ for $\mathbb{C}[z,\bar{z},Q]$ as a vector space on which the
same algebra acts via its left regular representation. In a similar manner, we
shall write $\mathcal{P}$ for $\mathbb{C}[z]$ understood as the corresponding
representation space. Clearly $\mathcal{P}$ is a subalgebra of $\mathcal{Q}$.
Also, every element in $\mathcal{Q}$ can be written uniquely as a finite sum
of polynomials homogeneous in $\bar{z}$, namely as
$\sum_{n}a_{n}\bar{z}^{n}\varphi_{n}$ with $a_{n}\in\mathbb{C}$ and
$\varphi_{n}\in\mathcal{P}$. Singling out the subalgebra
$\mathcal{P}=\mathbb{C}[z]$ for special consideration, instead of
$\mathbb{C}[\bar{z}]$ which we could have done, breaks the previously
mentioned symmetry between $z$ and $\bar{z}$.

We shall assume here that H3 holds for $\mathcal{H}$.

######  Proposition 2.16.

There exists a unique quadratic form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
on $\mathcal{Q}$ which extends the scalar product on $\mathcal{P}$ and such
that the left regular representation of $\mathbb{C}[z,\bar{z},Q]$ on
$\mathcal{Q}$ is symmetric with respect to that form. It is determined by

(2.51)
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\varphi|\bar{z}^{m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\sum_{k,l\geq
0}\prescript{n}{k}{\lozenge}^{m}_{l}\langle Z^{l}\varphi|Z^{k}\psi\rangle$

for $n,m\in\mathbb{N}$ and $\varphi,\psi\in\mathcal{P}$.

###### Proof.

Let us suppose that
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
exists. Then

$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\varphi|\bar{z}^{m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\varphi|z^{n}\bar{z}^{m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\sum_{k,l\geq
0}\prescript{n}{k}{\lozenge}^{m}_{l}\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\varphi|\bar{z}^{l}z^{k}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\sum_{k,l\geq
0}\prescript{n}{k}{\lozenge}^{m}_{l}\langle Z^{l}\varphi|Z^{k}\psi\rangle$

so indeed (2.51) holds and hence
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
must be unique. To prove its existence, we define it on $\bar{z}$-homogenous
polynomials by (2.51), extending it to $\mathcal{Q}\times\mathcal{Q}$ by using
bi-additivity. From
$\prescript{0}{k}{\lozenge}^{0}_{l}=\delta_{k0}\delta_{l0}$ it follows that
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
extends $\langle|\rangle$. It is also hermitian symmetric because of (2.46):

$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\varphi|\bar{z}^{m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}^{*}=\sum_{k,l\geq
0}\overline{\prescript{n}{k}{\lozenge}^{m}_{l}}\langle
Z^{k}\psi|Z^{l}\varphi\rangle=\sum_{k,l\geq
0}\prescript{m}{l}{\lozenge}^{n}_{k}\langle
Z^{k}\psi|Z^{l}\varphi\rangle=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{m}\psi|\bar{z}^{n}\varphi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}.$

So we have a quadratic form on $\mathcal{Q}$. We next have to prove that $z$
and $\bar{z}$ are mutually formally adjoint with respect to
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$.
Again, we verify this on polynomials homogeneous in $\bar{z}$ by using (2.45),
(2.49) and (2.51):

$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\varphi|\bar{z}\bar{z}^{m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\sum_{k,l\geq
0}\prescript{n}{k}{\lozenge}^{m+1}_{l}\langle
Z^{l}\varphi|Z^{k}\psi\rangle=\\!\\!\\!\sum_{k,s,a,b\geq
0}\\!\\{\prescript{n}{s}{\lozenge}^{1}_{a}\times\prescript{s}{k}{\lozenge}^{m}_{b}\\}\langle
Z^{a+b}\varphi|Z^{k}\psi\rangle=\\\ =\\!\\!\\!\sum_{k,s,a,b\geq
0}\\!\\!\\!\prescript{s}{k}{\lozenge}^{m}_{b}\,\langle
Z^{b}\bigl{\\{}\prescript{1}{a}{\lozenge}^{n}_{s}\,Z^{a}\varphi\bigr{\\}}|Z^{k}\psi\rangle=\\!\\!\sum_{a,s\geq
0}\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{s}\bigl{\\{}\prescript{1}{a}{\lozenge}^{n}_{s}Z^{a}\varphi\bigr{\\}}|\bar{z}^{m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle
z\bar{z}^{n}\varphi|\bar{z}^{m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}.$

So $z$ and $\bar{z}$ are formally adjoint relative to
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$.
Since they generate $\mathbb{C}[z,\bar{z},Q]$ this property extends to the
whole polynomial algebra. In other words we get that

(2.52)
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle
F\varphi|\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\varphi|\overline{F}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$

for every $F\in\mathbb{C}[z,\bar{z},Q]$ and $\varphi,\psi\in\mathcal{Q}$. ∎

######  Remark 2.28.

The form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
will not be positive in general. The positivity problem for the form, in the
context of the commutative algebra $\mathbb{C}[z,\bar{z}]$, relates to the
moment problem — the existence of a finite measure $\mu$ on $\mathbb{C}$
reproducing the scalar product via the standard $L^{2}$ inner product

(2.53)
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle
f|g\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\int_{\mathbb{C}}\bar{f}g\,d\mu(z,\bar{z}).$

It is easy to see that if a measure $\mu$ on $\mathbb{C}$ reproduces the
scalar product $\langle|\rangle$ on $\mathcal{P}$, then it automatically
reproduces the extended form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
on $\mathbb{C}[z,\bar{z}]$, and in particular it will be positive (possibly
not strictly positive). On the other hand there exist interesting examples
where
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
is strictly positive, even though there will be no underlying measure. As we
shall see, such exotic situations do not occur if the scalar product
$\langle|\rangle$ on $\mathcal{P}$ is diagonal.

In resonance with this, we can introduce an integration functional for the
algebra $\mathbb{C}[z,\bar{z},Q]$. We shall use the same generic integration
symbol as for $\mathcal{A}$, and similarly to (2.21) define

(2.54) $\int
F=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle
0|F\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}.\quad\forall
F\in\mathbb{C}[z,\bar{z},Q].$

(Recall the Dirac notation is
$|0\rangle=\phi_{0}(z)=1\in\mathbb{C}[z,\bar{z},Q]$.) This is a hermitian
functional evaluating to $s_{nm}=\langle z^{n}|z^{m}\rangle$ on the monomials
$\bar{z}^{n}z^{m}$. Moreover,

(2.55)
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\rho|\phi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\int\overline{\rho}\,\phi$

for every $\rho,\phi\in\mathbb{C}[z,\bar{z},Q]$.

######  Proposition 2.17.

The following conditions are equivalent:

—The space $\mathcal{P}$ is orthocomplementable in $\mathcal{Q}$ or in other
words there exists a subspace $\mathcal{P}^{\bot}$ such that

(2.56) $\mathcal{Q}=\mathcal{P}\oplus\mathcal{P}^{\bot}.$

—There exists a linear map $\Pi\colon\mathcal{Q}\rightarrow\mathcal{Q}$
satisfying

(2.57)
$\Pi^{2}=\Pi\qquad\mathrm{im}(\Pi)=\mathcal{P}\qquad\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\Pi\psi|\varphi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\psi|\Pi\varphi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$

for every $\varphi,\psi\in\mathcal{Q}$.

— Algebraic H0 holds. In other words $\mathcal{P}$ is $Z^{*}$-invariant and
$Z^{*}|\mathcal{P}=z^{*}$ the formal adjoint of $z$ in $\mathcal{P}$.

If any of these (and hence all of these) hold, then
$\ker(\Pi)=\mathcal{P}^{\bot}$ and in particular $\Pi$ is the projection
associated to the orthogonal decomposition of $\mathcal{Q}$. Moreover, there
is a commutative diagram

(2.58) $\begin{CD}\mathcal{Q}@>{\mbox{$\bar{z}$}}>{}>\mathcal{Q}\\\
@A{}A{}A@V{}V{\mbox{$\Pi$}}V\\\
\mathcal{P}@>{}>{\mbox{$z^{*}$}}>\mathcal{P},\end{CD}$

where the left vertical arrow is the inclusion map and the top horizontal
arrow is multiplication by $\bar{z}$.

###### Proof.

If (2.56) holds, then $\Pi$ is defined to be the associated projection on
$\mathcal{P}$ and (2.57) is immediate. On the other hand if (2.57) holds, then
the kernel of $\Pi$ is an orthocomplement of $\mathcal{P}$ and so (2.56)
holds.

If $\Pi$ exists as in (2.57), we define $z^{*}$ by (2.58). Then for
$\varphi,\psi\in\mathcal{P}$ we have

$\langle\varphi|z^{*}\psi\rangle=\langle\varphi|\Pi\bar{z}\psi\rangle=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\Pi\varphi|\bar{z}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\varphi|\bar{z}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle
z\varphi|\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\langle
z\varphi|\psi\rangle.$

So the operator $z^{*}$ is indeed the formal adjoint of $z$ in $\mathcal{P}$
and so Algebraic H0 holds.

Finally, assume Algebraic H0. So the operator $z^{*}$ exists (i.e., $Z^{*}$
preserves $\mathcal{P}$). Then we can define $\Pi$ on $\bar{z}$-homogeneous
polynomials by

(2.59) $\Pi(\bar{z}^{n}\psi)=z^{*n}\psi\in\mathcal{P}$

for $\psi\in\mathcal{P}$ and $n\in\mathbb{N}$, and extend it to $\mathcal{Q}$
by additivity. By construction $\Pi$ is an idempotent projecting on
$\mathcal{P}$. It is also symmetric. Indeed if $\varphi$ is also from
$\mathcal{P}$, then

$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\Pi(\bar{z}^{n}\varphi)|\bar{z}^{m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle
z^{*n}\varphi|\bar{z}^{m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\langle
z^{m}z^{*n}\varphi|\psi\rangle=\langle z^{*n}\varphi|z^{*m}\psi\rangle=\\\
{}=\langle\varphi|z^{n}z^{*m}\psi\rangle=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\varphi|z^{*m}\psi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\varphi|\Pi(\bar{z}^{m}\psi)\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$

and by bi-additivity the identity extends to the whole $\mathcal{Q}$.
Therefore all the identities in (2.57) have been proved. ∎

Here is an explicit way to calculate the projection map $\Pi$ in terms of the
canonical orthonormal polynomials
$\phi_{n}(z)\in\mathbb{C}[z]\subset\mathbb{C}[z,\bar{z},Q]$.

######  Proposition 2.18.

Assume that $\Pi$ exists as in Proposition 2.17. We then have for every
$F\in\mathbb{C}[z,\bar{z},Q]$ that

(2.60)
$\Pi(F)=\sum_{k=0}^{\infty}\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\phi_{k}(z)|F\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}\phi_{k}(z),$

There are only finitely many non-zero terms in this infinite sum.

###### Proof.

For $F\in\mathcal{P}=\mathbb{C}[z]$ equation (2.60) reduces to saying that the
polynomials $\phi_{n}(z)$ are an orthonormal basis of the vector space
$\mathbb{C}[z]$, in which case only finitely many terms are non-zero. For
$F\in\mathcal{P}^{\bot}=\ker(\Pi)$ the left side of (2.60) is zero and every
term on the right side is also zero since $\phi_{n}(z)\in\mathcal{P}$. Then
(2.60) follows for every
$F\in\mathcal{P}\oplus\mathcal{P}^{\bot}=\mathbb{C}[z,\bar{z},Q]$ by
additivity. ∎

######  Remark 2.29.

In Dirac notation (which technically does not apply, since we are not in a
Hilbert space setting) we can write (2.60) as

(2.61)
$\Pi=\sum_{k=0}^{\infty}|k\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle
k|.$

The quadratic form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
will fail in general to be non-degenerate. In other words, we might encounter
a non-trivial null-space $\mathcal{N}=\mathcal{Q}^{\bot}$. This next result is
immediate.

######  Proposition 2.19.

The null-space $\mathcal{N}$ is an invariant subspace for the left regular
representation of $\mathbb{C}[z,\bar{z},Q]$ in $\mathcal{Q}$. Moreover $\Pi$,
if it exists, maps $\mathcal{N}$ into $\\{0\\}$. Thus, the whole module
structure, quadratic form and $\Pi$ naturally project down to the factor space
$\mathcal{R}=\mathcal{Q}/\mathcal{N}$, preserving all the basic formulas. The
projected
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
is non-degenerate. We have $\mathcal{P}\cap\mathcal{N}=\\{0\\}$, and so
$\mathcal{P}$ is naturally a subspace of $\mathcal{R}$. ∎

######  Definition 2.30.

For any $f\in\mathbb{C}[z,\bar{z},Q]$ we define the Toeplitz operator with
symbol $f$, denoted as
$\mathbf{T}(f)=\mathbf{T}_{f}:\mathcal{P}\to\mathcal{P}$, by
$\mathbf{T}_{f}\phi:=\Pi(f\phi)$ for all $\phi\in\mathcal{P}$. Notice that the
product $f\phi$ of the two elements $f,\phi\in\mathbb{C}[z,\bar{z},Q]$ is
again an element in the algebra $\mathbb{C}[z,\bar{z},Q]$. Then the projection
$\Pi$ maps this product to an element of $\mathcal{P}$. In this way we obtain
a linear map $\mathbf{T}\colon\mathbb{C}[z,\bar{z},Q]\rightarrow\mathcal{A}$,
the *-algebra of operators generated by $z$ and $z^{*}$. We say that
$\mathbf{T}$ is the Toeplitz quantization.

Even though $\mathbf{T}$ is a linear map from one algebra to another, it is
not expected nor desired to be multiplicative, that is, a map of algebras. We
will come back to this point. However, $\mathbf{T}$ does preserve the identity
element, namely $\mathbf{T}_{1}$ is the identity operator of $\mathcal{P}$.

As a direct consequence of the above definition of $\mathbf{T}$, we find that

(2.62) $\mathbf{T}(\bar{z}^{n}z^{m})=z^{*}{}^{n}z^{m}.$

So the recipe to calculate the operators $\mathbf{T}_{f}$ is quite simple and
can be used as an alternative definition of $\mathbf{T}$. Just replace $z$ and
$\bar{z}$ by their counterparts $z$ and $z^{*}$ in $\mathcal{A}$ in the
polynomial expression for $f$, assuming that $f$ is written in anti-Wick form,
that is, all $\bar{z}$’s are moved to the left of $z$’s. This is the reason
for saying that Toeplitz quantization is an anti-Wick quantization. Equation
(2.62) works out so nicely in part because the definition of $z^{*}$ in
diagram (2.58) now reads as $\mathbf{T}(\bar{z})=z^{*}{}$. On the other hand
the identity $\mathbf{T}(z)=z$ follows immediately from the fact that
$\mathcal{P}=\mathbb{C}[z]$ is $z$-invariant. Equation (2.62) also shows that
all the operators $\mathbf{T}_{f}$ are indeed in $\mathcal{A}$.

######  Remark 2.31.

In general, $\mathbf{T}$ will not be surjective. Its image is the
$\mathbb{C}[\bar{z}]$–$\mathbb{C}[z]$ bimodule in $\mathcal{A}$ generated by
$1\in\mathcal{A}$. So $\mathcal{A}$ is always generated by the image of
$\mathbf{T}$. In this context, we shall refer to $\mathcal{A}$ as the algebra
of Toeplitz operators.

If we trivially extend the operators in $\mathcal{P}$ to $\mathcal{R}$ by
requiring that they vanish on $\mathcal{P}^{\bot}$, then we can write

(2.63) $\mathbf{T}_{f}=\Pi f\Pi$

where on the right side $f$ is interpreted as the left regular representation
operator. The expression on the right side of (2.63) is known as the
compression by the projection $\Pi$ of the operator defined by $f$

We see that $\mathbf{T}$ intertwines the *-structures on
$\mathbb{C}[z,\bar{z},Q]$ and $\mathcal{A}$. It also connects the integration
functionals on both algebras.

######  Proposition 2.20.

We have

(2.64) $\int\mathbf{T}=\int.$

###### Proof.

It is sufficient to check the identity on the monomials $\bar{z}^{n}z^{m}$.
And then both the left and the right hand side evaluate to $\langle
z^{n}|z^{m}\rangle=s_{nm}$. ∎

######  Lemma 2.3.

For every $\rho,\phi\in\mathcal{P}$ and $n\in\mathbb{N}$ we have

(2.65)
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\rho|(\bar{z}-z^{*})\phi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\langle\rho|\mathbf{T}(z^{n}\bar{z})\phi\rangle-\langle\rho|z^{n}z^{*}\phi\rangle.$

In particular, the map $\mathbf{T}$ is multiplicative only in the trivial
scenario $\mathcal{R}=\mathcal{P}$.

###### Proof.

We compute

$\displaystyle\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\rho|(\bar{z}-z^{*})\phi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
$\displaystyle=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\rho|\bar{z}\phi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}-\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{n}\rho|z^{*}\phi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\rho|z^{n}\bar{z}\phi\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}-\langle\rho|z^{n}z^{*}\phi\rangle$
$\displaystyle=\langle\rho|\mathbf{T}(z^{n}\bar{z})\phi\rangle-\langle\rho|z^{n}z^{*}\phi\rangle.$

We see that $\mathbf{T}(z^{n}\bar{z})=z^{n}z^{*}$ if and pnly if $\bar{z}$
acts as $z^{*}$ on $\mathcal{P}$. Since $\mathcal{P}$ is cyclic for $\bar{z}$
in $\mathcal{R}$ this is only possible when the two spaces coincide. ∎

######  Remark 2.32.

Therefore, the complement $\mathcal{P}^{\bot}$ of $\mathcal{P}$ in
$\mathcal{R}$ can be viewed as a subtle measure of difference between
$\mathbb{C}[z,\bar{z},Q]$ and $\mathcal{A}$. It is also worth observing that
$(\bar{z}-z^{*})\phi\in\mathcal{P}^{\bot}$ always.

### The Fifth Element

To sum things up so far, our construction produces a canonical
$\mathbb{C}[z,\bar{z},Q]$ representation space $\mathcal{R}$ extending
$\mathcal{P}$ and equipped with a non-degenerate hermitian form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$.
If Algebraic H0 holds, there is a symmetric idempotent $\Pi$ projecting
$\mathcal{R}$ onto $\mathcal{P}$ and realizing the operators $z$ and $z^{*}$
of $\mathcal{A}$ as compressions of $z$ and $\bar{z}$ of
$\mathbb{C}[z,\bar{z},Q]$ by $\Pi$. This is our main algebro-geometric setting
for the Toeplitz quantization.

The whole construction can be performed without essential changes with
$\mathcal{W}$ instead of $\mathcal{P}$. The advantage is that $\mathcal{W}$ is
always invariant under both $z$ and  $z$ , as far as we stay within Harmony
Two. In this case the space $\mathcal{Q}$ is redefined as

(2.66)
$\mathcal{Q}=\mathbb{C}[z,\bar{z},Q]\otimes_{\mathbb{C}[z]}\mathcal{W}\leftrightarrow\mathbb{C}[\bar{z}]\otimes\mathcal{W}.$

This brings us naturally to our final harmony property. It is about a mutual
relationship between the two principal algebras $\mathbb{C}[z,\bar{z},Q]$ and
$\mathcal{A}$, related via the Toeplitz quantization.

######  Definition 2.33.

We shall say that Harmony Four (or simply H4) holds if the form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
is positive.

In the framework of H4 the space $\mathcal{R}$ is a pre-Hilbert space and can
be completed into a Hilbert space $\mathcal{J}$. Moreover, $\mathcal{H}$ can
be viewed as a subspace of $\mathcal{J}$. The projection $\Pi$ extends to an
orthogonal projection $\Pi\colon\mathcal{J}\rightarrow\mathcal{H}$, where we
use the same notation.

######  Remark 2.34.

This is all much in the spirit of the Stinespring construction [21] for
completely positive maps between C*-algebras.

### Classification

The algebra $\mathbb{C}[z,\bar{z},Q]$ falls into one of four distinguished
classes, corresponding to four canonical forms of $Q$, obtained after making a
linear substitution $z\rightsquigarrow az+b$ where $a,b\in\mathbb{C}$ and
$a\neq 0$. Under such a substitution the matrix $Q$ transforms as

(2.67) $\begin{pmatrix}q_{00}&q_{01}\\\
q_{10}&q_{11}\end{pmatrix}\rightsquigarrow\begin{pmatrix}q(\bar{b},b)/|a|^{2}&(q_{01}+q_{11}\bar{b})/\bar{a}\\\
(q_{10}+q_{11}b)/a&q_{11}\end{pmatrix}$

where $q(\bar{b},b)=q_{00}+q_{01}b+q_{10}\bar{b}+q_{11}|b|^{2}$ as follows
from the defining relation (2.43). Let us briefly describe these classes.

#### Principal Flipping Type

This is defined by $q=1+q_{11}\neq 0,1$. After the appropriate linear
substitution, we arrive at

(2.68) $z\bar{z}=q\bar{z}z+h\qquad h\in\\{-|q|,0,1\\}\qquad
Q=\begin{pmatrix}h&0\\\ 0&q_{11}\end{pmatrix}.$

There are 6 important special subcases. The first one is given by $h=0$. This
is the Manin $q$-plane. A detailed analysis of this example and the general
theory of its Toeplitz quantization can be found in [10].

The classical part of the corresponding quantum space (given by the characters
of $\mathbb{C}[z,\bar{z},Q]$) is just one point–the character evaluating to 0
on $z$ and $\bar{z}$. An important part of the analysis is the question of
realizability of the algebra by operators in a Hilbert space, the case $q>0$
is realizable, and we can always assume $0<q<1$ since if $q>1$ then by
interchanging $z$ and $\bar{z}$ we have $q\rightsquigarrow 1/q$. As we shall
explain below, there is a natural realization of this algebra in a Hilbert
space of holomorphic functions in $\mathbb{C}-\\{0\\}$. The case $q<0$ is,
clearly, not realizable by operators in a Hilbert space.

Next, there are two essentially different cases with $q>0$ and $h=-q,1$. If
$0<q<1$ and $h=-q$ or $q>1$ and $h=1$ then we obtain equivalent realizations
of a $q$-variant of the standard quantum plane. If, on the other hand, $0<q<1$
and $h=1$ or equivalently $q>1$ and $h=-q$, then the algebra will represent a
Poincaré model for a quantum hyperbolic plane [9]. The horizon of infinity is
traced by a classical circle centered at $0$ of radius $\sqrt{h/1-q}$. These
can be morphed into

$\bar{z}z=qz\bar{z}+1-q$

with $0<q<1$, to fit the unitary disk $\mathbb{D}$.

If $q$ is negative, there are also two essentially different situations. The
first one is given by $h=q$ and corresponds to a character-free *-algebra non-
realizable in a Hilbert space. If $h=1$ then we are again within the quantum
hyperbolic planes, the structure can be morphed into the same generating
expression as above, but now with $-1<q<0$.

#### Parabolic Type

This corresponds to $q_{11}=0$ with $q_{01}=\overline{q_{10}}\neq 0$, which
can be transformed into

(2.69) $\bar{z}z=z\bar{z}+z+\bar{z}\qquad\quad Q=-\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$

The classical points are naturally labeled by purely imaginary numbers. The
generators $z$ and $\bar{z}$ linearly span a non-commutative 2-dimensional Lie
algebra. Abstractly, there is only one such a structure and one concrete
realization is the (complexified) Lie algebra of the group $t\longmapsto at+b$
where $a,b\in\mathbb{R}$ with $a>0$. These are orientation preserving affine
transformations of $\mathbb{R}$. The algebra $\mathbb{C}[z,\bar{z},Q]$ is then
viewable as the universal envelope of this Lie algebra.

#### Orthodox Quantum Plane

This corresponds to $q_{11}=q_{01}=q_{10}=0$ and $q_{00}\neq 0$. In this case
$q_{00}$ can be scaled to $-1$ or $1$ and we obtain the Heisenberg-Weyl
algebra

(2.70) $z\bar{z}-\bar{z}z=\pm 1\qquad\quad Q=\pm\begin{pmatrix}1&0\\\
0&0\end{pmatrix}.$

#### Classical Euclidean Plane

The classical commutative polynomial algebra $\mathbb{C}[z,\bar{z}]$ is
obtained when $q_{11}=q_{01}=q_{10}=q_{00}=0$.

### Diagonal Scalar Product

In this sub-section we will assume that the initial scalar product is
diagonal, in which case all calculations significantly simplify. As we have
explained already, property H0 always holds in this context. Also, recall the
definition (2.12) of the sequence $s_{n}:=s_{nn}=\langle
z^{n}|z^{n}\rangle>0$, which characterizes $\langle|\rangle$ in this case.

######  Lemma 2.4.

In terms of the sequence $s_{n}$, the quadratic form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
is determined by

(2.71)
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{i}z^{j}|\bar{z}^{n}z^{m}\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\sum_{k\geq
0}\prescript{i}{k}{\lozenge}^{n}_{k+m-j}s_{k+m}.$

###### Proof.

This is straightforward from the definitions of $s_{n}$ and the 4-array
$\prescript{}{}{\lozenge}$:

$\displaystyle\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{i}z^{j}|\bar{z}^{n}z^{m}\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
$\displaystyle=\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle
z^{j}|z^{i}\bar{z}^{n}z^{m}\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=\sum_{k,l\geq
0}\prescript{i}{k}{\lozenge}^{n}_{l}\langle
z^{j+l},z^{k+m}\rangle=\sum_{k,l\geq
0}\prescript{i}{k}{\lozenge}^{n}_{l}s_{j+l}\delta_{j+l,k+m}$
$\displaystyle=\sum_{k\geq 0}\prescript{i}{k}{\lozenge}^{n}_{k+m-j}s_{k+m}.$

We have also used the fact that the *-operation exchanges $z$ and $\bar{z}$. ∎

######  Remark 2.35.

In particular the quadratic form for the commutative polynomial algebra
$\mathbb{C}[z,\bar{z}]$ is determined by

(2.72)
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{i}z^{j}|\bar{z}^{n}z^{m}\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=s_{i+m}\,\delta_{i-j,n-m},$

where we used (2.48). This also follows easily by direct verification without
appealing to the above Lemma. Notice even in this highly simplified case that
the vector space basis $\bar{z}^{n}z^{m}$ of $\mathbb{C}[z,\bar{z}]$ is not
orthogonal. Formula (2.72) with $s_{n}=n!$ is readily available in the setting
of Bargmann’s seminal paper [3], even though it is not explicitly given there.

Let us now discuss Harmony One and Two. As for the interpretability of all the
elements of $\mathcal{H}$ as entire functions, this is not always possible. It
requires a special asymptotic behavior of the $\langle|\rangle$ defining
sequence $s_{n}$.

######  Proposition 2.21.

A necessary and sufficient condition for Harmony One is that

(2.73) $\lim\,s_{n}^{1/n}=+\infty.$

In this case the reproducing kernel for $\mathcal{H}$ is given by

(2.74) $K(\bar{w},z)=\sum_{n\geq 0}\frac{\bar{w}^{n}z^{n}}{s_{n}}.$

###### Proof.

The formula (2.74) for the reproducing kernel is a special case of (2.23) with
$\phi_{n}(z)$ given by (2.13). Let us now apply Proposition 2.4. The series
(2.29) becomes

$\sum_{n=0}^{\infty}|\phi_{n}(z)|^{2}=\sum_{n=0}^{\infty}\frac{|z|^{2n}}{s_{n}}.$

For this to be normally convergent on $\mathbb{C}$ a necessary and sufficient
condition is indeed (2.73), as the Cauchy-Hadamard formula reveals. Next, the
$\infty$-linear independence always holds here, because all power series
encode the values of their coefficients. ∎

######  Proposition 2.22.

A necessary and sufficient condition for Harmony Two, in the framework of
Harmony One, is that the sequence

(2.75) $t_{n}=\Bigl{(}\frac{s_{n+1}}{s_{n}}\Bigr{)}^{1/n}$

be bounded.

###### Proof.

If $Z^{*}$ is normally continuous by H2, then by using (2.13) and (2.16) we
have

$\sum_{n\geq 0}c_{n}z^{n}\longmapsto\sum_{n\geq 0}(c_{n+1}s_{n+1}/s_{n})z^{n}$

consistently defines the extension of $Z^{*}$ on the whole
$\mathrm{H}(\mathbb{C})$. This means that the radius of convergence for the
resulting power series on the right must always be infinite, that is, the same
as the radius of convergence of the initial power series on the left. This
property can be rephrased in terms of the sequences as follows. For any
sequence of complex numbers $r_{n}$ converging to $0$, the sequence
$r_{n}t_{n}$ must also converge to zero. This means that the sequence $t_{n}$
belongs to the multiplier algebra of the algebra $\mathrm{C}_{0}(\mathbb{N})$
of all sequences having limit zero. This multiplier algebra is precisely all
bounded sequences $\mathrm{B}(\mathbb{N})$. So $t_{n}$ must be bounded.

Conversely, it is a matter of a direct verification that the boundedness of
$t_{n}$ implies that the above formula defines a (necessarily unique)
continuous extension of $Z^{*}$ on the whole $\mathrm{H}(\mathbb{C})$. ∎

######  Proposition 2.23.

A particularly effective scenario for the applicability of the above criterion
occurs when

(2.76) $\lim s_{n+1}/s_{n}=+\infty\qquad\quad\lim
s_{n}s_{n+2}/s_{n+1}^{2}<+\infty.$

###### Proof.

A direct application of the classical convergence criterion. If
$s_{n+1}/s_{n}$ is convergent then $s_{n}^{1/n}$ is convergent with the same
limit. Similarly if the second sequence is convergent then $t_{n}$ will be
convergent with the same limit. ∎

### The Two Frameworks Intersection

Let us now elaborate more on the special situation, interesting in its own
light, where the quadratic algebra $\mathbb{C}[z,\bar{z},Q]$ is itself
viewable as the Toeplitz operator algebra $\mathcal{A}$ for some Hilbert space
of entire functions $\mathcal{H}$ satisfying Harmony Two.

The Heisenberg-Weyl algebra can be viewed in this way, via the Segal-Bargmann
space, where $s_{n}=n!$ and $K(\bar{w},z)=\exp(\bar{w}z)$. (See [3].) Another
class of examples is given by the $q$-variation

(2.77) $z\bar{z}=q\bar{z}z-q\qquad\quad 0<q<1$

of the Heisenberg-Weyl algebra. Indeed, if we define the scalar product by the
sequence

(2.78) $s_{n}=!_{q}(n):=\prod_{k=0}^{n-1}(1+1/q+\cdots+1/q^{k-1})$

then

$s_{n+1}/s_{n}=1+1/q+\cdots+1/q^{n}\rightsquigarrow\infty\qquad\quad\frac{s_{n}s_{n+2}}{s_{n+1}^{2}}=\frac{1}{q}\frac{1-q^{n+2}}{1-q^{n+1}}\rightsquigarrow\frac{1}{q}$

and thus H2 holds. A direct calculation then reveals that $z\longmapsto Z$ and
$\bar{z}\longmapsto Z^{*}$ extends to an isomorphism between
$\mathbb{C}[z,\bar{z},Q]$ and $\mathcal{A}$.

Furthermore, it turns out that the corresponding reproducing kernel is given
by

(2.79) $K(\bar{w},z)=E_{q}(\bar{w}z)$

where $E_{q}$ is the $q$-exponential function. Let us recall ([13]–Appendix
II) that the $q$-exponential function is defined as

(2.80)
$E_{q}(z)=(qz-z|q)_{\infty}=\sum_{n=0}^{\infty}\frac{(1-q)^{n}}{(q|q)_{n}}q^{\textstyle{\binom{n}{2}}}z^{n}=\sum_{n=0}^{\infty}\frac{z^{n}}{!_{q}(n)}$

where

(2.81)
$(a|q)_{n}=\prod_{k=0}^{n-1}(1-q^{k}a)\qquad(a|q)_{\infty}=\prod_{k=0}^{\infty}(1-q^{k}a).$

It is also worth recalling ([2]–Chapter 10, Section 2) the second Euler
$q$-identity

(2.82)
$\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\textstyle{\binom{n}{2}}}}{(q|q)_{n}}z^{n}=(z|q)_{\infty}$

valid for $|q|<1$ and all $z\in\mathbb{C}$.

Let us now calculate the induced metric on $\mathbb{C}$. Applying the formula
(2.32) we obtain

(2.83)
$\mathrm{d}s^{2}=\sum_{n=0}^{\infty}\frac{(1-q)q^{n}\>\mathrm{d}w\mathrm{d}\bar{w}}{(1+(1-q)q^{n}|w|^{2})^{2}}.$

With the help of a $q$-logarithm function

(2.84) $\displaystyle\log_{q}(1+z)$ $\displaystyle=(1-q)\sum_{n\geq
0}\frac{zq^{n}}{1+zq^{n}}$ $\displaystyle\log_{q}(1+z)$
$\displaystyle=\sum_{k\geq 0}\frac{(-1)^{k}z^{k+1}}{1+\cdots+q^{k}}$

where the first formula is valid for arbitrary $z\in\mathbb{C}$ and the second
in the unit disk $|z|<1$, the metric can also be expressed as

(2.85)
$\mathrm{d}s^{2}=\log_{q}^{\prime}\bigl{[}1+(1-q)|w|^{2}\bigr{]}\>\mathrm{d}w\mathrm{d}\bar{w}.$

We see that in the limit $q\rightsquigarrow 1^{-}$ this reproduces the
classical Euclidean metric, and the first quantum correction looks elliptic

(2.86)
$\mathrm{d}s^{2}\approx\bigl{(}1-2\frac{1-q}{1+q}|w|^{2}\bigr{)}\,\mathrm{d}w\mathrm{d}\bar{w}\qquad|w|^{2}\ll\frac{1}{1-q}.$

We present more formulae related to these spaces in Appendix B.

######  Proposition 2.24.

Modulo equivalence transformations of the quadratic form $Q$, the only
algebras $\mathbb{C}[z,\bar{z},Q]$ realizable as a Toeplitz operators algebra
$\mathcal{A}$ are precisely the Heisenberg-Weyl algebra and its above
described $q$-variants.

###### Proof.

As we have just explained, the Heisenberg-Weyl algebra and $q$-variations are
realizable as some $\mathcal{A}$. To complete the proof, let us observe that
the *-algebra $\mathcal{A}$ can not possess characters, in accordance with
Proposition 2.13. This effectively excludes all other algebras
$\mathbb{C}[z,\bar{z},Q]$. ∎

Let us now consider the Manin $q$-plane

(2.87) $z\bar{z}=q\bar{z}z\qquad 0<q<1.$

The formula (2.71) for the quadratic form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
simplifies into

(2.88)
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle\bar{z}^{i}z^{j}|\bar{z}^{n}z^{m}\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}=q^{in}s_{i+m}\,\delta_{i-j,n-m}$

a $q$-deformed version of (2.72).

There is only one classical point here. It is the unique character evaluating
to $0$ on $z$ and $\bar{z}$, which refers to the center $0$ of the classical
plane. The algebra therefore can not be realized within our principal
framework. But if we remove $0$ from consideration and allow Laurent series
with singularity in $0$ as constituents of our Hilbert space $\mathcal{H}$,
then a faithful representation is possible.

Let us consider the space $\mathcal{R}$ of generalized polynomials with all
integer powers of $z$, and define the scalar product by requiring mutual
orthogonality of the monomials $z^{n}$ and also

(2.89) $\langle z^{n}|z^{n}\rangle=q^{-\textstyle{\binom{n}{2}}}$

for all $n\in\mathbb{Z}$. If we interpret $z$ as the multiplication operator
by the coordinate $z$ and $\bar{z}$ as its formal adjoint, then the above non-
commutation relation for the Manin $q$-plane is fulfilled, and we have a
faithful representation.

This space $\mathcal{R}$ closes into a Hilbert space $\mathcal{J}$ of
holomorphic functions over $\mathbb{C}-\\{0\\}$. Its reproducing kernel is
given by

(2.90)
$K(\bar{w},z)=\sum_{n\in\mathbb{Z}}q^{\textstyle{\binom{n}{2}}}(\bar{w}z)^{n}.$

Let us recall ([2]–Chapter 10, Section 4) the classical triple product
identity

(2.91)
$\sum_{n\in\mathbb{Z}}(-1)^{n}q^{\textstyle{\binom{n}{2}}}z^{n}=(z|q)_{\infty}(q/z|q)_{\infty}(q|q)_{\infty}$

which holds for $z\neq 0$ and $|q|<1$. Applying this to our reproducing
kernel, we obtain

(2.92)
$K(\bar{w},z)=(-\bar{w}z|q)_{\infty}(-q/\bar{w}z|q)_{\infty}(q|q)_{\infty}.$

The induced metric on $\mathbb{C}-\\{0\\}$ is calculated by applying (2.32) to
this infinite product. The first two symbols on the right side of (2.92) are
transformed by the logarithm into infinite sums over $\mathbb{N}$, which after
applying the derivation $\partial^{2}/\partial w\partial\bar{w}$ merge into a
single sum over $\mathbb{Z}$. The third symbol $(q|q)_{\infty}$ does not
contribute to the metric, as it is a multiplicative constant transformed by
$\log$ into an additive constant. Explicitly, we get

(2.93)
$\mathrm{d}s^{2}=\sum_{n\in\mathbb{Z}}\frac{q^{n}\,\mathrm{d}w\,\mathrm{d}\bar{w}}{(1+q^{n}\bar{w}w)^{2}}.$

We see that the unique classical point represented by $0$ acts as a true
geometrical singularity, as the metric diverges when $|w|$ tends to $0$. On
the other hand, far away from $0$ the metric becomes asymptotically Euclidean.

Let $\Pi\colon\mathcal{J}\rightarrow\mathcal{J}$ be the orthogonal projector
on the positive part, the Hilbert space $\mathcal{H}$ generated by the
standard polynomials $\mathbb{C}[z]$. Clearly $\Pi(\mathcal{R})=\mathcal{P}$
and in such a way, for the sequence $s_{n}$ given by the positive part of
(2.89) we obtain a non-commutative system satisfying Harmony Three. We can see
this explicitly from

$\frac{s_{n+1}}{s_{n}}=q^{-n}\qquad\quad\frac{s_{n+1}^{2}}{s_{n}s_{n+2}}=q.$

This construction, which went ‘backwards’ relative to our main considerations,
ensures that Harmony Four holds, too. The space $\mathcal{H}$ consists of all
entire functions of $\mathcal{J}$ with the reproducing kernel given by

(2.94) $K(\bar{w},z)=\sum_{n\geq
0}q^{\textstyle{\binom{n}{2}}}(\bar{w}z)^{n}.$

The induced metric is hence

(2.95) $\mathrm{d}s^{2}=\sum_{n\geq
0}\frac{q^{n}\,\mathrm{d}w\,\mathrm{d}\bar{w}}{(1+q^{n}\bar{w}w)^{2}}=\log_{q}^{\prime}(1+|w|^{2})\frac{\mathrm{d}w\,\mathrm{d}\bar{w}}{1-q}.$

The operators $Z$ and $Z^{*}$ satisfy the following non-commutation relation

(2.96) $ZZ^{*}=qZ^{*}Z-q|0\rangle\langle 0|.$

From this it is easy to see that the whole matrix algebra
$\mathrm{M}_{\infty}(\mathbb{C})$ is included in $\mathcal{A}$, and that the
above relation is in fact a generating relation for $\mathcal{A}$. The
following short exact sequence holds:

(2.97)
$0\rightarrow\mathrm{M}_{\infty}(\mathbb{C})\longrightarrow\mathcal{A}\longrightarrow\mathbb{C}[z,\bar{z},Q]\rightarrow
0.$

So, basically, this Toeplitz quantization smooths out the singularity at $0$.
However, the change appears ‘mild’: The classical point at $0$ remains as the
unique character of $\mathcal{A}$. This is a counterexample for the possible
removal of the generators condition in Proposition 2.13.

It is also interesting to observe that the constructed non-commutative system
can be viewed as the Toeplitz quantization of the classical Euclidean plane in
the sense that harmony property H4 holds in this context, too. This follows
from the positivity analysis for the Stieltjes moment condition presented in
Appendix A together with this explicit calculation of the determinant
generated by the full defining sequence:

(2.98) $\det\left|\begin{matrix}&&\\\
&\smash[t]{q^{-{\textstyle\binom{i+j}{2}}}}&\\\
&&\end{matrix}\right|=q^{-{\textstyle\binom{n}{2}\frac{4n-5}{3}}}\prod_{k=1}^{n-1}(1-q^{k})^{n-k}.$

Here the integer indexes $i,j$ run from $0$ to $n-1$.

A different Toeplitz quantization of the Manin $q$-plane is obtained if we
choose the sequence (2.78). Then, as explained in Appendix B property H4
holds, too. It is interesting to observe that one and the same non-commutative
system is viewed as a quantization of the classical plane as well as of the
Manin $q$-plane.

To complete our analysis of different types of quadratic algebras
$\mathbb{C}[z,\bar{z},Q]$, let us focus on the parabolic type given by (2.69).
Although the framework of entire functions does not work here, the algebra
admits an interesting realization within a Hilbert space of holomorphic
functions in the positive half-plane $\Re(z)>0$. Let us fix a positive number
$h$. There exists a unique scalar product on the polynomial algebra
$\mathbb{C}[z]$ such that the constant function $1=|0\rangle$ has unit norm
and

(2.99) $z^{*}|0\rangle=\frac{h}{2}|0\rangle.$

If we define (using the Euler gamma function $\Gamma$) the polynomials as the
shifted factorials

(2.100)
$\xi_{n}(z)=(z+\frac{h}{2})_{n}\qquad(z)_{n}:=\prod_{j=0}^{n-1}(z+j)=\frac{\Gamma(z+n)}{\Gamma(z)},$

then it is easy to see that

(2.101) $z^{*}\xi_{n}(z)=(\frac{h}{2}+n)\xi_{n}(z)$

for every $n\in\mathbb{N}$. Clearly $|0\rangle=\xi_{0}(z)$ and these vectors
form a basis of $\mathbb{C}[z]$ with
$\mathbb{C}_{n}[z]=\mathrm{span}\\{\xi_{0}(z),\xi_{1}(z),\dots,\xi_{n}(z)\\}$
Furthermore, a direct calculation reveals

(2.102) $\langle\xi_{n}(z)|\xi_{m}(z)\rangle=(h)_{n+m}.$

Orthonormalization of these polynomials by the Gram-Schmidt procedure gives
for the canonical orthonormal basis

(2.103)
$|n\rangle=\phi_{n}(z)=\prod_{j=0}^{n-1}\frac{z-j-h/2}{\sqrt{(j+1)(h+j)}}=\frac{(-1)^{n}}{\sqrt{n!\,(h)_{n}}}(\frac{h}{2}-z)_{n}.$

By using the Weierstrass product formula and Stirling asymptotic formula

$\frac{1}{\Gamma(z)}=ze^{\gamma z}\prod_{n\geq
0}\Bigl{\\{}(1+\frac{z}{n})e^{-z/n}\Bigr{\\}}\qquad\qquad\Gamma(x)\sim\sqrt{2\pi}x^{x-1/2}e^{-x}\quad
x\rightsquigarrow+\infty$

it is easy to see that the kernel diagonal series is normally convergent on
the half-plane $\Re(z)>0$. The polynomials $\phi_{n}(z)$ are also
$\infty$-independent, so the conditions for Harmony One are here—although in
this quite different context of the positive half-plane. Actually, the kernel
can be explicitly summed to

(2.104)
$K(\bar{w},z)=\frac{\Gamma(h)\Gamma(\bar{w}+z)}{\Gamma(\bar{w}+h/2)\Gamma(z+h/2)}.$

This follows directly from the classical Gauss summation formula

$\sum_{n\geq
0}\frac{(a)_{n}(b)_{n}}{n!\,(c)_{n}}=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$

with $a=h/2-\bar{w}$, $b=h/2-z$ and $c=h$. The induced metric (2.32) on the
positive half-plane is especially simple, namely

(2.105)
$\mathrm{d}s^{2}=\mathrm{d}w\,\mathrm{d}\bar{w}\times\digamma^{\prime}(\bar{w}+w)$

where $\digamma(z)=\Gamma^{\prime}(z)/\Gamma(z)=[\log\Gamma(z)]^{\prime}$ is
the digamma function. Then taking into account the expansion around zero

$\digamma(z)=-\frac{1}{z}-\gamma-\sum_{n\geq 1}(-1)^{n}\zeta(n+1)z^{n}$

as well as the classical positive half-plane hyperbolic metric
$\mathrm{d}s^{2}={\mathrm{d}z\,\mathrm{d}\bar{z}}/{\Re(z)^{2}}$, we conclude
that the quantum metric asymptotically behaves like the classical hyperbolic
metric at the limit of the ‘infinity horizon’ of space represented by
imaginary numbers. Not surprisingly, as a matter of fact, this ‘geometrical
heaven’ is also interpretable as the classical part of this quantum space as
has been observed in our initial purely algebraic analysis.

##  3\. Some Algebraic Aspects of Toeplitz Quantization

In this section we shall describe an abstract algebraic underlying structure
for the Toeplitz quantization. In particular, this will provide the
foundations to fine-tune our principal construction of the extended space
$\mathcal{R}$ equipped with the quadratic form
$\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleleft$}\kern-2.8pt\langle|\rangle\kern-3.0pt\raisebox{1.2pt}{$\scriptscriptstyle\blacktriangleright$}$
and the projection $\Pi$, to Hilbert spaces of entire functions and harmony
conditions, where it is natural to use the more elaborate module
$\mathcal{W}$, instead of polynomials.

Let us assume that $\mathcal{P}$ is an everywhere dense linear subspace of a
Hilbert space $\mathcal{H}$. Let $\mathcal{C}$ be a unital subalgebra of the
*-algebra $M(\mathcal{P})$ of formally adjointable linear operators in
$\mathcal{P}$. Then the conjugate algebra $\bar{\mathcal{C}}$ is also of the
same category. Let $\mathcal{A}\subseteq M(\mathcal{P})$ be the *-subalgebra
generated by $\mathcal{C}$ and $\bar{\mathcal{C}}$.

The space $\mathcal{P}$ can be viewed as a left $\mathcal{A}$-module, and in
particular as both left $\mathcal{C}$-module as well as
$\bar{\mathcal{C}}$-module. The algebra $\bar{\mathcal{C}}$ can be purely
algebraically described as the opposite algebra of $\mathcal{C}$ equipped with
the conjugate vector space structure. In this interpretation, the *-operation
between $\mathcal{C}$ and $\bar{\mathcal{C}}$ is just the identity map. It
naturally extends to *-involutions on the vector spaces
$\mathcal{C}\otimes\bar{\mathcal{C}}$ and
$\bar{\mathcal{C}}\otimes\mathcal{C}$ by

(3.1)
$(\alpha\otimes\bar{\beta})^{*}=\beta\otimes\bar{\alpha}\qquad\quad(\bar{\alpha}\otimes\beta)^{*}=\bar{\beta}\otimes\alpha.$

In a similar way, the *-operation extends to arbitrary tensor products of
$\mathcal{C}$ and $\bar{\mathcal{C}}$. In particular, it mixes
$\mathcal{C}\otimes\mathcal{C}$ and
$\bar{\mathcal{C}}\otimes\bar{\mathcal{C}}$. Let
$m\colon\mathcal{C}\otimes\mathcal{C}\rightarrow\mathcal{C}$ and
$\bar{m}\colon\bar{\mathcal{C}}\otimes\bar{\mathcal{C}}\rightarrow\bar{\mathcal{C}}$
be the corresponding product maps. Clearly, we have this commutative diagram:

(3.2) $\begin{CD}\mathcal{C}\otimes\mathcal{C}@>{\mbox{$m$}}>{}>\mathcal{C}\\\
@V{\mbox{$*$}}V{}V@V{}V{\mbox{$*$}}V\\\
\bar{\mathcal{C}}\otimes\bar{\mathcal{C}}@>{}>{\mbox{$\bar{m}$}}>\mathcal{C}\end{CD}$

Let us assume that a linear map
$\sigma\colon\mathcal{C}\otimes\bar{\mathcal{C}}\rightarrow\bar{\mathcal{C}}\otimes\mathcal{C}$
is given such that the diagram

(3.3)
$\begin{CD}\mathcal{C}\otimes\bar{\mathcal{C}}@>{\mbox{$\sigma$}}>{}>\bar{\mathcal{C}}\otimes\mathcal{C}\\\
@V{\mbox{$*$}}V{}V@V{}V{\mbox{$*$}}V\\\
\mathcal{C}\otimes\bar{\mathcal{C}}@>{}>{\mbox{$\sigma$}}>\bar{\mathcal{C}}\otimes\mathcal{C}\\\
\end{CD}$

is commutative. Furthermore, let us assume that the following pentagonal
diagrams

(3.4)
$\qquad\quad\begin{CD}\mathcal{C}\otimes\mathcal{C}\otimes\bar{\mathcal{C}}@>{\mbox{$m\otimes\mathrm{id}$}}>{}>\mathcal{C}\otimes\bar{\mathcal{C}}@<{\mbox{$\mathrm{id}\otimes\bar{m}$}}<{}<\mathcal{C}\otimes\bar{\mathcal{C}}\otimes\bar{\mathcal{C}}\\\
@V{\mbox{$(\sigma\otimes\mathrm{id})(\mathrm{id}\otimes\sigma)$}}V{}V@V{}V{\mbox{$\sigma$}}V@V{}V{\mbox{$(\mathrm{id}\otimes\sigma)(\sigma\otimes\mathrm{id})$}}V\\\
\bar{\mathcal{C}}\otimes\mathcal{C}\otimes\mathcal{C}@>{}>{\mbox{$\mathrm{id}\otimes
m$}}>\bar{\mathcal{C}}\otimes\mathcal{C}@<{}<{\mbox{$\bar{m}\otimes\mathrm{id}$}}<\bar{\mathcal{C}}\otimes\bar{\mathcal{C}}\otimes\mathcal{C}\end{CD}$

are commutative too.

######  Remark 3.1.

It is easy to see that the commutativity of each of these pentagonal diagrams
implies that of the other, if we assume the conjugational symmetry (3.3). The
pentagonal diagrams as such, ensure that the space
$\bar{\mathcal{C}}\otimes\mathcal{C}$ can be equipped with a natural
associative product, such that

(3.5)
$(\bar{\gamma}\otimes\alpha)(\bar{\beta}\otimes\delta)=\bar{\gamma}\sigma(\alpha\otimes\bar{\beta})\delta$

and $1\otimes 1$ is the unit element of this algebra. This implies that always
$\sigma(\alpha\otimes 1)=1\otimes\alpha$ and
$\sigma(1\otimes\bar{\beta})=\bar{\beta}\otimes 1$. In particular
$\sigma(1\otimes 1)=1\otimes 1$. The conjugational symmetry then is equivalent
to the statement that the introduced $*$ is the *-structure on this algebra.
Indeed, if
$\sigma(\alpha\otimes\bar{\beta})=\sum_{k}\bar{\alpha_{k}}\otimes\beta_{k}$
then (3.3) is equivalent to

$\sigma(\beta\otimes\bar{\alpha})=\sum_{k}\bar{\beta_{k}}\otimes\alpha_{k}\quad\Leftrightarrow\quad[(1\otimes\alpha)(\bar{\beta}\otimes
1)]^{*}=(1\otimes\beta)(\bar{\alpha}\otimes 1),$

which in turn ensures that the *-operation on
$\bar{\mathcal{C}}\otimes\mathcal{C}$ is anti-multiplicative. Let us denote by
$\mathcal{B}$ the resulting *-algebra. By construction both $\mathcal{C}$ and
$\bar{\mathcal{C}}$ are subalgebras of $\mathcal{B}$, and they generate
$\mathcal{B}$ with the commutation rule given by $\sigma$.

Let us now define an extended representation space

(3.6)
$\mathcal{Q}=\mathcal{B}\otimes_{\mathcal{C}}\mathcal{P}\leftrightarrow\bar{\mathcal{C}}\otimes\mathcal{P}.$

Because of the above identification, the space $\mathcal{Q}$ is a left
$\mathcal{B}$-module, in a natural way: the module structure is induced simply
by left multiplication. By construction, the space $\mathcal{P}$ is naturally