#  Operator NC functions

Meric Augat  John E. McCarthy Partially supported by National Science
Foundation Grant DMS 2054199

###### Abstract

We establish a theory of NC functions on a class of von Neumann algebras with
a particular direct sum property, e.g. $B({\mathcal{H}})$. In contrast to the
theory’s origins, we do not rely on appealing to results from the matricial
case. We prove that the $k^{\mathrm{th}}$ directional derivative of any NC
function at a scalar point is a $k$-linear homogeneous polynomial in its
directions. Consequences include the fact that NC functions defined on domains
containing scalar points can be uniformly approximated by free polynomials as
well as realization formulas for NC functions bounded on particular sets, e.g.
the non-commutative polydisk and non-commutative row ball.

##  1 Introduction

Non-commutative function theory, as first proposed in the seminal work of J.L.
Taylor [12, 13] and developed for example in the monograph [6] by Kaliuzhnyi-
Verbovetskyi and Vinnikov, is a matricial theory, that is a theory of
functions of $d$-tuples of matrices. Let ${\mathbb{M}}_{n}$ denote the
$n$-by-$n$ square matrices, and let

${\mathbb{M}}^{[d]}\ :=\ \cup_{n=1}^{\infty}{\mathbb{M}}_{n}^{d}.$

A non-commutative function $f$ defined on a domain $\Omega$ in
${\mathbb{M}}^{[d]}$ is a function that satisfies the following two
properties.

(i) The function is graded: if $x\in{\mathbb{M}}_{n}^{d}$ then
$f(x)\in{\mathbb{M}}_{n}$.

(ii) It preserves intertwining: if $L:\mathbb{C}^{m}\to\mathbb{C}^{n}$ is
linear, $x=(x^{1},\dots,x^{d})\in{\mathbb{M}}_{m}^{d}$ and
$y=(y^{1},\dots,y^{d})$ are both in $\Omega$ and $Lx=yL$ (this means
$Lx^{r}=y^{r}L$ for each $1\leq r\leq d$), then $Lf(x)=f(y)L$.

The theory has been very successful, and can be thought of as extending free
polynomials in $d$ variables to non-commutative holomorphic functions.
However, the negative answer to Connes’s embedding conjecture [5] shows
evaluating non-commutative polynomials on tuples of matrices is not sufficient
to fully capture certain types of information, e.g. trace positivity of a free
polynomial evaluated on tuples of self-adjoint contractions [7]. Thus, there
is an incentive to understand non-commutative functions applied not to
matrices, but to operators on an infinite dimensional Hilbert space
${\mathcal{H}}$. Accordingly, it seems natural to exploit the fact that there
are (non-canonical) identifications of a matrix of operators with an
individual operator, and so one is led to to consider functions that map
elements of $B({\mathcal{H}})^{d}$ to $B({\mathcal{H}})$ and preserve
intertwining.

Such functions were studied in [1] and [9]. A key assumption in those papers,
however, was that the function was also sequentially continuous in the strong
operator topology. This assumption was needed in order to prove that the
derivatives at $0$ were actually free polynomials, by invoking this property
from the matricial theory and using the density of finite rank operators in
the strong operator topology. The main purpose of this note is to develop a
theory of non-commutative functions of operator tuples that does not depend on
the matricial theory.

For the rest of this paper, the following will be fixed. We shall let
${\mathcal{H}}$ be an infinite-dimensional Hilbert space. Let ${\mathcal{A}}$
be a unital sub-algebra of $B({\mathcal{H}})$ that is closed in the norm
topology. Let ${\mathcal{T}}_{n}({\mathcal{A}})$ denote the upper triangular
$n$-by-$n$ matrices with entries from ${\mathcal{A}}$. We shall assume that
${\mathcal{A}}$ has the following direct sum property:

$\forall n\geq 1,\
\exists\,U_{n}:\oplus_{j=1}^{n}{\mathcal{H}}\to{\mathcal{H}},\ {\rm unitary,\
with\ }\\\
U_{n}({\mathcal{T}}_{n}({\mathcal{A}}))U_{n}^{*}\subseteq{\mathcal{A}}.$ (1.1)

Examples of such an ${\mathcal{A}}$ include $B({\mathcal{H}})$; the upper
triangular matrices in $B({\mathcal{H}})$ with respect to a fixed basis; and
any von Neumann algebra that can be written as a tensor product of a
$I_{\infty}$ factor with something else.

We shall let $d$ be a positive integer, and it will denote the number of
variables. For a $d$-tuple $x\in{\mathcal{A}}^{d}$, we shall write its
coordinates with superscripts: $x=(x^{1},\dots,x^{d})$. We shall toplogize
${\mathcal{A}}^{d}$ with the relative norm topology from
$B({\mathcal{H}})^{d}$.

######  Definition 1.2.

A set $\Omega\subseteq{\mathcal{A}}^{d}$ is called an NC domain if it is open
and bounded, and closed with respect to finite direct sums in the following
sense: for each $n\geq 2$ there exists a unitary
$U_{n}:{\mathcal{H}}^{(n)}\to{\mathcal{H}}$ so that whenever
$x_{1},\dots,x_{n}\in\Omega$, then

$U_{n}\ \begin{bmatrix}x_{1}&0&\cdots&0\\\ 0&x_{2}&\cdots&0\\\ &&\ddots\\\
0&0&\cdots&x_{n}\end{bmatrix}U_{n}^{*}\ \in\ \Omega.$ (1.3)

######  Example 1.4.

The prototypical examples of NC domains are balls. The reader is welcome to
assume that $\Omega$ is either a non-commutative polydisk, that is of the form

$\mathcal{P}({\mathcal{A}})\ =\ \\{x\in{\mathcal{A}}^{d}:\max_{1\leq r\leq
d}\|x^{r}\|<1\\},$ (1.5)

or a non-commutative row ball, that is

$\mathcal{R}({\mathcal{A}})\ =\
\\{x\in{\mathcal{A}}^{d}:x^{1}(x^{1})^{*}+\dots+x^{d}(x^{d})^{*}<1\\}.$ (1.6)

More examples are given in Section 6.

######  Definition 1.7.

Let $\Omega\subseteq{\mathcal{A}}^{d}$ be an NC domain. A function
$F:\Omega\to B({\mathcal{H}})$ is intertwining preserving if whenever
$x,y\in\Omega$ and $L:{\mathcal{H}}\to{\mathcal{H}}$ is a bounded linear
operator that satisfies $Lx=yL$ (i.e., $Lx^{r}=y^{r}L$ for each $r$) then
$LF(x)=F(y)L$.

We say $F$ is an NC function if it is intertwining preserving and locally
bounded on $\Omega$.

Remark 1.8. For any positive integer $b$ we may similarly define an NC mapping
${\mathcal{F}}:\Omega\to B({\mathcal{H}})^{b}$ where
${\mathcal{F}}=(F^{1},\dots,F^{b})$ and each $F^{i}:\Omega\to
B({\mathcal{H}})$ is an NC function. Many of our results can be reinterpreted
for NC mappings with little to no overhead.

.

In Section 2 we show that every NC function is Fréchet holomorphic. Our first
main result is proved in Theorem 3.11. A scalar point $a$ is a point each of
whose components is a scalar multiple of the identity.

######  Theorem 1.9.

Suppose $\Omega$ is an NC domain containing a scalar point $a$, and $F$ is NC
on $\Omega$. Then for each $k$, the $k^{\rm th}$ derivative
$D^{k}F(a)[h_{1},\dots,h_{k}]$ is a symmetric homogeneous free polynomial of
degree $k$ in $h_{1},\dots,h_{k}$.

We derive several consequences of this result. In Theorem 4.3, we show that if
$\Omega$ is a balanced NC domain, then a function $F$ on $\Omega$ is an NC
function if and only if it can be uniformly approximated by free polynomials
on every finite set. In Theorem 6.3 we show that NC functions on most balanced
domains are automatically sequentially strong operator continuous. This allows
us to prove that every non-commutative function on the non-commutative matrix
polydisk (resp. row ball) has a unique extension to an NC function on
$\mathcal{P}(B({\mathcal{H}}))$ (resp. $\mathcal{R}(B({\mathcal{H}}))$).

Similarity preserving maps of matrices were studied by C. Procesi [10], who
showed they were all trace polynomials. In the matricial case, this can be
used to prove the analogue of Theorem 1.9 [8]. In the infinite dimensional
case, we cannot use this theory, which makes the proof of Theorem 1.9 more
complicated. However, we can then use the theorem to prove that the only
intertwining preserving bounded $k$-linear maps are the obvious ones, the free
polynomials. In Theorem 5.1 we prove:

######  Theorem 1.10.

Let $\Omega$ be an NC domain. Let $\Lambda:[\Omega]^{k}\to B({\mathcal{H}})$
be NC and $k$-linear. Then $\Lambda$ is a homogeneous free polynomial of
degree $k$.

##  2 Preliminaries

Throughout this section, we assume that $\Omega$ is an NC domain in
${\mathcal{A}}^{d}$, and $F:\Omega\to B({\mathcal{H}})$ is an NC function. Let
${\mathbb{N}}^{+}$ denote the positive integers.

For each $n\in{\mathbb{N}}^{+}$, define the unitary and similarity envelopes
by

$\displaystyle\widehat{\Omega}_{n}$ $\displaystyle\ :=\ $
$\displaystyle\\{U^{*}xU\ |\ U:{\mathcal{H}}^{(n)}\to{\mathcal{H}},{\rm\
unitary},\ x\in\Omega\\}$ $\displaystyle\widetilde{\Omega}_{n}$
$\displaystyle\ :=\ $ $\displaystyle\\{S^{-1}xS\ |\
S:{\mathcal{H}}^{(n)}\to{\mathcal{H}},{\rm\ invertible},\ x\in\Omega\\}.$

Notably, for $x_{1},\dots,x_{n}\in\Omega$,
$\oplus_{j=1}^{n}x_{j}\in\widehat{\Omega}_{n}$. We can extend $F$ to
$\widetilde{\Omega}=\cup_{n=1}^{\infty}\widetilde{\Omega}_{n}$ by

$\widetilde{F}(\tilde{x})\ =\ SF(x)S^{-1},$ (2.1)

where $\tilde{x}=S^{-1}xS$ for some $x\in\Omega$.

It is straightforward to prove the following from the intertwining preserving
property of $F$. Nevertheless, we include a proof to showcase the simplicity
of working with $\widetilde{F}$ in lieu of $F$.

######  Proposition 2.2.

The function $\widetilde{F}$ defined by (2.1) is well-defined, and if
$\tilde{x}\in\widetilde{\Omega}_{m}$ and $\tilde{y}\in\widetilde{\Omega}_{m}$
satisfy $\tilde{L}\tilde{x}=\tilde{y}\tilde{L}$ for some linear
$\tilde{L}:\text{ }{\begin{smallmatrix}{\mathcal{H}}\\\ \otimes\\\
\mathbb{C}^{m}\end{smallmatrix}}\text{ }\to\text{
}{\begin{smallmatrix}{\mathcal{H}}\\\ \otimes\\\
\mathbb{C}^{n}\end{smallmatrix}}\text{ }$, then
$\tilde{L}\widetilde{F}(\tilde{x})=\widetilde{F}(\tilde{y})\tilde{L}$.

In particular, if $x_{j}\in\Omega$ for $1\leq j\leq n$, then
$\widetilde{F}(\oplus x_{j})=\oplus F(x_{j})$.

###### Proof.

Let $\tilde{x}=S^{-1}xS$ and $\tilde{y}=T^{-1}yT$ for $x,y\in\Omega$. Define
$L:{\mathcal{H}}\to{\mathcal{H}}$ by $L=T\tilde{L}S^{-1}$ and consider the
following intetwining:

$\displaystyle Lx$
$\displaystyle=T\tilde{L}S^{-1}\hat{x}=T\tilde{L}S^{-1}S\tilde{x}S^{-1}$
$\displaystyle=T\tilde{L}\tilde{x}S^{-1}=T\tilde{y}\tilde{L}S^{-1}=TT^{-1}yT\tilde{L}S^{-1}$
$\displaystyle=yL.$

Thus, $LF(x)=F(y)L$ and consequently

$\displaystyle\tilde{L}\widetilde{F}(x)$
$\displaystyle=\tilde{L}S^{-1}F(x)S=T^{-1}LF(x)S=T^{-1}F(y)LS=\widetilde{F}(\tilde{y})T^{-1}LS$
$\displaystyle=\widetilde{F}(\tilde{y})\tilde{L}.$

Finally, let $P_{j}:{\mathcal{H}}\to{\mathcal{H}}^{(n)}$ be the inclusion of
${\mathcal{H}}$ onto the $j^{\text{th}}$-coordinate of ${\mathcal{H}}^{(n)}$.
Observe that $(\oplus_{i=1}^{n}x_{i})P_{j}=P_{j}x_{j}$. Hence,
$\widetilde{F}(\oplus_{i=1}^{n}x_{i})P_{j}=P_{j}\widetilde{F}(x_{i})=P_{j}F(x_{j})$.
The intertwining with $P_{j}^{*}$ has
$x_{j}P_{j}^{*}=P_{j}^{*}(\oplus_{i=1}^{n}x_{i})$. Thus,
$P_{j}^{*}F(x_{j})=P_{j}^{*}\widetilde{F}(\oplus_{i=1}^{n}x_{i})$ and
combining these two intertwining shows that
$\widetilde{F}(\oplus_{i=1}^{n}x_{i})$ is a diagonal block operator and

$\widetilde{F}(\oplus_{i=1}^{n}x_{i})=\oplus_{i=1}^{n}F(x_{i}).$

∎

For later use, let us give a sort of converse.

######  Lemma 2.3.

Suppose that $\Omega$ is an NC domain, and $F:\Omega\to B({\mathcal{H}})$
satisfies

$F(S^{-1}[x\oplus y]S)\ =\ S^{-1}[F(x)\oplus F(y)]S$ (2.4)

whenever $S:{\mathcal{H}}\to{\mathcal{H}}^{(2)}$ and $x,y,S^{-1}[x\oplus
y]S\in\Omega$. Then $F$ is intertwinining preserving.

###### Proof.

Suppose $Lx=yL$. Let

$S=\begin{bmatrix}1&L\\\ 0&1\end{bmatrix},$

and (2.4) implies that $LF(x)=F(y)L$. ∎

Recall that $F$ is Fréchet holomorphic if, for every $x\in\Omega$, there is an
open neighborhood $G$ of $0$ in ${\mathcal{A}}^{d}$ so that the Taylor series

$F(x+h)\ =\ F(x)+\sum_{k=1}^{\infty}D^{k}F(x)[h,\dots,h]$ (2.5)

converges uniformly for $h$ in $G$.

Using (1.1), it follows that if $x_{1},\dots,x_{n}\in\Omega$, then
$\exists\varepsilon>0$ so that if $y\in{\mathcal{T}}_{n}({\mathcal{A}})$ and
$\|y-\oplus x_{j}\|<\varepsilon$, then $U_{n}yU_{n}^{*}\in\Omega$. The
following is proved in [1], and, in the form stated, in [3]*Section 16.1.

######  Proposition 2.6.

If $\Omega\subset{\mathcal{A}}^{d}$ is an NC domain and $F$ is an NC function
on $\Omega$ then

  1. (i)

The function $F$ is Fréchet holomorphic.

  2. (ii)

For $x\in\Omega$, $h\in{\mathcal{A}}$,

$\widetilde{F}\left(\begin{bmatrix}x&h\\\ 0&x\end{bmatrix}\right)\ =\
\begin{bmatrix}F(x)&DF(x)[h]\\\ 0&F(x)\end{bmatrix}.$

We wish to prove that when $x$ is a scalar point, each derivative in (2.5) is
actually a free polynomial in $h$. This is straightforward for the first
derivative.

######  Lemma 2.7.

Suppose $a=(a^{1},\dots,a^{d})$ is a $d$-tuple of scalar matrices in $\Omega$.
Then $F(a)$ is a scalar, and $DF(a)[c]$ is scalar for any scalar $d$-tuple
$c$.

###### Proof.

For any $L\in B({\mathcal{H}})$, since $La=aL$, we have $LF(a)=F(a)L$.
Therefore $F(a)$ is a scalar. For all $t$ sufficiently close to $0$, $a+tc$ is
in $\Omega$ and $F(a+tc)-F(a)$ is scalar, therefore $DF(a)[c]$ is scalar. ∎

######  Lemma 2.8.

Suppose $a=(a^{1},\dots,a^{d})$ is a $d$-tuple of scalar matrices in $\Omega$.
Then $DF(a)[h]$ is a linear polynomial in $h$.

###### Proof.

First assume that $h=(h_{1},0,\dots,0)$. Let $\varepsilon>0$ be such that the
closed $\max(\varepsilon,\varepsilon\|h_{1}\|)$ ball around $a\oplus a$ is in
$\widetilde{\Omega}$. Let $J=(1,0,\dots,0)$ be the scalar $d$-tuple with first
entry $1$, the others $0$. As

$\begin{bmatrix}1&0\\\ 0&h_{1}\end{bmatrix}\ \begin{bmatrix}a&\varepsilon h\\\
0&a\end{bmatrix}\ =\ \begin{bmatrix}a&\varepsilon J\\\ 0&a\end{bmatrix}\
\begin{bmatrix}1&0\\\ 0&h_{1}\end{bmatrix},$

we get from Proposition 2.2 that

$DF(a)[h]=DF(a)[J]\ h_{1}.$ (2.9)

By Lemma 2.7, $DF(a)[J]$ is a scalar, $c_{1}$ say, so we get

$DF(a)[(h_{1},0,\dots,0)]\ =\ c_{1}h_{1}.$

Permuting the coordinates and using the fact that $DF(a)[h]$ is linear in $h$,
we get that for any $h$

$DF(a)[h]\ =\ \sum_{r=1}^{d}c_{r}h_{r}$

for some constants $c_{r}$. ∎

##  3 Derivatives of NC functions are free polynomials

The derivatives are defined inductively, by

$\displaystyle D^{k}F(x)[h_{1},\dots,h_{k}]\ =\ $ (3.10)
$\displaystyle\lim_{\lambda\to 0}\frac{1}{\lambda}\left(D^{k-1}F(x+\lambda
h_{k})[h_{1},\dots,h_{k-1}]-D^{k-1}F(x)[h_{1},\dots,h_{k-1}]\right).$

The $k^{\rm th}$ derivative is $k$-linear in $h_{1},\dots,h_{k}$. To extend
Lemma 2.8 to higher derivatives, we need to introduce some other operators,
called nc difference-differential operators in [6].

$\Delta^{k}F(x_{1},\dots,x_{k+1})[h_{1},\dots,h_{k}]$ is defined to be the
$(1,k+1)$ entry in the matrix

$\widetilde{F}\left(\begin{bmatrix}x_{1}&h_{1}&0&0&\dots&0\\\
0&x_{2}&h_{2}&0&\dots&0\\\ \vdots&\vdots&\vdots&\vdots&&\vdots\\\
0&0&0&0&\dots&x_{k+1}\end{bmatrix}\right)$ (3.2)

We shall show in Lemma 3.6 that it is $k$-linear in $[h_{1},\dots,h_{k}]$.

The $\Delta^{k}$ occur when applying $\widetilde{F}$ to a bidiagonal matrix.
This is proved in [6]*Thm. 3.11.

######  Lemma 3.3.

Let $F$ be NC. Then

$\displaystyle\widetilde{F}\left(\begin{bmatrix}x_{1}&h_{1}&0&\dots&0\\\
0&x_{2}&h_{2}&\dots&0\\\ \vdots&\vdots&\vdots&\dots&\vdots\\\
0&0&0&\dots&x_{k+1}\end{bmatrix}\right)$ (3.5)
$\displaystyle=\begin{bmatrix}F(x_{1})&\Delta^{1}F(x_{1},x_{2})[h_{1}]&\dots&\Delta^{k}F(x_{1},\dots,x_{k+1})[h_{1},\dots,h_{k}]\\\
0&F(x_{2})&\dots&\Delta^{k-1}F(x_{2},\dots,x_{k+1})[h_{2},\dots,h_{k}]\\\
\vdots&\vdots&\dots&\vdots\\\ 0&0&\dots&F(x_{k+1})\end{bmatrix}.$

###### Proof.

We will prove this by induction. For $k=1$, it is the definition of
$\Delta^{1}$. Assume it is proved for $k-1$. Let $I_{k}$ denote the $k$-by-$k$
matrix with diagonal entries the identity, and off-diagonal entries $0$. As

$\begin{bmatrix}x_{1}&h_{1}&0&\dots&0\\\ 0&x_{2}&h_{2}&\dots&0\\\
\vdots&\vdots&\vdots&\dots&\vdots\\\ 0&0&0&\dots&x_{k+1}\end{bmatrix}\
\begin{bmatrix}I_{k}\\\ 0\end{bmatrix}_{(k+1)\times k}\ =\
\begin{bmatrix}I_{k}\\\ 0\end{bmatrix}_{(k+1)\times k}\
\begin{bmatrix}x_{1}&h_{1}&0&\dots&0\\\ 0&x_{2}&h_{2}&\dots&0\\\
\vdots&\vdots&\vdots&\dots&\vdots\\\ 0&0&0&\dots&x_{k}\end{bmatrix}$

we conclude that the first $k$ columns of (3.5) agree with those of (3.5).
Similarly, intertwining by $\begin{bmatrix}0&I_{k}\end{bmatrix}$ we get that
the bottom $k$ rows agree. Finally, the $(1,(k+1))$ entry is the definition of
$\Delta^{k}$. ∎

A key property we need is that $\Delta^{k}$ is $k$-linear in the directions.
In the nc case, this is proved in [6]*3.5.

######  Lemma 3.6.

Let $x_{1},\dots,x_{k+1}\in\Omega$. Then
$\Delta^{k}F(x_{1},\dots,x_{k+1})[h_{1},\dots,h_{k}]$ is $k$-linear in
$h_{1},\dots,h_{k}$.

###### Proof.

Let us write $\Delta^{k}[h_{1},\dots,h_{k}]$ for
$\Delta^{k}F(x_{1},\dots,x_{k+1})[h_{1},\dots,h_{k}]$.

(i) First, we show this is linear with respect to $h_{1}$. Homogeneity follows
from observing that

$\begin{bmatrix}x_{1}&ch_{1}&0&\dots\\\ 0&x_{2}&h_{2}&\dots\\\
\vdots&\vdots&\vdots&\dots\\\ \end{bmatrix}\begin{bmatrix}c&0&0\\\ 0&1&\\\
\vdots&\vdots&\ddots\end{bmatrix}\ =\ \begin{bmatrix}c&0&0\\\ 0&1&\\\
\vdots&\vdots&\ddots\end{bmatrix}\begin{bmatrix}x_{1}&h_{1}&0&\dots\\\
0&x_{2}&h_{2}&\dots\\\ \vdots&\vdots&\vdots&\dots\\\ \end{bmatrix}$

and using the intertwining preserving property Prop. 2.2.

To show additivity, let $p\geq 1$ and $q\geq 0$ be integers. Let $Y$ be the
$(p+k+q)\times(p+k+q)$ matrix

$Y\ =\ \begin{bmatrix}x_{1}&0&\dots&&h_{1}&\dots&&0&\dots\\\
0&x_{1}&\dots&&0&\dots&&0&\dots\\\ &&\ddots&&&\dots&&&\dots\\\
0&\dots&&x_{1}&h_{1}^{\prime}&\dots&&0&\dots\\\
&&&0&x_{2}&h_{2}&\dots&0&\dots\\\ &&&&&\ddots&&&\\\ &&&&&&x_{k+1}&0&\\\
&&&&&&&x_{k+1}&\\\ &&&&&&&&\ddots\end{bmatrix}.$

Let $L$ be the $(k+1)\times(p+k+q)$ matrix

$L\ =\ \begin{bmatrix}1&0&\dots&0&\dots&&&0&\dots&0\\\
0&0&\dots&1&0&\dots&&0&&\\\ 0&\dots&&0&1&&&0&&\\\ 0&\dots&&&&\ddots&&&&\\\
0&\dots&&&&&1&0&\dots&1\\\ \end{bmatrix}$

Let $X$ be the $(k+1)\times(k+1)$ matrix

$X\ =\ \begin{bmatrix}x_{1}&h_{1}&0&0&\dots&0\\\ 0&x_{2}&h_{2}&0&\dots&0\\\
\vdots&\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&0&\dots&x_{k+1}\end{bmatrix}$

Then

$LY\ =\ \begin{bmatrix}x_{1}&\dots&h_{1}&0&\dots&&0&\dots\\\
0&\dots&x_{2}&h_{2}&\dots&&&\\\ &&&\ddots&&\\\
0&\dots&0&\dots&&x_{k+1}&0&\dots&x_{k+1}\end{bmatrix}\ =\ XL.$

Therefore the $(1,p+k+q)$ entry of $\widetilde{F}(Y)$ is
$\Delta^{k}[h_{1},\dots,h_{k}]$.

Let $L^{\prime}$ be the matrix obtained by replacing the first row of $L$ with
the row that is $1$ in the $p^{\rm th}$ entry and $0$ elsewhere. Then
$L^{\prime}Y=X^{\prime}L^{\prime}$, where $X^{\prime}$ is $X$ with $h_{1}$
replaced by the $d$-tuple $h_{1}^{\prime}$. This gives that the $(p,p+k+q)$
entry of $\widetilde{F}(Y)$ is $\Delta^{k}[h_{1}^{\prime},\dots,h_{k}]$.

Now let $L^{\prime\prime}$ be the matrix that replaces the first row of $L$
with a $1$ in both the first and $p^{\rm th}$ entry, and $X^{\prime\prime}$ be
$X$ with $h_{1}$ replaced by $h_{1}+h_{1}^{\prime}$. Then
$L^{\prime\prime}Y=X^{\prime\prime}L^{\prime\prime}$, and we conclude that

$\Delta^{k}[h_{1},\dots,h_{k}]+\Delta^{k}[h_{1}^{\prime},\dots,h_{k}]\ =\
\Delta^{k}[h_{1}+h_{1}^{\prime},\dots,h_{k}].$

Therefore $\Delta^{k}$ is linear in the first entry.

(ii) To prove that $\Delta^{k}$ is linear in the $i^{\rm th}$ entry, for
$i\geq 2$, choose $p,q$ so that

$p+i-1\ =\ k-i+1+q.$

Then $Y$ decomposes into a $2\times 2$ block of $(p+i-1)\times(p+i-1)$
matrices.

$Y\ =\ \begin{bmatrix}A&B\\\ 0&D\end{bmatrix}.$

Moreover $B$ is the matrix whose bottom left-hand entry is $h_{i}$, and
everything else is $0$. Therefore

$\widetilde{F}(Y)\ =\
\begin{bmatrix}\widetilde{F}(A)&\Delta^{1}\widetilde{F}(A,D)[B]\\\
0&\widetilde{F}(D)\end{bmatrix},$

and $\Delta^{1}\widetilde{F}(A,D)[B]$ is linear in $B$ (and hence in $h_{i}$)
by part (i). Therefore the $(1,p+k+q)$ entry of $\widetilde{F}(Y)$, which we
have established is $\Delta^{k}[h_{1},\dots,h_{k}]$, is linear in $h_{i}$, as
desired. ∎

######  Lemma 3.7.

Suppose $a=(a_{1},\dots,a_{k+1})$ is a $(k+1)$-tuple of points in $\Omega$,
each of which is a $d$-tuple of scalars. Then
$\Delta^{k}F(a_{1},\dots,a_{k+1})[h_{1},\dots,h_{k}]$ is a free polynomial in
$h$, homogeneous of degree $k$.

###### Proof.

Let us write $\Delta^{k}[h_{1},\dots,h_{k}]$ for
$\Delta^{k}F(a_{1},\dots,a_{k+1})[h_{1},\dots,h_{k}]$. By Lemma 3.6, we know
that $\Delta^{k}[h_{1},\dots,h_{k}]$ is $k$-linear. So we can assume that each
$h_{i}$ is a $d$-tuple with only one non-zero entry. Say
$h_{i}=H_{i}e_{j_{i}}$, where $e_{j_{i}}$is the $d$-tuple that is $1$ in the
$j_{i}$ slot, $0$ else, and $H_{i}$ is an operator.

Claim:

$\Delta^{k}[H_{1}e_{j_{1}},H_{2}e_{j_{2}},\dots]\ =\ H_{1}H_{2}\dots
H_{k}\Delta^{k}[e_{j_{1}},e_{j_{2}},\dots,e_{j_{k}}].$ (3.8)

This follows from the intertwining

$\displaystyle\begin{bmatrix}H_{1}H_{2}\dots H_{k}&0&0&\dots\\\ 0&H_{2}\dots
H_{k}&0&\dots&\\\ &&\ddots&\\\
&&&1\end{bmatrix}\begin{bmatrix}a_{1}&e_{j_{1}}&0&\dots\\\
0&a_{2}&e_{j_{2}}&\dots\\\ &&\ddots&\\\ &&&a_{k+1}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}a_{1}&e_{j_{1}}H_{1}&0&\dots\\\
0&a_{2}&e_{j_{2}}H_{2}&\dots\\\ &&\ddots&\\\
&&&a_{k+1}\end{bmatrix}\begin{bmatrix}H_{1}H_{2}\dots H_{k}&0&0&\dots\\\
0&H_{2}\dots H_{k}&0&\dots&\\\ &&\ddots&\\\ &&&1\end{bmatrix}$

Let

$X\ =\ \begin{bmatrix}a_{1}&e_{j_{1}}H_{1}&0&\dots\\\
0&a_{2}&e_{j_{2}}H_{2}&\dots\\\ &&\ddots&\\\ &&&a_{k+1}\end{bmatrix}$

As $X$ is a $d$-tuple of $(k+1)\times(k+1)$ matrices of scalars, it commutes
with any $(k+1)\times(k+1)$ matrix that has a constant operator $L$ on the
diagonal. Therefore $L$ commutes with
$\Delta^{k}[e_{j_{1}},e_{j_{2}},\dots,e_{j_{k}}]$. As $L$ is arbitrary, it
follows that $\Delta^{k}[e_{j_{1}},e_{j_{2}},\dots,e_{j_{k}}]$ is a scalar. So
from (3.8), we get that $\Delta^{k}[H_{1}e_{j_{1}},H_{2}e_{j_{2}},\dots]$ is a
constant times $H_{1}H_{2}\dots H_{k}$, and by linearity we are done. ∎

Now we relate $\Delta^{k}$ to $D^{k}$.

######  Lemma 3.9.

Let $F$ be NC. Then

$\Delta^{k}F(x,\dots,x)[h,\dots,h]\ =\ \frac{1}{k!}D^{k}F(x)[h,\dots,h]$

###### Proof.

Let $T$ be the upper-triangular Toeplitz matrix given by

$T\
=\begin{bmatrix}1&\frac{1}{\lambda}&\frac{1}{2!\lambda^{2}}&\dots&\frac{1}{k!\lambda^{k}}\\\
\\\ 0&1&\frac{1}{\lambda}&\dots&\frac{1}{(k-1)!\lambda^{k-1}}\\\
\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&\dots&1\end{bmatrix}.$

Its inverse is

$T^{-1}\ =\
\begin{bmatrix}1&\frac{-1}{\lambda}&\frac{1}{2!\lambda^{2}}&\dots&\frac{(-1)^{k}}{k!\lambda^{k}}\\\
\\\ 0&1&\frac{1}{\lambda}&\dots&\frac{(-1)^{k-1}}{(k-1)!\lambda^{k-1}}\\\
\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&\dots&1\end{bmatrix}.$

We have, componentwise in $x$ and $h$,

$T\ \begin{bmatrix}x&0&0&\dots&0\\\ 0&x+\lambda h&0&\dots&0\\\
\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&\dots&x+k\lambda h\end{bmatrix}\ T^{-1}\
=\ \begin{bmatrix}x&h&0&\dots&0\\\ 0&x+\lambda h&h&\dots&0\\\
\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&\dots&x+k\lambda h\end{bmatrix}$

Therefore

$\Delta^{k}F(x,x+\lambda h,\dots,x+k\lambda h)[h,h,\dots,h]\ =\
\frac{(-1)^{k}}{k!\lambda^{k}}\sum_{j=0}^{k}(-1)^{j}{k\choose j}f(x+j\lambda
h).$

Take the limit as $\lambda\to 0$ and the right-hand side converges to

$\frac{1}{k!}D^{k}F(x)[h,h,\dots,h].$

By continuity, the left-hand side converges to
$\Delta^{k}F(x,\dots,x)[h,\dots,h]$. ∎

Derivatives of NC functions are symmetric. The case $k=2$ was proved in [4].

######  Proposition 3.10.

Suppose $F$ is an NC function and $k\geq 1$ is an integer. If $\sigma$ is any
permutation in ${\textfrak S}_{k}$ then

$D^{k}F(x)[h_{1},\dots,h_{k}]=D^{k}F(x)[h_{\sigma(1)},\dots,h_{\sigma(2)}]$

for any $x$ in the domain of $F$ and for all
$h_{1},\dots,h_{k}\in{\mathcal{A}}^{d}$.

###### Proof.

The case $k=1$ is trivial, and $k=2$ was proved in [4]. Assume $k\geq 3$ and
that the result holds for $k-1$. If we can show that we can swap the last two
entries

$D^{k}F(x)[h_{1},\dots,h_{k-1},h_{k}]=D^{k}F(x)[h_{1},\dots,h_{k},h_{k-1}]$
(3.9)

and also permute the first $k-1$ entries

$D^{k}F(x)[h_{1},\dots,h_{k-1},h_{k}]=D^{k}F(x)[h_{\sigma(1)},\dots,h_{\sigma(k-1)},h_{k}]$
(3.10)

then the result follows. Set $G=D^{k-2}F$ and consider it as a function of
$x,h_{1},\dots,h_{k-2}$. Then $G$ is an NC function, and by the $k=2$ case,

$\displaystyle D^{2}$ $\displaystyle
G(x,h_{1},\dots,h_{k-2})[(\ell_{0},\dots,\ell_{k-2}),(\tilde{\ell}_{0},\dots,\tilde{\ell}_{k-2})]$
$\displaystyle=D^{2}G(x,h_{1},\dots,h_{k-2})[(\tilde{\ell}_{0},\dots,\tilde{\ell}_{k-2}),(\ell_{0},\dots,\ell_{k-2})].$

Since

$D^{2}G(x,h_{1},\dots,h_{k_{2}})[h_{k-1},0,\dots,0,h_{k},0,\dots,0]=D^{k}F(x)[h_{1},\dots,h_{k}],$

we see that Equation (3.9) holds. The induction hypothesis says that

$D^{k-1}F(x)[h_{1},\dots,h_{k-1}]=D^{k-1}F(x)[h_{\sigma(1)},\dots,h_{\sigma(k-1)}].$
(3.11)

If $G^{\prime}=D^{k-1}F$ is treated as function in $x,h_{1},\dots,h_{k-1}$,
then applying Equation (3.11), we have

$\displaystyle D^{k}F(x)[h_{1},\dots,h_{k-1},h_{k}]$
$\displaystyle=DG^{\prime}(x,h_{1},\dots,h_{k-1})[h_{k},0,\dots,0]$
$\displaystyle=DG^{\prime}(x,h_{\sigma(1)},\dots,h_{\sigma(k-1)})[h_{k},0,\dots,0]$
$\displaystyle=D^{k}F(x)[h_{\sigma(1)},\dots,h_{\sigma(k-1)},h_{k}].$

Thus, both Equation (3.9) and Equation (3.10) hold. Therefore, the
$k^{\text{th}}$ derivative of $F$ is symmetric in its arguments. ∎

Combining Lemma 3.7, Lemma 3.9 and Proposition 3.10, we get our first main
result.

######  Theorem 3.11.

Suppose $\Omega$ is an NC domain that contains a scalar point $a$ and $F$ is
an NC function on $\Omega$. Then for each $k$, the $k^{\rm th}$ derivative
$D^{k}F(a)[h_{1},\dots,h_{k}]$ is a homogeneous polynomial of degree $k$, it
is $k$-linear, and it is symmetric with respect to the action of ${\textfrak
S}_{k}$.

###### Proof.

We know that $D^{k}F(a)[h_{1},\dots,h_{k}]$ is $k$-linear, so we can assume
that each $h_{i}$ is a $d$-tuple with only one entry; we can write
$h_{i}=H_{i}e_{j_{i}}$, as in the proof of Lemma 3.7. We want to show that

$D^{k}F(a)[H_{1}e_{j_{1}},\dots,H_{k}e_{j_{k}}]$ (3.12)

is a homogeneous polynomial of degree $k$ in the operators
$H_{1},\dots,H_{k}$. Let $s_{i}$ be scalars for $1\leq i\leq k$, and consider

$D^{k}F(a)[s_{1}H_{1}e_{j_{1}}+\dots+s_{k}H_{k}e_{j_{k}},s_{1}H_{1}e_{j_{1}}+\dots+s_{k}H_{k}e_{j_{k}},\dots].$
(3.13)

Since all the arguments are the same, by Lemma 3.9 this agrees with $k!$ times
$\Delta^{k}$, which by Lemma 3.7 is a homogeneous polynomial of degree $k$.
Group the terms in (3.13) by what the commutative monomial in
$s_{1},\dots,s_{k}$ is, and consider the sum of the terms in (3.13) that are a
multiple of $s_{1}\dots s_{k}$. These correspond to

$\sum_{\sigma\in{\textfrak
S}_{k}}D^{k}F(a)[H_{\sigma(1)}e_{j_{\sigma}(1)},\dots,H_{\sigma}(k)e_{j_{\sigma}(k)}].$
(3.14)

By Proposition 3.10, (3.14) is just $k!$ times (3.12), and hence this is a
homogeneous polynomial in $H_{1},\dots,H_{k}$, as desired. ∎

##  4 Approximating NC functions by free polynomials

The results in this section are in improvement over those in [1], as they do
not need the a priori assumption that the function is sequentially strong
operator continuous. Recall that a set $\Omega$ in a vector space is balanced
if $\alpha\Omega\subseteq\Omega$ whenever $\alpha$ is a complex number of
modulus less than or equal to $1$. Importantly, $\mathcal{P}(A)$ and
$\mathcal{R}(A)$ are balanced.

If $\Omega$ contains a scalar point $\alpha$, and $F$ is NC on $\Omega$, then
$F$ is given by a convergent series of free Taylor polynomials near $\alpha$.
For convenience, we assume $\alpha=0$.

######  Lemma 4.1.

Let $\Omega$ be an NC domain containing $0$, and let $F$ be an NC function on
$\Omega$. Then there is an open set $\Upsilon\subset\Omega$ containing $0$,
and homogeneous free polynomials $p_{k}$ of degree $k$ so that

$F(x)\ =\ \sum_{k=0}^{\infty}p_{k}(x)\quad\forall\ x\in\Upsilon,$ (4.2)

and the convergence is uniform in $\Upsilon$.

###### Proof.

By Proposition 2.6, we know that $F$ is Fréchet holomorphic at $0$, and by
Theorem 3.11, we know that the $k^{\rm th}$ derivative is a homogeneous
polynomial $p_{k}$ of degree $k$. Therefore (4.2) holds. ∎

######  Theorem 4.3.

Let $\Omega$ be a balanced NC domain, and $F:\Omega\to B({\mathcal{H}})$. The
following statements are equivalent.

  1. (i)

The function $F$ is NC.

  2. (ii)

There is a power series expansion $\sum_{k=0}^{\infty}p_{k}(x)$ that converges
absolutely and locally uniformly at each point $x\in\Omega$ to $F(x)$, and
such that each $p_{k}$ is a homogeneous free polynomial of degree $k$.

  3. (iii)

For any triple of points in $\Omega$, there is a sequence of free polynomials
that converge uniformly to $F$ on a neighborhood of each point in the triple.

###### Proof.

$(i)\Rightarrow(ii):$ By Lemma 4.1, $F$ is given by a power series expansion
(4.2) in a neighborhood of $0$. We must show that this series converges
absolutely on all of $\Omega$.

Let $x\in\Omega$. Since $\Omega$ is open and balanced, there exists $r>1$ so
that $\mathbb{D}(0,r)x\subseteq\Omega$. Define a function
$f:\mathbb{D}(0,r)\to B({\mathcal{H}})$ by

$f(\zeta)\ =\ F(\zeta x).$

Then $f$ is holomorphic, and so norm continuous [11]*Thm 3.31. Therefore

$\sup\left\\{\|f(\zeta)\|\,:\,|\zeta|=\frac{1+r}{2}\right\\}\ =:\ M\ \
<\infty.$

By the Cauchy integral formula,

$\|p_{k}(x)\|\ =\
\frac{1}{k!}\left\|\frac{d^{k}}{d\zeta^{k}}f(\zeta)\big{\lvert}_{0}\right\|\
\leq\ M\left(\frac{2}{1+r}\right)^{k}.$

Therefore the power series $\sum p_{k}(x)$ converges absolutely, to
$f(1)=F(x)$.

Since $F$ is NC, it is bounded on some neighborhood of $x$, and by the Cauchy
estimate again, the convergence of the power series is uniform on that
neighborhood.

$(ii)\Rightarrow(iii):$ Let $x_{1},x_{2},x_{3}\in\Omega$. Let
$q_{k}=\sum_{j=0}^{k}p_{k}$. Then $q_{k}(x)$ converges uniformly to $F(x)$ on
an open set containing $\\{x_{1},x_{2},x_{3}\\}$.

$(iii)\Rightarrow(i):$ Since $F$ is locally uniformly approximable by free
polynomials, it is locally bounded. To see that it is also intertwining
preserving, we shall show that it satisfies the hypotheses of Lemma 2.3. Let
$S:{\mathcal{H}}\to{\mathcal{H}}^{(2)}$ be invertible, and assume that $x,y$
and $z=S^{-1}[x\oplus y]S$ are all in $\Omega$. Let $q_{k}$ be a sequence of
free polynomials that approximate $F$ on $\\{x,y,z\\}$. Then

$\displaystyle F\left(S^{-1}\begin{bmatrix}x&0\\\ 0&y\end{bmatrix}S\right)$
$\displaystyle\ =\ $
$\displaystyle\lim_{k}q_{k}\left(S^{-1}\begin{bmatrix}x&0\\\
0&y\end{bmatrix}S\right)$ $\displaystyle=$
$\displaystyle\lim_{k}\left(S^{-1}\begin{bmatrix}p_{k}(x)&0\\\
0&p_{k}(y)\end{bmatrix}S\right)$ $\displaystyle=$
$\displaystyle\left(S^{-1}\begin{bmatrix}F(x)&0\\\
0&F(y)\end{bmatrix}S\right).$

So by Lemma 2.3, $F$ is intertwining preserving. ∎

The requirement that $F$ be intertwining preserving forces $F(x)$ to always
lie in the double commutant of $x$. But if $F$ is also locally bounded on a
balanced domain containing $x$, we get a much stronger conclusion as a
corollary of Theorem 4.3.

######  Corollary 4.4.

Suppose $F$ is an NC function on a balanced NC domain $\Omega$. Then $F(x)$ is
in the norm closed unital algebra generated by $\\{x^{1},\dots,x^{d}\\}$.

##  5 $k$-linear NC functions

In the following theorem, we assume that $\Lambda$ is NC as a function of all
$dk$ variables at once, and is $k$-linear if they are broken up into
$d$-tuples. If we had an independent proof of Theorem 5.1, we could use it to
prove Theorem 3.11 with the aid of Lemma 5.4. Instead, we deduce it as a
consequence of Theorem 3.11.

######  Theorem 5.1.

Let $\Omega$ be an NC domain. Let $\Lambda:[\Omega]^{k}\to B({\mathcal{H}})$
be NC and $k$-linear. Then $\Lambda$ is a homogeneous free polynomial of
degree $k$.

###### Proof.

Let ${\mathfrak{h}}=(h_{1},\dots,h_{k})$ be a $k$-tuple of $d$-tuples in
$\Omega$. Calculating, and using $k$-linearity, we get

$\displaystyle D\Lambda(x)[{\mathfrak{h}}]$ $\displaystyle\ =\ $
$\displaystyle\lim_{\lambda\to 0}\frac{1}{\lambda}[\Lambda(x+\lambda
h)-\Lambda(x)]$ $\displaystyle=$
$\displaystyle\Lambda(h_{1},x_{2},\dots,x_{k})+\Lambda(x_{1},h_{2},\dots,x_{k})+\dots.$

Repeating this calculation, we get that
$D^{2}\Lambda(x)[{\mathfrak{h}},{\mathfrak{h}}]$ is $2!$ times the sum of
$\Lambda$ evaluated at every $k$-tuple that has $k-2$ entries from
$(x_{1},\dots,x_{k})$ and $2$ entries from ${\mathfrak{h}}$. Continuing, we
get

$D^{k}\Lambda(x)[{\mathfrak{h}},\dots,{\mathfrak{h}}]\ =\ k!\
\Lambda(h_{1},\dots,h_{k}).$ (5.2)

By Theorem 3.11, the left-hand side of (5.2) is a homogeneous free polynomial
of degree $k$, so the right-hand side is too. ∎

It is worth singling out a special case of Theorem 5.1.

######  Corollary 5.3.

Let $\Lambda:[B({\mathcal{H}})]^{dk}\to B({\mathcal{H}})$ be $k$-linear,
intertwining preserving, and bounded. Then $\Lambda$ is a homogeneous nc
polynomial of degree $k$.

######  Lemma 5.4.

The $k^{\rm th}$ derivative $D^{k}F(x)[h_{1},\dots,h_{k}]$ is NC on
$\Omega\times{\mathcal{A}}^{dk}$. If $a\in\Omega$ is a scalar point, then
$D^{k}F(a)[h_{1},\dots,h_{k}]$ is NC on ${\mathcal{A}}^{dk}$.

###### Proof.

The first assertion follows from induction, and the observation that
difference quotients preserve intertwining. The second assertion follows from
the fact that if $a$ is scalar,

$D^{k}F(a)[S^{-1}h_{1}S,\dots,S^{-1}h_{k}S]\ =\
D^{k}F(S^{-1}aS)[S^{-1}h_{1}S,\dots,S^{-1}h_{k}S].$

∎

##  6 Realization formulas

One can generalize Example 1.4. For $\delta$ a matrix of free polynomials, let

$B_{\delta}({\mathcal{A}})\ =\ \\{x\in{\mathcal{A}}^{d}:\|\delta(x)\|<1\\}.$

These sets are all NC domains. If

$\delta(x)\ =\ \begin{bmatrix}x^{1}&0&\dots&0\\\ 0&x^{2}&\dots&0\\\
&&\ddots\\\ 0&0&\dots&x^{d}\end{bmatrix},$

then $B_{\delta}({\mathcal{A}})$ is $\mathcal{P}({\mathcal{A}})$ from (1.5).
If we set

$\delta(x)\ =\ (x^{1}\ x^{2}\ \cdots\ x^{d}),$

then $B_{\delta}({\mathcal{A}})$ is $\mathcal{R}({\mathcal{A}})$ from (1.6).

The sets $B_{\delta}({\mathcal{A}})$ are closed not just under finite direct
sums, but countable direct sums, in the following sense.

######  Definition 6.1.

A family $\\{E_{k}\\}_{k=1}^{\infty}$ is an exhaustion of $\Omega$ if

  1. 1.

$E_{k}\subseteq\ {\rm int}(E_{k+1})$ for all $k$;

  2. 2.

$\Omega=\bigcup_{k=1}^{\infty}E_{k}$;

  3. 3.

each $E_{k}$ is bounded;

  4. 4.

each $E_{k}$ is closed under countable direct sums: if $x_{j}$ is a sequence
in $E_{k}$, then there exists a unitary
$U:{\mathcal{H}}\to{\mathcal{H}}^{(\infty)}$ such that

$U^{-1}\ \begin{bmatrix}x_{1}&0&\cdots&\\\ 0&x_{2}&\cdots\\\
\cdots&\cdots&\ddots\end{bmatrix}U\ \in\ E_{k}.$ (6.2)

If we set

$E_{k}\ =\ \\{x\in B_{\delta}({\mathcal{A}}):\|\delta(x)\|\leq 1-1/k,\ {\rm
and\ }\|x\|\leq k\\},$

then $E_{k}$ is an exhaustion of $B_{\delta}({\mathcal{A}})$.

We have the following automatic continuity result for NC functions on balanced
domains that have an exhaustion.

######  Theorem 6.3.

Suppose $\Omega\subseteq{\mathcal{A}}^{d}$ is a balanced NC domain that has an
exhaustion $(E_{k})$, and $F:\Omega\to B({\mathcal{H}})$ is NC and bounded on
each $E_{k}$. Suppose for some $k$, there is a sequence $(x_{j})$ in $E_{k}$
that converges to $x\in E_{k}$ in the strong operator topology. Then
$F(x_{j})$ converges to $F(x)$ in the strong operator topology.

###### Proof.

Let $U:{\mathcal{H}}\to{\mathcal{H}}^{(\infty)}$ be a unitary so that
$U^{-1}[\oplus x_{j}]U=z\in E_{k}$. Let
$\Pi_{j}:{\mathcal{H}}^{\infty}\to{\mathcal{H}}$ be projection onto the
$j^{\rm th}$ component. Let $L_{j}=\Pi_{j}U$. Then $L_{j}z=x_{j}L_{j}$.
Therefore $F(z)\ =\ U^{-1}[\oplus F(x_{j})]U.$

Let $v$ be any unit vector, and $\varepsilon>0$. By Theorem 4.3, there is a
free polynomial $p$ so that $\|p(x)-F(x)\|<\varepsilon/3$ and
$\|p(z)-F(z)\|<\varepsilon/3$. Therefore $\|p(x_{j})-F(x_{j})\|<\varepsilon/3$
for each $j$.

Now choose $N$ so that $j\geq N$ implies $\|[p(x)-p(x_{j})]v\|<\varepsilon/3$,
which we can do because multiplication is continuous on bounded sets in the
strong operator topology. Then we get for $j\geq N$ that

$\|[F(x)-F(x_{j})]v\|\ \leq\
\|F(x)-p(x)\|+\|[p(x)-p(x_{j})]v\|+\|p(x_{j})-F(x_{j})\|\ \leq\ \varepsilon.$

∎

######  Definition 6.4.

Let $\delta$ be an $I\times J$ matrix of free polynomials, and
$F:B_{\delta}({\mathcal{A}})\to B({\mathcal{H}})$. A realization for $F$
consists of an auxiliary Hilbert space ${\mathcal{M}}$ and an isometry

$\begin{bmatrix}A&B\\\ C&D\end{bmatrix}\
:\mathbb{C}\oplus{\mathcal{M}}^{I}\to\mathbb{C}\oplus{\mathcal{M}}^{J}$ (6.5)

such that for all $x$ in $B_{\delta}({\mathcal{A}})$

$F(x)\ =\ \text{ }{\begin{smallmatrix}A\\\ \otimes\\\
1\end{smallmatrix}}\text{ }+\text{ }{\begin{smallmatrix}B\\\ \otimes\\\
1\end{smallmatrix}}\text{ }\text{ }{\begin{smallmatrix}1\\\ \otimes\\\
\delta(x)\end{smallmatrix}}\text{ }\left[1-\text{ }{\begin{smallmatrix}D\\\
\otimes\\\ 1\end{smallmatrix}}\text{ }\text{ }{\begin{smallmatrix}1\\\
\otimes\\\ \delta(x)\end{smallmatrix}}\text{ }\right]^{-1}\text{
}{\begin{smallmatrix}C\\\ \otimes\\\ 1\end{smallmatrix}}\text{ }.$ (6.6)

We write tensors vertically in (6.6) just to enhance readability.

In [1] it was shown that if $B_{\delta}(B({\mathcal{H}}))$ is connected and
contains $0$, then every sequentially strong operator continuous function (in
the sense of Theorem 6.3) NC function from $B_{\delta}(B({\mathcal{H}}))$ that
is bounded by $1$ has a realization. The strong operator continuity was needed
to pass from a realization of $B_{\delta}$ in the matricial case given in [2]
to a realization for operators. In light of Proposition 6.3, though, this
hypothesis is automatically fulfilled. So we get:

######  Corollary 6.7.

Let $\delta$ be an $I\times J$ matrix of free polynomials, and
$F:B_{\delta}(B({\mathcal{H}}))\to B({\mathcal{H}})$ satisfy $\sup\|F(x)\|\leq
1$. Assume that $B_{\delta}(B({\mathcal{H}}))$ is balanced. Then $F$ is NC if
and only if it has a realization.

As another consequence, we get that every bounded non-commutative function on
$B_{\delta}({\mathbb{M}})$ (by which we mean
$\\{x\in{\mathbb{M}}^{[d]}:\|\delta(x)\|<1\\}$), has a unique extension to an
NC function on $B_{\delta}(B({\mathcal{H}}))$, where we embed
${\mathbb{M}}^{[d]}$ into $B({\mathcal{H}})^{d}$ by choosing a basis of
${\mathcal{H}}$ and identifying an $n$-by-$n$ matrix with the finite rank
operator that is $0$ outside the first $n$-by-$n$ block.

######  Corollary 6.8.

Assume $B_{\delta}(B({\mathcal{H}}))$ is balanced. Then every non-commutative
bounded function $f$ on $B_{\delta}({\mathbb{M}})$ has a unique extension to
an NC function on $B_{\delta}(B({\mathcal{H}}))$.

###### Proof.

Suppose $F_{1}$ and $F_{2}$ are both extensions of $f$, and let
$F=F_{1}-F_{2}$. As $0\in B_{\delta}(B({\mathcal{H}}))$ and $\delta$ is
continuous, there exists $r>0$ so that
$r\mathcal{P}(B({\mathcal{H}}))\subseteq B_{\delta}(B({\mathcal{H}}))$.

Let $x\in r\mathcal{P}(B({\mathcal{H}}))$. Then there exists a sequence
$(x_{j})$ in $r\mathcal{P}({\mathbb{M}})$ that converges to $x$ in the strong
operator topology. As $F(x_{j})=0$ for each $j$, by Theorem 6.3 we get
$F(x)=0$. Therefore $F$ vanishes on an open subset of
$B_{\delta}(B({\mathcal{H}}))$. As $F$ is holomorphic, and
$B_{\delta}(B({\mathcal{H}}))$ is connected, we conclude that $F$ is
identically zero. ∎

######  Question 6.9.

Are the previous results true if $B_{\delta}(B({\mathcal{H}}))$ is not
balanced?

If one has a realization formula (Equation 6.6) for
$B_{\delta}({\mathcal{A}})$, then it automatically extends to
$B_{\delta}(B({\mathcal{H}}))$. We do not know how different choices of
algebra ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ satisfying (1.1) affect
the set of NC functions on their balls.

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