A possible scenario of the Pioneer anomaly in the framework of Finsler
geometry
Xin Li∗,‡<EMAIL_ADDRESS>and Zhe Chang†,‡<EMAIL_ADDRESS>
∗Institute of Theoretical Physics, Chinese Academy of Sciences, 100190
Beijing, China
†Institute of High Energy Physics, Chinese Academy of Sciences, 100049
Beijing, China
‡Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences

###### Abstract

The weak field approximation of geodesics in Randers-Finsler space is
investigated. We show that a Finsler structure of Randers space corresponds to
the constant and sunward anomalous acceleration demonstrated by the Pioneer 10
and 11 data. The additional term in the geodesic equation acts as “electric
force”, which provides the anomalous acceleration.

PACS numbers: 02.40.-k, 04.50.Kd, 95.10.Ce

Newton’s theory of gravitation was proposed almost three hundred and fifty
years ago. Einstein’s general relativity reveals the intrinsic geometric
property of gravity. Newton’s theory of gravitation is the main guideline of
the celestial mechanics, especially for the solar system. General relativity
provides small corrections. It is well-known that the Newton and Einstein’s
theory of gravitation still faces problems. One of them is that the flat
rotation curves of spiral galaxies violate the prediction of Newton and
Einstein’s gravity. Another is related with recent astronomical
observations[1]. Our universe is acceleratedly expanding. This result can not
be obtained directly from Einstein’s gravity and his cosmological principle.
In fact, new puzzle has also arisen in the solar system. That is the Pioneer
anomaly. The Pioneer spacecrafts are excellent tools for dynamical astronomy
studies in the solar system. The radio metric data from the Pioneer 10/11
spacecraft indicate the presence of a small, anomalous, Doppler frequency
drift over the range of 20–70 astronomical units[2]. The drift is blue-shift,
uniformly changing with a rate of $~{}6\times 10^{-9}Hz/s$[3]. It has revealed
an anomalous constant sunward acceleration, $a_{P}=(8.74\pm 1.33)\times
10^{-10}m/s^{2}$.

The Pioneer data have been studied in three different navigational computer
programs. Namely: the JPL’s Orbit Determination Program (ODP), the Aerospace
Corporation’s CHASPM code extended for deep space navigation[2], and a code
written in the Goddard Space Flight Center[4]. These data analysis all confirm
the existence of anomalous acceleration with the following basic properties:
the direction of the anomalous acceleration of the spacecraft towards the Sun,
the anomalous acceleration appears close to $20AU$ and up to $70AU$, the
anomalous acceleration seems to be a constant with 10% order of temporal and
spatial variations of the anomaly’s magnitude. Turyshev et al.[5] summarized
recent results on researches of the anomaly. Several conventional physical
mechanisms have been proposed to explain the anomaly, such as the unknown
systematic–the gas leaks from the propulsion system or a recoil force due to
the on-board thermal power inventory, and the conventional gravitational force
due to a known mass distribution in the outer solar system–the Kuiper Belt
Objects or dust, and the expansion of the universe motivated by the numerical
coincidence $a_{P}\simeq cH_{0}$. However, it was pointed out that these
conventional physical mechanisms can not be the answer of the anomalous
accelerations[2, 6].

The failure of the conventional physical mechanisms imply that the Pioneer
anomaly may correspond to ‘new physics’. One of the most popular ‘new physics’
is the dark matter hypothesis. A specific distribution of dark matter in the
solar system would yield the wanted result[7]. However, this special
distribution of dark matter is not like the consequence of gravity. Thus, to
explain the Pioneer anomaly, dark matter hypothesis still need more work.
Several modified gravitational theories also was suggested to explain the
Pioneer anomaly, such as the scalar-tensor vector gravity (STVG)theory[8],
brane-world models with large extra dimensions[9], and conformal gravity with
dynamic mass generation[2]. These modified gravitational theories seem
appealing, however, most of them either much more complicated or involves too
much hypothesis which does not verified by experiments.

Finsler geometry, which takes Riemann geometry as its special case, is a good
candidate to solve the facing problems of the theory of gravitation. The
gravity in Finsler space has been studied for a long time[10, 11, 12, 13]. In
our previous paper[14], a modified Newton’s gravity was obtained as the weak
field approximation of the Einstein’s equation in Finsler space of Berwald
type. We have shown that the prediction of the modified Newton’s gravity is in
good agreement with the rotation curves of spiral galaxies without invoking
dark matter hypothesis.

Randers space, as a special kind of Finsler space, was first proposed by G.
Randers[15]. Within the framework of Finsler geometry, modified dispersion
relation of free particle in Randers space has been discussed[16]. A modified
Friedmann model in Randers space is proposed. It is showed that the
accelerated expanding universe is guaranteed by a constrained Randers-Finsler
structure without invoking dark energy[17].

In this Letter, in the framework of Finsler geometry we will try to give a
simple and clear description of the Pioneer anomaly. As well-known, the length
in Riemann geometry is a function of positions. However, this is not the case
in Finsler geometry. In Finsler geometry, the length is a function of both
position and velocity. Finsler geometry is base on the so called Finsler
structure $F$ with the property $F(x,\lambda y)=\lambda F(x,y)$, where $x$
represents position and $y$ represents velocity. The Finsler metric is given
as[18]

$\displaystyle g_{\mu\nu}\equiv\frac{\partial}{\partial
y^{\mu}}\frac{\partial}{\partial y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (1)

The Randers metric is a Finsler structure $F$ on $TM$ of the form

$\displaystyle F(x,y)\equiv\alpha(x,y)+\beta(x,y)~{},$ (2)

where

$\displaystyle\alpha(x,y)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\tilde{a}_{\mu\nu}(x)y^{\mu}y^{\nu}}$
$\displaystyle\beta(x,y)$ $\displaystyle\equiv$
$\displaystyle\tilde{b}_{\mu}(x)y^{\mu}.$ (3)

Here $\tilde{\alpha}$ is a Riemannian metric on the manifold $M$. In this
Letter, the indices decorated with a tilde are lowered and raised by
$\tilde{\alpha}_{\mu\nu}$ and its inverse matrix $\tilde{\alpha}^{\mu\nu}$,
otherwise lower and raise the indices are carried by $g_{\mu\nu}$ and
$g^{\mu\nu}$. We will show that the Finsler structure in Randers space with
$\tilde{b}$ taking the specific form $\tilde{b}_{\mu}=\\{-kr,0,0,0\\}$
corresponds to the Pioneer anomaly. The above form of $\tilde{b}$ is given in
spherical coordinate and $k$ is a constant.

The parallel transport in Finsler space has been studied in terms of Cartan
connection[20, 21, 22]. The notation of parallel transport in Finsler manifold
means that the length $F\left(\frac{d\sigma}{d\tau}\right)$ is constant.
Following the calculus of variations, one gets the autoparallel equation in
Finsler space as[18]

$\displaystyle\frac{d^{2}\sigma^{\lambda}}{d\tau^{2}}+\gamma^{\lambda}_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}=0.$
(4)

The autoparallel equation (4) is directly derived from the integral length of
$\sigma$

$\displaystyle L=\int F\left(\frac{d\sigma}{d\tau}\right)d\tau,$ (5)

the inner product
$\left(\sqrt{g_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}}=F\left(\frac{d\sigma}{d\tau}\right)\right)$
of two parallel transported vectors is preserved. To get a modified Newton’s
gravity, we consider a particle moving slowly in a weak stationary
gravitational field[19]. Here, we suppose that the Riemannian metric
$\tilde{\alpha}$ is close to Minkowskian metric, and
$|\tilde{b}_{\mu}\tilde{b}^{\mu}|$ is very small

$\displaystyle\tilde{a}_{\mu\nu}(x)=\tilde{\eta}_{\mu\nu}+\tilde{h}_{\mu\nu}(x),$
(6)

where $\tilde{\eta}_{\mu\nu}$ is the Minkowskian metric and
$|\tilde{h}_{\mu\nu}|\ll 1$. Deducing from (4), we obtain the geodesic of
Randers space with constant Riemanian speed (namely,
$\alpha(\frac{d\sigma}{d\tau})$ is constant)

$\displaystyle\frac{d^{2}\sigma^{\lambda}}{d\tau^{2}}+\tilde{\gamma}^{\lambda}_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}+\tilde{a}^{\lambda\mu}f_{\mu\nu}\alpha\left(\frac{d\sigma}{d\tau}\right)\frac{d\sigma^{\nu}}{d\tau}=0,$
(7)

where $f_{\mu\nu}\equiv\frac{\partial\tilde{b}_{\mu}}{\partial
x^{\nu}}-\frac{\partial\tilde{b}_{\nu}}{\partial x^{\mu}}$.

Randers[15] has already found that the Randers metric is related to five
dimensional Riemannian geometry. The five dimensional Riemannian metric
$\gamma_{mn}$ ($m,n=1,2,3,4,5;\mu,\nu=1,2,3,4$) is given as

$\gamma_{\mu\nu}=\tilde{a}_{\mu\nu}-\tilde{b}_{\mu}\tilde{b}_{\nu};~{}\gamma_{\mu
5}=\gamma_{5\mu}=\tilde{b}_{\mu};~{}\gamma_{55}=-1.$ (8)

And the geodesic equation (7) of Randers metric is a solution of the five
dimensional Einstein’s field equation. The five dimensional Einstein tensor is
expressed as

$\displaystyle G^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\tilde{R}^{\mu\nu}-\frac{1}{2}\tilde{a}^{\mu\nu}\tilde{R}\right)+\frac{1}{2}\tilde{E}^{\mu\nu},$
(9) $\displaystyle G_{5}^{~{}\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}f^{\nu\mu}_{~{}~{}~{};\mu},$ (10) $\displaystyle
G_{55}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tilde{R}-\frac{3}{4}f^{\mu\nu}f_{\mu\nu}\right),$
(11)

where
$\tilde{E}^{\mu\nu}=-f^{\mu\lambda}f_{\lambda}^{~{}\nu}+\frac{1}{4}\tilde{a}^{\mu\nu}f^{\lambda\theta}f_{\lambda\theta}$,
$\tilde{R}^{\mu\nu}$ is the Ricci tensor of the four dimensional Riemannian
metric $\alpha$, and the covariant derivative of four dimensional Riemannian
metric $\alpha$ is denoted by “;”. In a geometrical viewpoint, the Randers
metric arised from the Zermelo navigation problem [23]. It aims to find the
paths of shortest travel time in a Riemannian manifold under the influence of
a drift (“wind”). Shen [24] has shown that these minimum time trajectories are
exactly the geodesics of a particular Finsler geometry-Randers metric. The map
between the Randers metric to a Riemannian space in the viewpoint of Zermelo
navigation problem is investigated in the paper [25].

In weak field approximation, the second term of the left side of the equation
(7) represents the Newtonian gravitational acceleration. And the third term
may induce the anomalous acceleration. One should notice that the trajectory
of the Pioneer 10 spacecraft is different from that of the Pioneer 11
spacecraft. The basic property of the Pioneer anomaly that the direction of
the anomalous acceleration of the spacecraft towards the Sun tells us that the
non vanish components of $f_{\mu\nu}$ is $f_{0i}$. The term $f_{\mu\nu}$ acts
as electromagnetic force. In dealing with the Pioneer anomaly, one need take
only the “electric force” into account. Also, due to physical consideration,
the “electric force” should be static. Thus, in the approximation of moving
slowly and weak field, the geodesic equation (7) reduces to

$\displaystyle\frac{d^{2}t}{d\tau^{2}}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{d^{2}\sigma^{i}}{d\tau^{2}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\partial
h_{00}}{\partial\sigma^{i}}\left(\frac{dt}{d\tau}\right)^{2}-\frac{\partial
b_{0}}{\partial\sigma^{i}}\alpha\left(\frac{d\sigma}{d\tau}\right)\left(\frac{dt}{d\tau}\right).$
(12)

The solution of the first equation in (S0.Ex2) is $dt/d\tau=const.$. Dividing
the second equation in (S0.Ex2) by $(dt/d\tau)^{2}$, we obtain

$\displaystyle\frac{d\sigma^{i}}{dt}=-\frac{1}{2}\frac{\partial
h_{00}}{\partial\sigma^{i}}-\frac{\partial b_{0}}{\partial\sigma^{i}}.$ (13)

In spherical coordinate, the above equation changes as

$\displaystyle a=\nabla\varphi+k,$ (14)

where $\varphi\equiv-\frac{GM_{\odot}}{r}$ is the Newtonian gravitational
potential. Then, from the equation (14) one can see clearly that the anomalous
acceleration

$\displaystyle a_{p}=k.$ (15)

Taking the average value of $a_{p}$, we can set the constant $k$ as
$9.71\times 10^{-25}m^{-1}$.

At a position of 20AU far from the Sun, the Newtonian gravitational potential
$\varphi=-4.43\times 10^{7}m^{2}/s^{2}$ and the perturbation of Minkowskian
metric $h_{00}=9.85\times 10^{-10}$, and $b_{0}=2.91\times 10^{-14}$. Thus,
the Finsler structure of Randers space is a good description for metric
fluctuation around the Minkowskian one. At a position of 1AU far from the Sun,
the Newtonian gravitational potential $\varphi=-8.87\times 10^{8}m^{2}/s^{2}$
and the perturbation of Minkowskian metric $h_{00}=1.97\times 10^{-8}$, and
$b_{0}=1.46\times 10^{-15}$. Einstein’s relativity offers high order
correction for Newtonian mechanics (post-Newtonian approximation)[19], the
corresponding metric correction approximately equals $h_{00}^{2}$. Here, we
can see that $h_{00}^{2}$ is very close to $b_{0}$. While the Pioneer is not
far from the earth, it is hard to distinguish the effect of general
relativity(or Riemann geometry) and Finsler geometry. This is a reason for why
the anomaly appears in the position of 20AU far from the Sun.

The equation (14) implies that the modified gravitational potential is

$\varphi_{P}=krc^{2},$ (16)

where $c$ is the speed of light. Since the parameter $k$ is set as $9.71\times
10^{-25}m^{-1}$, the ration $|\frac{\varphi_{P}}{\varphi}|$ is less than
$10^{-8}$ for the solar system. The classical tests of general relativity are
carried in solar system. Thus, the geodesic equation (7) also predict the same
astrophysical phenomena that Einstein’s general relativity are able to
predict. One also could directly obtain this fact from the field equation (9),
for the tensor $\tilde{E}^{\mu\nu}$ in it is the second order in $f^{\mu\nu}$.

The existence of the Pioneer anomaly suggests the Newton’s theory of
gravitation and general relativity need to be modified even in the solar
system. Here, we have suggested that Finsler geometry could give a clear and
simply description of the Pioneer anomaly. The specific Finsler structure of
the Randers space corresponds to the Pioneer anomaly. We hope that the gravity
anomalies mentioned in the beginning of the Letter can be solved
systematically in the framework of Finsler geometry.

Acknowledgements

We would like to thank Prof. C. J. Zhu, H. Y. Guo and C. G. Huang for useful
discussions. The work was supported by the NSF of China under Grant No.
10525522 and 10875129.

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[3]

# Rational Cuntz states peak on the free disk algebra

Robert T.W. Martin Supported by NSERC grant 2020-05683 University of Manitoba
Eli Shamovich Ben-Gurion University of the Negev

###### Abstract

We apply realization theory of non-commutative rational multipliers of the
Fock space, or _free Hardy space_ of square–summable power series in several
non-commuting variables to the convex analysis of states on the Cuntz algebra.
We show, in particular, that a large class of Cuntz states which arise as the
‘non-commutative Clark measures’ of isometric NC rational multipliers are peak
states for Popescu’s free disk algebra in the sense of Clouâtre and Thompson.

##  1 Introduction

This paper applies non-commutative (NC) analysis to a question in non-
commutative convexity. Our main result constructs states on the Cuntz algebra,
${\mathcal{O}}_{d}$, which peak at non-commutative rational inner functions in
Popescu’s _free disk algebra_ , $\mathbb{A}_{d}$. We also provide a novel
characterization of unital quantum channels in terms of NC rational inner
functions.

Non-commutative convexity first appeared in the seminal work of Arveson [5].
In [5], Arveson extended classical Choquet theory to a non-commutative,
operator–algebraic setting. Classical Choquet theory studies the extreme
boundary of compact convex sets and representing measures. Namely, let $K$ be
a compact convex set; a probability measure $\mu$ on $K$ represents a point
$x\in K$ if the restriction of the corresponding state from $C(K)$, the
continuous functions on $K$, to the space of continuous affine functions on
$K$, is the functional of evaluation at $x$. Bauer characterized the extreme
points of $K$ as those that admit only one representing measure (necessarily
$\delta_{x}$). This characterization lends itself to an extension to a more
general setting of unital subspaces $1\in M\subset C(X)$, where $X$ is a
compact Hausdorff space. The Choquet boundary of such a subspace is the
collection of all points $x\in X$, such that $\delta_{x}|_{M}$ admits a unique
extension to $C(X)$. One way to obtain points in the Choquet boundary is to
find peak points. A point $x\in X$ is an $M-$peak point, if there exists an
$f\in M$, such that $|f(x)|=\|f\|$ and $|f(y)|<\|f\|$ for any $y\neq x$. In
particular, if $1\in A\subset C(X)$ is a uniform algebra, and $X$ is
metrizable, then an important result of Bishop states that the Choquet
boundary of $A$ is precisely the set of peak points of $A$.

The field of non-commutative convexity has expanded quickly with contributions
from Wittstock, Effros and his collaborators, and many others. The reader is
referred to the monograph of Davidson and Kennedy [16] and the references
therein for further details on non-commutative convexity theory. Of particular
interest is the non-commutative Choquet theory first introduced and developed
by Arveson [5, 2, 4, 3]. Suppose $B$ is a unital $C^{*}$-algebra and $1\in
A\subset B$ is an operator algebra. In this case, we say that an irreducible
representation $\pi\colon B\to B({\mathcal{H}})$ is a boundary representation
if $\pi|_{A}$ has a unique extension to $B$. The image of the direct sum of
all boundary representations gives the _$C^{*}-$ envelope_, $C^{*}_{min}(A)$
of $A$. This is a $C^{*}-$algebra in which $A$ embeds, completely
isometrically, and it is universal and minimal in the sense that if $A$ embeds
completely isometrically into any $C^{*}-$algebra, $B$, then there is a
$*-$homomorphism from $B$ onto $C^{*}_{min}(A)$ which intertwines the
embeddings. Existence and construction of the $C^{*}-$envelope via boundary
representations was a long-unsolved problem in operator algebra theory until
it was resolved in full generality by Davidson and Kennedy [15, 2, 5, 26, 22].
Arveson also introduced the concept of a peaking representation in [3]. His
ideas were further extended by Clouâtre [11], Clouâtre and Thompson [12, 13],
and Davidson and Passer [18]. In particular, Clouâtre and Thompson propose to
study ‘peaking states’. Precise definitions and details on peaking states can
be found in Subsection 2.3. Clouâtre and Thompson [13] suggest that studying
peaking phenomena on the Cuntz algebra will be interesting. We will provide a
family of examples of peaking states and corresponding representations in this
setting.

In more detail, we will study states on the Cuntz algebra ${\mathcal{O}}_{d}$,
which peak at elements of Popescu’s free or non-commutative disk algebra,
$\mathbb{A}_{d}$. Here, $\mathscr{E}_{d}:=C^{*}\\{I,L_{1},\cdots,L_{d}\\}$, is
the _Cuntz–Toeplitz algebra_ , the unital $C^{*}-$algebra generated by the
left creation operators, $L_{k}$, on the full Fock space,
$\mathbb{H}^{2}_{d}$. Here, $\mathbb{H}^{2}_{d}$ can be defined as the Hilbert
space of complex square–summable power series in several NC variables,
equipped with the $\ell^{2}-$inner product of the complex power series
coefficients. In this viewpoint the $L_{k}=M^{L}_{\mathfrak{z}_{k}}$ act as
isometric left multiplications by the $d$ independent formal NC variables,
$\mathfrak{z}=(\mathfrak{z}_{1},\cdots,\mathfrak{z}_{d})$. This algebra
contains the compact operators, $\mathscr{K}(\mathbb{H}^{2}_{d})$, on
$\mathbb{H}^{2}_{d}$ and the Cuntz algebra is then defined as the quotient
$C^{*}-$algebra,
${\mathcal{O}}_{d}:=\mathscr{E}_{d}/\mathscr{K}(\mathbb{H}^{2}_{d})$.
Popescu’s free disk algebra $\mathbb{A}_{d}:=\mathrm{Alg}\\{I,L_{1},\cdots
L_{d}\\}$ is the unital norm-closed algebra generated by the left creation
operators. Moreover, $\mathbb{A}_{d}$ can be identified, completely
isometrically with $\mathrm{Alg}\\{I,S_{1},\cdots,S_{d}\\}$, where the $S_{j}$
denote the generators of the Cuntz algebra.

Our primary tools are free analysis and non-commutative function theory. These
are rapidly growing fields in modern analysis. Free or non-commutative
analysis was initially motivated by the study of analytic functional calculus
of several commuting and non-commuting operators, as pioneered by Taylor [46,
47], Voiculescu’s free probability theory [48, 49], and Takesaki’s extension
of Gelfand duality to arbitrary, non-commutative $C^{*}-$algebras in [45].
Popescu [37, 39, 41, 42] studied non-commutative functions to extend the
classical Sz. Nagy–Foias theory of dilations and von Neumann’s inequality to
the multivariable setting. He introduced the full Fock space as an analog of
the classical Hardy Hilbert space of analytic function in the complex unit
disk, $\mathbb{D}$. We provide the necessary definitions and background theory
in Subsection 2.1.

A particularly well-studied class of NC functions is the set of all non-
commutative rational functions. These functions arise naturally in many
different branches of pure and applied mathematics, such as the theory of
localizations and quasideterminants in non-commutative rings [1, 14, 25], the
theory of formal languages [9, 27], and systems theory [6, 7]. This paper will
focus on non-commutative rational functions that are elements of the weak
operator topology (WOT)-closed algebra generated by the left creation
operators on the full Fock space. Such functions were extensively studied in
[31, 32]. In this paper, there is a critical interplay between non-commutative
rational inner functions, _i.e._ NC rational functions which define isometric
(inner) multipliers on the Fock space, states on the Cuntz algebra which peak
on the free disk algebra, and finite-dimensional row coisometries. A row
contraction is a contractive linear map from several copies of a Hilbert space
into one copy. The necessary background and details are in Subsection 2.2.

Given an irreducible and finite-dimensional row coisometry,
$T=(T_{1},\cdots,T_{d})$, and a unit vector $x$, we can define a linear
functional on $\mathbb{A}_{d}$ via $\mu(f)=\langle x,f(T)x\rangle$. One can
apply a Gelfand–Naimark–Segal construction to $(\mu,\mathbb{A}_{d})$, and this
yields a GNS-Hilbert space, $\mathbb{H}^{2}_{d}(\mu)$ and a $*-$representation
of the Cuntz–Toeplitz $C^{*}-$algebra, $\pi_{\mu}$. Since $T$ is a row-
coisometry, one can show that $\Pi_{\mu}:=\pi_{\mu}(L)$ is a Cuntz, _i.e._
surjective row isometry, so that $\mu$ admits a unique extension to a state on
${\mathcal{O}}_{d}$, $\hat{\mu}$ by [33, Proposition 5.11]. This state,
$\hat{\mu}$, is a finitely–correlated state, as introduced by Bratteli and
Jørgensen [10]. Our main result is Theorem 3.5, which states that every such
finitely–correlated state is an $\mathbb{A}_{d}$-peak state. In particular, by
results of Clouâtre, this implies that every such state is an exposed extreme
point of the state space of the non-commutative or free disk operator system.
The proof of the theorem constructs a non-commutative rational inner, such
that the state peaks at it. However, this inner is constructed from
$(T^{t}_{1},\cdots,T^{\mathrm{t}}_{d})$, the coordinate-wise transpose of $T$.
This duality leads us to Theorem 3.8, which shows that a non-commutative
rational inner $\mathfrak{b}$ arises from a quantum channel if and only if the
‘transpose’ $\mathfrak{b}^{t}$, obtained by reversing the order of all
products in monomials in the power series of $\mathfrak{b}$, is an inner as
well. En route to these results, we obtain some results on spectra of non-
commutative rational functions regular at the origin, which refine and extend
our previous results in [32].

##  2 Background

###  2.1 Non-commutative Hardy space and its multipliers

Given $d\in\mathbb{N}$, the _full Fock space_ is the Hilbert space direct sum
$\mathbb{H}^{2}_{d}=\oplus_{n=0}^{\infty}\left(\mathbb{C}^{d}\right)^{\otimes
n}$, with the usual convention of $\left(\mathbb{C}^{d}\right)^{\otimes
0}\cong\mathbb{C}$. We will call $\mathbb{H}^{2}_{d}$ the _free_ or _non-
commutative_ (NC) _Hardy space_ as it has many properties in common with the
classical Hardy space, $H^{2}$, of square–summable power series in the complex
unit disk. In particular, the elements of $\mathbb{H}^{2}_{d}$ can be viewed
as NC functions on the NC unit ball

$\mathbb{B}^{d}_{\mathbb{N}}=\bigsqcup_{n=1}^{\infty}\mathbb{B}_{n}^{d},\text{
where
}\mathbb{B}^{d}_{n}=\left\\{X\in\mathscr{B}(\mathbb{C}^{n}\otimes\mathbb{C}^{d},\mathbb{C}^{n})\mid
XX^{*}<I\right\\}.$

The interpretation of $\mathbb{H}^{2}_{d}$ as a space of non-commutative
functions was first considered by Popescu in [41]. A natural way to see
$\mathbb{H}^{2}_{d}$ as a space of functions is by noting that
$\mathbb{H}^{2}_{d}$ is the completion of the free algebra,
$\mathbb{C}\langle\mathfrak{z}_{1},\ldots,\mathfrak{z}_{d}\rangle=\mathbb{C}\langle\mathbb{\mathfrak{z}}\rangle$,
of NC or free polynomials, with respect to the inner product that makes the
monomials orthonormal. The NC monomials correspond to words in the alphabet
$\\{1,\ldots,d\\}$. Given a word $\alpha=i_{1}\cdots i_{n}$,
$i_{j}\in\\{1,\cdots,d\\}$, we write
$\mathfrak{z}^{\alpha}=\mathfrak{z}_{i_{1}}\cdots\mathfrak{z}_{i_{n}}$. If the
word is empty, we define $\mathfrak{z}^{\emptyset}:=1$. We will also denote
the length of $\alpha=i_{1}\cdots i_{n}$ by $|\alpha|=n$. For a $d$-tuple of
operators $T=(T_{1},\ldots,T_{d})$, we set $T^{\alpha}=T_{i_{1}}\cdots
T_{i_{n}}$ and $T^{\emptyset}=I$. Note that for any free polynomial,
$p\in\mathbb{C}\langle\mathbb{\mathfrak{z}}\rangle$, we can evaluate $p$ on
any $d$-tuple of matrices $Z=(Z_{1},\cdots,Z_{d})\in\mathbb{C}^{n\times
n}\otimes\mathbb{C}^{1\times d}=:\mathbb{C}^{d}_{n}$ and obtain a function
with the following properties: $p(Z)=p(Z_{1},\cdots,Z_{d})$

  * (i)

is graded: for every $X\in\mathbb{C}^{d}_{n}$, $p(X)\in\mathbb{C}^{n\times
n}$,

  * (ii)

respects direct sums: for every $X\in\mathbb{C}^{d}_{n}$ and
$Y\in\mathbb{C}^{d}_{m}$, we have

$p(X\oplus Y)=p\left(\begin{pmatrix}X&0\\\
0&Y\end{pmatrix}\right)=\begin{pmatrix}p(X)&0\\\
0&p(Y)\end{pmatrix}=p(X)\oplus p(Y),$

  * (iii)

respects similarities: for every $X\in\mathbb{C}^{d}_{n}$ and
$S\in\mathrm{GL}_{n}$, we have

$p(S^{-1}XS)=p(S^{-1}X_{1}S,\ldots,S^{-1}X_{d}S)=S^{-1}p(X)S.$

The elements of $\mathbb{H}^{2}_{d}$, thus, can be viewed as power series in
non-commuting variables. The power series converge uniformly and absolutely on
all tuples of matrices of norm $\leq r$, for every $0<r<1$. The properties
above hold, except that the third property needs to be modified to hold only
if $S^{-1}XS\in\mathbb{B}_{\mathbb{N}}^{d}$. Therefore, we obtain a space of
NC functions on $\mathbb{B}_{\mathbb{N}}^{d}$.

In direct analogy with classical Hardy space theory, (left) multiplication by
any of the $d$ independent variables define isometries,
$L_{j}=M^{L}_{\mathfrak{z}_{j}}$ on $\mathbb{H}^{2}_{d}$ with pairwise
orthogonal ranges. The ‘row’ operator,

$L=(L_{1},\ldots,L_{d})\colon\mathbb{H}^{2}_{d}\otimes\mathbb{C}^{d}\to\mathbb{H}^{2}_{d},$

is then a _row isometry_ , _i.e._ an isometry from several copies of a Hilbert
space into one copy. The operators $L_{j}$ are called the left creation
operators or the left free shifts. The WOT-closed algebra
$\mathbb{H}^{\infty}_{d}$, the _free Hardy algebra_ , generated by the $L_{j}$
was extensively studied by Popescu [41, 42, 39] and Davidson–Pitts [21, 19,
20]. The free Hardy algebra is completely isometrically isomorphic to the
algebra of all uniformly bounded NC functions on $\mathbb{B}_{\mathbb{N}}^{d}$
with the uniform norm.

$\|f\|=\sup_{X\in\mathbb{B}_{\mathbb{N}}^{d}}\|f(X)\|.$

The space $\mathbb{H}^{2}_{d}$ is an NC reproducing kernel Hilbert space
(RKHS) in the sense of Ball, Marx, and Vinnikov [8]. The free Hardy algebra
can then be identified as the algebra of left multipliers of this NC-RKHS.

Popescu and Davidson–Pitts have generalized the classical inner-outer
factorization of bounded analytic functions on the disk to
$\mathbb{H}^{\infty}_{d}$. Here, as in classical $H^{2}$ theory, an inner NC
function in $\mathbb{H}^{\infty}_{d}$ defines an isometric left multiplier on
$\mathbb{H}^{2}_{d}$ and an outer NC function is one that defines a left
multiplier with a dense range.

Recall that the free disk algebra, $\mathbb{A}_{d}$, is the unital norm-closed
algebra generated by the left creation operators on the full Fock space. We
assume throughout that $d\geq 2$. This algebra is a very close NC analogue of
the classical disk algebra, $\mathbb{A}_{1}=A(\mathbb{D})$. The Cuntz algebra
is the universal $C^{*}$-algebra of a surjective row isometry. That is, if
$S=(S_{1},\ldots,S_{d})$ denotes the row isometry of generators of the Cuntz
algebra, then $\sum_{j=1}^{d}S_{j}S_{j}^{*}=I$. The quotient map from the
Cuntz–Toeplitz algebra, $\mathscr{E}_{d}$ onto ${\mathcal{O}}_{d}$ restricts
to a complete isometry on $\mathbb{A}_{d}$, and since the Cuntz algebra is
simple, it is the $C^{*}$-envelope of $\mathbb{A}_{d}$. Classically,
$C({\mathbb{T}})$ is the $C^{*}$-envelope of $A(\mathbb{D})$. Moreover,
$C({\mathbb{T}})=\overline{A(\mathbb{D})+A(\mathbb{D})^{*}}$. In the NC
setting, $\mathscr{A}_{d}:=(\mathbb{A}_{d}+\mathbb{A}_{d}^{*})^{-\|\cdot\|}$
is an operator system that embeds, completely isometrically, into
${\mathcal{O}}_{d}$, but does not coincide with the Cuntz algebra. In
particular, $\mathscr{A}_{d}$ is not an algebra or a $C^{*}-$algebra. Namely,
by [40, Theorem 3.1], if $S_{1},\cdots,S_{d}$ denote the generators of
${\mathcal{O}}_{d}$, then the free disk algebra
$\mathbb{A}_{d}:=\mathrm{Alg}\\{I,L_{1},\cdots,L_{d}\\}^{-\|\cdot\|}$ and
$\mathbb{A}_{d}(S):=\mathrm{Alg}\\{I,S_{1},\cdots,S_{d}\\}^{-\|\cdot\|}$ are
completely isometrically isomorphic. Moreover, by [36, Proposition 3.5], the
_free disk system_ ,
$\mathscr{A}_{d}:=(\mathbb{A}_{d}+\mathbb{A}_{d}^{*})^{-\|\cdot\|}$ and the
operator system
$\mathscr{A}_{d}(S)=(\mathbb{A}_{d}(S)+\mathbb{A}_{d}(S)^{*})^{-\|\cdot\|}$
are then completely isometrically isomorphic. For the remainder of the paper,
we identify $\mathbb{A}_{d}$ with $\mathbb{A}_{d}(S)$ and $\mathscr{A}_{d}$
with $\mathscr{A}_{d}(S)$ so that the free disk algebra and the free disk
system are viewed as subspaces of the Cuntz algebra, ${\mathcal{O}}_{d}$. An
important property of $\mathbb{A}_{d}$ is that it is semi-Dirichlet, namely
that $\mathbb{A}_{d}^{*}\mathbb{A}_{d}\subset\mathscr{A}_{d}$. This property
enables one to perform a Gelfand–Naimark–Segal (GNS)-type construction
directly from positive linear functionals on $\mathscr{A}_{d}$. Jury and the
first author [28] have developed an NC extension of the classical
Alexandrov–Clark measure theory. In this NC theory, the positive linear
functionals on the free disk system, denoted by
$(\mathscr{A}_{d}^{\dagger})_{+}$, play the role of positive measures on the
unit circle. There is, in particular, a one-to-one correspondence between
states on $\mathscr{A}_{d}$ and contractive functions in
$\mathbb{H}^{\infty}_{d}$ that vanish at $0$. We will be primarily interested
in the positive _finitely–correlated states_ or ‘NC measures’ which arise as
the ‘NC Clark measures’ of NC rational multipliers of the Fock space, and we
will introduce these in more detail in the following section.

###  2.2 Non-commutative rational functions

The theory of non-commutative rational functions has been developed
independently in pure and applied disciplines ranging from pure algebra to
computational and systems theory. In particular, the algebra of all NC
rational functions is the _free skew field_ as constructed by Amitsur [1] and
Cohn [14] and is denoted by $\mathbb{C}\
\mathclap{\,<}{\left(\right.}\,\mathfrak{z}\,\mathclap{\,\,\,\,>}{\left.\right)}$.
(In NC algebra, Amitsur and Cohn proved that this ‘free skew field’ is the
universal ‘field of fractions’ of the free algebra,
$\mathbb{C}\langle\mathbb{\mathfrak{z}}\rangle$, of free or non-commutative
complex polynomials in the $d$ NC variables,
$\mathfrak{z}=(\mathfrak{z}_{1},\cdots,\mathfrak{z}_{d})$.) The domain of an
NC rational function is roughly the largest collection of $d$-tuples of
matrices to which our NC rational function can be continued. We will denote
the domain of $\mathfrak{r}$ by $\mathrm{Dom}\,\mathfrak{r}$. This paper will
focus on NC rational functions that are defined and bounded on
$\mathbb{B}^{d}_{\mathbb{N}}$. This assumption simplifies much of the theory.
The interested reader should consult [34, 35, 50] and the references therein
for more detail on the theory of NC rational functions.

Since $0\in\mathbb{B}^{d}_{\mathbb{N}}$, we will assume that our NC rational
functions always have $0$ in their domains, and we will denote the algebra of
all such NC rational functions by $\mathbb{C}_{0}\
\mathclap{\,<}{\left(\right.}\,\mathfrak{z}\,\mathclap{\,\,\,\,>}{\left.\right)}$.
This assumption simplifies the definition of an NC rational function somewhat.
We will say that an NC rational function in $\mathbb{C}_{0}\
\mathclap{\,<}{\left(\right.}\,\mathfrak{z}\,\mathclap{\,\,\,\,>}{\left.\right)}$
is any expression of the form

$\mathfrak{r}(\mathfrak{z})=c^{*}\left(I-\sum_{j=1}^{d}\mathfrak{z}_{j}A_{j}\right)^{-1}b.$

Here $A_{1},\ldots,A_{d}\in\mathbb{C}^{d}_{n}$ and $b,c\in\mathbb{C}^{n}$. The
evaluation of such a function on a tuple
$Z_{1},\ldots,Z_{d}\in\mathbb{C}^{d}_{m}$ is performed via tensor products,
i.e.,

$\mathfrak{r}(Z)=(I_{m}\otimes c)^{*}\left(I_{m}\otimes
I_{n}-\sum_{j=1}^{d}Z_{j}\otimes A_{j}\right)^{-1}(I_{m}\otimes b).$ (2.1)

One should note that this is not the original definition of an NC rational
function but rather a result in the sense that an NC rational function is
usually defined as a certain equivalence class of valid ‘NC rational
expressions’ obtained by applying the arithmetic operations ‘$+,\cdot$, and,
-1’, to the free algebra, $\mathbb{C}\langle\mathbb{\mathfrak{z}}\rangle$. One
can then prove that any such NC rational function in $\mathbb{C}_{0}\
\mathclap{\,<}{\left(\right.}\,\mathfrak{z}\,\mathclap{\,\,\,\,>}{\left.\right)}$
obeys a ‘realization formula’ as in Equation (2.1) above. Such a triple,
$(A,b,c)\in\mathbb{C}^{d}_{n}\times\mathbb{C}^{n}\times\mathbb{C}^{n}$ is
called a descriptor realization of the NC rational function $\mathfrak{r}$.
For every NC rational function, there exist many such descriptor realizations.
However, there exists one with $n$ minimal, the _minimal realization_. The
article “the” is justified because two realizations of $r$ with minimal $n$
are jointly similar. Namely, given two minimal realizations $(A,b,c)$ and
$(\widetilde{A},\widetilde{b},\widetilde{c})$, if

$\tilde{c}^{*}\left(I-\sum_{j=1}^{d}\mathfrak{z}_{j}\tilde{A}_{j}\right)^{-1}\tilde{b}=\mathfrak{r}(\mathfrak{z})=c^{*}\left(I-\sum_{j=1}^{d}\mathfrak{z}_{j}A_{j}\right)^{-1}b,$

then there exists $S\in\mathrm{GL}_{n}$, such that $\tilde{A}_{j}=S^{-1}AS$,
for $1\leq j\leq d$, $\tilde{b}=Sb$, and $\tilde{c}=S^{-1*}c$ [9, Theorem
2.4]. Moreover, a realization $(A,b,c)$ of size $n$ is minimal if and only if
$c$ is $A-$cyclic and $b$ is $A^{*}-$cyclic in the sense that

$\bigvee_{\omega\in{\mathbb{F}}^{d}}A^{\omega}c=\mathbb{C}^{n}=\bigvee
A^{*\omega}b,$

where ${\mathbb{F}}^{d}$ denotes the free monoid of all words in the $d$
letters $\\{1,\cdots,d\\}$. We will write
$L_{A}(\mathfrak{z}):=I-\sum_{j=1}^{d}\mathfrak{z}_{j}\tilde{A}_{j}$ whose
inverse appears in the above expression. This object, $L_{A}(Z)$, is called a
_linear pencil_ (it is affine linear). It is a result of Vinnikov–Kaliuzhnyi-
Verbovetskyi [34] and Volčič [50] that the domain of our function
$\mathfrak{r}$ can be described as the collection of all $Z$, such that $\det
L_{A}(Z)\neq 0$, where $A$ comes from the minimal realization.

NC rational functions in $\mathbb{H}^{2}_{d}$ and $\mathbb{H}^{\infty}_{d}$
were studied by Jury, and the authors in [31]. In fact,
$\mathfrak{r}\in\mathbb{H}^{2}_{d}$ if and only if the tuple
$A=(A_{1},\ldots,A_{d})$ appearing in its minimal realization has joint
spectral radius strictly less than $1$. Here, the joint spectral radius of $A$
was defined by Popescu [43] as a natural multivariate analogue of Beurling’s
spectral radius formula,

$\rho(A)=\lim_{n\to\infty}\left\|\sum_{|\alpha|=n}A^{\alpha}A^{\alpha*}\right\|^{\dfrac{1}{2n}}.$

It further follows, by Popescu’s multi-variable Rota–Strang theorem, that
$\mathfrak{r}\in\mathbb{H}^{2}_{d}$ if and only if it has a row ball of radius
strictly greater than $1$ in its domain [43]. In particular, if this is the
case, then $\mathfrak{r}\in\mathbb{A}_{d}\subset\mathbb{H}^{\infty}_{d}$. See
[31, Theorem A] for several characterizations equivalent to membership of an
NC rational function in the full Fock space.

Every NC rational contractive function can be associated (essentially)
uniquely to an NC Clark measure, i.e., a positive functional on
$\mathscr{A}_{d}$. Jury and the authors in [32] characterized such linear
functionals or ’NC rational Clark measures’. It turns out that the NC Clark
measures that arise from inner NC rational functions in
$\mathbb{H}^{\infty}_{d}$ with $\mathfrak{r}(0)=0$ are precisely the
finitely–correlated states studied by Bratteli and Jørgensen [10] and later by
Davidson, Kribs, and Shpigel [17]. Here, note that an NC Clark measure,
$\mu_{b}$, corresponding to any $b\in[\mathbb{H}^{\infty}_{d}]_{1}$ can be any
positive linear functional on the free disk system, and this functional will
be a state, _i.e._ $\mu_{b}(I)=1$, if and only if $b(0)=0$. Moreover, [32,
Theorem 4.1] provides a complete description of all NC rational inners in
terms of finite–dimensional row coisometries. Since we will apply this
description, we will recall it: Let $\mathfrak{b}$ be a contractive NC
rational function in the unit row-ball with $\mathfrak{b}(0)=0$. (Again,
$\mathfrak{b}(0)=0$ ensures that its NC Clark measure, $\mu_{\mathfrak{b}}$,
is a state.) Then, such an $\mathfrak{b}$ is inner if and only if there exists
a row coisometry
$T=(T_{1},\cdots,T_{d})\colon\mathbb{C}^{n}\otimes\mathbb{C}^{d}\to\mathbb{C}^{n}$
and a unit vector $x\in\mathbb{C}^{n}$, such that $x$ is $T$ and
$T^{*}-$cyclic and $\mathfrak{b}$ is given by the realization formula,

$\mathfrak{b}(\mathfrak{z})=(P_{0}x)^{*}\left(I-\sum_{j=1}^{d}\mathfrak{z}_{j}T_{0,j}^{*}\right)^{-1}\left(\sum_{j=1}^{d}z_{j}T_{j}^{*}x\right).$
(2.2)

Moreover, in this case,
$\mu_{\mathfrak{b}}(L^{\omega})=\mu_{T,x}(L^{\omega}):=x^{*}T^{*\omega}x$.
Here we set ${\mathcal{H}}_{0}=\bigvee_{\alpha\neq\emptyset}T^{*\alpha}x$,
$P_{0}$ is the orthogonal projection on ${\mathcal{H}}_{0}$, and for $1\leq
j\leq d$, $T_{0,j}^{*}=T_{j}^{*}(I-xx^{*})|_{{\mathcal{H}}_{0}}$. In
particular, if $T$ is irreducible (the co-ordinate matrices of $T$ generate
$\mathbb{C}^{n\times n}$ as an algebra), then every unit vector will give rise
to such a realization and ${\mathcal{H}}_{0}=\mathbb{C}^{n}$. This form of
realization is called a Fornasini–Marchesini (FM) realization. More generally,
an FM realization is one of the form

$\mathfrak{r}(\mathfrak{z})=D+C^{*}\left(I-\sum_{j=1}^{d}\mathfrak{z}_{j}A_{j}\right)^{-1}\left(\sum_{j=1}^{d}\mathfrak{z}_{j}B_{j}\right).$

Note that $D=\mathfrak{r}(0)$, so that if we assume that $\mathfrak{r}(0)=0$,
we obtain the preceding form of Equation (2.2). We will denote the FM
realization as $(A,B,C,D)$. One can pass from a descriptor realization to an
FM one quite easily. The following lemma is well-known to experts, but since
we do not have a reference, we chose to include it.

######  Lemma 2.1.

Let $\mathfrak{r}\in\mathbb{C}_{0}\
\mathclap{\,<}{\left(\right.}\,\mathfrak{z}\,\mathclap{\,\,\,\,>}{\left.\right)}$.
If $(A,B,C,D)$ is a minimal FM realization of $\mathfrak{r}$, then

$\mathrm{Dom}\,\mathfrak{r}=\left\\{Z\in\mathbb{C}^{d}_{\mathbb{N}}|\
\mathrm{det}\,L_{A}(Z)\neq 0\right\\}.$

###### Proof.

By [50, Theorem 3.5], the domain of $\mathfrak{r}$ is the complement of the
singularity locus of the linear pencil $L_{\hat{A}}(Z)$, where $(\hat{A},b,c)$
is a minimal descriptor realization of $\mathfrak{r}$. A minimal FM
realization, $(A^{\prime},B^{\prime},C^{\prime},D^{\prime})$ of $\mathfrak{r}$
can then be constructed by setting
${\mathcal{H}}^{\prime}:=\bigvee_{\omega\neq\emptyset}\hat{A}^{\omega}c$,
$A^{\prime}:=\hat{A}|_{{\mathcal{H}}^{\prime}}$, $B^{\prime}:=\hat{A}c$,
$C^{\prime}=(P_{{\mathcal{H}}^{\prime}}b)^{*}$ and
$D^{\prime}:=\mathfrak{r}(0)$. By uniqueness of minimal realizations
(uniqueness also holds for minimal FM realizations), we can assume that
$(A^{\prime},B^{\prime},C^{\prime},D^{\prime})=(A,B,C,D)$. Since
${\mathcal{H}}^{\prime}$ has codimension at most $1$, if
${\mathcal{H}}^{\prime}\subsetneq{\mathcal{H}}$, then $\hat{A}$ decomposes as

$\hat{A}=\begin{pmatrix}A&*\\\ 0&a\end{pmatrix},$

where $a\in\mathbb{C}^{d}$. Again, by [50, Theorem 3.5], if
$Z\in\mathrm{Dom}\,\mathfrak{r}$ then

$0\neq\mathrm{det}\,L_{\hat{A}}(Z)=\mathrm{det}(L_{A}(Z))\mathrm{det}\,L_{a}(Z),$

so that $\mathrm{det}\,L_{A}(Z)\neq 0$. Conversely if $\mathrm{det}\,L_{A}(Z)$
is not $0$ for some $Z\in\mathbb{C}^{d}_{n}$, then $\mathfrak{r}(Z)$ is well-
defined as the transfer function,

$DI_{n}+I_{n}\otimes CL_{A}(Z)^{-1}Z\otimes B,$

so that $Z\in\mathrm{Dom}\,\mathfrak{r}$. ∎

If $\mathfrak{r}\in\mathbb{C}_{0}\
\mathclap{\,<}{\left(\right.}\,\mathfrak{z}\,\mathclap{\,\,\,\,>}{\left.\right)}$
with minimal descriptor realization $(A,b,c)$, we will allow the domain of
$\mathfrak{r}$ to include $d-$tuples of operators in an infinite dimensional
Hilbert space. We will denote such operator $d-$tuples by
$\mathbb{C}^{d}_{\infty}$ or $\mathscr{B}({\mathcal{H}})^{d}$, where
${\mathcal{H}}$ is a separable Hilbert space. Namely, given
$Z=(Z_{1},\cdots,Z_{d})\in\mathbb{C}^{d}_{\infty}$, we will say that
$Z\in\mathrm{Dom}\,\mathfrak{r}$ if $L_{A}(Z)$ is invertible. It will be
convenient to introduce some basic notations; we view any
$Z\in\mathbb{C}^{d}_{n}$, $n\in\mathbb{N}$ as a row $d-$tuple of operators,
$Z=(Z_{1},\cdots,Z_{d})$, $Z_{j}\in\mathbb{C}^{n\times n}$. Any such row
defines a bounded linear map from $\mathbb{C}^{n}\otimes\mathbb{C}^{d}$ into
$\mathbb{C}^{n}$. Hence, by $Z^{*}$, we mean the ‘column operator’,
$Z^{*}:=\left(\begin{smallmatrix}Z_{1}^{*}\\\ \vdots\\\
Z_{d}^{*}\end{smallmatrix}\right):\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}\otimes\mathbb{C}^{d}$,
obtained as the Hilbert space adjoint of the linear map $Z$. Similarly,
$Z^{\mathrm{t}}:=\left(\begin{smallmatrix}Z_{1}^{\mathrm{t}}\\\ \vdots\\\
Z_{d}^{\mathrm{t}}\end{smallmatrix}\right)$ denotes the transpose of the row
operator, $Z$, with respect to the standard bases of $\mathbb{C}^{n}$ and
$\mathbb{C}^{n}\otimes\mathbb{C}^{d}$. We will, however, also have occasion to
consider the row operator
$\mathrm{row}(Z^{*}):=(Z_{1}^{*},\cdots,Z_{d}^{*}):\mathbb{C}^{n}\otimes\mathbb{C}^{d}\rightarrow\mathbb{C}^{n}$
obtained as the component–wise adjoint of $Z$. The row operator
$\mathrm{row}(Z^{\mathrm{t}})$ is defined similarly. We will also consider the
operation $\overline{(\cdot)}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$,
defined by $x\,\mapsto\,\overline{x}$, where $\overline{x}$ denotes entry–wise
complex conjugation with respect to the standard basis of $\mathbb{C}^{n}$. If
$Z=(Z_{1},\cdots,Z_{d})\in\mathbb{C}^{d}_{n}$, we define
$\overline{Z}:=(\overline{Z}_{1},\cdots,\overline{Z}_{d})$, where
$\overline{Z}_{j}:=\overline{(\cdot)}\circ Z_{j}\circ\overline{(\cdot)}$, so
that $\overline{Z}_{j}$ is obtained by entry–wise complex conjugation of the
matrix $Z_{j}$ and $\overline{Z}=(Z^{*})^{\mathrm{t}}=(Z^{\mathrm{t}})^{*}$ is
a row $d-$tuple of matrices.

The following lemmas give us a useful condition on the spectra of NC rational
functions with the origin in their domains.

######  Lemma 2.2.

Let ${\mathcal{H}}$ and ${\mathcal{K}}$ be Hilbert spaces. Let $A\in
B({\mathcal{H}})$ and $D\in B({\mathcal{K}})$ be invertible and $B\in
B({\mathcal{K}},{\mathcal{H}})$ and $C\in B({\mathcal{H}},{\mathcal{K}})$.
Then, the Schur complement, $A-BD^{-1}C$, has a non-trivial kernel if and only
if $D-CA^{-1}B$ has a non-trivial kernel. Moreover, the map,
$D^{-1}C|_{\ker(A-BD^{-1}C)}\colon\ker(A-BD^{-1}C)\to\ker(D-CA^{-1}B)$, is an
isomorphism.

###### Proof.

Since the claim is symmetric, we will prove only the forward implication. Let
us assume that $0\neq v\in\ker(A-BD^{-1}C)$. Then, $0\neq Av=BD^{-1}Cv$. In
particular, $D^{-1}Cv\neq 0$. Therefore,

$(D-CA^{-1}B)D^{-1}Cv=Cv-CA^{-1}BD^{-1}Cv=CA^{-1}(Av-BD^{-1}Cv)=0.$

To prove the last part of the claim, we let
$\Phi=D^{-1}C|_{\ker(A-BD^{-1}C)}$. As we saw above, $\Phi$ is well-defined
and injective. Similarly, let $\Psi=A^{-1}B|_{\ker(D-CA^{-1}B)}$. Then, for
every $v\in\ker(A-BD^{-1}C)$, we have that

$\Psi\Phi v=A^{-1}BD^{-1}Cv=A^{-1}Av=v.$

Similarly, in the other direction. Hence, we obtain our isomorphism. ∎

######  Proposition 2.3.

Let $\mathfrak{r}\in\mathbb{C}_{0}\
\mathclap{\,<}{\left(\right.}\,\mathfrak{z}\,\mathclap{\,\,\,\,>}{\left.\right)}$
with minimal FM realization $(A,B,C,D)$. If $Z\in\mathrm{Dom}\,\mathfrak{r}$
then $\lambda\neq\mathfrak{r}(0)$ is an eigenvalue of $\mathfrak{r}(Z)$ if and
only if $1$ is an eigenvalue of $Z\otimes A^{(\lambda)}$, where

$A^{\lambda}_{j}:=A_{j}+(\lambda-\mathfrak{r}(0))^{-1}B_{j}C.$

If $v$ is an eigenvector of $Z\otimes A^{(\lambda)}$ corresponding to the
eigenvalue $1$, then $I\otimes Cv$ is an eigenvector of $\mathfrak{r}(Z)$ with
corresponding eigenvalue $1$.

###### Proof.

Consider the following matrix

$\begin{pmatrix}(\lambda-\mathfrak{r}(0))I&I\otimes C^{*}\\\
\sum_{j=1}^{d}Z_{j}\otimes B_{j}&I-\sum_{j=1}^{d}Z_{j}\otimes
A_{j}\end{pmatrix}.$

Note that one Schur complement is

$\lambda-\mathfrak{r}(0)I-(I\otimes C^{*})\left(I-\sum_{j=1}^{d}Z_{j}\otimes
A_{j}\right)^{-1}\left(\sum_{j=1}^{d}Z_{j}\otimes
A_{j}\right)=\lambda-\mathfrak{r}(Z).$

The other is

$I-\sum_{j=1}^{d}Z_{j}\otimes
A_{j}-\frac{1}{\lambda-\mathfrak{r}(0)}\sum_{j=1}^{d}Z_{j}\otimes
B_{j}C^{*}=I-\sum_{j=1}^{d}Z_{j}\otimes A^{(\lambda)}_{j}.$

The claim now follows from the preceding lemma. ∎

The previous proposition extends and refines [32, Proposition 5.5]. In [32,
Proposition 5.6], we applied these spectral results to prove that any NC
rational inner, $\mathfrak{b}$, has eigenvalues of modulus $1$ when evaluated
at certain points on the boundary of the unit row-ball,
$\mathbb{B}^{d}_{\mathbb{N}}$. Here, recall that NC rational inner multipliers
of the Fock space, $\mathfrak{b}$, are in bijective correspondence with pairs
$(T,x)$, where $T\in\mathbb{C}^{d}_{n}$ is a row coisometry and $x$ is any
vector which is both $T$ and $T^{*}$ cyclic. In particular, if
$\mathfrak{b}=\mathfrak{b}_{T,x}$ is the unique NC rational inner
corresponding to such a pair, then for any $\zeta\in\partial\mathbb{D}$, the
NC rational inner $\overline{\zeta}\mathfrak{b}$ corresponds to a pair
$(T(\zeta),x)$, where $T(\zeta)$ is a rank–one row coisometric perturbation of
$T$, see Equation (2.2) and [32, Theorem 4.1]. Proposition 5.6 of [32] then
states:

######  Proposition.

Given $\zeta\in\partial\mathbb{D}$, let $A_{\zeta}^{*}$ be a column–isometric
restriction of $T(\zeta)^{*}$ to an invariant subspace. Then $\zeta$ is an
eigenvalue of $\mathfrak{b}_{T,x}(A_{\zeta}^{\mathrm{t}})$.

There is a minor error in the proof of [32, Proposition 5.6]. Fortunately, the
proposition statement is still correct, and the proof can be readily fixed:

###### Proof.

([32, Proposition 5.6]) We have that
$\zeta\in\partial\mathbb{D}\cap\sigma(\mathfrak{b}(Z))$ if and only if

$\mathrm{det}\left(I\otimes I-\sum_{j=1}^{d}Z_{j}\otimes
T(\zeta)_{j}^{*}\right)=0.$

Taking adjoints, this happens if and only if

$\mathrm{det}\left(I\otimes I-\sum_{j=1}^{d}Z_{j}^{*}\otimes
T(\zeta)_{j}\right)=0.$

By vectorization, this happens if and only if there is a matrix, $X$, so that

$X-\sum_{j=1}^{d}T(\zeta)_{j}X\overline{Z}_{j}=0.$

If

$T(\zeta)^{*}=:\begin{pmatrix}A^{*}&B^{*}\\\ &C^{*}\end{pmatrix},$

then take

$Z=\begin{pmatrix}A^{\mathrm{t}}&0\\\ 0&0\end{pmatrix},$

and if $A\in\mathbb{C}^{d}_{k}$, let $X=I_{k}$ so that

$\displaystyle X-T(\zeta)X\overline{Z}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}I_{k}&0\\\ 0&0\end{pmatrix}-\begin{pmatrix}A&0\\\
B&C\end{pmatrix}\begin{pmatrix}I_{k}&0\\\
0&0\end{pmatrix}\begin{pmatrix}A^{*}&0\\\ 0&0\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}I_{k}&0\\\
0&0\end{pmatrix}-\begin{pmatrix}AA^{*}&0\\\
BA^{*}&0\end{pmatrix}=\begin{pmatrix}I_{k}&0\\\
0&0\end{pmatrix}-\begin{pmatrix}I_{k}&0\\\ 0&0\end{pmatrix}=0.$

Here we have used the fact that $T(\zeta)$ is a row coisometry. ∎

###  2.3 Peaking states and representations

In this section we follow the work of Clouâtre and Thompson in non-commuative
convexity in operator algebra and operator system theory [13]. Let $A$ be a
unital operator algebra and let $B$ be a $C^{*}$-cover of $A$. Namely, we have
a unital, completely isometric embedding $\iota\colon A\to B$, such that
$B=C^{*}(\iota(A))$. Let us denote by $K(B)$ the state space of $B$. Clouâtre
and Thompson say that $\mu\in K(B)$ is _$A-$ peaking_, or an _$A-$ peak
state_, if there exists a contraction $a\in A$, such that $\mu(a^{*}a)=1$ and
$\nu(a^{*}a)<1$, for all $\nu\in K(B)\setminus\\{\mu\\}$. If $\mu$ is
$A$-peaking, then it is quite immediate that $\mu$ is pure and $\mu$ has the
unique extension property (UEP), i.e., the functional $\mu|_{A}$ has a unique
Hahn-Banach extension to $B$. In fact, any
$\mu\in(\mathscr{A}_{d}^{\dagger})_{+}$ with the property that its GNS row
isometry is a Cuntz row isometry has the UEP by [33, Proposition 5.11]. If
$\pi_{\mu}\colon B\to B({\mathcal{H}}_{\mu})$ is a GNS representation of $\mu$
with cyclic vector $\xi_{\mu}$, then, by Cauchy-Schwarz,
$\pi_{\mu}(a)\xi_{\mu}=\xi_{\mu}$. This implies that there is a finite-
dimensional subspace $F\subset{\mathcal{H}}_{\mu}$, such that
$\|P_{F}\pi_{\mu}(a)|_{F}\|=1$, where $P_{F}$ is the projection onto $F$.
Similarly, in [11], if $B$ is a $C^{*}-$algebra and $\mathscr{S}\subseteq B$
is an operator system, a state, $\mu\in K(B)$ is said to be
$\mathscr{S}-$peaking if there is a self-adjoint element $s\in\mathscr{S}$ so
that $\|s\|=1$ and $|\lambda(s)|<\mu(s)=1$ for every $\lambda\in
K(B)\setminus\\{\mu\\}$.

An irreducible representation $\pi\colon B\to B({\mathcal{H}}_{\pi})$ is
called local $A$-peak by Clouâtre and Thompson, if there exists
$n\in\mathbb{N}$, $T\in M_{n}(A):=A\otimes M_{n}$ with $\|T\|=1$, such that
for every irreducible representation $\sigma\colon B\to
B({\mathcal{H}}_{\sigma})$ unitarily inequivalent to $\pi$ and every finite-
dimensional subspace $G\subset{\mathcal{H}}_{\sigma}^{\oplus n}$,
$\|P_{G}\sigma^{(n)}(T)|_{G}\|<1$. This implies, that there is a finite-
dimensional subspace $F\subset{\mathcal{H}}_{\pi}^{\oplus n}$, such that
$\|P_{F}\pi^{(n)}(T)|_{F}\|=1$ [13, Lemma 2.2]. By [13, Theorem 2.7], if
$\mu\in K(B)$ is $A$-peaking, then $\pi_{\mu}$ is local $A$-peak. The
converse, however, is false [13, Example 2]. One can say slightly more about
the connection between the two notions. This is the goal of the following two
lemmas.

######  Lemma 2.4.

Let $B$ be a unital $C^{*}$-algebra and $A\subset B=C^{*}(A)$ be a unital
operator algebra. Let $\pi\colon B\to B({\mathcal{H}})$ be a representation,
such that there exists a contraction $a\in A$ and a finite-dimensional
subspace $G\subset{\mathcal{H}}$ with $\|P_{G}\pi(a)|_{G}\|=1$. Then, there
exists a vector $\xi\in G$, such that $\pi(a^{*}a)\xi=\xi$.

###### Proof.

Let $\varphi$ be the ucp $\varphi(b)=P_{G}\pi(b)|_{G}\in B(G)$. By
Kadison–Schwarz, $\varphi(a)^{*}\varphi(a)\leq\varphi(a^{*}a)$. Hence,

$1=\|\varphi(a)\|^{2}\leq\|\varphi(a^{*}a)\|\leq 1$

In particular, $\|\varphi(a^{*}a)\|=1$ and since it is a selfadjoint operator
on a finite-dimensional space, there exists $\xi\in G$, such that
$\varphi(a^{*}a)\xi=\xi$. Since $\pi(a^{*}a)$ is a contraction, we have that
$\pi(a^{*}a)\xi=\xi$. ∎

######  Lemma 2.5.

Let $B$ be a unital $C^{*}$-algebra and $1\in A\subset B$ an operator algebra,
such that $B=C^{*}(A)$. Let $a\in A$ be a contraction and set

$F_{a}=\left\\{\mu\in K(B)\mid\mu(a^{*}a)=1\right\\}.$

Then, $F_{a}$ is a closed face of $K(B)$. Moreover, if $\pi\colon A\to
B({\mathcal{H}})$ is local $A$-peak with witness $a$, then
$F_{a}\neq\emptyset$ and all the extreme points of $F_{a}$ are states arising
from $\pi$. Lastly, $\partial_{e}F_{a}$ is in one to one correspondence with
vectors $\eta\in{\mathcal{H}}$, such that $\pi(a^{*}a)\eta=\eta$.

###### Proof.

Since $a^{*}a$ defines a weak*-continuous contractive real functional on
$(B_{sa})^{*}$, then $F_{a}$ is the closed face where this functional attains
its maximum on $K(B)$. Let $\pi$ be a local $A$-peak representation with
witness $a$. Then, by the previous lemma, there exists $\xi\in G$, such that
$\pi(a^{*}a)\xi=\xi$. Hence, the state $\mu(b)=\langle\xi,\pi(b)\xi\rangle$ is
in $F_{a}$. Since $F_{a}$ is a closed face, its extreme points are pure
states. Let $\varphi\in F_{a}$ be a pure state and
$(\sigma,{\mathcal{K}},\eta)$ be its GNS representation. Then, it is immediate
that $\sigma(a^{*}a)\eta=\eta$. Thus, if $G^{\prime}\subset{\mathcal{K}}$ is
the space spanned by $\eta$ and $\sigma(a)\eta$, then
$\|P_{G^{\prime}}\sigma(a)|_{G^{\prime}}\|=1$ and thus $\sigma$ is unitarily
equivalent to $\pi$ ∎

In this paper, we are interested in the case when $A=\mathbb{A}_{d}$ and
$B={\mathcal{O}}_{d}$. Since $\mathbb{A}_{d}$ is semi-Dirichlet, we can say
more about $\mathbb{A}_{d}$ -peak states and connect them to peaking states
for operator systems.

######  Lemma 2.6.

If $\mu\in K({\mathcal{O}}_{d})$ is an $\mathbb{A}_{d}$-peak state, then there
exists $b\in\mathscr{A}_{d}$ positive, such that $\mu(b)=1$ and $\nu(b)<1$,
for all states $\nu\neq\mu$. In particular, $\mu$ is an $\mathscr{A}_{d}-$peak
state, $\mu|_{\mathscr{A}_{d}}$ is a weak* exposed extreme point of the unit
ball of $({\mathcal{S}}_{d}^{*})_{sa}$ and it has the unique extension
property.

###### Proof.

Let $\mu\in K({\mathcal{O}}_{d})$ be an $\mathbb{A}_{d}$-peak state. Then,
there is a contraction $a\in\mathscr{A}_{d}$, such that $\mu(a^{*}a)=1$ and
for every state $\nu\neq\mu$, $\nu(a^{*}a)<1$. However, since $A$ is semi-
Dirichlet, $a^{*}a\in\mathscr{A}_{d}$. Hence, we are done. The second part
follows from [11, Theorem 3.2]. ∎

One could try and argue the converse claim. Let $b\in\mathscr{A}_{d}$ be a
positive element, such that $\mu(b)=1$ and for every $\nu\in
K({\mathcal{O}}_{d})\setminus\\{\mu\\}$, $\nu(b)<1$. By replacing $b$ by
$\frac{1}{2}(1+b)$, we may assume that $b$ is invertible. Thus, $b$ is
factorizable in the sense of Popescu [38]. Namely, there exists
$c\in\mathbb{H}^{\infty}_{d}$, such that $b=c^{*}c$. However, we do not know
that $c\in\mathbb{A}_{d}$. This is true if we assume that $b$ is the real part
of an NC rational function. In this case, $c$ is NC rational by the NC
rational Fejér–Riesz theorem of [30, Theorem 6.5]. However, we do not need
this result for our purposes and only require the following observation to
construct a large class of examples of $\mathbb{A}_{d}$-peaking states.

######  Lemma 2.7.

Let $b\in\mathbb{A}_{d}$ be inner. Let $\mu\in K({\mathcal{O}}_{d})$ be such
that $\mu(b)=1$ and for all $\nu\in K({\mathcal{O}}_{d})\setminus\\{\mu\\}$,
$|\nu(b)|<1$. Then, $\mu$ is $\mathbb{A}_{d}-$peaking.

###### Proof.

Since $b$ is inner, we have that:

$\frac{1}{4}(1+b^{*})(1+b)=\frac{1}{4}(2+2\mathrm{Re}(b))=\frac{1}{2}(1+\mathrm{Re}(b)).$

Let $a=\frac{1}{2}(1+\mathrm{Re}(b))$, then $a\geq 0$ and $\mu(a)=1$.
Moreover, if $\nu\in K({\mathcal{O}}_{d})\setminus\\{\mu\\}$ is such that
$\nu(a)=1$, then $\mathrm{Re}\nu(b)=1$. However, this contradicts the
assumption that $|\nu(b)|<1$. ∎

The above observation suggests the following strategy: To construct examples
of states on the Cuntz algebra which peak on the free disk algebra, we will
show that if $\mathfrak{b}\in\mathbb{H}^{\infty}_{d}$ is NC rational and
inner, then there are finite points, $A$, on the boundary of the unit row-ball
so that $\mathfrak{b}(A)$ has $1$ as an eigenvalue of multiplicity one.

##  3 Main result

A relationship between spectra and intertwiners is encoded in the following
lemmas. Let $Z\in\mathbb{C}^{d}_{n}$ be an irreducible row contraction of row
norm $1$. Let $Y\in B({\mathcal{H}})^{d}=\mathbb{C}^{d}_{\infty}$ be another
row contraction on a separable Hilbert space. Let $\psi_{Y,Z}\colon
B(\mathbb{C}^{n},{\mathcal{H}})\to B(\mathbb{C}^{n},{\mathcal{H}})$ be the map
$\psi_{Y,Z}(T)=\sum_{j=1}^{d}Y_{j}TZ_{j}^{*}$. Assume that
$1\in\sigma_{p}(\psi_{Y,Z})$. Then, there exists $0\neq T\in
B(\mathbb{C}^{n},{\mathcal{H}})$, such that $\psi_{Y,Z}(T)=T$. We may assume
that $\|T\|=1$. Then, since $n<\infty$, there is a unit vector
$v\in\mathbb{C}^{n}$, such that $\|Tv\|=1$ and hence

$1=\|Tv\|^{2}=\langle\psi_{Y,Z}(T)v,Tv\rangle=\langle(I\otimes
T)Z^{*}v,Y^{*}Tv\rangle.$

Since $T$, $Z$, and $Y$ are contractions, we conclude from the equality clause
of Cauchy-Schwarz that $Y^{*}Tv=(I\otimes T)Z^{*}v$. Therefore, for every
$1\leq n\leq d$. The same argument can be applied to the self-compositions
$\psi_{Y,Z}^{\circ n}$ to conclude that for every
$p\in\mathbb{C}\langle\mathbb{\mathfrak{z}}\rangle$, $Tp(Z^{*})v=p(Y^{*})Tv$.
Now since $Z$ is irreducible, so is $Z^{*}$. This implies that $v$ is cyclic.
Therefore, for every $w\in\mathbb{C}^{n}$, there exists an NC polynomial $p$,
such that $p(Z^{*})v=w$. Hence, for every $1\leq j\leq d$,

$TZ_{j}^{*}w=TZ_{j}^{*}p(Z^{*})v=Y_{j}^{*}p(Y^{*})Tv=Y_{j}^{*}Tp(Z^{*})v=Y_{j}^{*}Tw.$

We conclude that $T$ is a homomorphism of
$\mathbb{C}\langle\mathbb{\mathfrak{z}}\rangle$-modules from $\mathbb{C}^{n}$
to ${\mathcal{H}}$. In particular, since the kernel of a homomorphism is a
submodule. $T$ must be injective. Thus, we have obtained the following lemma:

######  Lemma 3.1.

Let $Z\in\mathbb{C}^{d}_{n}$ be an irreducible row contraction of row norm $1$
and $Y\in B({\mathcal{H}})^{d}$ a row contraction, such that
$1\in\sigma_{p}(\psi_{Y,Z})$. Then, there exists a $Y$-coinvariant
$n$-dimensional subspace ${\mathcal{K}}\subset{\mathcal{H}}$, such that
$Y^{*}|_{{\mathcal{K}}}$ is similar to $Z^{*}$.

We can do slightly better, assuming that both $Z$ and $Y$ are coisometries.
Under the assumptions of the lemma, let $T$ be the intertwinner obtained
above. Then,

$Z(I\otimes T^{*}T)Z^{*}=T^{*}YY^{*}T=T^{*}T.$

However, since $Z$ is irreducible, a result of Farenick [24, Theorem 2]
implies that the map $A\mapsto Z(I\otimes A)Z^{*}$ is irreducible and thus, by
the Perron-Frobenius theorem for positive maps of Evans and Høegh-Krohn [23,
Theorem 2.3], we know that there is a unique (up to scalar multiplication)
eigenvector of this map that corresponds to eigenvalue $1$. However, since $Z$
is a coisometry, the corresponding map is a ucp. Hence, $T^{*}T$ is a scalar
multiple of the identity of norm $1$. Therefore, $T^{*}T=I$ and $T$ is an
isometry. We summarize

######  Lemma 3.2.

Let $Z\in\mathbb{C}^{d}_{n}$ be an irreducible row coisometry and let $Y\in
B({\mathcal{H}})^{d}$ be a row coisometry. If $1\in\sigma_{p}(\psi_{Y,Z})$,
then there exists a unique isometry $V\colon\mathbb{C}^{n}\to{\mathcal{H}}$,
such that $Y^{*}V=(I\otimes V)Z^{*}$.

Note that we can canonically identify $B(\mathbb{C}^{n},{\mathcal{H}})$ with
$(\mathbb{C}^{n})^{*}\otimes{\mathcal{H}}$. The identification sends
$\varphi\otimes\xi$ to the linear map $v\mapsto\varphi(v)\xi$. Now let $Y\in
B({\mathcal{H}})^{d}$ and $X\in\mathbb{C}^{d}_{n}$. Then, we can define an
operator $\sum_{j=1}^{d}X_{j}^{t}\otimes Y_{j}\in
B((\mathbb{C}^{n})^{*}\otimes{\mathcal{H}})$ acting via

$\left(\sum_{j=1}^{d}X_{j}^{t}\otimes
Y_{j}\right)(\varphi\otimes\xi)=\sum_{j=1}^{d}(\varphi\circ X_{j})\otimes
Y_{j}\xi.$

Now via identification, the right-hand map is

$\left(\sum_{j=1}^{d}(\varphi\circ X_{j})\otimes
Y_{j}\xi\right)v=\sum_{j=1}^{d}\varphi(X_{j}v)Y_{j}\xi=\left(\sum_{j=1}^{d}Y_{j}(\varphi\otimes\xi)X_{j}\right)v.$

Therefore, we get that the map $\psi_{Y,X}$ corresponds to the product
$\sum_{j=1}^{d}\overline{X_{j}}\otimes Y_{j}$. Hence,
$1\in\sigma_{p}(\psi_{Y,X})$ if and only if
$1\in\sigma_{p}\left(\sum_{j=1}^{d}\overline{X_{j}}\otimes Y_{j}\right)$,

Assume that $T\in\mathbb{C}^{d}_{n}$ is an irreducible row co-isometry. By
[32, Lemma 2.1], $\mathrm{row}(T^{\mathrm{t}})$ has joint spectral radius $1$
and by [44, Lemma 4.10], $\mathrm{row}(T^{\mathrm{t}})$ is jointly similar to
an irreducible row co-isometry, $Z\in\mathbb{B}^{d}_{n}$. Fix a unit vector,
$x\in\mathbb{C}^{n}$. Then there is a unique NC rational inner function,
$\mathfrak{b}=\mathfrak{b}_{T,x}$, corresponding to the pair $(T,x)$ by [32,
Theorem 4.1]. Let $S\in\mathrm{GL}_{n}$ be an invertible matrix, such that
$S^{-1}T^{t}S=Z$. Since $\mathfrak{b}$ is NC, we have that

$\mathfrak{b}(Z)=\mathfrak{b}(S^{-1}T^{t}S)=S^{-1}\mathfrak{b}(T^{t})S.$

In particular, the dimension of the eigenspace corresponding to $1$ of
$\mathfrak{b}(T^{t})$ and $\mathfrak{b}(Z)$ are the same.

######  Lemma 3.3.

In the above setting, $\mathfrak{b}(Z)$ has an eigenspace of dimension $1$
corresponding to eigenvalue $1$.

###### Proof.

Consider the map $\psi_{T}(X)=\sum_{j=1}^{d}T_{j}XT_{j}^{*}$. This map
corresponds to the tensor $\sum_{j=1}^{d}\overline{T_{j}}\otimes T_{j}$. Since
the map is unital, we have that
$1\in\sigma\left(\sum_{j=1}^{d}\overline{T_{j}}\otimes T_{j}\right)$.
Moreover, $1$ is the Perron-Frobenius eigenvalue of $\psi_{T}$, and since $T$
is irreducible, the dimension of the corresponding eigenspace is $1$. Taking
the adjoint, we get the tensor $\sum_{j=1}^{d}T_{j}^{t}\otimes T_{j}^{*}$. By
Lemma 2.3, we know that the dimension of the eigenspace corresponding to $1$
of $\mathfrak{b}(T^{t})$ is $1$, as well. The claim for $Z$ now follows from
the observation preceding the lemma. ∎

Let $y$ be a unit vector in this one-dimensional eigenspace of
$\mathfrak{b}(Z)$ to eigenvalue $1$ and define the finitely–correlated Cuntz
state $\mu=\mu_{Z,y}\in(\mathscr{A}_{d}^{\dagger})_{+}$ by
$\mu(L^{\omega}):=y^{*}Z^{\omega}y$. Let $V$ denote the minimal row isometric
dilation of $Z$ on ${\mathcal{H}}\supseteq\mathbb{C}^{n}$. By [17, Theorem
6.5], $V$ is an irreducible Cuntz row isometry. Given any
$\mu\in(\mathscr{A}_{d}^{\dagger})_{+}$, we can, as in [32], apply a
Gelfand–Naimark–Segal (GNS) construction to $(\mu,\mathbb{A}_{d})$ to obtain a
GNS-Hilbert space, $\mathbb{H}^{2}_{d}(\mu)$, and a row isometry, $\Pi_{\mu}$,
acting on $\mathbb{H}^{2}_{d}(\mu)$, where $\mathbb{H}^{2}_{d}(\mu)$ is
defined as the completion of the free algebra,
$\mathbb{C}\langle\mathbb{\mathfrak{z}}\rangle$, modulo vectors of
zero–length, with respect to the pre-inner product,

$\langle{p},{q}\rangle_{\mu}:=\mu(p(L)^{*}q(L)).$

Equivalence classes of free polynomials, $p+N_{\mu}$, where $N_{\mu}$ denotes
the left ideal of zero-length vectors with respect to $\|\cdot\|_{\mu}$, are
dense in $\mathbb{H}^{2}_{d}(\mu)$ by construction. This construction also
comes equipped with a row isometry, $\Pi_{\mu}$, defined by left
multiplications by the independent variables on $\mathbb{H}^{2}_{d}(\mu)$,
$\Pi_{\mu;j}p+N_{\mu}:=\mathfrak{z}_{j}p+N_{\mu}$. Also note that defining
$\pi_{\mu}(L_{k}):=\Pi_{\mu;k}$, $1\leq k\leq d$, yields a $*-$representation
of the Cuntz–Toeplitz algebra, $\mathscr{E}_{d}$, and vice versa. Moreover,
$\Pi_{\mu}$ is a Cuntz (surjective) row isometry if and only if
$\mu\in(\mathscr{A}_{d}^{\dagger})_{+}$ has a unique positive extension to
${\mathcal{O}}_{d}$ by [33, Proposition 5.11]. Hence, any
$\mu\in(\mathscr{A}_{d}^{\dagger})_{+}$ with GNS row isometry $\Pi_{\mu}$, of
Cuntz type can be uniquely identified with a positive state, $\hat{\mu}$, on
${\mathcal{O}}_{d}$, and applying the GNS construction to
$(\mu,\mathbb{A}_{d})$ or to $(\hat{\mu},{\mathcal{O}}_{d})$ yields the same
GNS-Hilbert space, $\mathbb{H}^{2}_{d}(\mu)$, and the same Cuntz
representation, $\pi_{\mu}$.

By [32, Proposition 3.6], if $\mu=\mu_{Z,y}$ and we define

${\mathcal{H}}_{\mu}:=\bigvee\Pi_{\mu}^{*\omega}1+N_{\mu},\quad\quad
Z_{\mu}:=(\Pi_{\mu}^{*}|_{{\mathcal{H}}_{\mu}})^{*},$

then the pair $(Z,V)$ are jointly unitarily equivalent to
$(Z_{\mu},\Pi_{\mu})$ by a unitary which sends $y$ to $1+N_{\mu}$. In
particular, ${\mathcal{H}}_{\mu}$ is finite–dimensional, $Z_{\mu}$ is an
irreducible row co-isometry, and $\Pi_{\mu}$ is its minimal row isometric
dilation, and this is irreducible and Cuntz.

######  Lemma 3.4.

Let $\pi\colon{\mathcal{O}}_{d}\to B({\mathcal{H}})$ be a representation so
that $1\in\sigma_{p}(\pi(\mathfrak{b}))$. Then, there exists a unique isometry
$V\colon\mathbb{C}^{n}\to{\mathcal{H}}$, such that $\pi(S)^{*}V=(I\otimes
V)Z^{*}$.

###### Proof.

Since $\xi=\pi(\mathfrak{b})\xi=\mathfrak{b}(\pi(S))\xi$, by Proposition 2.3,
we have that $1\in\sigma_{p}\left(\sum_{j=1}^{d}\pi(S_{j})\otimes
T_{j}^{*}\right)$. Now applying the interchange unitary, we have that
$1\in\sigma_{p}\left(\sum_{j=1}^{d}T_{j}^{*}\otimes\pi(S_{j})\right)$. The
latter corresponds to the map $\psi\colon B(\mathbb{C}^{n},{\mathcal{H}})\to
B(\mathbb{C}^{n},{\mathcal{H}})$ given by
$\psi(X)=\sum_{j=1}^{d}\pi(S_{j})X\overline{T_{j}}$. We note that
$\bar{T}=(T^{t})^{*}$. However, we also know that
$\overline{T_{j}}=S^{*}Z_{j}^{*}S^{-1*}$. Thus,

$\psi(X)=\sum_{j=1}^{d}\pi(S_{j})X\overline{T_{j}}=(I\otimes
S^{*})\left(\sum_{j=1}^{d}\pi(S_{j})XZ_{j}^{*}\right)(I\otimes S^{-1*}).$

Hence, if we set $\varphi(X)=\sum_{j=1}^{d}\pi(S_{j})XZ_{j}^{*}$. Then,
$1\in\sigma_{p}(\varphi)$. Therefore, by Lemma 3.2, we get that there exists a
unique isometry $V\colon\mathbb{C}^{n}\to{\mathcal{H}}$, such that
$\pi(S)^{*}V=(I\otimes V)Z^{*}$. ∎

######  Theorem 3.5.

Let $T\in\mathbb{C}^{d}_{n}$ be an irreducible row co-isometry and
$x\in\mathbb{C}^{n}$ a unit vector so that $\mathfrak{b}=\mathfrak{b}_{T,x}$
is the unique NC rational inner function corresponding to $(T,x)$. Let $Z$ be
the irreducible row co-isometry which is jointly similar to
$\mathrm{row}(T^{\mathrm{t}})$ via an invertible matrix, $S$, and let
$y\in\mathbb{C}^{n}$ be the unique unit eigenvector of $\mathfrak{b}(Z)$ to
eigenvalue $1$. Then the finitely–correlated Cuntz state
$\mu:=\mu_{Z,y}\in(\mathscr{A}_{d}^{\dagger})_{+}$ is an $\mathbb{A}_{d}-$peak
state which peaks at $\mathfrak{b}\in\mathbb{A}_{d}$.

###### Proof.

As described above, since $\mu(L^{\omega}):=y^{*}Z^{\omega}y$ and
$\mathfrak{b}(Z)y=y$, where the unit vector, $y$, spans the eigenspace for
$\mathfrak{b}(Z)$ corresponding to eigenvalue $1$, it follows that
$\mu(\mathfrak{b})=y^{*}\mathfrak{b}(Z)y=1$. Note that $\mu$ is a pure Cuntz
state since $\pi_{\mu}$ is an irreducible representation of the Cuntz algebra.
By the equality in the Cauchy–Schwarz inequality, it follows as before that
$\pi_{\mu}(\mathfrak{b})1+N_{\mu}=\mathfrak{b}(\Pi_{\mu})1+N_{\mu}=1+N_{\mu}$.
Assume that $h\in\mathbb{H}^{2}_{d}(\mu)$ is any other element such that
$\mathfrak{b}(\Pi_{\mu})h=h$. Note that ${\mathcal{H}}_{\mu}$ is
$\Pi_{\mu}-$co-invariant by construction so that ${\mathcal{H}}_{\mu}^{\perp}$
is invariant. Consider the block decomposition of $\Pi_{\mu}$ and $h$ with
respect to
$\mathbb{H}^{2}_{d}(\mu)={\mathcal{H}}_{\mu}\oplus{\mathcal{H}}_{\mu}^{\perp}$:

$\mathfrak{b}(\Pi_{\mu})h=\begin{pmatrix}\mathfrak{b}(Z_{\mu})&0\\\
*&*\end{pmatrix}\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}=\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix},$

and we conclude that $\mathfrak{b}(Z_{\mu})h_{1}=h_{1}$. Since $y$ is the
unique (up to scalars) eigenvector of $\mathfrak{b}(Z)$ to eigenvalue $1$,
$1+N_{\mu}$ is the unique eigenvector of $\mathfrak{b}(Z_{\mu})$ so that
$h_{1}=\alpha 1+N_{\mu}$ for some $\alpha\in\mathbb{C}$. It further follows
that $h_{2}=h-\alpha(1+N_{\mu})\in{\mathcal{H}}_{\mu}^{\perp}$ is an
eigevector of $\mathfrak{b}(\Pi_{\mu})|_{{\mathcal{H}}_{\mu}^{\perp}}$ to
eigenvalue $1$. By [17, Corollary 5.3],
${\mathcal{H}}_{\mu}^{\perp}\simeq\mathbb{H}^{2}_{d}\otimes\mathbb{C}^{k}$ and
$\Pi_{\mu}|_{{\mathcal{H}}_{\mu}^{\perp}}\simeq L\otimes I_{k}$. However,
since
$P_{{\mathcal{H}}_{\mu}^{\perp}}\mathfrak{b}(\Pi_{\mu})|_{{\mathcal{H}}_{\mu}^{\perp}}\simeq\mathfrak{b}(L)\otimes
I_{k}$, this is a pure isometry and we conclude that $h_{2}=0$. That is,
$1+N_{\mu}$ is the unique eigenvector (up to non-zero scalars) of
$\mathfrak{b}(\Pi_{\mu})$ to eigenvalue $1$.

Now let $\lambda\in K({\mathcal{O}}_{d})$ be another Cuntz state which peaks
at $\mathfrak{b}$, $\lambda(\mathfrak{b})=1$. As before equality in the
Cauchy–Schwarz inequality implies that
$\mathfrak{b}(\Pi_{\lambda})1+N_{\lambda}=1+N_{\lambda}$. By Lemma 3.4 and
Lemma 3.2, there is a unique isometry,
$V:\mathbb{C}^{n}\rightarrow\mathbb{H}^{2}_{d}(\lambda)$ so that

$VZ^{\omega*}=\Pi_{\lambda}^{\omega*}V.$

Hence,

$V^{*}\Pi_{\lambda}^{\omega}V=Z^{\omega},$

and $\Pi_{\lambda}$ is a row-isometric dilation of $Z$, which must be minimal
as $\Pi_{\lambda}$ is irreducible. Hence, by uniqueness of the minimal
dilation, $\Pi_{\lambda}\simeq\Pi_{\mu}$. Let
$U:\mathbb{H}^{2}_{d}(\lambda)\rightarrow\mathbb{H}^{2}_{d}(\mu)$ be the
unitary implementing this equivalence,
$U\Pi_{\lambda}^{\alpha}=\Pi_{\mu}^{\alpha}U$. Then,

$\displaystyle U1+N_{\lambda}$ $\displaystyle=$ $\displaystyle
U\mathfrak{b}(\Pi_{\lambda})1+N_{\lambda}$ $\displaystyle=$
$\displaystyle\mathfrak{b}(\Pi_{\mu})U1+N_{\lambda},$

so that $U1+N_{\lambda}=:h$ is a unit eigenvector of $\mathfrak{b}(\Pi_{\mu})$
to eigenvalue $1$ so that by the previous arguments,
$U1+N_{\lambda}=\zeta(1+N_{\mu})$ for some $\zeta\in\partial\mathbb{D}$. This
proves that $\lambda=\mu$ so that $\mu=\mu_{Z,y}$ is an $\mathbb{A}_{d}-$peak
state. ∎

The following corollary follows immediately from the preceding theorem and
Lemma 2.6.

######  Corollary 3.6.

Every finitely–correlated state $\mu$ on $\mathscr{A}_{d}$ that arises from an
irreducible finite-dimensional row coisometry and a unit vector is an exposed
extreme point of the state space of $\mathscr{A}_{d}$.

######  Corollary 3.7.

If $\mu=\mu_{T,x}$, for a finite irreducible row coisometry $T$ and unit
vector $x$, let $Z$ be the unique irreducible finite row co-isometry which is
jointly similar to $\mathrm{row}(T^{\mathrm{t}})$ and let $y$ be the
eigenvector of $\mathfrak{b}_{T,x}(Z)$ corresponding to the multiplicity one
eigenvalue, $1$. Then the finitely–correlated state $\mu_{Z,y}$ peaks at the
NC rational inner $\mathfrak{b}_{T,x}$ and $\mu_{T,x}$ peaks at the NC
rational inner $\mathfrak{b}_{Z,y}$.

If we define the unital, completely positive map,

$\mathrm{Ad}_{T,T^{*}}(A):=\sum_{j=1}^{d}T_{j}AT_{j}^{*},$

then this map is a unital quantum channel, _i.e._ $\mathrm{Ad}_{T^{*},T}$ is
also unital, if and only if $\mathrm{row}(T^{*})$ is also a row coisometry,
or, again equivalently, $\mathrm{row}(T^{\mathrm{t}})$ is a row coisometry. We
require another definition to describe the class of NC rational inner
functions associated with unital quantum channels. Let $\alpha=i_{1}\cdots
i_{n}$ be a word in the alphabet $\\{1,\cdots,d\\}$. We set
$\alpha^{t}=i_{n}i_{n-1}\cdots i_{1}$. Namely, $\alpha^{t}$ is the reversal of
$\alpha$. We define a unitary on $\mathbb{H}^{2}_{d}$ by
$(\mathfrak{z}^{\alpha})^{t}=\mathfrak{z}^{\alpha^{t}}$. This unitary is
important in realization theory of NC rational functions [31]. It is proved in
[32, Lemma 2.2] that if $\mathfrak{r}$ is an NC rational function, so is
$\mathfrak{r}^{\mathrm{t}}$. However, it need not be the case that if
$\mathfrak{b}$ is an NC rational inner, that $\mathfrak{b}^{t}$ is inner. A
simple example is
$\mathfrak{b}(\mathfrak{z})=(1+\mathfrak{z_{1}})\mathfrak{z_{2}}$. It is an
immediate calculation, that $\mathfrak{b}(L)^{*}\mathfrak{b}(L)=I$. However,
$\mathfrak{b}^{t}(\mathfrak{z})=\mathfrak{z_{2}}(1+\mathfrak{z_{1}})$. Here
the inner part of $\mathfrak{b}^{t}(L)$ is $L_{2}$, and the outer part is
$1+L_{1}$. In particular, it is easily checked that
$\|\mathfrak{b}^{\mathrm{t}}(L)\|=\sqrt{2}$ so that
$\mathfrak{b}^{\mathrm{t}}\in\mathbb{H}^{\infty}_{d}$ is not even contractive,
see [29, Example 3.4]. The following theorem identifies the class of all NC
rational functions $\mathfrak{b}$, such that $\mathfrak{b}^{t}$ is also inner,
as precisely those that arise from quantum channels.

######  Theorem 3.8.

Let $\mathfrak{b}:=\mathfrak{b}_{T,x}$ be the NC rational inner generated by
the pair $(T,x)$, where $T$ is a finite–dimensional row co-isometry on
${\mathcal{H}}$ and $x\in{\mathcal{H}}$ is both $T$ and $T^{*}-$cyclic. Then,
$\mathrm{row}(T^{\mathrm{t}})$ is also a row co-isometry if and only if
$\mathfrak{b}^{\mathrm{t}}$ is also NC rational inner and in this case
$\mathfrak{b}^{\mathrm{t}}=\mathfrak{b}_{\mathrm{row}(T^{\mathrm{t}}),\overline{x}}$.
If $x$ is a unit vector and both $T$ and $T^{\mathrm{t}}$ are irreducible row
coisometries then the NC rational Clark states $\mu_{T,x}$ and
$\mu_{T^{\mathrm{t}},\bar{x}}$ peak at the NC rational inners
$\mathfrak{b}^{\mathrm{t}}$ and $\mathfrak{b}$, respectively.

###### Proof.

By [32, Theorem 3.2, Remark 3.4], $(T^{*},x,x)$ is a minimal descriptor
realization of $\mathfrak{G}:=(1-\mathfrak{b})^{-1}$, so that the Taylor
coefficients of $\mathfrak{G}$ at $0$ are
$\hat{\mathfrak{G}}_{\omega}=x^{*}T^{*\omega}x$. Hence the Taylor coefficients
of $\mathfrak{G}^{\mathrm{t}}=(1-\mathfrak{b}^{\mathrm{t}})^{-1}$ are equal to

$\displaystyle\mathfrak{G}^{\mathrm{t}}_{\omega}$ $\displaystyle=$
$\displaystyle\hat{\mathfrak{G}}_{\omega^{\mathrm{t}}}=x^{*}T^{*\omega^{\mathrm{t}}}x$
$\displaystyle=$ $\displaystyle x^{*}(T^{\omega})^{*}x$ $\displaystyle=$
$\displaystyle(T^{\omega}x)^{*}x$ $\displaystyle=$
$\displaystyle\overline{x}^{*}\overline{T}^{\omega}\overline{x}$
$\displaystyle=$
$\displaystyle\overline{x}^{*}T^{\mathrm{t}*\omega}\overline{x}.$

This shows that $\mathfrak{G}^{\mathrm{t}}$ has the minimal descriptor
realization $(\overline{T},\overline{x},\overline{x})$, where
$T^{\mathrm{t}*}=\overline{T}$. If $\mathrm{row}(T^{\mathrm{t}})$ is also
finite row coisometry then by [32, Theorem 4.1], it follows that
$\mathfrak{b}^{\mathrm{t}}$ is also NC rational inner with minimal FM
realization:

$A_{j}:=\overline{T}_{j}(I-\overline{x}\overline{x}^{*}),\quad
B_{j}:=\overline{T}_{j}\overline{x},\quad
C:=\overline{x}^{*},\quad\mbox{and}\quad D:=\mathfrak{b}^{\mathrm{t}}(0)=0.$

Conversely, as above, given any NC rational inner
$\mathfrak{b}=\mathfrak{b}_{T,x}$, a minimal descriptor realization of
$(1-\mathfrak{b}^{\mathrm{t}})^{-1}=\mathfrak{G}^{\mathrm{t}}$ is given by
$(\overline{T},\overline{x},\overline{x})$. Assuming that
$\mathfrak{b}^{\mathrm{t}}$ is also NC rational inner, [32, Theorem 3.2,
Remark 3.4] implies that there is a finite row co-isometry, $W$, and a vector,
$y$, which is both $W$ and $W^{*}-$cyclic so that $(W,y,y)$ is also minimal
descriptor realization of $\mathfrak{G}^{\mathrm{t}}$ so that for any word,
$\omega\in{\mathbb{F}}^{d}$,

$y^{*}W^{*\omega}y=\overline{x}^{*}\overline{T}^{\omega}\overline{x}.$

Equivalently, if $\|T^{\mathrm{t}}\|_{row}$ is the row-norm of
$\mathrm{row}(T^{\mathrm{t}})$, then for any word,
$\omega\in{\mathbb{F}}^{d}$,

$\frac{1}{\|T^{\mathrm{t}}\|_{row}^{|\omega|}}y^{*}W^{*\omega}y=\frac{1}{\|T^{\mathrm{t}}\|_{row}^{|\omega|}}\overline{x}^{*}\overline{T}^{\omega}\overline{x}.$

If $\|\mathrm{row}(T^{\mathrm{t}})\|>1$, then
$\frac{1}{\|T^{\mathrm{t}}\|_{row}}\mathrm{row}(T^{\mathrm{t}})$ and
$\frac{1}{\|T^{\mathrm{t}}\|_{row}}W$ are both row contractions. In either
case, [32, Proposition 3.6, Lemma 3.9] implies that
$\mathrm{row}(T^{\mathrm{t}})$ and $W$ are jointly unitarily equivalent via a
unitary $U$ which sends $y$ to $\overline{x}$. Hence
$\mathrm{row}(T^{\mathrm{t}})$ is a row coisometry. ∎

######  Example 3.9.

The examples [32, Example 4.4, Example 4.5] both give examples of NC rational
inners arising from finite, irreducible row coisometries, $T$. Namely,

$T=\left(\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\right)$

and

$S=\frac{1}{\sqrt{2}}\left(\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}\right).$

It is easily checked that $\mathrm{row}(T^{\mathrm{t}})$ and
$\mathrm{row}(S^{\mathrm{t}})$ are both row coisometries so that, by the
previous theorems, if $x$ is any unit vector then
$\mathfrak{b}_{T,x},\mathfrak{b}^{\mathrm{t}}_{T,x}=\mathfrak{b}_{\mathrm{row}(T^{\mathrm{t}}),\overline{x}}$
and $\mathfrak{b}_{S,x},\mathfrak{b}^{\mathrm{t}}_{S,x}$ are all NC rational
inner, $\mu_{T,x},\mu_{\mathrm{row}(T^{\mathrm{t}}),\overline{x}}$ and
$\mu_{S,x},\mu_{\mathrm{row}(S^{\mathrm{t}}),\overline{x}}$ are all Cuntz
states which peak at
$\mathfrak{b}^{\mathrm{t}}_{T,x},\mathfrak{b}_{T,x},\mathfrak{b}^{\mathrm{t}}_{S,x}$
and $\mathfrak{b}_{S,x}$, respectively.

The following example illustrates what happens if we drop the assumption that
$T$ is irreducible but require still that its minimal isometric dilation is
irreducible.

######  Example 3.10.

Consider the following coisometry:

$T_{1}=\dfrac{1}{2}\begin{pmatrix}-1&0&-1\\\ -1&0&1\\\
-1&0&-1\end{pmatrix},\quad T_{2}=\dfrac{1}{2}\begin{pmatrix}1&-1&0\\\
-1&-1&0\\\ -1&1&0\end{pmatrix}.$

Let us write $e_{1}$, $e_{2}$, and $e_{3}$ for the vectors of the standard
basis of $\mathbb{C}^{3}$. I is now easy to check that
$T_{1}e_{1}=-\frac{1}{2}(e_{1}+e_{2}+e_{3})$,
$T_{2}e_{1}=\frac{1}{2}(e_{1}-e_{2}-e_{3})$,
$T_{2}T_{1}e_{1}=\frac{1}{2}e_{2}$. This implies that $e_{1}$ is $T$-cyclic.
Similarly, $T_{1}^{*}e_{1}=-\frac{1}{2}(e_{1}+e_{3})$,
$T_{2}^{*}e_{1}=\frac{1}{2}(e_{1}-e_{2})$, and
$T_{1}^{*}T_{2}^{*}e_{1}=-\frac{1}{2}e_{3}$. This implies that $e_{1}$ is also
$T^{*}$-cyclic and, moreover, that
$\bigvee_{\alpha\neq\emptyset}T^{*\alpha}e_{1}=\mathbb{C}^{3}$. Therefore, by
[32], the following NC rational function is an inner

$\mathfrak{r}(\mathfrak{z}_{1},\mathfrak{z}_{2})=e_{1}^{*}\left(I-\mathfrak{z}_{1}T_{1,0}^{*}-\mathfrak{z}_{2}T_{2,0}^{*}\right)^{-1}\left(\mathfrak{z}_{1}T_{1}^{*}e_{1}+\mathfrak{z}_{2}T_{2}^{*}e_{1}\right).$

Here,

$T_{1,0}^{*}=T_{1}^{*}(I-e_{1}e_{1}^{*})\text{ and
}T_{2,0}^{*}=T_{2}^{*}(I-e_{1}e_{1}^{*}).$

Therefore, we have the following expression for the pencil

$I-\mathfrak{z}_{1}T_{1,0}^{*}-\mathfrak{z}_{2}T_{2,0}^{*}=\begin{pmatrix}1&\frac{1}{2}(\mathfrak{z}_{1}+\mathfrak{z}_{2})&\frac{1}{2}(\mathfrak{z}_{1}+\mathfrak{z}_{2})\\\
0&1+\frac{1}{2}\mathfrak{z}_{2}&-\frac{1}{2}\mathfrak{z}_{2}\\\
0&-\frac{1}{2}\mathfrak{z}_{1}&1+\frac{1}{2}\mathfrak{z}_{1}\end{pmatrix}.$

Since we know $T_{1}^{*}e_{1}$ and $T_{2}^{*}e_{1}$, we conclude that the
expression for our function is

$\mathfrak{r}(\mathfrak{z}_{1},\mathfrak{z}_{2})=\frac{1}{2}(\mathfrak{z}_{2}-\mathfrak{z}_{1})+\frac{1}{4}(\mathfrak{z}_{1}+\mathfrak{z}_{2})\begin{pmatrix}1&1\end{pmatrix}\begin{pmatrix}1+\frac{1}{2}\mathfrak{z}_{2}&-\frac{1}{2}\mathfrak{z}_{2}\\\
-\frac{1}{2}\mathfrak{z}_{1}&1+\frac{1}{2}\mathfrak{z}_{1}\end{pmatrix}^{-1}\begin{pmatrix}\mathfrak{z}_{2}\\\
\mathfrak{z}_{1}\end{pmatrix}.$

In particular, we have that $r(-1,0)=1$.

Now we observe that $T_{1}^{*}$ and $T_{2}^{*}$ have a common eigenvector. In
fact, set

$U=\begin{pmatrix}\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\\\ 0&1&0\\\
\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\end{pmatrix}.$

Then,

$UT_{1}U=\begin{pmatrix}-1&0&0\\\ 0&0&-\frac{1}{\sqrt{2}}\\\
0&0&0\end{pmatrix}\text{ and }UT_{2}U=\begin{pmatrix}0&0&0\\\
-\frac{1}{2\sqrt{2}}&-\frac{1}{2}&-\frac{1}{2\sqrt{2}}\\\
\frac{1}{2}&-\frac{1}{\sqrt{2}}&\frac{1}{2}\end{pmatrix}.$

We note that $\mathfrak{r}(T^{t})$ has eigenvalue $1$ of multiplicity $1$. The
corresponding eigenvector is $e_{1}+e_{3}$. The above calculation shows that
only the semi-simple part of $T^{t}$ is in the closed ball. The similarity
orbit of $T^{t}$ itself never intersects the closed ball.

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