# Iteration theory of noncommutative maps

Serban T. Belinschi  CNRS - IMT  and  Eli Shamovich<EMAIL_ADDRESS>Ben-
Gurion University of the Negev

###### Abstract.

This note aims to study the iteration theory of noncommutative self-maps of
bounded matrix convex domains. We prove a version of the Denjoy-Wolff theorem
for the row ball and the maximal quantization of the unit ball of
${\mathbb{C}}^{d}$. For more general bounded matrix convex sets, we prove a
version of Wolff’s theorem inspired by the results of Abate. Lastly, we use
iteration and fixed point theory to generalize the commutative results of
Davidson, Ramsey, and Shalit to quotients of the free semigroup algebra by WOT
closed ideals.

This work was initiated in November 2019 while STB was a visiting fellow
supported by the Faculty of Natural Sciences Distinguished Scientist Visitors
Program of the Ben Gurion University of the Negev, Beer-Sheva, Israel.

##  1\. Introduction

This note aims to study the iteration theory of noncommutative functions on
matrix convex sets and, more generally, noncommutative domains. Noncommutative
analysis is a rapidly growing area of research. The origins of noncommutative
analysis can be traced back to the works of Takesaki [38], Taylor [39, 40],
and Voiculescu [44, 45, 46]. Though their motivations varied greatly, the
notion of a noncommutative (nc for short) function is apparent in all of these
works. Additional impetus to the field was given by results in noncommutative
convexity with a view towards applications (see, for example, [20, 21, 22]).
Arveson [9] pioneered a different approach to noncommutative convexity.
Arveson has generalized the classical notion of Choquet boundary to the
setting of operator systems. In particular, Arveson’s extension theorem led
Wittstock to prove an extension theorem on completely bounded maps and
introduce matrix convex sets [47, 48]. The reader should consult the monograph
[15] and the references therein for more information on noncommutative
convexity. A basic introduction to nc domains and functions is given in
Section 2. We refer the reader to the books [8, 24] for more details on nc
function theory.

Classically, open convex domains in ${\mathbb{C}}^{d}$ are a primary example
of hyperbolic spaces in the sense of Kobayashi [25]. A fundamental result of
Lempert [26, 27] states that the Caratheodory and Kobayashi metrics on a
convex bounded domain coincide (see also [5] and [33]). This hyperbolic metric
is an essential tool in the study of iteration theory and fixed points of
holomorphic self-maps of convex domains [1, 2, 3, 12, 42, 43]. Nc hyperbolic
metrics were introduced by the first author and Vinnikov in [13]. The
definition and basic properties of the nc Lempert function and hyperbolic
metrics appear in Section 3.

A particularly nice example of a convex set is the unit ball; in the
noncommutative case, it is the row ball. Popescu has extensively studied nc
function theory on the row ball (see, for example, [28, 29, 30, 31, 32]).
Davidson and Pitts [18, 16, 17] have studied the free semigroup algebra and,
in doing so, studied nc bounded analytic functions on the row ball (see also
[35, 36]). In particular, Popescu proved a noncommutative version of Wolff’s
theorem for the row ball [32, Theorem 3.1]. A noncommutative version of a
fixed point theorem of Rudin [34] and Hervé [23] was obtained by the second
author in [37]. The primary motivation for the proof of the latter fixed point
theorem was the classification of quotients of the free semigroup algebra by
WOT closed ideals up to completely isometric isomorphism [35, 36].

In Section 4, we prove a version of the Denjoy-Wolff theorem for the row ball
(Theorem 4.4 and the maximal quantization of the unit ball
${\mathbb{B}}_{d}\subset{\mathbb{C}}^{d}$ (Corollary 4.8). In particular, this
answers a question raised by Popescu in [32] on the Denjoy-Wolff theorem for
the row ball. One of the key ingredients of the proof of these results are
forms of the nc maximum modulus principle. Popescu obtained one version of the
maximum modulus principle for the row ball [28, Theorem 3.3], and a different
version (more applicable to the problem at hand) was obtained in [35, Lemma
6.11]. In this paper, we obtain a maximum modulus principle for the maximal
quantization of ${\mathbb{B}}_{d}$ (see Theorem 4.7). It is interesting
whether such a strong form of maximum modulus principle is available for every
quantization of the unit ball.

Section 5 contains an nc version of the classical Vigué and Bedford theorems
on iterations of holomorphic functions (Theorems 5.1 and 5.4). We use these
results to obtain an extension of [37, Theorem 4.1] that is the full nc
version of a result of Davidson, Ramsey, and Shalit [19, Theorem 4.4].

######  Theorem A (Theorem 5.7).

Let ${\mathfrak{V}}\subset{\mathfrak{B}}_{d}$ and
${\mathfrak{W}}\subset{\mathfrak{B}}_{e}$ be two nc analytic varieties. If
$H^{\infty}({\mathfrak{V}})$ and $H^{\infty}({\mathfrak{W}})$ are completely
isometrically isomorphic, then there exists $k\in{\mathbb{N}}$ and an
automorphism $F\colon{\mathfrak{B}}_{k}\to{\mathfrak{B}}_{k}$, such that
${\mathfrak{V}},{\mathfrak{W}}\subset{\mathfrak{B}}_{k}$ and $F$ maps
${\mathfrak{V}}$ onto ${\mathfrak{W}}$.

Lastly, in Section 6, we provide an nc version of Wolff’s theorem in the
spirit of the result of Abate [1]. Namely, we define nc analogs of big and
small horospheres centered at a scalar point on the boundary of a bounded
matrix convex set and prove that iterations of an nc analytic self-map send
the small one into the big one.

##  2\. Noncommutative functions and domains

Let $E$ be a vector space, we set
${\mathbb{M}}(E)=\bigsqcup_{n=1}^{\infty}M_{n}(E)$, where
$M_{n}(E)=M_{n}\otimes E$. We write
${\mathbb{M}}_{d}={\mathbb{M}}({\mathbb{C}}^{d})$ and treat ${\mathbb{M}}_{d}$
as the set of all $d$-tuples of matrices of all sizes. Note that
$\operatorname{{\mathbf{G}L}}_{n}$ acts on $M_{n}(E)$ by
$S\cdot(A\otimes\xi)=(S^{-1}AS)\otimes\xi$. We will denote this action for
${\mathbf{X}}\in M_{n}(E)$ simply by $S^{-1}{\mathbf{X}}S$.

Additionally, for ${\mathbf{X}}\in M_{n}(E)$ and ${\mathbf{Y}}\in M_{m}(E)$ we
will write

${\mathbf{X}}\oplus{\mathbf{Y}}=\begin{pmatrix}{\mathbf{X}}&0\\\
0&{\mathbf{Y}}\end{pmatrix}\in M_{n+m}(E).$

A subset $\Omega\subset{\mathbb{M}}(E)$ is called an nc set, if for every
${\mathbf{X}},{\mathbf{Y}}\in\Omega$,
${\mathbf{X}}\oplus{\mathbf{Y}}\in\Omega$.

If $E$ is an operator space, then we set:

${\mathfrak{B}}_{E}=\bigsqcup_{n=1}^{\infty}\left\\{{\mathbf{X}}\in
M_{n}(E)\mid\|{\mathbf{X}}\|_{n}<1\right\\}.$

We endow ${\mathbb{C}}^{d}$ with the row norm and write ${\mathfrak{B}}_{d}$
for ${\mathfrak{B}}_{{\mathbb{C}}^{d}}$. We will write
${\mathbb{B}}_{d}={\mathfrak{B}}_{d}(1)$ for the unit ball of
${\mathbb{C}}^{d}$ and ${\mathbb{D}}$ for the disc. A subset
$\Omega\subset{\mathbb{M}}(E)$ will be called an nc domain if $\Omega$ is an
nc set, level-wise open, and invariant under unitary similarities.

An nc set $\Omega\subset{\mathbb{M}}(E)$ is called matrix convex if, for every
${\mathbf{X}}\in\Omega(n)$ and $V\colon{\mathbb{C}}^{m}\to{\mathbb{C}}^{n}$ an
isometry, we have that $V^{*}{\mathbf{X}}V\in\Omega(m)$.

Let $\Omega\subset{\mathbb{M}}(E)$ be an nc domain. We say that a function
$f\colon\Omega\to{\mathbb{M}}(F)$ is an nc function, if

  * •

$f$ is graded, i.e, $f(\Omega(n))\subset M_{n}(F)$, for every
$n\in{\mathbb{N}}$. We write $f_{n}=f|_{\Omega(n)}$.

  * •

$f$ respects direct sums, i.e,
$f({\mathbf{X}}\oplus{\mathbf{Y}})=f({\mathbf{X}})\oplus f({\mathbf{Y}})$.

  * •

$f$ respects similarities, for every $n\in{\mathbb{N}}$, every
${\mathbf{X}}\in\Omega(n)$, and every $S\in\operatorname{{\mathbf{G}L}}_{n}$,
if $S^{-1}{\mathbf{X}}S\in\Omega(n)$, then
$f(S^{-1}{\mathbf{X}}S)=S^{-1}f({\mathbf{X}})S$.

It is a remarkable result of Kalyuzhnyi-Verbovetskyi and Vinnikov that a
locally bounded nc function is both Gateaux and Frechet differentiable (see
also [6]). Hence, we will say an analytic nc function when we mean locally
bounded. The key to proving this result is the nc difference-differential
operator. Namely, for every ${\mathbf{X}}\in\Omega(n)$,
${\mathbf{Y}}\in\Omega(m)$, there exists a linear map $\Delta
f({\mathbf{X}},{\mathbf{Y}})\colon M_{n,m}\otimes E\to M_{n,m}\otimes E$, such
that for every ${\mathbf{Z}}\in M_{n,m}\otimes E$, such that
$\left(\begin{smallmatrix}{\mathbf{X}}&{\mathbf{Z}}\\\
0&{\mathbf{Y}}\end{smallmatrix}\right)\in\Omega(n+m)$,

$f\left(\begin{pmatrix}{\mathbf{X}}&{\mathbf{Z}}\\\
0&{\mathbf{Y}}\end{pmatrix}\right)=\begin{pmatrix}f({\mathbf{X}})&\Delta
f({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})\\\ 0&f({\mathbf{Y}})\end{pmatrix}.$

In fact, $\Delta f({\mathbf{X}},{\mathbf{X}})$ is the Frechet derivative of a
locally bounded nc function at ${\mathbf{X}}$. Moreover, we have the identity

$\Delta
f({\mathbf{X}},{\mathbf{Y}})({\mathbf{X}}-{\mathbf{Y}})=f({\mathbf{X}})-f({\mathbf{Y}}).$

For the properties of the nc difference-differential operator, we refer the
reader to [24].

The nc sets that we will focus on are described in the following definition.

######  Definition 2.1.

Let ${\mathcal{H}}$, ${\mathcal{K}}$ be a Hilbert space. Let
$\Omega\subset{\mathbb{M}}(E)$ be an nc domain and
$Q\colon\Omega\to{\mathbb{M}}(B({\mathcal{H}},{\mathcal{K}}))$ be an nc
analytic function. We define
${\mathbb{D}}_{Q}=\left\\{{\mathbf{Z}}\in\Omega\mid
Q({\mathbf{Z}})Q({\mathbf{Z}})^{*}<I\right\\}$.

######  Remark 2.2.

For ${\mathbf{X}}\in\Omega(n)$, ${\mathbf{Y}}\in\Omega(m)$, and
${\mathbf{Z}}\in M_{n,m}\otimes B({\mathcal{H}})$, define the nc affine kernel
$G({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})={\mathbf{Z}}-Q({\mathbf{X}}){\mathbf{Z}}Q({\mathbf{Y}}^{*})^{*}$.
Then, ${\mathbb{D}}_{Q}$ coincides with a generalized noncommutative ball, as
defined in [13]. Domains of the form ${\mathbb{D}}_{Q}$ were originally
defined by Agler and McCarthy for $Q$, a matrix of nc polynomials (see, for
example, [6, 7]). In this form, the sets ${\mathbb{D}}_{Q}$ first appeared in
the works of Ball, Marx, and Vinnikov [10, 11].

Similarly, we have a generalized analog of the half-plane, which by the
Effros-Winkler separation theorem, includes all matrix convex sets.

######  Definition 2.3.

Let ${\mathcal{H}}$ be a Hilbert space. Let $\Omega\subset{\mathbb{M}}(E)$ be
an nc domain invariant under pointwise conjugation and
$L\colon\Omega\to{\mathbb{M}}(B({\mathcal{H}}))$ be an injective nc analytic
function. We define
${\mathbb{H}}_{L}=\left\\{{\mathbf{Z}}\in\Omega\mid\operatorname{Re}(L({\mathbf{Z}}))<I\right\\}$.
Following [11], we also have a representation $\Omega={\mathbb{D}}_{Q}$, where
$Q$ is obtained from $L$ using the Cayley transform.

Note that as observed by Taylor in [39, Section 6], for every nc domain
$\Omega\subset{\mathbb{M}}_{d}$, the algebra ${\mathcal{T}}(\Omega)$ of
holomorphic nc functions on $\Omega$ is a nuclear Frechet algebra. The
topology on ${\mathcal{T}}(\Omega)$ is the topology of uniform convergence on
levelwise compacta. Note that the map that takes an nc function to its
restriction to the various levels gives a continuous embedding
${\mathcal{T}}(\Omega)\hookrightarrow\prod_{n=1}^{\infty}{\mathcal{O}}(\Omega(n))\otimes
M_{n}$, where we write ${\mathcal{O}}(\Omega(n))$ for the algebra of
holomorphic functions on $\Omega(n)$ with the topology of uniform convergence
on compacta. Since ${\mathcal{O}}(\Omega(n))\otimes M_{n}$ is nuclear for
every $n\in{\mathbb{N}}$, the product is also nuclear by [41, Proposition
50.1]. Moreover, since the conditions of respecting direct sums and
similarities are closed, we immediately see that ${\mathcal{T}}(\Omega)$ is a
closed subspace and is also nuclear. Hence, by [41, Proposition 50.2], every
bounded subset of a nuclear space is precompact, and thus, in particular, in a
Frechet space, every bounded sequence has a convergent subsequence. In other
words, we can apply Montel’s theorem to sequences of nc functions and maps.
The obtained convergence is, of course, uniform convergence on compacta on
each level. There are stronger versions of Montel’s theorem (see, for example,
[8, Theorem 13.14] and [28, Theorem 5.2]). However, this is the one that will
be used almost exclusively throughout this paper. We will apply Popescu’s
version of Montel’s theorem for the row ball [28, Theorem 5.2] to obtain a
stronger Denjpy-Wolff theorem in this setting.

##  3\. Noncommutative hyperbolic metrics

Noncommutative hyperbolic metrics and the nc Lempert function were introduced
by the first author and Vinnikov in [13]. For a contraction $T\in
B({\mathcal{H}},{\mathcal{K}})$, we will write $D_{T}=1-TT^{*}$ and
$D_{T^{*}}=1-T^{*}T$ for the defect operators. The nc Lempert function for an
nc domain $\Omega\subset{\mathbb{M}}(E)$ is defined by

$\delta({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})=\left[\sup\left\\{t\in[0,\infty]\mid\forall
s\in[0,t),\begin{pmatrix}{\mathbf{X}}&s{\mathbf{Z}}\\\
0&{\mathbf{Y}}\end{pmatrix}\in\Omega\right\\}\right]^{-1}$

Here ${\mathbf{X}}\in\Omega(n)$, ${\mathbf{Y}}\in\Omega(m)$, and
${\mathbf{Z}}\in M_{n,m}\otimes E$. We also set for
${\mathbf{X}},{\mathbf{Y}}\in\Omega(n)$.

$\widetilde{\delta}({\mathbf{X}},{\mathbf{Y}})=\delta({\mathbf{X}},{\mathbf{Y}})({\mathbf{X}}-{\mathbf{Y}}).$

In the case of ${\mathbb{D}}_{Q}$ we have a special form for the nc Lempert
function, which is a particular case of the equality appearing in [13,
Equation 21].

######  Lemma 3.1.

For ${\mathbf{X}}\in{\mathbb{D}}_{Q}(n)$,
${\mathbf{Y}}\in{\mathbb{D}}_{Q}(m)$, and ${\mathbf{Z}}\in M_{n,m}\otimes E$
we have

$\delta({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})=\left\|D_{Q({\mathbf{X}})}^{-1/2}\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})D_{Q({\mathbf{Y}})^{*}}^{-1/2}\right\|.$
$\widetilde{\delta}({\mathbf{X}},{\mathbf{Y}})=\left\|D_{Q({\mathbf{X}})}^{-1/2}\left(Q({\mathbf{X}})-Q({\mathbf{Y}})\right)D_{Q({\mathbf{Y}})^{*}}^{-1/2}\right\|$

###### Proof.

Let $t>0$ be such that $\left(\begin{smallmatrix}{\mathbf{X}}&t{\mathbf{Z}}\\\
0&{\mathbf{Y}}\end{smallmatrix}\right)\in{\mathbb{D}}_{Q}$. Then

$\begin{pmatrix}Q({\mathbf{X}})&t\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})\\\
0&Q({\mathbf{Y}})\end{pmatrix}\begin{pmatrix}Q({\mathbf{X}})^{*}&0\\\ t\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})^{*}&Q({\mathbf{Y}})^{*}\end{pmatrix}<I.$

Multiplying the matrices and applying Schur’s complement, we see that

$D_{Q({\mathbf{X}})}-t^{2}\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})^{*}>\\\ t^{2}\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})Q({\mathbf{Y}})^{*}D_{Q({\mathbf{Y}})}^{-1}Q({\mathbf{Y}})\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})^{*}.$

Let $T\in B({\mathcal{H}},{\mathcal{K}})$ be a strict row contraction. Writing
$D_{T}^{-1}$ as a power series in $TT^{*}$, we get that
$T^{*}D_{T}^{-1}T=D_{T^{*}}^{-1}-I$. Hence,

$D_{Q({\mathbf{X}})}>t^{2}\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})D_{Q({\mathbf{Y}})^{*}}^{-1}\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})^{*}.$

Thus

$\frac{1}{t^{2}}>D_{Q({\mathbf{X}})}^{-1/2}\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})D_{Q({\mathbf{Y}})^{*}}^{-1}\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})^{*}D_{Q({\mathbf{X}})}^{-1/2}.$

Running the argument back, we see that this condition is necessary and
sufficient. Hence we conclude that

$\delta({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})=\left\|D_{Q({\mathbf{X}})}^{-1/2}\Delta
Q({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})D_{Q({\mathbf{Y}})^{*}}^{-1/2}\right\|$

The second equality follows from the nc difference-differential formula [24,
Theorem 2.10]. ∎

We have a similar result for generalized half-planes.

######  Lemma 3.2.

For ${\mathbf{X}}\in{\mathbb{H}}_{L}(n)$,
${\mathbf{Y}}\in{\mathbb{H}}_{L}(m)$, and ${\mathbf{Z}}\in
M_{n,m}\otimes{\mathbb{C}}^{d}$ we have

$\delta({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})=\frac{1}{2}\left\|\left(I-\operatorname{Re}(L({\mathbf{X}}))\right)^{-1/2}\Delta
L({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})\left(I-\operatorname{Re}(L({\mathbf{Y}}))\right)^{-1/2}\right\|.$
$\widetilde{\delta}({\mathbf{X}},{\mathbf{Y}})=\frac{1}{2}\left\|\left(I-\operatorname{Re}(L({\mathbf{X}}))\right)^{-1/2}\left(L({\mathbf{X}})-L({\mathbf{Y}})\right)\left(I-\operatorname{Re}(L({\mathbf{Y}}))\right)^{-1/2}\right\|$

######  Remark 3.3.

As stated in [13], $\tilde{\delta}$ is symmetric. We include a brief proof of
this fact for the sake of completeness. Let $\Omega$ be a unitarily invariant
nc domain and let $t\in(0,\pi/2)$. For every
${\mathbf{X}},\,{\mathbf{Y}}\in\Omega(n)$, we have

$\begin{pmatrix}\cos t&-\sin t\\\ -\sin t&-\cos
t\end{pmatrix}\begin{pmatrix}{\mathbf{Y}}&\frac{\cos t}{\sin
t}({\mathbf{Y}}-{\mathbf{X}})\\\
0&{\mathbf{X}}\end{pmatrix}\begin{pmatrix}\cos t&-\sin t\\\ -\sin t&-\cos
t\end{pmatrix}=\begin{pmatrix}{\mathbf{X}}&\frac{\cos t}{\sin
t}({\mathbf{X}}-{\mathbf{Y}})\\\ 0&{\mathbf{Y}}\end{pmatrix}.$

Hence,
$\widetilde{\delta}({\mathbf{X}},{\mathbf{Y}})=\widetilde{\delta}({\mathbf{Y}},{\mathbf{X}})$.

We record the following simple observation, that is, however, using the power
of the nc assumption (see also [4, Theorem 5]).

######  Lemma 3.4.

Let $\Omega\subset{\mathbb{M}}(E)$ be a bounded matrix convex nc domain that
contains $0$. Let $f\colon\Omega\to\Omega$ be an nc analytic map. If there
exists $0<r<1$, such that $f(\Omega)\subset r\Omega$, then $f$ has a unique
fixed point in $\Omega(1)$. Namely, there exists ${\mathbf{X}}\in\Omega(1)$,
such that $f({\mathbf{X}})={\mathbf{X}}$ and the only fixed points of $f$ are
ampliations of ${\mathbf{X}}$.

###### Proof.

Let $f_{n}=f|_{\Omega(n)}$. By the Earle-Hamilton theorem,
$f_{n}\colon\Omega(n)\to\Omega(n)$ has a unique fixed point. In particular,
let ${\mathbf{X}}\in\Omega(1)$ be the unique fixed point on this level, then
${\mathbf{X}}^{\oplus n}$ is fixed by $f_{n}$ since $f$ respects direct sums.
Thus, by uniqueness, the fixed point on level $n$ is ${\mathbf{X}}^{\oplus
n}$. ∎

######  Lemma 3.5.

Let $\Omega={\mathbb{D}}_{Q}$. Then for every $n$,
$\rho_{{\mathbf{X}}}({\mathbf{Z}})=\delta({\mathbf{X}},{\mathbf{X}})({\mathbf{Z}})$
is a seminorm. Furthermore, $\rho_{{\mathbf{X}}}$ is a norm if and only if
$\Delta Q({\mathbf{X}},{\mathbf{X}})$ is injective.

###### Proof.

Obvious from Lemma 3.1 and the linearity of $\Delta
Q({\mathbf{X}},{\mathbf{X}})$. ∎

Note that for every ${\mathbf{X}}\in{\mathbb{D}}_{Q}(n)$, we can define a
collection of seminorms on $M_{k}(M_{n}(E))$, via
$\rho_{k,{\mathbf{X}}}({\mathbf{Z}})=\delta(I_{k}\otimes{\mathbf{X}},I_{k}\otimes{\mathbf{X}})({\mathbf{Z}})$.

######  Proposition 3.6.

Let $Q$ be such that for some ${\mathbf{X}}\in{\mathbb{D}}_{Q}(n)$, $\Delta
Q({\mathbf{X}},{\mathbf{X}})$ is injective. Then $\rho_{k,{\mathbf{X}}}$ form
an operator space structure on $M_{n}(E)$.

###### Proof.

By our assumption and the properties of the nc derivative $\Delta
Q(I_{k}\otimes{\mathbf{X}},I_{k}\otimes{\mathbf{X}})$ is injective for every
$k\in{\mathbb{N}}$. Thus, each $\rho_{k}$ is a norm by the previous lemma.
Hence, we will verify Ruan’s axioms.

Let ${\mathbf{Z}}\in{\mathbb{M}}_{kn}(E)$ and ${\mathbf{W}}\in M_{\ell n}(E)$.
By the properties of $\delta$, we have that

$\rho_{k+\ell}({\mathbf{Z}}\oplus{\mathbf{W}})=\delta(I_{k+\ell}\otimes{\mathbf{X}},I_{k+\ell}\otimes{\mathbf{X}})({\mathbf{Z}}\oplus{\mathbf{W}})=\max\\{\rho_{k}({\mathbf{Z}}),\rho_{\ell}({\mathbf{W}})\\}.$

Now let $\alpha\in M_{\ell,k}$, $\beta\in M_{k,\ell}$, and $Z\in
M_{k}(M_{n}(E))$. Write $I=I_{M_{n}(E)}$ and observe that since $\Delta
Q({\mathbf{X}},{\mathbf{X}})$ is linear and satisfies for every
$m\in{\mathbb{N}}$, $\Delta
Q(I_{m}\otimes{\mathbf{X}},I_{m}\otimes{\mathbf{X}})=I_{M_{m}}\otimes\Delta
Q({\mathbf{X}},{\mathbf{X}})$, we get that

$\Delta
Q(I_{\ell}\otimes{\mathbf{X}},I_{\ell}\otimes{\mathbf{X}})((\alpha\otimes
I){\mathbf{Z}}(\beta\otimes I))=(\alpha\otimes I)\Delta
Q(I_{k}\otimes{\mathbf{X}},I_{k}\otimes{\mathbf{X}})({\mathbf{Z}})(\beta\otimes
I).$

Hence,

$\rho_{\ell,{\mathbf{X}}}\left((\alpha\otimes I){\mathbf{Z}}(\beta\otimes
I)\right)\leq\|\alpha\|\|\beta\|\rho_{k,{\mathbf{X}}}({\mathbf{Z}}).$

This completes the proof. ∎

######  Remark 3.7.

If $Q\colon{\mathbb{M}}(E)\to{\mathbb{M}}(B({\mathcal{H}},{\mathcal{K}}))$ is
an injective analytic nc function, it is not hard to check that $\Delta
Q({\mathbf{X}},{\mathbf{X}})$ is injective for every
${\mathbf{X}}\in{\mathbb{M}}(E)$. Indeed, for every ${\mathbf{Z}}\in
M_{n}(E)$, such that $\Delta Q({\mathbf{X}},{\mathbf{X}})({\mathbf{Z}})=0$,

$Q\left(\begin{pmatrix}{\mathbf{X}}&{\mathbf{Z}}\\\
0&{\mathbf{X}}\end{pmatrix}\right)=\begin{pmatrix}Q({\mathbf{X}})&0\\\
0&Q({\mathbf{X}})\end{pmatrix}=Q({\mathbf{X}}\oplus{\mathbf{X}}).$

Hence, for every point ${\mathbf{X}}\in{\mathbb{D}}_{Q}$, we have an operator
space structure on the “tangent space” at ${\mathbf{X}}$.

######  Corollary 3.8.

Let $f\colon{\mathbb{D}}_{Q}\to{\mathbb{D}}_{P}$ be an nc function. Then, for
every ${\mathbf{X}}\in{\mathbb{D}}_{Q}$, the linear map $\Delta
f({\mathbf{X}},{\mathbf{X}})$ is a complete contraction.

###### Proof.

The claim follows immediately from [13, Proposition 3.2] and the properties of
$\delta$ and the nc difference-differential operator. ∎

Recall from [13] that there are several ways to define distances on an nc
domain $\Omega$ using $\widetilde{\delta}$. The simplest one is a level-
dependent distance. Let ${\mathbf{X}},{\mathbf{Y}}\in\Omega(n)$, we set:

$\widetilde{d}_{\Omega}({\mathbf{X}},{\mathbf{Y}})=\inf\\{\sum_{k=1}^{m}\widetilde{\delta}({\mathbf{Z}}_{k-1},{\mathbf{Z}}_{k})\mid
m\in{\mathbb{N}},\,{\mathbf{X}}={\mathbf{Z}}_{0},{\mathbf{Z}}_{1},\ldots,{\mathbf{Z}}_{m}={\mathbf{Y}}\in\Omega(n)\\}$

One can remove the dependence on levels by altering the definition slightly

$\widetilde{d}_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})=\inf\\{\sum_{k=1}^{m}\widetilde{\delta}({\mathbf{Z}}_{k-1},{\mathbf{Z}}_{k})\mid
m,\ell\in{\mathbb{N}},\,I_{\ell}\otimes{\mathbf{X}}={\mathbf{Z}}_{0},{\mathbf{Z}}_{1},\ldots,{\mathbf{Z}}_{m}=I_{\ell}\otimes{\mathbf{Y}}\in\Omega(\ell
n)\\}$

Lastly, one can define a distance by measuring the lengths of paths connecting
the two points.

$d_{\Omega}({\mathbf{X}},{\mathbf{Y}})=\inf\\{\int_{0}^{1}\delta(\gamma(t),\gamma(t))(\gamma^{\prime}(t))dt\mid\gamma\in
C^{1}_{pw}([0,1],\Omega(n))\\}.$
$d_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})=\inf\\{\int_{0}^{1}\delta(\gamma(t),\gamma(t))(\gamma^{\prime}(t))dt\mid\gamma\in
C^{1}_{pw}([0,1],\Omega(\ell n)),\ell\in{\mathbb{N}}\\}.$

Here $C^{1}_{pw}([0,1],\Omega(n))$ stands for the piecewise continuously
differentiable paths in $\Omega(n)$. Namely $\gamma\colon[0,1]\to\Omega(n)$,
such that there exists finitely many $0=t_{1}<t_{2}<\cdots<t_{k}=1$, such that
$f$ is $C^{1}$ on every open segment $(t_{i},t_{i+1})$ and the one-sided
derivatives exist at all the $t_{i}$.

######  Lemma 3.9.

Let $\Omega\subset{\mathbb{M}}(E)$ be an open bounded matrix convex nc domain.
Let $V\colon{\mathbb{C}}^{k}\to{\mathbb{C}}^{n}$ and
$W\colon{\mathbb{C}}^{\ell}\to{\mathbb{C}}^{m}$ be isometries. Then, for every
${\mathbf{X}}\in\Omega(n)$, ${\mathbf{Y}}\in\Omega(m)$, and ${\mathbf{Z}}\in
M_{n,m}(E)$, we have

$\delta(V^{*}{\mathbf{X}}V,W^{*}{\mathbf{Y}}W)(V^{*}{\mathbf{Z}}W)\leq\delta({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}}).$

In particular, if ${\mathbf{X}},{\mathbf{Y}}\in\Omega(n)$, then for every
isometry $V\colon{\mathbb{C}}^{k}\to{\mathbb{C}}^{n}$:

$\widetilde{\delta}(V^{*}{\mathbf{X}}V,V^{*}{\mathbf{Y}}V)\leq\widetilde{\delta}({\mathbf{X}},{\mathbf{Y}}).$

###### Proof.

Note that for $t>0$

$\begin{pmatrix}V^{*}&0\\\
0&W^{*}\end{pmatrix}\begin{pmatrix}{\mathbf{X}}&t{\mathbf{Z}}\\\
0&{\mathbf{Y}}\end{pmatrix}\begin{pmatrix}V&0\\\
0&W\end{pmatrix}=\begin{pmatrix}V^{*}{\mathbf{X}}V&tV^{*}{\mathbf{Z}}W\\\
0&W^{*}{\mathbf{Y}}W\end{pmatrix}.$

Hence, if $\left(\begin{smallmatrix}{\mathbf{X}}&t{\mathbf{Z}}\\\
0&{\mathbf{Y}}\end{smallmatrix}\right)\in\Omega(n+m)$, then since $\Omega$ is
matrix convex and $V\oplus W$ is an isometry, we conclude that
$\left(\begin{smallmatrix}V^{*}{\mathbf{X}}V&tV^{*}{\mathbf{Z}}W\\\
0&W^{*}{\mathbf{Y}}W\end{smallmatrix}\right)\in\Omega(n+m)$. Hence

$A=\left\\{t\in[0,\infty]\mid\begin{pmatrix}{\mathbf{X}}&s{\mathbf{Z}}\\\
0&{\mathbf{Y}}\end{pmatrix}\in\Omega(n+m),\,\text{for all
}s\in[0,t)\right\\}\subseteq\\\
\left\\{t\in[0,\infty]\mid\begin{pmatrix}V^{*}{\mathbf{X}}V&sV^{*}{\mathbf{Z}}W\\\
0&W^{*}{\mathbf{Y}}W\end{pmatrix}\in\Omega(n+m),\,\text{for all
}s\in[0,t)\right\\}=B.$

Since
$\delta(V^{*}{\mathbf{X}}V,W^{*}{\mathbf{Y}}W)(V^{*}{\mathbf{Z}}W)=\left(\sup
B\right)^{-1}$ and $\delta({\mathbf{X}},{\mathbf{Y}})({\mathbf{Z}})=\left(\sup
A\right)^{-1}$, we get the desired inequality.

The second inequality follows immediately from the first. ∎

######  Corollary 3.10.

Let $\Omega$ be an open matrix convex domain and let
${\mathbf{X}},{\mathbf{Y}}\in\Omega(n)$, then

$\widetilde{d}_{\Omega}({\mathbf{X}},{\mathbf{Y}})=\widetilde{d}_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}}),\,d_{\Omega}({\mathbf{X}},{\mathbf{Y}})=d_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}}).$

###### Proof.

By definition
$\widetilde{d}_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})\leq\widetilde{d}_{\Omega}({\mathbf{X}},{\mathbf{Y}})$
and $d_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})\leq
d_{\Omega}({\mathbf{X}},{\mathbf{Y}})$ (see also [13]). Now let $\epsilon>0$
be arbitrary. We can find $\ell,m\in{\mathbb{N}}$ and
$I_{\ell}\otimes{\mathbf{X}}={\mathbf{Z}}_{0},{\mathbf{Z}}_{1},\ldots,{\mathbf{Z}}_{m}=I_{\ell}\otimes{\mathbf{Y}}\in\Omega(\ell
n)$, such that

$\sum_{k=1}^{m}\widetilde{\delta}({\mathbf{Z}}_{k-1},{\mathbf{Z}}_{k})<\widetilde{d}_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})+\epsilon.$

Let $V\colon{\mathbb{C}}^{n}\to{\mathbb{C}}^{\ell n}$ be the isometry that
embeds ${\mathbb{C}}^{n}$ as the subspace spanned by the first $n$ standard
basis vectors. Then $V^{*}{\mathbf{Z}}_{0}V={\mathbf{X}}$,
$V^{*}{\mathbf{Z}}_{m}V={\mathbf{Y}}$, and since $\Omega$ is matrix convex for
every $1\leq k\leq m-1$, $V^{*}{\mathbf{Z}}_{k}V\in\Omega(n)$. By Lemma 3.9 we
have that for every $1\leq k\leq m$,
$\widetilde{\delta}(V^{*}{\mathbf{Z}}_{k-1}V,V^{*}{\mathbf{Z}}_{k}V)\leq\widetilde{\delta}({\mathbf{Z}}_{k-1},{\mathbf{Z}}_{k})$
and thus

$\sum_{k=1}^{m}\widetilde{\delta}(V^{*}{\mathbf{Z}}_{k-1}V,V^{*}{\mathbf{Z}}_{k}V)<\widetilde{d}_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})+\epsilon.$

This proves that
$\widetilde{d}_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})=\widetilde{d}_{\Omega}({\mathbf{X}},{\mathbf{Y}})$.

Similarly, let $\gamma\in C^{1}_{pw}([0,1],\Omega(\ell n))$ be such that
$\gamma(0)=I_{\ell}\otimes{\mathbf{X}}$,
$\gamma(1)=I_{\ell}\otimes{\mathbf{Y}}$, and

$\int_{0}^{1}\delta(\gamma(t),\gamma(t))(\gamma^{\prime}(t))<d_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})+\epsilon.$

Let $V$ be as above and set $\beta=V^{*}\gamma V\in
C^{1}_{pw}([0,1],\Omega(n))$. Then, $\beta(0)={\mathbf{X}}$,
$\beta(1)={\mathbf{Y}}$, and $\beta^{\prime}(t)=V^{*}\gamma^{\prime}(t)V$.
Thus by Lemma 3.9,

$\int_{0}^{1}\delta(\beta(t),\beta(t))(\beta^{\prime}(t))\leq\int_{0}^{1}\delta(\gamma(t),\gamma(t))(\gamma^{\prime}(t))<d_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})+\epsilon.$

Thus
$d_{\Omega,\infty}({\mathbf{X}},{\mathbf{Y}})=d_{\Omega}({\mathbf{X}},{\mathbf{Y}})$.
∎

######  Lemma 3.11.

Let $\Omega$ be an open matrix convex domain. Let
${\mathbf{X}},{\mathbf{Y}}\in\Omega(n)$ and ${\mathbf{A}}\in M_{n}(E)$. Then
for every $0<t<1$

$\delta(t{\mathbf{X}}+(1-t){\mathbf{Y}},t{\mathbf{X}}+(1-t){\mathbf{Y}})({\mathbf{A}})\leq\frac{1}{2\sqrt{t(1-t)}}\max\\{\delta({\mathbf{X}},{\mathbf{Y}})({\mathbf{A}}),\delta({\mathbf{Y}},{\mathbf{X}})({\mathbf{A}})\\}.$
$\delta(t{\mathbf{X}}+(1-t){\mathbf{Y}},t{\mathbf{X}}+(1-t){\mathbf{Y}})({\mathbf{A}})\leq\max\\{\delta({\mathbf{X}},{\mathbf{X}})({\mathbf{A}}),\delta({\mathbf{Y}},{\mathbf{Y}})({\mathbf{A}})\\}.$

###### Proof.

Let us define for every $0<t<1$, an isometry
$V_{t}\colon{\mathbb{C}}^{n}\to{\mathbb{C}}^{2n}$:

$V_{t}=\begin{pmatrix}\sqrt{t}I_{n}\\\ \sqrt{1-t}I_{n}\end{pmatrix}$

Then

$t{\mathbf{X}}+(1-t){\mathbf{Y}}=V_{t}^{*}({\mathbf{X}}\oplus{\mathbf{Y}})V_{t}$

Applying Lemma 3.9 and [13, Lemma 3.5], we get

$\delta(V_{t}^{*}({\mathbf{X}}\oplus{\mathbf{Y}})V_{t},V_{t}^{*}({\mathbf{X}}\oplus{\mathbf{Y}})V_{t})\left(\frac{1}{2\sqrt{t(1-t)}}V_{t}^{*}\begin{pmatrix}0&{\mathbf{A}}\\\
{\mathbf{A}}&0\end{pmatrix}V_{t}\right)\leq\\\
\frac{1}{2\sqrt{t(1-t)}}\delta({\mathbf{X}}\oplus{\mathbf{Y}},{\mathbf{X}}\oplus{\mathbf{Y}})\left(\begin{pmatrix}0&{\mathbf{A}}\\\
{\mathbf{A}}&0\end{pmatrix}\right)=\frac{1}{2\sqrt{t(1-t)}}\max\\{\delta({\mathbf{X}},{\mathbf{Y}})({\mathbf{A}}),\delta({\mathbf{Y}},{\mathbf{X}})({\mathbf{A}})\\}.$

The proof of the second inequality is identical. We only note that
${\mathbf{A}}=V_{t}^{*}({\mathbf{A}}\oplus{\mathbf{A}})V_{t}$. ∎

######  Theorem 3.12.

Let $\Omega$ be an open matrix convex domain. Then for every
${\mathbf{X}},{\mathbf{Y}}\in U(n)$, we have that

$\widetilde{d}_{\Omega}({\mathbf{X}},{\mathbf{Y}})\leq
d_{\Omega}({\mathbf{X}},{\mathbf{Y}})\leq\pi\widetilde{d}_{\Omega}({\mathbf{X}},{\mathbf{Y}}).$

In other words, the metrics $d_{\Omega}$ and $\widetilde{d}_{\Omega}$ are
equivalent.

###### Proof.

By [13], $\widetilde{d}_{\Omega}({\mathbf{X}},{\mathbf{Y}})\leq
d_{\Omega}({\mathbf{X}},{\mathbf{Y}})$. Now let $\epsilon>0$ and find
${\mathbf{X}}={\mathbf{Z}}_{0},{\mathbf{Z}}_{1},\ldots,{\mathbf{Z}}_{m}={\mathbf{Y}}\in\Omega(n)$,
such that

$\sum_{k=1}^{m}\widetilde{\delta}({\mathbf{Z}}_{k-1},{\mathbf{Z}}_{k})<\widetilde{d}_{\Omega}({\mathbf{X}},{\mathbf{Y}})+\frac{\epsilon}{\pi}.$

Now consider the paths
$\beta_{k}(t)=t{\mathbf{Z}}_{k}+(1-t){\mathbf{Z}}_{k-1}$ and concatenate them
to obtain a piecewise smooth path $\beta\colon[0,1]\to\Omega(n)$, such that
$\beta(0)={\mathbf{X}}$ and $\beta(1)={\mathbf{Y}}$. By Lemma 3.11 we have
that

$\int_{0}^{1}\delta(\beta_{k}(t),\beta_{k}(t))(\beta_{k}^{\prime}(t))dt\leq\widetilde{\delta}({\mathbf{Z}}_{k-1},{\mathbf{Z}}_{k})\int_{0}^{1}\frac{dt}{2\sqrt{t(1-t)}}=\pi\widetilde{\delta}({\mathbf{Z}}_{k-1},{\mathbf{Z}}_{k})$

Thus,

$\int_{0}^{1}\delta(\beta(t),\beta(t))(\beta^{\prime}(t))dt<\pi\widetilde{d}_{\Omega}({\mathbf{X}},{\mathbf{Y}})+\epsilon.$

Since $\epsilon$ was arbitrary, we get the result. ∎

Now we collect a few useful properties of the distance $d_{\Omega}$.

######  Proposition 3.13.

Let $\Omega$ be a uniformly bounded and open matrix convex domain and
${\mathbf{X}},{\mathbf{Y}},{\mathbf{Z}},{\mathbf{W}}\in\Omega(n)$. Then

  * (i)

For $V\in U_{n}$ a unitary, we have
$d_{\Omega}(V^{*}{\mathbf{X}}V,V^{*}{\mathbf{Y}}V)=d_{\Omega}({\mathbf{X}},{\mathbf{Y}})$.

  * (ii)

If $V\colon{\mathbb{C}}^{k}\to{\mathbb{C}}^{n}$ is an isometry, then
$d_{\Omega}(V^{*}{\mathbf{X}}V,V^{*}{\mathbf{Y}}V)\leq
d_{\Omega}({\mathbf{X}},{\mathbf{Y}})$

  * (iii)

$d_{\Omega}({\mathbf{X}}\oplus{\mathbf{Y}},{\mathbf{Z}}\oplus{\mathbf{W}})\geq\max\\{d_{\Omega}({\mathbf{X}},{\mathbf{Z}}),d_{\Omega}({\mathbf{Y}},{\mathbf{W}})\\}$.

###### Proof.

Given $\epsilon>0$, let $\gamma\colon[0,1]\to\Omega(n)$ be a piecewise
continuously differentiable path, such that $\gamma(0)={\mathbf{X}}$,
$\gamma(1)={\mathbf{Y}}$, and

$\int_{0}^{1}\delta(\gamma(t),\gamma(t))(\gamma^{\prime}(t))dt<d_{\Omega}({\mathbf{X}},{\mathbf{Y}})+\epsilon.$

Now note that
$\delta(V^{*}\gamma(t)V,V^{*}\gamma(t)V)(V^{*}\gamma^{\prime}(t)V)=\delta(\gamma(t),\gamma(t))(\gamma^{\prime}(t))$
by [13]. Thus taking the path $V^{*}\gamma V$, we get that
$d_{\Omega}(V^{*}{\mathbf{X}}V,V^{*}{\mathbf{Y}}V)<d_{\Omega}({\mathbf{X}},{\mathbf{Y}})+\epsilon$.
Since $\epsilon>0$ was arbitrary, we get that
$d_{\Omega}(V^{*}{\mathbf{X}}V,V^{*}{\mathbf{Y}}V)\leq
d_{\Omega}({\mathbf{X}},{\mathbf{Y}})$. Since $\operatorname{Ad}_{V^{*}}$ is
invertible, we get the converse inequality.

The second item follows immediately from Lemma 3.9. For the third item, let
$V\colon{\mathbb{C}}^{n}\to{\mathbb{C}}^{2n}$ be an isometry that embeds
${\mathbb{C}}^{n}$ as the span of the first $n$ standard basis vectors. Then

$d_{\Omega}({\mathbf{X}},{\mathbf{Z}})=d_{\Omega}(V^{*}({\mathbf{X}}\oplus{\mathbf{Y}})V,V^{*}({\mathbf{Z}}\oplus{\mathbf{W}})V)\leq
d_{\Omega}({\mathbf{X}}\oplus{\mathbf{Y}},{\mathbf{Z}}\oplus{\mathbf{W}}).$

Similarly $d_{\Omega}({\mathbf{Y}},{\mathbf{W}})\leq
d_{\Omega}({\mathbf{X}}\oplus{\mathbf{Y}},{\mathbf{Z}}\oplus{\mathbf{W}})$. ∎

##  4\. Noncommutative Denjoy-Wolff theorems

Let $f\colon{\mathbb{D}}\to{\mathbb{D}}$ be a holomorphic function that is not
an automorphism and has no fixed points. In order to prove the classical
Denjoy-Wolff theorem, one applies the Montel theorem to obtain convergent
subsequences of the sequence $f_{n}$ of iterations of $f$. We can then split
the rest of the proof into two pieces. In the first part, we apply the maximum
modulus principle to show that a partial limit of the sequence $f_{n}$ is
either constant or maps ${\mathbb{D}}$ into itself. Then one applies the Wolff
lemma to show that in the constant case, the value is the Wolff point of $f$
and works to rule out the second option. This observation leads us to try and
extend the Denjoy-Wolff-type result of Abate and Raissy to the noncommutative
setting.

######  Definition 4.1.

Let $\Omega\subset{\mathbb{M}}_{d}$ be an nc domain. We will say that $\Omega$
has a simple nc boundary if for every nc holomorphic function
$f\colon\Omega^{\prime}\subset{\mathbb{M}}_{k}\to\overline{\Omega}$, if
$f(\Omega^{\prime})\cap\partial\Omega\neq\emptyset$, then $f$ is constant.

######  Example 4.2.

By the maximum modulus principle [35, Lemma 6.1], the row ball
${\mathfrak{B}}_{d}$ has a simple nc boundary.

The following is an extension of the result of Abate and Raissy [3, Theorem 2]
to the noncommutative setting.

######  Theorem 4.3.

Let $\Omega\subset{\mathbb{M}}_{d}$ be a uniformly bounded matrix convex
domain with a simple nc boundary. Assume that $\Omega(1)$ is strictly convex.
Let $f\colon\Omega\to\Omega$ be an nc map with no fixed points. Then there
exists $\zeta\in\partial\Omega(1)$, such that the sequence of iterates
$f^{\circ\,n}$ converges to the constant function $\zeta$, where

$f^{\circ\,n}=\underbrace{f\circ f\circ\cdots\circ f}_{n\text{ times}}.$

###### Proof.

By [3, Theorem 2], the sequence of iterates
$(f^{\circ\,n})_{1}=f_{1}^{\circ\,n}$ converges to the constant function
$\zeta$. Now applying Montel’s theorem to $f^{\circ\,n}$, we know that the
sequence has convergent subsequences. Let $f^{\circ\,n_{k}}$ be a convergent
subsequence and $g$ its limit. Since $g_{1}=\zeta$, by the fact that $\Omega$
has a simple nc boundary, $g$ is constant and equals $\zeta$. Hence, every
convergent subsequence converges to the same function, and thus,
$f^{\circ\,n}$ converges to $\zeta$ uniformly on every compact. ∎

######  Theorem 4.4.

Let $f\colon{\mathfrak{B}}_{d}\to{\mathfrak{B}}_{d}$ be an nc function without
fixed points, then the sequence of iterates $f^{\circ\,n}$ converges uniformly
on subball $r\overline{{\mathfrak{B}}_{d}}$ to the constant function $\zeta$,
where $\zeta$ is the Denjoy-Wolff point of $f_{1}=f|_{{\mathfrak{B}}_{d}(1)}$.

###### Proof.

By Popescu’s Montel theorem [28, Theorem 5.2], the sequence $f^{\circ\,n}$ has
subsequences that converge uniformly on subballs. Hence, we can argue as
above, but the convergence is on subballs. ∎

###  4.1. Maximal quantization of the unit ball

Let $K\subset{\mathbb{C}}^{d}$ be a bounded open convex set and assume for
simplicity that $0\in K$. By the Hahn-Banach separation theorem, for every
$z\in{\mathbb{C}}^{d}\setminus K$, there exists $w\in{\mathbb{C}}^{d}$ and
$s\in{\mathbb{R}}$, such that for every $u\in K$, $\operatorname{Re}\langle
u,w\rangle<s\leq\operatorname{Re}\langle z,w\rangle$. Since necessarily $s>0$,
we may replace $w$ by $\frac{1}{s}w$ and set $s=1$. Hence, $K$ can be
identified as an intersection of open half-planes. Let us write
$K^{\circ}=\left\\{w\in{\mathbb{C}}^{d}\mid\forall u\in
K,\,\operatorname{Re}\langle u,w\rangle\leq 1\right\\}$. As in [14] (in the
compact case), we define the maximal matrix convex set that contains $K$ by
$\operatorname{Max}(K)=\sqcup_{n\in{\mathbb{N}}}\operatorname{Max}(K)(n)$,
where

$\operatorname{Max}(K)(n)=\left\\{{\mathbf{Z}}\in{\mathbb{M}}_{d}(n)\mid\forall
w\in
K^{\circ},\,\operatorname{Re}\left(\sum_{j=1}^{d}\overline{w_{j}}Z_{j}\right)<1\right\\}.$

From now will focus on $\operatorname{Max}({\mathbb{B}}_{d})$. Let
$\overline{\operatorname{Max}({\mathbb{B}}_{d})}$ be the levelwise closure of
$\operatorname{Max}({\mathbb{B}}_{d})$. Note that by continuity of the
functionals,
$\overline{\operatorname{Max}({\mathbb{B}}_{d})}\subset\operatorname{Max}(\overline{{\mathbb{B}}_{d}})$.
Conversely, if $Z\in\operatorname{Max}(\overline{{\mathbb{B}}_{d}})$, then for
every $0<r<1$, $rZ\in\operatorname{Max}({\mathbb{B}}_{d})$ and, thus, these
sets coincide. We aim to show that the Denjoy-Wolff theorem holds for
$\operatorname{Max}({\mathbb{B}}_{d})$. One of the key ingredients is the
maximal modulus principle. We need some preparation to prove a version of the
maximum modulus principle for $\operatorname{Max}({\mathbb{B}}_{d})$.

######  Lemma 4.5.

Let $\Omega\subset{\mathbb{M}}_{d}$ be an open nc set and
$f\colon\Omega\to{\mathbb{M}}_{1}$ be a locally bounded nc function. If
$\operatorname{Re}f({\mathbf{W}})=I$ for all ${\mathbf{W}}\in\Omega$, then $f$
is constant.

###### Proof.

Since $f$ is levelwise holomorphic and $\Delta
f({\mathbf{W}},{\mathbf{W}})({\mathbf{Z}})$ is the directional derivative at
${\mathbf{W}}$. It suffices to prove that for every ${\mathbf{W}}\in\Omega$
and ${\mathbf{Z}}\in{\mathbb{M}}_{d}$, $\Delta
f({\mathbf{W}},{\mathbf{W}})({\mathbf{Z}})=0$. Note that since $\Omega$ is
open there exists $t>0$, such that
$\left(\begin{smallmatrix}{\mathbf{W}}&t{\mathbf{Z}}\\\
0&{\mathbf{W}}\end{smallmatrix}\right)\in\Omega$. Hence,

$\begin{pmatrix}I&0\\\
0&I\end{pmatrix}=\operatorname{Re}f\left(\begin{pmatrix}{\mathbf{W}}&{\mathbf{Z}}\\\
0&{\mathbf{W}}\end{pmatrix}\right)=\begin{pmatrix}I&\frac{1}{2}\Delta
f({\mathbf{W}},{\mathbf{W}})({\mathbf{Z}})\\\ \frac{1}{2}\Delta
f({\mathbf{W}},{\mathbf{W}})({\mathbf{Z}})^{*}&I\end{pmatrix}.$

∎

######  Lemma 4.6.

Let ${\mathbf{X}}\in\operatorname{Max}(\overline{{\mathbb{B}}_{d}})$ and
assume that $X_{1}=I$. Then $X_{2}=\cdots=X_{d}=0$.

###### Proof.

Since $\operatorname{Max}(\overline{{\mathbb{B}}_{d}})$ is matrix convex, for
every unit vector $u\in{\mathbb{C}}^{n}$, we have that
$u^{*}{\mathbf{X}}u=(1,\langle X_{2}u,u\rangle,\cdots,\langle
X_{d}u,u\rangle)\in\overline{{\mathbb{B}}_{d}}$. Which implies that the
numerical ranges of $X_{2},\,\ldots,\,X_{d}$ are $\\{0\\}$ and thus these
matrices are zero.

∎

The following theorem is a version of the maximum modulus from [35] for the
set $\operatorname{Max}({\mathbb{B}}_{d})$.

######  Theorem 4.7.

Let $\Omega\subset{\mathbb{M}}_{k}$ be an open unitarily-invariant nc set and
let $f\colon\Omega\to\operatorname{Max}(\overline{{\mathbb{B}}_{d}})$ be a
locally bounded holomorphic function. If for some ${\mathbf{W}}_{0}\in\Omega$,
$f({\mathbf{W}}_{0})\in\partial\operatorname{Max}(\overline{{\mathbb{B}}_{d}})$,
then $f$ is constant.

###### Proof.

Assume ${\mathbf{W}}_{0}\in\Omega(n)$. Since
$f({\mathbf{W}}_{0})\in\partial\operatorname{Max}(\overline{{\mathbb{B}}_{d}})$,
there exists $\lambda\in{\mathbb{C}}^{d}$ with $\|\lambda\|=1$, such that
$\operatorname{Re}\left(\sum_{j=1}^{d}\overline{\lambda_{j}}f_{j}({\mathbf{W}}_{0})\right)$
has $1$ in its spectrum. Set
$g({\mathbf{W}})=\sum_{j=1}^{d}\overline{\lambda_{j}}f_{j}({\mathbf{W}})$ and
let $u$ be a unit vector, such that $\operatorname{Re}g({\mathbf{W}}_{0})u=u$.
Let $k\in{\mathbb{N}}$ and define $\varphi_{k}\colon\Omega(nk)\to{\mathbb{R}}$
by $\varphi_{k}({\mathbf{W}})=\operatorname{Re}\langle
g({\mathbf{W}})u^{(k)},u^{(k)}\rangle$, where
$u^{(k)}=\frac{1}{\sqrt{k}}\left(\begin{smallmatrix}u\\\ \vdots\\\
u\end{smallmatrix}\right)\in{\mathbb{C}}^{nk}$. Since
$\operatorname{Re}g({\mathbf{W}})\leq I$ for all ${\mathbf{W}}\in\Omega$, we
have that $\varphi_{k}({\mathbf{W}})\leq 1$. Moreover, by assumption
$\varphi_{k}(I_{k}\otimes{\mathbf{W}}_{0})=1$. Since $\varphi_{k}$ is
pluriharmonic, we can apply the classical maximum modulus to deduce that
$\varphi_{k}$ is constant. Hence, for every ${\mathbf{W}}\in\Omega(nk)$, we
have that $\operatorname{Re}\langle g({\mathbf{W}})u,u\rangle=1$. By unitary
invariance of $\Omega$, we have that for every unitary $nk\times nk$ matrix
$V$, we have that

$\operatorname{Re}\langle
g({\mathbf{W}})Vu^{(k)},g({\mathbf{W}})Vu^{(k)}\rangle=\operatorname{Re}\langle
g(V^{*}{\mathbf{W}}V)u^{(k)},u^{(k)}\rangle=1.$

In particular, the numerical range of $\operatorname{Re}g({\mathbf{W}})$ is
the singleton $1$, i.e, $\operatorname{Re}g({\mathbf{W}})=I$. Now we can apply
Lemma 4.5 to deduce that the restriction of $g$ to levels that are multiples
of $n$ is constant. However, for every $m$ not divisible by $n$, we have that
$\Omega(m)$ can be embedded into $\Omega(nm)$ via direct sums, and thus $g$ is
constant.

Now note that if $U\in M_{d}$ is a unitary, then for
${\mathbf{W}}\in\operatorname{Max}(\overline{{\mathbb{B}}_{d}})$, we have
$U({\mathbf{W}})\in\operatorname{Max}(\overline{{\mathbb{B}}_{d}})$. Hence,
applying a unitary, we may assume that $\lambda$ is the first vector of the
standard basis. Hence, we have that $f_{1}=1$. By Lemma 4.6, we have that
$f_{2}=\cdots=f_{d}=0$. ∎

######  Corollary 4.8.

Let
$f\colon\operatorname{Max}({\mathbb{B}}_{d})\to\operatorname{Max}({\mathbb{B}}_{d})$
be an nc map without fixed points. Then, the sequence of iterates of $f$
converges to the constant function $\zeta$, where $\zeta$ is the Denjoy-Wolff
point of $f_{1}$.

###### Proof.

The above theorem shows that $\operatorname{Max}({\mathbb{B}}_{d})$ has a
simple nc boundary. Thus, the result is a consequence of Theorem 4.3 ∎

######  Question 4.9.

Does every quantization of ${\mathbb{B}}_{d}$ have a simple nc boundary?

##  5\. Noncommutative Vigué and Bedford theorems

In [43], Vigué has proved that for a bounded convex domain
$U\subset{\mathbb{C}}^{d}$, the set of fixed points of an analytic self-map is
a smooth analytic subvariety. Moreover, a smooth analytic subvariety $V\subset
U$ is the set of fixed points of some analytic self-map if and only if $V$ is
a holomorphic retract of $U$, i.e., there exists $\psi\colon U\to V$ analytic,
such that $\psi|_{V}=\mathrm{id}_{V}$. Our goal here is to show that Vigué’s
methods extend almost verbatim to the setting of bounded matrix convex sets.

By [2, Proposition 2.1.8] [2, Corollary 2.1.11], every bounded convex domain
$W\subset{\mathbb{C}}^{n}$ is both taut and tautly embedded in
${\mathbb{C}}^{n}$, Hence, given a sequence of maps $f_{n}\colon W\to W$ that
converges uniformly on compacta to $f$, we have that $f(W)\subset W$ or
$f(W)\subset\partial W$.

Note that if $\Omega\subset{\mathbb{M}}_{d}$ is matrix convex, then for
decomposable point in $\Omega$, the restriction to the invariant subspaces and
the compression to coinvariant subspaces are in $\Omega$. We say that an nc
subset $\Xi\subset\Omega$ is (relatively) bifull if it is closed under
similarities, restrictions to invariant subspaces, and compressions to
coinvariant subspaces (see [11]). To be more precise,“closed under
similarities” means that for every $n\in{\mathbb{N}}$, $X\in\Xi(n)$, and every
$S\in\mathrm{GL}_{n}({\mathbb{C}})$, such that $S^{-1}XS\in\Omega$, we have
that $S^{-1}XS\in\Xi$.

######  Theorem 5.1 (Noncommutative Vigué’s theorem).

Let $\Omega\subset{\mathbb{M}}_{d}$ be a bounded matrix convex set. Let
${\mathfrak{V}}\subset\Omega$ be an nc relatively bifull subset, such that for
every $n\in{\mathbb{N}}$, ${\mathfrak{V}}(n)$ is a smooth analytic subvariety
of $\Omega(n)$. Then, ${\mathfrak{V}}$ is the fixed point set of an nc map
$f\colon\Omega\to\Omega$, if and only if ${\mathfrak{V}}$ is the image on an
nc holomorphic retraction, i.e., there exists $\psi\colon\Omega\to\Omega$,
such that $\psi\circ\psi=\psi$ and $\psi(\Omega)={\mathfrak{V}}$.

###### Proof.

Just like classically, one direction is easy. If there exists an nc
holomorphic retraction $\psi\colon\Omega\to{\mathfrak{V}}$, then
${\mathfrak{V}}$ is the fixed point set of $\psi$. Moreover, every fixed point
set is level-wise smooth by the classical theorem of Vigué and is bifull by
basic properties of nc functions.

For the second direction, we can apply Vigué’s proof of his theorem and only
verify small details. Let $f$ be an nc map, so ${\mathfrak{V}}$ is the fixed
point set of $f$. Write $f_{0}=\mathrm{id}_{U}$ and $f_{n}=f\circ f_{n-1}$,
for $n\in{\mathbb{N}}$. Consider the sequence
$\varphi_{m}=\frac{1}{m}\sum_{k=0}^{m-1}f_{k}$. Since each $\Omega(n)$ is
convex, each $\varphi_{m}$ is a self-map of $\Omega$. Passing to a
subsequence, we may assume that $\varphi_{m}$ converges level-wise on compacta
to an nc function $\varphi$. The fixed point set of $\varphi$ contains
${\mathfrak{V}}$. Again, since each $\Omega(n)$ is convex and bounded, it is
taut, and thus $\varphi$ is also a self-map of $\Omega$. Now let $\psi$ be a
partial limit of the sequence of iterates of $\varphi$. Refining the
subsequence, if necessary, we see that $\psi$ is a retraction. Arguing as in
the proof of [43, Theorem 6.5], we see that level-wise $\psi_{n}$ maps
$\Omega(n)$ into ${\mathfrak{V}}(n)$ and we are done. ∎

######  Remark 5.2.

One can also argue level-wise instead of discussing nuclearity. Since on each
level the Montel theorem holds, if we have a sequence of nc functions
$f_{n}\colon\Omega\to\Omega$, then we can choose a subsequence
$f_{n_{k}^{(1)}}$ that converges on the first level and a subsequence of it
$f_{n_{k}^{(2)}}$ that converges on the second level and so on. Now we set
$g_{k}=f_{n_{k}^{(k)}}$, and this sequence converges on all levels. The limit
is, of course, an nc function.

######  Corollary 5.3.

Let $\Omega$ be a bounded matrix convex set, and let $f\colon\Omega\to\Omega$
be a nc map. Then the fixed point set of $f$ contains scalar points. In other
words, if $f$ has no fixed points on the first level, then $f$ has no fixed
points.

###### Proof.

Let ${\mathfrak{V}}$ be the collection of fixed points of $f$. By the nc Vigué
theorem, we have that ${\mathfrak{V}}$ is the image of some nc holomorphic
retraction. Since nc maps are graded, we have that
${\mathfrak{V}}(1)\neq\emptyset$. ∎

In [12], Bedford proved an important dichotomy for converging iterates of
self-maps of a taut domain. The following result is a straightforward
generalization of Bedford’s theorem to the setting of uniformly bounded matrix
convex domains.

######  Theorem 5.4 (Noncommutative Bedford theorem).

Let $\Omega\subset{\mathbb{M}}_{d}$ be a uniformly bounded matrix convex
domain. Let $f\colon\Omega\to\Omega$ be an nc self-map. Let $f^{\circ
n}=\underbrace{f\circ f\circ\cdots\circ f}_{n\text{ times}}$ be the sequence
of iterates of $f$. Suppose the sequence $f^{\circ n_{k}}$ converges to $g$
level-wise on compacta. In that case, we have that either for every
$n\in{\mathbb{N}}$, $g(\Omega)\subset\partial\Omega$ or there exists an nc
analytic subvariety of ${\mathfrak{V}}\subset\Omega$, an nc automorphism
$\varphi\colon{\mathfrak{V}}\to{\mathfrak{V}}$ and a holomorphic retraction
$\psi\colon\Omega\to{\mathfrak{V}}$, such that $g=\varphi\circ\psi$.

###### Proof.

Due to tautness on every level, $g_{n}(\Omega(n))$ is either a subset of
$\partial\Omega(n)$ or of $\Omega(n)$. Now assume that there exists
$n_{0}\in{\mathbb{N}}$, such that
$g_{n_{0}}(\Omega(n_{0}))\subset\partial\Omega(n_{0})$. Let
${\mathbf{X}}\in\Omega(1)$, then $I_{n_{0}}\otimes
g({\mathbf{X}})=g(I_{n_{0}}\otimes{\mathbf{X}})\in\partial\Omega(n_{0})$.
Since $\Omega$ is closed under direct sums and direct summands,
$g({\mathbf{X}})\in\partial\Omega(1)$. By tautness
$g(\Omega(1))\subset\partial\Omega(1)$. Thus, for every $n\in{\mathbb{N}}$,
$I_{n}\otimes g({\mathbf{X}})\in\partial\Omega(n)$ and applying tautness again
we conclude that $g_{n}(\Omega(n))\subset\partial\Omega(n)$.

Assume that $g(\Omega)\subset\Omega$. The proof proceeds as the proof of [12,
Theorem 1.1]. We record the proof here for the sake of completeness. Passing
to subsequences, we may assume that $m_{k}=n_{k+1}-n_{k}$ and
$\ell_{k}=n_{k+1}-2n_{k}$ are strictly increasing, and the sequences $f^{\circ
m_{k}}$ and $f^{\circ\ell_{k}}$ converge (using the fact that the space of all
level-wise holomorphic nc functions is a Montel space). Denote

$\lim_{k\to\infty}f^{\circ
m_{k}}=\psi,\,\lim_{k\to\infty}f^{\circ\ell_{k}}=\tau.$

Since $f^{\circ m_{k}}\circ f^{\circ n_{k}}=f^{\circ n_{k+1}}=f^{\circ
n_{k}}\circ f^{\circ m_{k}}$, we have that $\psi\circ g=g=g\circ\psi$.
Similarly, $\tau\circ g=\psi=g\circ\tau$. In particular, $\psi$ fixes the
image of $g$. Applying the argument above we conclude that
$\psi(\Omega)\subset\Omega$ and $\tau(\Omega)\subset\Omega$. Set
${\mathfrak{V}}=\operatorname{Fix}(\psi)$ and note that
$g(\Omega)\subset{\mathfrak{V}}$. For every $n\in{\mathbb{N}}$ and every
${\mathbf{Z}}\in\Omega(n)$, we have $\ker Dg_{n}({\mathbf{Z}})=\ker
D\psi_{n}({\mathbf{Z}})$ since by the chain rule
$Dg_{n}(\psi_{n}({\mathbf{Z}}))D\psi_{n}({\mathbf{Z}})=Dg_{n}({\mathbf{Z}})$
and
$D\psi_{n}({\mathbf{Z}})=D\tau_{n}(g_{n}({\mathbf{Z}}))Dg_{n}({\mathbf{Z}})$.
Note that ${\mathfrak{W}}(n)=\psi_{n}^{-1}({\mathfrak{V}}(n))$ is an analytic
subvariety of $\Omega(n)$ cut out by
$\psi(\psi({\mathbf{Z}}))-\psi({\mathbf{Z}})$. The dimension of the fiber of
${\mathfrak{W}}(n)$ over every point ${\mathbf{Z}}\in g_{n}(\Omega(n))$ is
precisely $dn^{2}-\dim\ker D\psi_{n}({\mathbf{Z}})=dn^{2}-\dim\ker
Dg_{n}({\mathbf{Z}})$. Thus ${\mathfrak{W}}(n)$ is of dimension $dn^{2}$, and
we conclude that $\psi$ is a holomorphic retract with image ${\mathfrak{V}}$.
Hence ${\mathfrak{V}}$ is a smooth nc analytic subvariety of $\Omega$.

Lastly note that for every ${\mathbf{Z}}\in{\mathfrak{V}}$,
$\tau(g({\mathbf{Z}}))=g(\tau({\mathbf{Z}}))=\psi({\mathbf{Z}})={\mathbf{Z}}$.
Hence $g({\mathfrak{V}})={\mathfrak{V}}$ and $g|_{{\mathfrak{V}}}$ is an
automorphism of ${\mathfrak{V}}$ with inverse $\tau|_{{\mathfrak{V}}}$. Set
$\varphi=g|_{{\mathfrak{V}}}$ and thus $g=\varphi\circ\psi$. ∎

###  5.1. Application to the isomorphism problem

Recall that $H^{\infty}({\mathfrak{B}}_{d})$ is the operator algebra of
bounded nc analytic functions on ${\mathfrak{B}}_{d}$. Moreover,
$H^{\infty}({\mathfrak{B}}_{d})$ is the multiplier algebra of the
noncommutative reproducing kernel Hilbert space of the nc Szego kernel. This
implies that $H^{\infty}({\mathfrak{B}}_{d})$ is a dual operator algebra [35].
The Hilbert space that $H^{\infty}({\mathfrak{B}}_{d})$ acts upon can be
identified with the full Fock space ${\mathcal{F}}^{2}_{d}$ [10, 35]. Thus,
$H^{\infty}({\mathfrak{B}}_{d})$ is unitarily equivalent to the weak-* closed
subalgebra of operators on the Fock space ${\mathcal{F}}^{2}_{d}$, generated
by the left creation operators (the regular free semigroup algebra in the
language of Davidson and Pitts [16, 17, 18]).

Given a subset of bounded nc functions $S\subset
H^{\infty}({\mathfrak{B}}_{d})$ we consider the nc analytic subvariety cut out
by $S$:

${\mathfrak{V}}(S)=\left\\{{\mathbf{Z}}\in{\mathfrak{B}}_{d}\mid\forall f\in
S,\,f({\mathbf{Z}})=0\right\\}.$

It is immediate that ${\mathfrak{V}}(S)={\mathfrak{V}}({\mathcal{I}})$, where
${\mathcal{I}}$ is the weak-* closed two-sided ideal generated by $S$.
Conversely, given a set of points ${\mathfrak{S}}\subset{\mathfrak{B}}_{d}$,
we define

${\mathcal{I}}({\mathfrak{S}})=\left\\{f\in H^{\infty}({\mathfrak{B}}_{d})\mid
f|_{{\mathfrak{S}}}=0\right\\}.$

Clearly, ${\mathcal{I}}({\mathfrak{S}})$ is weak-* closed two sided ideal in
$H^{\infty}({\mathfrak{B}}_{d})$. In this section, we say that
${\mathfrak{V}}\subset{\mathfrak{B}}_{d}$ is an nc analytic subvariety if
${\mathfrak{V}}({\mathcal{I}}({\mathfrak{V}}))={\mathfrak{V}}$ (in other
words, ${\mathfrak{V}}$ is Zariski closed in the appropriate sense).

Given a non-empty analytic subvariety
${\mathfrak{V}}\subset{\mathfrak{B}}_{d}$, we have that
$H^{\infty}({\mathfrak{B}}_{d})/{\mathcal{I}}({\mathfrak{V}})$ is the algebra
of bounded nc analytic functions on ${\mathfrak{V}}$. It can also be
identified with the left multiplier algebra of an appropriate nc reproducing
kernel Hilbert space on ${\mathfrak{V}}$. We will denote this algebra by
$H^{\infty}({\mathfrak{V}})$.

The following results are necessary for our discussion:

######  Theorem 5.5 ([35, Theorem 6.12, Corollary 6.14]).

Let ${\mathfrak{V}}\subset{\mathfrak{B}}_{d}$ and
${\mathfrak{W}}\subset{\mathfrak{B}}_{e}$ be two nc analytic varieties. Let
$\alpha\colon H^{\infty}({\mathfrak{V}})\to H^{\infty}({\mathfrak{W}})$ be a
completely isometric isomorphism. Then there exists nc maps
$g\colon{\mathfrak{B}}_{e}\to{\mathfrak{B}}_{d}$ and
$f\colon{\mathfrak{B}}_{d}\to{\mathfrak{B}}_{e}$, such that

$f\circ g|_{{\mathfrak{W}}}=\operatorname{id}_{{\mathfrak{W}}},\quad g\circ
f|_{{\mathfrak{V}}}=\operatorname{id}_{{\mathfrak{V}}}.$

Moreover, composition with $g$ implements $\alpha$.

######  Theorem 5.6 ([37, Theorem 3.12]).

Let $f\colon{\mathfrak{B}}_{d}\to{\mathfrak{B}}_{d}$ be an nc map, such that
$f(0)=0$. Let $\operatorname{Fix}(f)$ be the set of fixed points of $f$. If
$V\subset{\mathbb{C}}^{d}$ is a linear subspace, such that
$V\cap{\mathfrak{B}}_{d}(1)=\operatorname{Fix}(f)(1)$, then for every
$n\in{\mathbb{N}}$ $\operatorname{Fix}(f)(n)=\left(V\otimes
M_{n}\right)\cap{\mathfrak{B}}_{d}(n)$.

This theorem can be read differently. Recall that for an analytic self-map of
${\mathfrak{B}}_{d}(1)$ that fixes the origin, the fixed point set coincides
with the fixed subspace of $f^{\prime}(0)$ intersected with
${\mathfrak{B}}_{d}(1)$. Since for every $n\in{\mathbb{N}}$, $\Delta
f(0_{n},0_{n})=f^{\prime}(0)\otimes I_{M_{n}}$ is the derivative of $f_{n}$ at
$0_{n}$. We see that the fixed point set of $f_{n}$ is precisely the set of
fixed points of $\Delta f(0_{n},0_{n})$ intersected with
${\mathfrak{B}}_{d}(n)$.

Now with these theorems at hand, we can prove the following theorem, which is
a strengthening of both [37, Theorem 4.1] and [19, Theorem 4.5]. Recall that
for $S\subset{\mathbb{M}}_{d}$, the matrix span of $S$ is $\operatorname{mat-
span}(S)=\sqcup_{n\in{\mathbb{N}}}\operatorname{mat-span}(S)(n)$, where

$\operatorname{mat-span}(S)(n)=\operatorname{Span}\left\\{(I_{d}\otimes
T)({\mathbf{X}})\mid{\mathbf{X}}\in S(n),\,T\in{\mathcal{L}}(M_{n})\right\\}.$

######  Theorem 5.7.

Let ${\mathfrak{V}}\subset{\mathfrak{B}}_{d}$ and
${\mathfrak{W}}\subset{\mathfrak{B}}_{e}$ be two nc analytic varieties. If
$H^{\infty}({\mathfrak{V}})$ and $H^{\infty}({\mathfrak{W}})$ are completely
isometrically isomorphic, then there exists $k\in{\mathbb{N}}$ and an
automorphism $F\colon{\mathfrak{B}}_{k}\to{\mathfrak{B}}_{k}$, such that
${\mathfrak{V}},{\mathfrak{W}}\subset{\mathfrak{B}}_{k}$ and $F$ maps
${\mathfrak{V}}$ onto ${\mathfrak{W}}$.

###### Proof.

Applying [35, Lemma 8.2] we can may assume that for $n>>0$, we have that
$\operatorname{mat-span}({\mathfrak{V}})(n)={\mathbb{C}}^{d}\otimes M_{n}$ and
$\operatorname{mat-span}({\mathfrak{W}})(n)={\mathbb{C}}^{e}\otimes M_{n}$.
Let $g\colon{\mathfrak{B}}_{e}\to{\mathfrak{B}}_{d}$ and
$f\colon{\mathfrak{B}}_{e}\to{\mathfrak{B}}_{d}$ be the nc maps guaranteed by
Theorem 5.5. Set $h=g\circ f$ and observe that
${\mathfrak{V}}\subset\operatorname{Fix}(h)$. However, by Corollary 5.3,
$\operatorname{Fix}(h)(1)\neq\emptyset$. Pre- and post-composing with
automorphisms of ${\mathfrak{B}}_{d}$ (they induce unitary equivalence on
$H^{\infty}({\mathfrak{B}}_{d})$), we may assume that
$0\in\operatorname{Fix}(h)(1)$. Thus, by Theorem 5.6, there exists a subspace
$V\subset{\mathbb{C}}^{d}$, such that $\operatorname{Fix}(h)(n)=V\otimes
M_{n}$. As mentioned above, $\operatorname{Fix}(h)(n)$ coincides with the
fixed points of the linear operator $\Delta
f(0_{n},0_{n})=f^{\prime}(0)\otimes I_{M_{n}}$. Therefore, $\operatorname{mat-
span}({\mathfrak{V}})(n)\cap{\mathfrak{B}}_{d}(n)\subset\operatorname{Fix}(h)(n)$.
Thus, by assumption on ${\mathfrak{V}}$,
$\operatorname{Fix}(h)(n)={\mathbb{C}}^{d}\otimes M_{n}$, for every
$n\in{\mathbb{N}}$. Applying the same argument to $f\circ g$, we see that
$d=e$, and $f$ and $g$ are automorphisms of ${\mathfrak{B}}_{d}$ that map
${\mathfrak{V}}$ onto ${\mathfrak{W}}$ and vice versa. ∎

##  6\. Horospheres in generalized balls

Let us assume that $Q(0)=0$ and $Q$ is nc analytic. For every
${\mathbf{Z}}\in{\mathbb{D}}_{Q}$,

$\widetilde{\delta}(0,{\mathbf{Z}})=\left\|D_{Q({\mathbf{Z}})}^{-1/2}Q({\mathbf{Z}})\right\|.$

Let $T=Q({\mathbf{Z}})Q({\mathbf{Z}})^{*}$, then

$\widetilde{\delta}(0,{\mathbf{Z}})^{2}=\left\|T(I-T)^{-1}\right\|=\sup_{x\in\sigma(T)}\frac{x}{1-x}.$

Since the function $\frac{x}{1-x}$ is monotonically increasing on $[0,1)$,
$\widetilde{\delta}(0,{\mathbf{Z}})^{2}=\frac{\|T\|}{1-\|T\|}=\frac{\|Q({\mathbf{Z}})\|^{2}}{1-\|Q({\mathbf{Z}})\|^{2}}$.

######  Lemma 6.1.

Let $T$ be a positive operator with $\|T\|<1$, then $(1-\|T\|)(1-T)^{-1}\leq
1$.

###### Proof.

Since $T\leq\|T\|$, then $1-T\geq 1-\|T\|$. Now multiplying on left and right
by $(1-T)^{-1/2}$, we get $1\geq(1-\|T\|)(1-T)^{-1}$. ∎

######  Corollary 6.2.

For every ${\mathbf{Z}}\in{\mathbb{D}}_{Q}$, then
$\widetilde{\delta}(0,{\mathbf{W}})^{-1}\widetilde{\delta}({\mathbf{Z}},{\mathbf{W}})$
is bounded as ${\mathbf{W}}$ approaches a point on the boundary.

###### Proof.

Note that

$\widetilde{\delta}(0,{\mathbf{W}})^{-1}\widetilde{\delta}({\mathbf{Z}},{\mathbf{W}})=\\\
\sqrt{\frac{1-\|Q({\mathbf{W}})\|^{2}}{\|Q({\mathbf{W}})\|^{2}}}\left\|D_{Q({\mathbf{Z}})}^{-1/2}(Q({\mathbf{Z}})-Q({\mathbf{W}}))D_{Q({\mathbf{W}})^{*}}^{-1/2}\right\|.$

Note that $\|Q({\mathbf{W}})\|\to 1$, as ${\mathbf{W}}$ approaches the
boundary of ${\mathbb{D}}_{Q}$. By the preceding lemma applied to
$T=Q({\mathbf{W}})^{*}Q({\mathbf{W}})$,

$D_{Q({\mathbf{Z}})}^{-1/2}(Q({\mathbf{Z}})-Q({\mathbf{W}}))(1-\|Q({\mathbf{W}})\|^{2})D_{Q({\mathbf{W}})^{*}}(Q({\mathbf{Z}})^{*}-Q({\mathbf{W}})^{*})D_{Q({\mathbf{Z}})}^{-1/2}\leq\\\
D_{Q({\mathbf{Z}})}^{-1/2}(Q({\mathbf{Z}})-Q({\mathbf{W}}))(Q({\mathbf{Z}})^{*}-Q({\mathbf{W}})^{*})D_{Q({\mathbf{Z}})}^{-1/2}.$

The right-hand side is bounded in norm, and we are done. ∎

######  Definition 6.3.

Let $\zeta\in\partial{\mathbb{D}}_{Q}(1)$. We denote by $w$ points in
${\mathbb{D}}_{Q}(1)$. The big and small horospheres of radius $R>0$ at
$\zeta$ are

$F_{R}(\zeta)(n)=\left\\{{\mathbf{Z}}\in{\mathbb{D}}_{Q}\mid\liminf_{w\to\zeta}\widetilde{\delta}(0,w)^{-1}\widetilde{\delta}({\mathbf{Z}},w\otimes
I_{n})<R\right\\},$
$E_{R}(\zeta)(n)=\left\\{{\mathbf{Z}}\in{\mathbb{D}}_{Q}\mid\limsup_{w\to\zeta}\widetilde{\delta}(0,w)^{-1}\widetilde{\delta}({\mathbf{Z}},w\otimes
I_{n})<R\right\\}.$

Note that since ${\mathbb{D}}_{Q}$ is unitary invariant and closed under
direct sums, for every ${\mathbf{W}}\in{\mathbb{D}}_{Q}$, ${\mathbf{W}}\otimes
I_{n}\in{\mathbb{D}}_{Q}$, since it is up to a canonical shuffle
$I_{n}\otimes{\mathbf{W}}={\mathbf{W}}^{\oplus n}\in D_{Q}$. Moreover, since
$Q$ is nc we have that $Q({\mathbf{W}}\otimes I_{n})=Q({\mathbf{W}})\otimes
I_{n}$. For a point $w\in{\mathbb{D}}_{Q}(1)$, we have $w\otimes
I_{n}=I_{n}\otimes w$.

######  Lemma 6.4.

If ${\mathbb{D}}_{Q}$ is matrix convex, then for every $R>0$ and
$\zeta\in\partial{\mathbb{D}}_{Q}(1)$, $E_{R}(\zeta)$ is matrix convex.

###### Proof.

Let ${\mathbf{Z}}\in E_{R}(\zeta)(n)$ and
$V\colon{\mathbb{C}}^{m}\to{\mathbb{C}}^{n}$ an isometry. Then, for
$w\in{\mathbb{D}}_{Q}(1)$, $V^{*}(w\otimes I_{n})V=w\otimes I_{m}$ and thus,
by Lemma 3.9, we have that $\widetilde{\delta}(V^{*}{\mathbf{Z}}V,w\otimes
I_{m})\leq\widetilde{\delta}({\mathbf{Z}},w\otimes I_{n})$ and thus
$V^{*}{\mathbf{Z}}V\in E_{R}(\zeta)(n)$. Moreover, if ${\mathbf{Z}}_{1}\in
E_{R}(\zeta)(n)$ and ${\mathbf{Z}}_{2}\in E_{R}(\zeta)(m)$, then

$\widetilde{\delta}({\mathbf{Z}}_{1}\oplus{\mathbf{Z}}_{2},w\otimes
I_{n+m})=\max\left\\{\widetilde{\delta}({\mathbf{Z}}_{1},w\otimes
I_{n}),\widetilde{\delta}({\mathbf{Z}}_{2},w\otimes I_{m})\right\\}.$

Hence, $E_{R}(\zeta)$ is closed under direct sums. ∎

The following lemma lists some properties of the horospheres. These properties
follow immediately from the definition, and we include them for the sake of
completeness (see [1, 3]).

######  Lemma 6.5.

Fix a domain bounded ${\mathbb{D}}_{Q}$.

  1. (i)

For all $R>0$, $E_{R}(\zeta)\subset F_{R}(\zeta)$.

  2. (ii)

For $0<R_{1}<R_{2}$, $\overline{E_{R_{1}}(\zeta)}\subset E_{R_{2}}(\zeta)$ and
$\overline{F_{R_{1}}(\zeta)}\subset F_{R_{2}}(\zeta)$.

  3. (ii)

${\mathbb{D}}_{Q}=\bigcup_{R>0}E_{R}(\zeta)=\bigcup_{R>0}F_{R}(\zeta)$.

  4. (iii)

$\bigcap_{R>0}E_{R}(\zeta)=\bigcap_{R>0}F_{R}(\zeta)=\emptyset$.

######  Example 6.6.

In the classical case of the ball, we recover the classical definition of a
horosphere in the ball. Indeed, if
$\zeta=\left(\begin{smallmatrix}1&0&\ldots&0\end{smallmatrix}\right)$. In the
case of the ball $Q({\mathbf{W}})$ is the operator ${\mathbf{W}}$, hence for
$w\in{\mathbb{B}}_{d}$, we have $Q(w)Q(w)^{*}=\|w\|^{2}$ and $Q(w)^{*}Q(w)$ is
a rank one operator in $M_{d}$.

For $z,w\in{\mathbb{B}}_{d}$,

$\widetilde{\delta}(0,w)^{-2}\widetilde{\delta}(z,w)^{2}=\frac{1-\|w\|^{2}}{\|w\|^{2}(1-\|z\|^{2})}(Q(z)-Q(w))(1-Q(w)^{*}Q(w))^{-1}(Q(z)-Q(w))^{*}$

Write $v=Q(w)^{*}$, a column vector, and let $P_{v}$ be the orthogonal
projection on the subspace spanned by $v$. Note that
$P_{v}=\frac{1}{\|v\|^{2}}vv^{*}$. Therefore,

$(1-Q(w)^{*}Q(w))^{-1}=(1-vv^{*})^{-1}=(1-\|v\|^{2}P_{v})^{-1}=\\\
\sum_{n=0}^{\infty}\|v\|^{2n}P_{v}^{n}=I+\frac{\|v\|^{2}}{1-\|v\|^{2}}P_{v}.$

Now multiplying by $1-\|Q(w)\|^{2}$ we get that

$(1-\|Q(w)\|^{2})(1-Q(w)^{*}Q(w))^{-1}=(1-\|Q(w)\|^{2})+\|Q(w)\|^{2}P_{Q(w)^{*}}.$

Thus, as $w\to\zeta$, we get that
$\lim_{w\to\zeta}(1-\|Q(w)\|^{2})(1-Q(w)^{*}Q(w))^{-1}=P_{Q(\zeta)^{*}}$.
Therefore,

$\lim_{w\to\zeta}\widetilde{\delta}(0,w)^{-2}\widetilde{\delta}(z,w)^{2}=\frac{|1-z_{1}|^{2}}{1-\|z\|^{2}}.$

Therefore, $E_{R}(\zeta)(1)=F_{R}(\zeta)(1)$ in this case and are the
classical horospheres in the ball. Moreover, note that since the row $Q(w)$ is
scalar, we have that $Q(w)\otimes I_{n}=I_{n}\otimes Q(w)$. Hence we have

$(1-\|w\|)^{2}(Q({\mathbf{Z}})-Q(w)\otimes I_{n})(1-Q(w)^{*}Q(w)\otimes
I_{n})^{-1}(Q({\mathbf{Z}})-Q(w)\otimes I_{n})^{*}\stackrel{{\scriptstyle
w\to\zeta}}{{\longrightarrow}}\\\ (Q({\mathbf{Z}})-Q(\zeta)\otimes
I_{n})P_{\zeta}(Q({\mathbf{Z}})-Q(\zeta)\otimes
I_{n})^{*}=(I-Z_{1})(I-Z_{1})^{*}.$

Hence,

$\lim_{w\to\zeta}\widetilde{\delta}(0,w)^{-2}\widetilde{\delta}({\mathbf{Z}},w\otimes
I_{n})^{2}=\|D_{{\mathbf{Z}}}^{-1/2}(I-Z_{1})\|.$

In particular, the limit exists. Thus, $E_{R}(\zeta)=F_{R}(\zeta)$. Moreover,
for ${\mathbf{Z}}\in E_{r}(\zeta)$

$(I-Z_{1})(I-Z_{1})^{*}<R^{2}(I-{\mathbf{Z}}{\mathbf{Z}}^{*}).$

Thus, we recover Popescu’s definition [32] of a horosphere in the free ball.

######  Lemma 6.7.

For three distinct points ${\mathbf{X}},{\mathbf{Z}},{\mathbf{W}}\in
D_{Q}(n)$. Then,

$\delta({\mathbf{X}},{\mathbf{W}})^{-1}\delta({\mathbf{Z}},{\mathbf{W}})\leq\dfrac{\delta({\mathbf{Z}},{\mathbf{X}})}{\|Q({\mathbf{W}})\|-\|Q({\mathbf{X}})\|}+\|D_{Q({\mathbf{Z}})^{*}}^{-1/2}D_{Q({\mathbf{X}})^{*}}^{1/2}\|.$

###### Proof.

By the formula in Lemma 3.1

$\delta({\mathbf{Z}},{\mathbf{W}})=\|D_{Q({\mathbf{Z}})}^{-1/2}(Q({\mathbf{Z}})-Q({\mathbf{W}}))D_{Q({\mathbf{W}})^{*}}^{-1/2}\|\leq\\\
\|D_{Q({\mathbf{Z}})}^{-1/2}(Q({\mathbf{Z}})-Q({\mathbf{X}}))D_{Q({\mathbf{W}})^{*}}^{-1/2}\|+\|D_{Q({\mathbf{Z}})}^{-1/2}(Q({\mathbf{X}})-Q({\mathbf{W}}))D_{Q({\mathbf{W}})^{*}}^{-1/2}\|\leq\\\
\delta({\mathbf{Z}},{\mathbf{X}})\|D_{Q({\mathbf{X}})^{*}}^{1/2}D_{Q({\mathbf{W}})^{*}}^{-1/2}\|+\delta({\mathbf{X}},{\mathbf{W}})\|D_{Q({\mathbf{Z}})^{*}}^{-1/2}D_{Q({\mathbf{X}})^{*}}^{1/2}\|.$

Since the points are distinct $\delta({\mathbf{X}},{\mathbf{W}})\neq 0$.
Moreover, since $Q({\mathbf{X}})$ is a contraction, we have that
$\|D_{Q({\mathbf{X}})^{*}}^{1/2}\|\leq 1$. Hence, we obtain the following
inequality:

$\delta({\mathbf{X}},{\mathbf{W}})^{-1}\delta({\mathbf{Z}},{\mathbf{W}})\leq\delta({\mathbf{Z}},{\mathbf{X}})\frac{\|D_{Q({\mathbf{W}})^{*}}^{-1/2}\|}{\delta({\mathbf{X}},{\mathbf{W}})}+\|D_{Q({\mathbf{Z}})^{*}}^{-1/2}D_{Q({\mathbf{X}})^{*}}^{1/2}\|.$

We need to bound
$\frac{\|D_{Q({\mathbf{W}})^{*}}^{-1/2}\|}{\delta({\mathbf{X}},{\mathbf{W}})}$.
By functional calculus,
$\|D_{Q({\mathbf{W}})^{*}}^{-1/2}\|=\frac{1}{\sqrt{1-\|Q({\mathbf{W}})\|^{2}}}$.
Hence, we will bound
$\sqrt{1-\|Q({\mathbf{W}})\|^{2}}\delta({\mathbf{X}},{\mathbf{W}})$ from
bellow. Again, since $Q({\mathbf{X}})$ is a strict contraction,
$(1-Q({\mathbf{X}})Q({\mathbf{X}})^{*})^{-1}\geq 1$. Let us write

$T=D_{Q({\mathbf{X}})}^{-1/2}(Q({\mathbf{X}})-Q({\mathbf{W}}))D_{Q({\mathbf{W}})^{*}}^{-1/2},$

Then,

$T^{*}T\geq
D_{Q({\mathbf{W}})^{*}}^{-1/2}(Q({\mathbf{X}})^{*}-Q({\mathbf{W}})^{*})(Q({\mathbf{X}})-Q({\mathbf{W}}))D_{Q({\mathbf{W}})^{*}}^{-1/2}.$

Hence,

$\sqrt{1-\|Q({\mathbf{W}})\|^{2}}\delta({\mathbf{X}},{\mathbf{W}})=\sqrt{1-\|Q({\mathbf{W}})\|^{2}}\|T\|\geq\\\
\sqrt{1-\|Q({\mathbf{W}})\|^{2}}\|(Q({\mathbf{X}})-Q({\mathbf{W}}))D_{Q({\mathbf{W}})^{*}}^{-1/2}\|\geq\\\
\sqrt{1-\|Q({\mathbf{W}})\|^{2}}\left(\|Q({\mathbf{W}})D_{Q({\mathbf{W}})^{*}}^{-1/2}\|-\|Q({\mathbf{X}})D_{Q({\mathbf{W}})^{*}}^{-1/2}\|\right)$

Now, if $A=Q({\mathbf{W}})D_{Q({\mathbf{W}})^{*}}^{-1/2}$, then
$A^{*}A=Q({\mathbf{W}})^{*}Q({\mathbf{W}})D_{Q({\mathbf{W}})^{*}}^{-1}$. Thus,
by functional calculus,
$\|A\|=\frac{\|Q({\mathbf{W}})\|}{\sqrt{1-\|Q({\mathbf{W}})\|^{2}}}$. Now, for
the second summand, we have

$\sqrt{1-\|Q({\mathbf{W}})\|^{2}}\|Q({\mathbf{X}})D_{Q({\mathbf{W}})^{*}}^{-1/2}\|\leq\\\
\sqrt{1-\|Q({\mathbf{W}})\|^{2}}\|Q({\mathbf{X}})\|\|D_{Q({\mathbf{W}})^{*}}^{-1/2}\|=\|Q({\mathbf{X}})\|.$

In summary,

$\sqrt{1-\|Q({\mathbf{W}})\|^{2}}\delta({\mathbf{X}},{\mathbf{W}})\geq\|Q({\mathbf{W}})\|-\|Q({\mathbf{X}})\|.$

Thus,

$\delta({\mathbf{X}},{\mathbf{W}})^{-1}\delta({\mathbf{Z}},{\mathbf{W}})\leq\dfrac{\delta({\mathbf{Z}},{\mathbf{W}})}{\|Q({\mathbf{W}})\|-\|Q({\mathbf{X}})\|}+\|D_{Q({\mathbf{Z}})^{*}}^{-1/2}D_{Q({\mathbf{X}})^{*}}^{1/2}\|.$

∎

######  Theorem 6.8 (Noncommutative Wolff theorem).

Let $0\in\Omega={\mathbb{D}}_{Q}$ be a bounded matrix convex set. Let
$f\colon\Omega\to\Omega$ be an nc map with no fixed points. Then there exists
$\xi\in\partial\Omega(1)$, such that for every $R>0$ and $n\in{\mathbb{N}}$,
$f^{\circ n}(E_{R}(\xi))\subseteq F_{R}(\xi)$.

###### Proof.

Let $0<r_{m}<1$ be a sequence that monotonically increases to $1$. Let
$f_{m}({\mathbf{Z}})=f(r_{m}{\mathbf{Z}})$. By Lemma 3.4, $f_{m}$ has a unique
fixed point $w_{m}\in\Omega(1)$. Any accumulation point of $w_{m}$ in
$\Omega(1)$ will be a fixed point of $f$. Hence (passing to a subsequence, if
necessary), we may assume that $w_{m}\to\xi\in\partial\Omega$. Fix $R>0$, and
let ${\mathbf{Z}}\in E_{R}(\xi)(n)$. Then

$\limsup_{m\to\infty}\widetilde{\delta}(0,w_{m})^{-1}\widetilde{\delta}({\mathbf{Z}},w_{m}\otimes
I_{n})\leq\limsup_{w\to\xi}\widetilde{\delta}(0,w)^{-1}\widetilde{\delta}({\mathbf{Z}},w\otimes
I_{n})<R.$

Fix an arbitrary $\varepsilon>0$ small enough. Then, for every $m$ big enough,

$\widetilde{\delta}(0,w_{m})^{-1}\widetilde{\delta}({\mathbf{Z}},w_{m}\otimes
I_{n})<R-\varepsilon.$

By [13, Corollary 3.4] and the fact that $w_{m}$ is a fixed point of
$f_{m}^{\circ n}$, for every $n\in{\mathbb{N}}$, we conclude that for every
$m>m_{0}$,

$\widetilde{\delta}(f_{m}^{\circ n}({\mathbf{Z}}),f_{m}^{\circ
n}(w_{m})\otimes I_{n})=\widetilde{\delta}(f_{m}^{\circ
n}({\mathbf{Z}}),w_{m}\otimes
I_{n})\leq\widetilde{\delta}({\mathbf{Z}},w_{m}\otimes
I_{n})<\widetilde{\delta}(0,w_{m})(R-\varepsilon).$

Now, fix $n\in{\mathbb{N}}$. For every ${\mathbf{X}}\in\Omega$,
$\lim_{m\to\infty}f^{\circ n}_{m}({\mathbf{X}})=f^{\circ n}({\mathbf{X}})$. By
the continuity of $\widetilde{\delta}$, we have that
$\lim_{m\to\infty}\widetilde{\delta}(f^{\circ n}_{m}({\mathbf{X}}),f^{\circ
n}({\mathbf{X}}))=0$. By Lemma 6.7,

$\widetilde{\delta}(f^{\circ n}_{m}({\mathbf{Z}}),w_{m}\otimes
I_{n})^{-1}\widetilde{\delta}(f^{\circ n}({\mathbf{Z}}),w_{m}\otimes
I_{n})\leq\\\ \frac{\widetilde{\delta}(f^{\circ n}({\mathbf{Z}}),f^{\circ
n}_{m}({\mathbf{Z}}))}{\|Q(w_{m})\|-\|Q(f^{\circ
n}_{m}({\mathbf{Z}}))\|}+\|D_{Q(f^{\circ
n}({\mathbf{Z}}))^{*}}^{-1/2}D_{Q(f^{\circ
n}_{m}({\mathbf{Z}}))^{*}}^{1/2}\|.$

Since $\lim_{m\to\infty}\frac{1}{\|Q(w_{m})\|-\|Q(f^{\circ
n}_{m}({\mathbf{Z}}))\|}=\frac{1}{1-\|Q(f^{\circ n}({\mathbf{Z}}))\|}$, we get
that

$\limsup_{m\to\infty}\widetilde{\delta}(f^{\circ
n}_{m}({\mathbf{Z}}),w_{m}\otimes I_{n})^{-1}\widetilde{\delta}(f^{\circ
n}({\mathbf{Z}}),w_{m}\otimes I_{n})\leq 1.$

In particular, for $m>>0$,

$\widetilde{\delta}(f^{\circ n}({\mathbf{Z}}),w_{m}\otimes
I_{n})<(1+\varepsilon)\widetilde{\delta}(f^{\circ
n}_{m}({\mathbf{Z}}),w_{m}\otimes
I_{n})\leq(1+\varepsilon)\widetilde{\delta}({\mathbf{Z}},w_{m}\otimes I_{n}).$

Hence, if we take $0<\delta<\frac{\varepsilon}{R-\varepsilon}$, then again for
every $m$ big enough,

$\widetilde{\delta}(0,w_{m})^{-1}\widetilde{\delta}(f^{\circ
n}({\mathbf{Z}}),w_{m}\otimes I_{n})<(1+\delta)(R-\varepsilon)<R.$

This implies that

$\liminf_{w\to\zeta}\widetilde{\delta}(0,w)^{-1}\widetilde{\delta}(f^{\circ
n}({\mathbf{Z}}),w\otimes I_{n})\leq\\\
\liminf_{m\to\infty}\widetilde{\delta}(0,w_{m})^{-1}\widetilde{\delta}(f^{\circ
n}({\mathbf{Z}}),w_{m}\otimes I_{n})<R.$

Thus, by definition, $f^{\circ n}({\mathbf{Z}})\in F_{R}(\zeta)$. ∎

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