# An approximate unique extension property for completely positive maps

Ian Thompson  Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba, Canada R3T 2N2<EMAIL_ADDRESS>
###### Abstract.

We study the closure of the unitary orbit of a given point in the non-
commutative Choquet boundary of a unital operator space with respect to the
topology of pointwise norm convergence. This may be described more extensively
as the $*$-representations of the $\mathrm{C}^{*}$-envelope that are
approximately unitarily equivalent to one that possesses the unique extension
property. Although these $*$-representations do not necessarily have the
unique extension property themselves, we show that their unital completely
positive extensions display significant restrictions. When the underlying
operator space is separable, this allows us to connect our work to Arveson’s
hyperrigidity conjecture. Finally, as an application, we reformulate the
classical Šaškin Theorem and Arveson’s essential normality conjecture.

The author was partially supported by an NSERC CGS-D Scholarship.

##  1\. Introduction

The Weyl von-Neumann Theorem characterizes normal operators on a separable
Hilbert space as the diagonal operators modulo the compacts. In
[voiculescu1976non], Voiculescu unveiled a vast non-commutative generalization
regarding $*$-representations of separable $\mathrm{C}^{*}$-algebras on
separable Hilbert spaces. Namely, every $*$-representation
$\pi:{\mathfrak{A}}\rightarrow B({\mathcal{H}}_{\pi})$ on a separable Hilbert
space lies in the pointwise norm closed unitary orbit of some direct sum of
irreducible $*$-representations. That is, there is a collection of irreducible
$*$-representations $\\{\pi_{i}:{\mathfrak{A}}\rightarrow
B({\mathcal{H}}_{i}):i\in I\\}$ and a sequence of unitary operators
$u_{n}:\bigoplus_{i\in I}{\mathcal{H}}_{i}\rightarrow{\mathcal{H}}_{\pi}$ such
that

$\lim_{n\rightarrow\infty}\left\|u_{n}^{*}\pi(t)u_{n}-\left(\bigoplus_{i\in
I}\pi_{i}\right)(t)\right\|=0,\ \ \ \ \ \ \ t\in{\mathfrak{A}}.$

Moreover, we may assume that $u_{n}^{*}\pi(t)u_{n}$ is equal to the direct sum
of irreducible $*$-representations modulo the compacts for each $n$. Within
[voiculescu1976non], there are several results collectively labelled as
‘Voiculescu’s Theorem.’ For our purposes, by Voiculescu’s Theorem we mean that
$*$-representations of a separable $\mathrm{C}^{*}$-algebra acting on a
separable Hilbert space possess the same point-norm closed unitary equivalence
class precisely when the $*$-representations have pointwise equal rank.

In [hadwin1981nonseparable, Theorem 3.14], Hadwin provided a non-separable
adaption of Voiculescu’s Theorem, which we now describe. For this, recall that
the rank of a Hilbert space operator will refer to the dimension of the
closure of the range. We say that two $*$-representations $\pi,\sigma$ of a
$\mathrm{C}^{*}$-algebra ${\mathfrak{A}}$ are _approximately unitarily
equivalent_ if there is a net of unitary operators
$u_{\beta}:{\mathcal{H}}_{\sigma}\rightarrow{\mathcal{H}}_{\pi}$ such that
$\|u_{\beta}^{*}\pi(t)u_{\beta}-\sigma(t)\|\rightarrow 0$ for each
$t\in{\mathfrak{A}}$. Whenever ${\mathfrak{A}}$ and ${\mathcal{H}}_{\pi}$ are
separable, the net may be replaced by a sequence. Then, Hadwin showed that
$\pi$ and $\sigma$ are approximately unitarily equivalent if and only if
$\text{rank}(\pi(t))=\text{rank}(\sigma(t))$ for every $t\in{\mathfrak{A}}$.
When ${\mathcal{H}}_{\pi}$ fails to be separable, one may imagine difficulties
in capturing cardinality but our results do not require such intricacies. As a
consequence to Hadwin’s Theorem, one can generally expect that the approximate
unitary equivalence class of a $*$-representation is quite large. Indeed, if
$\pi$ and $\sigma$ are $*$-representations with the same kernel, then
$\pi^{(\kappa)}$ and $\sigma^{(\kappa)}$ are approximately unitarily
equivalent for a suitably large choice of cardinal $\kappa$.

Our work focuses on studying the approximate unitary equivalence class of a
particularly significant class of $*$-representations that are associated with
a unital operator space. To this end, an _operator space_ is a subspace
${\mathcal{M}}\subset B({\mathcal{H}})$ and, when ${\mathcal{M}}$ is self-
adjoint and unital, ${\mathcal{M}}$ is referred to as an _operator system_. We
let the smallest $\mathrm{C}^{*}$-algebra containing a unital operator space
${\mathcal{M}}$ in $B({\mathcal{H}})$ be denoted by
$\mathrm{C}^{*}({\mathcal{M}})$. A _$\mathrm{C}^{*}$ -cover_ is a pair
$({\mathfrak{A}},\iota)$ where $\iota:{\mathcal{M}}\rightarrow
B({\mathcal{K}})$ is a unital completely isometric linear map and
${\mathfrak{A}}=\mathrm{C}^{*}(\iota({\mathcal{M}}))$. The _$\mathrm{C}^{*}$
-envelope_ of ${\mathcal{M}}$ is a $\mathrm{C}^{*}$-cover
$(\mathrm{C}^{*}_{e}({\mathcal{M}}),\varepsilon)$ that satisfies the following
universal property: whenever $({\mathfrak{A}},\iota)$ is a
$\mathrm{C}^{*}$-cover of ${\mathcal{M}}$, there exists a $*$-representation
$\theta:{\mathfrak{A}}\rightarrow\mathrm{C}^{*}_{e}({\mathcal{M}})$ satisfying
$\theta\circ\iota=\varepsilon$. Existence of the $\mathrm{C}^{*}$-envelope is
non-trivial and was originally shown in [hamana1979injective]. Alternatively,
a large program of Arveson [arveson1969subalgebras] showed that the
$\mathrm{C}^{*}$-envelope always exists due to the existence of sufficiently
many boundary representations for ${\mathcal{M}}$
[arveson2008noncommutative],[davidson2015choquet],[dritschel2005boundary],[muhly1998algebraic].
We say that a $*$-representation $\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ has the _unique extension property_ with respect to
${\mathcal{M}}$ if, whenever $\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$, then we have that $\psi=\pi$.
If, in addition, $\pi$ is irreducible, then $\pi$ is said to be a _boundary
representation_ for ${\mathcal{M}}$. Arveson’s program, as well as many
subsequent developments
[arveson1998subalgebras],[arveson2011noncommutative],[clouatre2021finite],[clouatre2018multiplier],[clouatre2020finite],[clouatre2022minimal],[davidson2019noncommutative],[davidson2021choquet],[davidson2022strongly],[kakariadis2013vsilov],[kennedy2015essential],[kennedy2021noncommutative],[kleski2014korovkin],
have demonstrated that boundary representations encode many underlying
features of a unital operator space and may be thought of as a non-commutative
Choquet boundary [arveson2008noncommutative]. We take our main inspiration
from one of these developments [arveson2011noncommutative].

Motivated by classical results of Korovkin and Šaškin, Arveson proposed a non-
commutative approximation theory that is encoded in terms of boundary
representations [arveson2011noncommutative]. Initially, Korovkin had proven a
rigidity result concerning sequences of positive maps with domain and codomain
equal to $\mathrm{C}[0,1]$. Šaškin had then demonstrated that Korovkin’s
result is an example of a particular unital operator space
${\mathcal{M}}\subset\mathrm{C}[0,1]$ satisfying the property that all
irreducible $*$-representations of $\mathrm{C}[0,1]$ are boundary
representations for ${\mathcal{M}}$ [vsavskin1967mil]. Explicitly, Šaškin had
showed that, given a compact metric space $X$, every irreducible
$*$-representation for $\mathrm{C}(X)$ is a boundary representation for a
unital operator space ${\mathcal{M}}\subset\mathrm{C}(X)$ if and only if,
whenever $\psi_{n}:\mathrm{C}(X)\rightarrow\mathrm{C}(X)$ is a sequence of
unital positive maps such that $\|\psi_{n}(g)-g\|\rightarrow 0$ for every
$g\in{\mathcal{M}}$, then we have that $\|\psi_{n}(f)-f\|\rightarrow 0$ for
every $f\in\mathrm{C}(X)$. Decades later, Arveson provided a non-commutative
adaptation of this phenomenon [arveson2011noncommutative, Theorem 2.1]. Given
a separable unital operator space ${\mathcal{M}}\subset B({\mathcal{H}})$,
Arveson showed that every $*$-representation of
$\mathrm{C}^{*}({\mathcal{M}})$ has the unique extension property with respect
to ${\mathcal{M}}$ if and only if, whenever
$\omega:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{K}})$ is a
faithful $*$-representation and $\psi_{n}:B({\mathcal{K}})\rightarrow
B({\mathcal{K}})$ is a sequence of unital completely positive maps satisfying

$\lim_{n\rightarrow\infty}\|\psi_{n}(\omega(m))-\omega(m)\|=0,\ \ \ \ \ \ \
m\in{\mathcal{M}},$

then we necessarily have that

$\lim_{n\rightarrow\infty}\|\psi_{n}(\omega(t))-\omega(t)\|=0,\ \ \ \ \ \ \
t\in\mathrm{C}^{*}({\mathcal{M}}).$

A unital operator space ${\mathcal{M}}$ with the property that all
$*$-representations of $\mathrm{C}^{*}({\mathcal{M}})$ have the unique
extension property is said to be _hyperrigid_. Unfortunately, Arveson was
unable to frame hyperrigidity in terms of irreducible $*$-representations.
Accordingly, Arveson conjectured that every irreducible $*$-representation of
$\mathrm{C}^{*}({\mathcal{M}})$ is a boundary representation for a separable
unital operator space ${\mathcal{M}}$ if and only if every $*$-representation
of $\mathrm{C}^{*}({\mathcal{M}})$ has the unique extension property with
respect to ${\mathcal{M}}$. Despite significant effort
[arveson2011noncommutative],[clouatre2018non],[clouatre2018unperforated],[clouatre2018multiplier],[davidson2021choquet],[katsoulis2020hyperrigidity],[kennedy2015essential],[kim2021hyperrigidity],[kleski2014korovkin],[salomon2019hyperrigid],
an answer to Arveson’s hyperrigidity conjecture is still unknown, even in the
commutative case. Indeed, the reader should note that it is not clear whether
Šaškin’s result coincides with the commutative case of Arveson’s result due to
the difference in ranges for the completely positive maps.

For the non-trivial direction of Arveson’s conjecture, suppose that every
irreducible $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ is a
boundary representation for ${\mathcal{M}}$. It is well-known that the unique
extension property is preserved under direct sums [arveson2011noncommutative,
Proposition 4.4]. So, Voiculescu’s Theorem implies that every
$*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ that acts on a separable
Hilbert space is approximately unitarily equivalent to a $*$-representation
with the unique extension property. If Arveson’s conjecture were true, then
these $*$-representations would necessarily have the unique extension
property. Accordingly, the main purpose of our work is to study those
$*$-representations that are approximately unitarily equivalent to a
$*$-representation with the unique extension property.

In Section 2, we record preliminary information and basic observations
regarding the $*$-representations of $\mathrm{C}^{*}({\mathcal{M}})$ that are
approximately unitarily equivalent to a $*$-representation with the unique
extension property. In particular, we address whether _irreducible_
$*$-representations that are approximately unitarily equivalent to a boundary
representation are necessarily boundary representations. In the literature,
there are a couple of examples demonstrating that this question has a negative
answer (Example 2 and surrounding discussion) and we record the details of one
such example.

In Section 3, we study a variation on the unique extension property. Our first
main development (Theorem 3.6) shows that this variation is inherited among
those $*$-representations that are approximately unitarily equivalent to a
$*$-representation with the unique extension property.

######  Theorem 1.1.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation that is approximately unitarily equivalent to a
$*$-representation with the unique extension property. If
$\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ is a
unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$, then there is a net of unitary
operators $u_{\beta}\in B({\mathcal{H}}_{\pi})$ such that
$u_{\beta}^{*}\psi(t)u_{\beta}\rightarrow\pi(t)$ in the weak operator topology
for every $t\in\mathrm{C}^{*}({\mathcal{M}})$.

We say that a $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ that
satisfies the conclusion of Theorem 1.1 has the _approximate unique extension
property_ with respect to ${\mathcal{M}}$. Throughout Section 3, we study the
class of $*$-representations with the approximate unique extension property
and develop a collection of structural properties that are satisfied by these
$*$-representations (Propositions 3.5, 3.7, and 3.8). Consequently, this
allows us to relate our work to Arveson’s hyperrigidity conjecture (Theorems
3.11 and 4.8). For this, recall that a $\mathrm{C}^{*}$-algebra
${\mathfrak{A}}$ is _postliminal_ if the image of every irreducible
$*$-representation for ${\mathfrak{A}}$ contains the ideal of compact
operators.

######  Theorem 1.2.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a separable unital operator
space. Consider the following statements:

  1. (i)

The operator space ${\mathcal{M}}$ is hyperrigid.

  2. (ii)

Every irreducible $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ is a
boundary representation.

  3. (iii)

Every irreducible $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ is
approximately unitarily equivalent to a boundary representation.

  4. (iv)

Every $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ has the
approximate unique extension property.

Then, we have that (i)$\Rightarrow$(ii)$\Rightarrow$(iii)$\Rightarrow$(iv) and
(iii)$\not\Rightarrow$(ii). Moreover, if $\mathrm{C}^{*}({\mathcal{M}})$ is
postliminal, then we have that (iv)$\Rightarrow$(ii).

Since (iii)$\not\Rightarrow$(ii) in general, our results will imply all of the
same restrictions to a class of unital operator spaces that do not satisfy the
property that every irreducible $*$-representation is a boundary
representation. Thus, these operator spaces have the property that the space
of completely positive extensions for every $*$-representation is incredibly
restrictive, yet there are irreducible $*$-representations that fail to be
boundary representations (Example 2).

We now unveil the main result of our work, which provides a restriction on the
space of completely positive extensions of $*$-representations with the
approximate unique extension property (Theorems 4.1 and 4.4).

######  Theorem 1.3.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation that possesses the approximate unique extension property.
Suppose that $\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$. Then, the following statements
hold.

  1. (i)

There is a homomorphic conditional expectation
$\Gamma_{\psi}:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\text{Im}\pi$
satisfying $\Gamma_{\psi}\circ\psi=\pi$.

  2. (ii)

The conditional expectation $\Gamma_{\psi}$ has the unique extension property
with respect to $\text{Im}\psi$.

  3. (iii)

We have that $\ker\Gamma_{\psi}$ is the smallest closed two-sided ideal of
$\mathrm{C}^{*}(\text{Im}\psi)$ that contains $\text{Im}(\psi-\pi)$. In
particular, $\psi=\pi$ if and only if $\Gamma_{\psi}$ is isometric.

As a consequence, Theorem 1.3 demonstrates that there is a split exact
sequence of $\mathrm{C}^{*}$-algebras given by

${0}$${\langle\langle\text{Im}(\psi-\pi)\rangle\rangle}$${\mathrm{C}^{*}(\text{Im}\psi)}$${\text{Im}\pi}$${0}$

where the second term is the smallest closed two-sided ideal of
$\mathrm{C}^{*}(\text{Im}\psi)$ that contains $\text{Im}(\psi-\pi)$. The
subsequent results in Section 4 are consequences to the above developments. In
particular, this allows us to apply our machinery to obtain new results on the
structure of completely positive extensions of arbitrary isometric
$*$-representations of the $\mathrm{C}^{*}$-envelope (Corollary 4.2).

In Section 5, we consider two classes of concrete examples where our results
may apply. The first is a reformulation of Šaškin’s Theorem (Proposition 5.1)
that addresses commutativity and the approximate unique extension property.
For the second application, note that Voiculescu’s Theorem and Arveson’s
program guarantee that the collection of $*$-representations that possess the
approximate unique extension property is always quite large. Therefore, one
would expect that requiring a single $*$-representation to have the
approximate unique extension property to be a weak assumption. Nevertheless,
we apply our machinery to operator systems arising from quotient modules of
multipliers on the Drury-Arveson space and find that hyperrigidity is encoded
in a single $*$-representation possessing the approximate unique extension
property (Theorem 5.2). Due to work of Kennedy and Shalit
[kennedy2015essential], our development allows us to reframe the essential
normality conjecture of Arveson.

Acknowledgements. The author would like to thank their advisor, Raphaël
Clouâtre, for many helpful discussions during the preparation of this
manuscript.

##  2\. Approximate unitary equivalence and the unique extension property

In this section, we present a few preliminary facts as well as the fundamental
relationship between the unique extension property and its preservation under
approximate unitary equivalence. The results within Subsections 2.2 and 2.3
are likely known to experts, but a condensed treatment is unavailable.

###  2.1. Preliminaries

First, we record some well-known facts in the literature. We start with a
central tool for determining when two $*$-representations are approximately
unitarily equivalent. This was first shown in [hadwin1981nonseparable, Theorem
3.14].

######  Theorem 2.1.

Let ${\mathfrak{A}}$ be a $\mathrm{C}^{*}$-algebra. Suppose that
$\pi:{\mathfrak{A}}\rightarrow B({\mathcal{H}}_{\pi})$ and
$\sigma:{\mathfrak{A}}\rightarrow B({\mathcal{H}}_{\sigma})$ are
$*$-representations. Then, the following statements are equivalent.

  1. (i)

The $*$-representations $\pi$ and $\sigma$ are approximately unitarily
equivalent.

  2. (ii)

We have that $\text{rank}(\pi(t))=\text{rank}(\sigma(t))$ for every
$t\in{\mathfrak{A}}$.

Next, we record a few basic facts regarding $*$-representations with the
unique extension property.

######  Lemma 2.2.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and
$\\{\pi_{i}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{i}):i\in
I\\}$ be a collection of $*$-representations. Then, the following statements
are equivalent.

  1. (i)

For each $i\in I$, $\pi_{i}$ has the unique extension property with respect to
${\mathcal{M}}$.

  2. (ii)

We have that $\bigoplus_{i\in I}\pi_{i}$ has the unique extension property
with respect to ${\mathcal{M}}$.

###### Proof.

See [arveson2011noncommutative, Proposition 4.4] and [clouatre2020finite,
Lemma 2.8]. ∎

Finally, we require a couple known facts on hyperrigidity and boundary
representations. In particular, statement (iii) will imply that, in the
interest of the hyperrigidity conjecture, it is of no harm to assume that
${\mathcal{M}}$ generates its $\mathrm{C}^{*}$-envelope.

######  Theorem 2.3.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and
$(\mathrm{C}^{*}_{e}({\mathcal{M}}),\varepsilon)$ denote the
$\mathrm{C}^{*}$-envelope of ${\mathcal{M}}$. Let
$\theta:\mathrm{C}^{*}({\mathcal{M}})\rightarrow\mathrm{C}^{*}_{e}({\mathcal{M}})$
denote the $*$-representation that satisfies
$\theta\mid_{\mathcal{M}}=\varepsilon$. Then, the following statements hold.

  1. (i)

There is an isometric $*$-representation of
$\mathrm{C}^{*}_{e}({\mathcal{M}})$ that possesses the unique extension
property with respect to $\varepsilon({\mathcal{M}})$.

  2. (ii)

If $\beta:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\beta})$
is a boundary representation for ${\mathcal{M}}$, then there is a
$*$-representation
$\widetilde{\beta}:\mathrm{C}^{*}_{e}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\beta})$ such that $\beta=\widetilde{\beta}\circ\theta$.

  3. (iii)

If every irreducible $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ is
a boundary representation for ${\mathcal{M}}$, then
$\mathrm{C}^{*}({\mathcal{M}})$ is the $\mathrm{C}^{*}$-envelope of
${\mathcal{M}}$.

###### Proof.

(i): By [clouatre2020finite, Lemma 2.11], there is a collection $\Delta$
consisting of boundary representations for $\varepsilon({\mathcal{M}})$, that
are defined on $\mathrm{C}^{*}_{e}({\mathcal{M}})$, such that
$\bigoplus_{\delta\in\Delta}\delta$ is isometric. Thus, by Lemma 2.2,
$\bigoplus_{\delta\in\Delta}\delta$ is an isometric $*$-representation of
$\mathrm{C}^{*}_{e}({\mathcal{M}})$ that has the unique extension property
with respect to $\varepsilon({\mathcal{M}})$.

(ii): By [arveson2008noncommutative, Theorem 7.1] and [davidson2015choquet,
Theorem 3.3], there is a collection $\Sigma$ consisting of boundary
representations for ${\mathcal{M}}$, that are defined on
$\mathrm{C}^{*}({\mathcal{M}})$, such that $\bigoplus_{\sigma\in\Sigma}\sigma$
is completely isometric on ${\mathcal{M}}$ (alternatively, see
[clouatre2020finite, Theorem 2.10]). Define a closed two-sided ideal of
$\mathrm{C}^{*}({\mathcal{M}})$ by

${\mathfrak{J}}_{\Sigma}=\bigcap_{\sigma\in\Sigma}\ker\sigma.$

By [arveson1969subalgebras, Theorem 2.2.3], the ideal
${\mathfrak{J}}_{\Sigma}$ coincides with the intersection of the kernels of
all boundary representations for ${\mathcal{M}}$ that are defined on
$\mathrm{C}^{*}({\mathcal{M}})$. Then, by [arveson1969subalgebras, Proposition
2.2.4 and Theorem 2.2.5], we have that
$\mathrm{C}^{*}({\mathcal{M}})/{\mathfrak{J}}_{\Sigma}$ is the
$\mathrm{C}^{*}$-envelope of ${\mathcal{M}}$. Since
${\mathfrak{J}}_{\Sigma}\subset\ker\beta$, statement (ii) is immediate.

(iii): Again, let ${\mathfrak{J}}_{\Sigma}$ denote the intersection of the
kernels of all boundary representations for ${\mathcal{M}}$ that are defined
on $\mathrm{C}^{*}({\mathcal{M}})$. By [arveson1969subalgebras, Proposition
2.2.4 and Theorem 2.2.5],
$\mathrm{C}^{*}({\mathcal{M}})/{\mathfrak{J}}_{\Sigma}$ is the
$\mathrm{C}^{*}$-envelope of ${\mathcal{M}}$. So (iii) is immediate since
${\mathfrak{J}}_{\Sigma}=\\{0\\}$. ∎

We remark that the closed two-sided ideal ${\mathfrak{J}}_{\Sigma}$
constructed in the proof of Theorem 2.3 is referred to as the the _Shilov
ideal_ of ${\mathcal{M}}$ in $\mathrm{C}^{*}({\mathcal{M}})$. Since
$\mathrm{C}^{*}({\mathcal{M}})/{\mathfrak{J}}_{\Sigma}$ is the
$\mathrm{C}^{*}$-envelope of ${\mathcal{M}}$, the collection of all boundary
representations for ${\mathcal{M}}$ is typically thought of as the _non-
commutative Choquet boundary_ for ${\mathcal{M}}$.

###  2.2. Approximate unitary equivalence and the unique extension property

For our investigation, we start by showing that the collection of
$*$-representations that possess the unique extension property is rarely
closed under approximate unitary equivalence.

######  Proposition 2.4.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and
assume that $\mathrm{C}^{*}({\mathcal{M}})$ is the $\mathrm{C}^{*}$-envelope
of ${\mathcal{M}}$. Then, the following statements are equivalent.

  1. (i)

The operator space ${\mathcal{M}}$ is hyperrigid.

  2. (ii)

Whenever $\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$
is a $*$-representation that is approximately unitarily equivalent to a
$*$-representation with the unique extension property, then $\pi$ has the
unique extension property.

###### Proof.

It is trivial that (i)$\Rightarrow$(ii). For the converse, let
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation. We show that $\pi$ has the unique extension property.
Since ${\mathcal{M}}$ generates its $\mathrm{C}^{*}$-envelope, by Lemma 2.2
and Theorem 2.3 (i), we may assume that $\pi$ is injective. Define an infinite
cardinal $\kappa=\max\\{\aleph_{0},\dim{\mathcal{H}}_{\pi}\\}$. By considering
$\pi^{(\kappa)}$ if necessary, we may additionally assume that
$\text{rank}(\pi(t))=\kappa$ for each non-zero
$t\in\mathrm{C}^{*}({\mathcal{M}})$.

Now, by Theorem 2.3 (i), there is an injective $*$-representation $\sigma$ of
$\mathrm{C}^{*}({\mathcal{M}})$ that possesses the unique extension property
with respect to ${\mathcal{M}}$. Again, we may assume that
$\text{rank}(\sigma(t))=\kappa$ for every non-zero
$t\in\mathrm{C}^{*}({\mathcal{M}})$. By Theorem 2.1, we have that $\pi$ and
$\sigma$ are approximately unitarily equivalent. Whence, $\pi$ has the unique
extension property with respect to ${\mathcal{M}}$ by assumption. ∎

######  Remark 2.5.

Assume that $\mathrm{C}^{*}_{e}({\mathcal{M}})$ is separable and postliminal.
To study hyperrigidity, the reader may find it desirable to consider how the
unique extension property behaves with respect to direct integrals as this
machinery can be used to classify multiplicity-free $*$-representations up to
unitary equivalence [arveson2012invitation, Theorem 4.3.4]. However, due to
[hadwin1981nonseparable, Corollary 6.3] and Lemma 2.2, the unique extension
property is stable under direct integrals whenever it is stable under
approximate unitary equivalence. We opt to study the latter, but remark that
there has been some progress using direct integrals
[arveson2011noncommutative],[kleski2014korovkin].

Now, we discuss approximate unitary equivalence of _irreducible_
$*$-representations with the unique extension property. As the next two
examples demonstrate, the answer to this question is more subtle than the
previous one.

We first note that a variation of Proposition 2.4 fails even in the classical
setting of function spaces. For this, recall that postliminal
$\mathrm{C}^{*}$-algebras satisfy the property that irreducible
$*$-representations that possess the same kernel are necessarily unitarily
equivalent [diximier1977c, Corollary 4.1.10]. Thus, whenever the
$\mathrm{C}^{*}$-envelope is postliminal, the non-commutative Choquet boundary
is automatically stable under approximate unitary equivalence.

######  Example 1.

There is a compact metric space $X$ and a unital subalgbera
${\mathcal{A}}\subset\mathrm{C}(X)$ that separates points in $X$ with the
property that the Choquet boundary is a proper subset of the Shilov boundary
[phelps2001lectures, pg. 42]. In other words, not every irreducible
$*$-representation of $\mathrm{C}^{*}_{e}({\mathcal{A}})$ is a boundary
representation for ${\mathcal{A}}$ [gamelin2005uniform, Theorem II.11.3].

On the other hand, suppose that $\pi$ is a boundary representation for
${\mathcal{A}}$. If $\sigma$ is an irreducible $*$-representation of
$\mathrm{C}^{*}_{e}({\mathcal{A}})$ that is approximately unitarily equivalent
to $\pi$, then $\pi$ and $\sigma$ are unitarily equivalent by the previous
discussion. Whence, $\sigma$ is also a boundary representation for
${\mathcal{A}}$. So, the boundary representations for ${\mathcal{A}}$ are
stable under approximate unitary equivalence, yet not every irreducible
$*$-representation of $\mathrm{C}^{*}_{e}({\mathcal{A}})$ is a boundary
representation for ${\mathcal{A}}$.

The above conclusion can be achieved for any other unital operator space that
fails to be hyperrigid and has a postliminal $\mathrm{C}^{*}$-envelope.
However, when we remove the postliminal assumption, the non-commutative
Choquet boundary is not necessarily closed under approximate unitary
equivalence. Two examples that demonstrate the failure of this preservation
are the operator system generated by a pair of freely independent semicircular
operators [davidson2019noncommutative, Example 6.6.3] and the infinite Cuntz
algebra [muhly1998tensor, Continuation of Example 2.7]. We recall the details
of the latter example.

######  Example 2.

Let ${\mathcal{O}}_{\infty}$ denote the infinite Cuntz algebra with generators
$(V_{n})_{n\in{\mathbb{N}}},$ and consider the algebra ${\mathcal{A}}$ that is
generated by $\\{V_{n}:n\in{\mathbb{N}}\\}$. As
$\mathrm{C}^{*}({\mathcal{A}})\cong{\mathcal{O}}_{\infty}$ is simple, we have
that all irreducible $*$-representations of ${\mathcal{O}}_{\infty}$ are
approximately unitarily equivalent.

However, up to unitary equivalence, there is a unique irreducible
$*$-representation of ${\mathcal{O}}_{\infty}$ that is not a boundary
representation for ${\mathcal{A}}$. Indeed, the $*$-representations of
${\mathcal{O}}_{\infty}$ are in one-to-one correspondence with sequences of
isometries $(W_{n})_{n\geq 0}$ satisfying
$\sum_{n=1}^{\infty}W_{n}W_{n}^{*}\leq I$. An irreducible $*$-representation
is a boundary representation for ${\mathcal{A}}$ if and only if the associated
sequence of isometries satisfies $\sum_{n=1}^{\infty}W_{n}W_{n}^{*}=I$ (see
[muhly1998tensor] for details). By work of Popescu [popescu1989isometric,
Theorem 1.3], up to unitary equivalence, there is a unique such irreducible
$*$-representation with $\sum_{n=1}^{\infty}W_{n}W_{n}^{*}\neq I$.

We remark that the operator system generated by a pair of freely independent
semicircular operators possibly demonstrates an opposite extrema to that of
the infinite Cuntz algebra. Indeed, within [davidson2019noncommutative,
Example 6.6.3], an uncountable class of irreducible $*$-representations were
constructed where all but a finite number were boundary representations.
Simultaneously, one can obtain that all irreducible $*$-representations of the
$\mathrm{C}^{*}$-envelope are approximately unitarily equivalent for the same
reasoning as in Example 2. On the other hand, in Example 2, it was found that
there is a unique irreducible $*$-representation that fails to be a boundary
representation.

###  2.3. Asymptotics of the unique extension property

The unique extension property enjoys a well-known characterization in the
framework of pointwise convergent sequences. The separable version can be
found within [kleski2014korovkin, Proposition 2.2]. We record the non-
separable version, which is obtained by replacing sequences with nets.

######  Proposition 2.6.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and let
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation. Then, the following statements are equivalent.

  1. (i)

The $*$-representation $\pi$ has the unique extension property with respect to
${\mathcal{M}}$.

  2. (ii)

If $\psi_{\alpha}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a net of unital completely positive maps such that
$\psi_{\alpha}(m)\rightarrow\pi(m)$ in the weak operator topology for every
$m\in{\mathcal{M}}$, then $\psi_{\alpha}(t)\rightarrow\pi(t)$ in the weak
operator topology for every $t\in\mathrm{C}^{*}({\mathcal{M}}).$

  3. (iii)

If $\psi_{\alpha}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a net of unital completely positive maps such that
$\psi_{\alpha}(m)\rightarrow\pi(m)$ in the strong operator topology for every
$m\in{\mathcal{M}}$, then $\psi_{\alpha}(t)\rightarrow\pi(t)$ in the strong
operator topology for every $t\in\mathrm{C}^{*}({\mathcal{M}}).$

We present a reinterpretation of Proposition 2.6 that allows us to connect the
unique extension property with a notion of approximate unitary equivalence for
unital completely positive extensions. We will return to this philosophy in
future sections.

######  Proposition 2.7.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation. Then, the following statements are equivalent.

  1. (i)

The $*$-representation $\pi$ has the unique extension property with respect to
${\mathcal{M}}$.

  2. (ii)

If $\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ is a
unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$, then there exists a net of
unitary operators $w_{\beta}\in B({\mathcal{H}}_{\pi})$ such that
$w_{\beta}^{*}\pi(t)w_{\beta}\rightarrow\psi(t)$ in the strong operator
topology for every $t\in\mathrm{C}^{*}({\mathcal{M}})$.

###### Proof.

${\rm(i)}\Rightarrow{\rm(ii)}:$ In this case, $\psi=\pi$ and so we may take
$w_{\beta}$ to be the identity operator on ${\mathcal{H}}_{\pi}$.

${\rm(ii)}\Rightarrow{\rm(i)}:$ Let
$\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$ and let $w_{\beta}\in
B({\mathcal{H}}_{\pi})$ be a net of unitary operators such that
$w_{\beta}^{*}\pi(t)w_{\beta}\rightarrow\psi(t)$ in the strong operator
topology for each $t\in\mathrm{C}^{*}({\mathcal{M}})$. As multiplication over
bounded subsets is continuous in the strong operator topology, and as the net
$(w_{\beta}\pi(\cdot)w_{\beta})_{\beta}$ consists of $*$-representations, we
may conclude that $\psi(st)=\psi(s)\psi(t)$ for all
$s,t\in\mathrm{C}^{*}({\mathcal{M}}).$ Since
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$, we then obtain that
$\psi=\pi$. ∎

As we shall see in Section 3, the equivalence of Proposition 2.7 is no longer
true if one were to interchange the roles of the maps $\psi$ and $\pi$ in
statement (ii).

##  3\. Approximate unique extension property

In accordance with studying the approximate unitary equivalence class of a
$*$-representation possessing the unique extension property, we focus our
attention on a variation of the unique extension property.

######  Definition 3.1.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and let
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation. We say that $\pi$ has the _approximate unique extension
property_ with respect to ${\mathcal{M}}$ if, whenever
$\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ is a
unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$, then there is a net of
unitaries $u_{\beta}\in B({\mathcal{H}}_{\pi})$ such that
$u_{\beta}^{*}\psi(t)u_{\beta}\rightarrow\pi(t)$ in the weak operator topology
for every $t\in\mathrm{C}^{*}({\mathcal{M}})$.

When there is no confusion, we will refrain from referencing the operator
space and simply state that a $*$-representation possesses the approximate
unique extension property. Moreover, when ${\mathcal{M}}$ and
${\mathcal{H}}_{\pi}$ are separable, then the net of unitaries may be replaced
by a sequence.

We caution the reader that the approximate unique extension property differs
from the characterization of the unique extension property that was seen
within Proposition 2.7 (ii). For example, we will see that this follows from
Theorem 3.6. Additionally, we remark that Definition 3.1 is somewhat
reminiscent of different conditions seen in the literature but remains largely
independent [davidson2017dilations, Theorem 5.2],[hadwin1987completely,
Theorem 2.4].

Recall that boundary representations factor through the
$\mathrm{C}^{*}$-envelope by Theorem 2.3 (ii). Since the approximate unique
extension property appears to be a significant constraint on the space of
completely positive extensions, one may wonder whether similar behaviour is
present. While the author is unaware of such a restriction, we start with a
fact that is sufficient for our purposes.

######  Lemma 3.2.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and
${\mathfrak{A}}\subset B({\mathcal{K}})$ be a $\mathrm{C}^{*}$-algebra. Let
$q:\mathrm{C}^{*}({\mathcal{M}})\rightarrow{\mathfrak{A}}$ be a surjective
$*$-homomorphism and $\pi:{\mathfrak{A}}\rightarrow B({\mathcal{H}}_{\pi})$ be
a $*$-representation. If $\pi\circ q$ possesses the approximate unique
extension property with respect to ${\mathcal{M}}$, then $\pi$ possesses the
approximate unique extension property with respect to $q({\mathcal{M}})$.

###### Proof.

Suppose that $\psi:{\mathfrak{A}}\rightarrow B({\mathcal{H}}_{\pi})$ is a
unital completely positive map such that
$\psi\mid_{q({\mathcal{M}})}=\pi\mid_{q({\mathcal{M}})}$. Then
$\varphi:=\psi\circ q$ is a unital completely positive map satisfying
$\varphi\mid_{\mathcal{M}}=(\pi\circ q)\mid_{\mathcal{M}}$. By the assumption,
there is a net of unitary operators $u_{\beta}\in B({\mathcal{H}}_{\pi})$ such
that

$\lim_{\beta}u_{\beta}^{*}\varphi(t)u_{\beta}=(\pi\circ q)(t),\ \ \ \ \ \
t\in\mathrm{C}^{*}({\mathcal{M}}),$

in the weak operator topology. Thus,

$\lim_{\beta}u_{\beta}^{*}\psi(s)u_{\beta}=\pi(s),\ \ \ \ \ \
s\in{\mathfrak{A}},$

in the weak operator topology. So $\pi$ has the approximate unique extension
property. ∎

As a completely isometric representation of ${\mathcal{M}}$ induces a
$*$-isomorphism of $\mathrm{C}^{*}$-envelopes [paulsen2002completely, pg.
219], Lemma 3.2 implies that the $*$-representations of
$\mathrm{C}^{*}_{e}({\mathcal{M}})$ that possess the approximate unique
extension property are invariant under completely isometric isomorphism.

######  Proposition 3.3.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ and ${\mathcal{N}}\subset
B({\mathcal{K}})$ be unital operator spaces that generate their respective
$\mathrm{C}^{*}$-envelopes. Suppose that
$\iota:{\mathcal{M}}\rightarrow{\mathcal{N}}$ is a unital completely isometric
linear map and let
$\theta:\mathrm{C}^{*}_{e}({\mathcal{M}})\rightarrow\mathrm{C}^{*}_{e}({\mathcal{N}})$
be a $*$-isomorphism satisfying $\theta\mid_{\mathcal{M}}=\iota$. Then,
$\pi:\mathrm{C}^{*}_{e}({\mathcal{N}})\rightarrow B({\mathcal{H}}_{\pi})$
possesses the approximate unique extension property with respect to
${\mathcal{N}}$ if and only if $\pi\circ\theta$ possesses the approximate
unique extension property with respect to ${\mathcal{M}}$.

###### Proof.

If $\pi\circ\theta$ has the approximate unique extension property with respect
to ${\mathcal{M}}$, then $\pi$ has the approximate unique extension property
with respect to ${\mathcal{N}}$ by Lemma 3.2.

Conversely, if $\pi=(\pi\circ\theta)\circ\theta^{-1}$ has the approximate
unique extension property with respect to ${\mathcal{N}}$, then again
$\pi\circ\theta$ has the approximate unique extension property with respect to
${\mathcal{M}}$ by Lemma 3.2. ∎

Additionally, we remark that the approximate unique extension property has a
characterization that is reminiscent of Proposition 2.6.

######  Proposition 3.4.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and let
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation. Then, the following statements are equivalent.

  1. (i)

The $*$-representation $\pi$ possesses the approximate unique extension
property.

  2. (ii)

If $\psi_{\alpha}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a net of unital completely positive maps such that
$\psi_{\alpha}(m)\rightarrow\pi(m)$ in the weak operator topology for every
$m\in{\mathcal{M}}$, then there is a subnet $(\psi_{\alpha_{\eta}})_{\eta}$ of
unital completely positive maps and a net of unitary operators $u_{\beta}\in
B({\mathcal{H}}_{\pi})$ such that

$\lim_{\beta}\lim_{\eta}u_{\beta}^{*}\psi_{\alpha_{\eta}}(t)u_{\beta}=\pi(t),\
\ \ \ \ \ t\in\mathrm{C}^{*}({\mathcal{M}}),$

in the weak operator topology.

###### Proof.

(i)$\Rightarrow$(ii): Let
$\psi_{\alpha}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ be a net of unital completely positive maps such that
$\psi_{\alpha}(m)\rightarrow\pi(m)$ in the weak operator topology for every
$m\in{\mathcal{M}}$. By compactness, we may obtain a subnet
$(\psi_{\alpha_{\eta}})_{\eta}$ of unital completely positive maps that
converges to a unital completely positive map
$\varphi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ in
the pointwise weak operator topology [paulsen2002completely, Theorem 7.4]. In
other words, we have that $\psi_{\alpha_{\eta}}(t)\rightarrow\varphi(t)$ in
the weak operator topology for every $t\in\mathrm{C}^{*}({\mathcal{M}})$.
Consequently, we have that $\varphi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$.

Since $\pi$ has the approximate unique extension property, there is a net of
unitary operators $u_{\beta}\in B({\mathcal{H}}_{\pi})$ such that
$u_{\beta}^{*}\varphi(t)u_{\beta}\rightarrow\pi(t)$ in the weak operator
topology for every $t\in\mathrm{C}^{*}({\mathcal{M}})$. So we conclude that

$\lim_{\beta}\lim_{\eta}u_{\beta}^{*}\psi_{\alpha_{\eta}}(t)u_{\beta}=\lim_{\beta}u_{\beta}^{*}\varphi(t)u_{\beta}=\pi(t),\
\ \ \ \ \ t\in\mathrm{C}^{*}({\mathcal{M}}),$

as desired.

(ii)$\Rightarrow$(i): Trivial. ∎

We now work towards identifying a large class of $*$-representations that
possess the approximate unique extension property. For this, we require that
the approximate unique extension property be preserved under approximate
unitary equivalence.

######  Proposition 3.5.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and let
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation. Suppose that $\pi$ possesses the approximate unique
extension property. If $\sigma:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\sigma})$ is a $*$-representation that is approximately
unitarily equivalent to $\pi$, then $\sigma$ also possesses the approximate
unique extension property.

###### Proof.

The approximate unique extension property is easily seen to be stable under
unitary equivalence. Thus, since there is a unitary operator
$U:{\mathcal{H}}_{\sigma}\rightarrow{\mathcal{H}}_{\pi}$, we may assume for
simplicity that ${\mathcal{H}}_{\sigma}={\mathcal{H}}_{\pi}$. In this case,
there is a net of unitary operators $w_{\gamma}\in B({\mathcal{H}}_{\pi})$
such that

$\lim_{\gamma}\lVert w_{\gamma}^{*}\sigma(t)w_{\gamma}-\pi(t)\rVert=0,\ \ \ \
\ \ t\in\mathrm{C}^{*}({\mathcal{M}}).$

In particular, we also have that

$\lim_{\gamma}\|\sigma(t)-w_{\gamma}\pi(t)w_{\gamma}^{*}\|=0,\ \ \ \ \ \
t\in\mathrm{C}^{*}({\mathcal{M}}).$

Let $\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be
a unital completely positive map such that
$\psi\mid_{\mathcal{M}}=\sigma\mid_{\mathcal{M}}$. By compactness
[paulsen2002completely, Theorem 7.4], we may obtain a subnet
$(w_{\gamma_{\eta}}^{*}\psi(\cdot)w_{\gamma_{\eta}})_{\eta}$ of unital
completely positive maps and a unital completely positive map
$\varphi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ such
that

$\lim_{\eta}w_{\gamma_{\eta}}^{*}\psi(t)w_{\gamma_{\eta}}=\varphi(t),\ \ \ \ \
\ t\in\mathrm{C}^{*}({\mathcal{M}}),$

in the weak operator topology. In particular,

$\varphi(m)=\lim_{\eta}w_{\gamma_{\eta}}^{*}\psi(m)w_{\gamma_{\eta}}=\lim_{\eta}w_{\gamma_{\eta}}^{*}\sigma(m)w_{\gamma_{\eta}}=\pi(m),\
\ \ \ \ \ \ m\in{\mathcal{M}}.$

As $\pi$ has the approximate unique extension property, there is a net of
unitaries $u_{\beta}\in B({\mathcal{H}}_{\pi})$ such that

$\pi(t)=\lim_{\beta}u_{\beta}^{*}\varphi(t)u_{\beta}=\lim_{\beta}\lim_{\eta}u_{\beta}^{*}w_{\gamma_{\eta}}^{*}\psi(t)w_{\gamma_{\eta}}u_{\beta},\
\ \ \ \ \ t\in\mathrm{C}^{*}({\mathcal{M}}).$

Then we see that

$\lim_{\gamma}\lim_{\beta}\lim_{\eta}w_{\gamma}u_{\beta}^{*}w_{\gamma_{\eta}}^{*}\psi(t)w_{\gamma_{\eta}}u_{\beta}w_{\gamma}^{*}=\lim_{\gamma}w_{\gamma}\pi(t)w_{\gamma}^{*}=\sigma(t),\
\ \ \ \ \ t\in\mathrm{C}^{*}({\mathcal{M}}).$

Thus, there is a net of unitaries
$v_{\alpha}:=w_{\gamma_{\alpha}}u_{\beta_{\alpha}}^{*}w_{\gamma_{\eta_{\alpha}}}^{*}$
that corresponds to the above iterated limit (for example – see
[kelley2017general, pg. 69]) and so, we have that

$\lim_{\alpha}v_{\alpha}^{*}\psi(t)v_{\alpha}=\sigma(t),\ \ \ \ \ \
t\in\mathrm{C}^{*}({\mathcal{M}}).$

Therefore $\sigma$ has the approximate unique extension property. ∎

This allows us to obtain a particularly significant class of
$*$-representations that possess the approximate unique extension property.

######  Theorem 3.6.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space.
Suppose that $\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a $*$-representation that is approximately
unitarily equivalent to a $*$-representation with the unique extension
property. Then, we have that $\pi$ has the approximate unique extension
property.

###### Proof.

By Proposition 3.5, it suffices to show that $*$-representations with the
unique extension property have the approximate unique extension property.
Thus, we assume that $\pi$ has the unique extension property. In which case,
if $\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ is a
unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$, then $\psi=\pi$ and so we may
take $u_{\beta}$ to be the identity operator. ∎

We remark that one form of Voiculescu’s Theorem states that, whenever $\pi$
and $\sigma$ are approximately unitarily equivalent $*$-representations of a
separable $\mathrm{C}^{*}$-algebra ${\mathfrak{A}}$ on a separable Hilbert
space, then the limiting difference $\sigma(t)-u_{n}^{*}\pi(t)u_{n}$ can be
made compact for every $t\in{\mathfrak{A}}$ and every $n\in{\mathbb{N}}$
[voiculescu1976non, Theorem 1.5]. A similar statement has been obtained for
tuples of separable operators [davidson2017dilations, Theorem 5.2]. Thus, one
may wonder whether the difference in our setting, given by
$\pi(t)-u_{\beta}^{*}\psi(t)u_{\beta}$, can be made compact under separability
assumptions. However, we do not know if such a refinement is possible in our
context due to the application of compactness in Proposition 3.5.

Since the usual unique extension property is not preserved under approximate
unitary equivalence (Example 2), by Theorem 3.6 we obtain that the approximate
unique extension property is truly different than the usual unique extension
property. Moreover, note that whenever
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ has the
approximate unique extension property and
$\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ is a
completely positive extension of $\pi\mid_{\mathcal{M}}$, we may conclude by
[kleski2014korovkin, Lemma 2.1] that there is a net of unitaries $u_{\beta}\in
B({\mathcal{H}}_{\pi})$ such that
$u_{\beta}^{*}\psi(t)u_{\beta}\rightarrow\pi(t)$ in the _strong operator
topology_ for every $t\in\mathrm{C}^{*}({\mathcal{M}})$. In particular, one
may not interchange the roles of the maps $\psi$ and $\pi$ in Proposition 2.7.

Notwithstanding the above differences, the approximate unique extension
property still behaves well enough to form a well-behaved class of
$*$-representations.

######  Proposition 3.7.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space. If
$\\{\pi_{i}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{i}):i\in
I\\}$ is a collection of $*$-representations that possess the approximate
unique extension property, then $\pi=\bigoplus_{i\in I}\pi_{i}$ also possesses
the approximate unique extension property.

###### Proof.

Let $\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B(\bigoplus_{i\in
I}{\mathcal{H}}_{i})$ be a unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$. For each $i$, define a unital
completely positive map

$\psi_{i}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{i}),\ \ \
\ \ \ t\mapsto P_{{\mathcal{H}}_{i}}\psi(t)\mid_{{\mathcal{H}}_{i}},$

and note that $\psi_{i}\mid_{\mathcal{M}}=\pi_{i}\mid_{\mathcal{M}}$. In turn,
as each $\pi_{i}$ possesses the approximate unique extension property, we may
obtain a directed set $\Lambda_{i}$ and a corresponding net of unitary
operators $u_{\lambda_{i}}\in B({\mathcal{H}}_{i})$,
$\lambda_{i}\in\Lambda_{i},$ such that

$\lim_{\lambda_{i}\in\Lambda_{i}}u_{\lambda_{i}}^{*}\psi_{i}(t)u_{\lambda_{i}}=\pi_{i}(t),\
\ \ \ \ \ t\in\mathrm{C}^{*}({\mathcal{M}}),$

where convergence is in the weak operator topology.

Now, let ${\mathcal{F}}$ denote the directed set consisting of finite subsets
of $I$. Given $F=\\{i_{1},\ldots,i_{n}\\}\in{\mathcal{F}}$ and
$\lambda_{i_{k}}\in\Lambda_{i_{k}}$, define a unitary operator on
$\bigoplus_{i\in I}{\mathcal{H}}_{i}$ by

$u_{F,\lambda_{i_{1}},\ldots,\lambda_{i_{n}}}=\bigoplus_{i\in{\mathcal{F}}}u_{\lambda_{i}}\oplus\bigoplus_{i\notin{\mathcal{F}}}I_{{\mathcal{H}}_{i}}$

where $I_{{\mathcal{H}}_{i}}$ denotes the identity operator on
${\mathcal{H}}_{i}$. For $t\in\mathrm{C}^{*}({\mathcal{M}})$, we have that

$P_{{\mathcal{H}}_{i}}u_{F,\lambda_{i_{1}},\ldots,\lambda_{i_{n}}}^{*}\psi(t)u_{F,\lambda_{i_{1}},\ldots,\lambda_{i_{n}}}\mid_{{\mathcal{H}}_{i}}=\begin{cases}u_{\lambda_{i}}^{*}\psi_{i}(t)u_{\lambda_{i}},&i\in
F\\\ \psi_{i}(t),&i\notin F.\end{cases}$

Thus, we must have that the iterated limit

$\lim_{\lambda_{i_{1}}\in\Lambda_{i_{1}}}\lim_{\lambda_{i_{2}}\in\Lambda_{i_{2}}}\ldots\lim_{\lambda_{i_{n}}\in\Lambda_{i_{n}}}P_{{\mathcal{H}}_{i}}u_{F,\lambda_{i_{1}},\ldots,\lambda_{i_{n}}}^{*}\psi(t)u_{F,\lambda_{i_{1}},\ldots,\lambda_{i_{n}}}\mid_{{\mathcal{H}}_{i}}$

exists in the weak operator topology for any $i\in I$ and any
$F=\\{i_{1},\ldots,i_{n}\\}\in{\mathcal{F}}$. Indeed, if $i\in{\mathcal{F}}$,
then the limit equals $\pi_{i}(t)$ and, when $i\notin{\mathcal{F}}$, we have
that the limit is equal to $\psi_{i}(t)$. So

$\lim_{\\{i_{1},\ldots,i_{n}\\}\in{\mathcal{F}}}\lim_{\lambda_{i_{1}}\in\Lambda_{i_{1}}}\lim_{\lambda_{i_{2}}\in\Lambda_{i_{2}}}\ldots\lim_{\lambda_{i_{n}}\in\Lambda_{i_{n}}}P_{{\mathcal{H}}_{i}}u_{F,\lambda_{i_{1}},\ldots,\lambda_{i_{n}}}^{*}\psi(t)u_{F,\lambda_{i_{1}},\ldots,\lambda_{i_{n}}}\mid_{{\mathcal{H}}_{i}}=\pi_{i}(t)$

for every $i\in I$. Thus, by [kelley2017general, pg. 69], there is a net of
unitary operators $u_{\beta}\in B(\bigoplus_{i\in I}{\mathcal{H}}_{i})$ such
that

$\lim_{\beta}P_{{\mathcal{H}}_{i}}u_{\beta}^{*}\psi(t)u_{\beta}\mid_{{\mathcal{H}}_{i}}=\pi_{i}(t),\
\ \ \ \ \ t\in\mathrm{C}^{*}({\mathcal{M}}),$

for every $i\in I$.

By compactness, we may extract a subnet
$(u_{\beta_{\eta}}^{*}\psi(\cdot)u_{\beta_{\eta}})_{\eta}$ and a unital
completely positive map $\varphi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B(\bigoplus_{i\in I}{\mathcal{H}}_{i})$ such that

$\lim_{\eta}u_{\beta_{\eta}}^{*}\psi(t)u_{\beta_{\eta}}=\varphi(t),\ \ \ \ \ \
\ t\in\mathrm{C}^{*}({\mathcal{M}}),$

in the weak operator topology [paulsen2002completely, Theorem 7.4]. From here,
the argument is a direct adaptation of [arveson2011noncommutative, Proposition
4.4]. Indeed, observe that
$P_{{\mathcal{H}}_{i}}\varphi(t)P_{{\mathcal{H}}_{i}}=\pi(t)P_{{\mathcal{H}}_{i}}$
for each $i\in I$ and so, by the Schwarz inequality,

$\displaystyle
P_{{\mathcal{H}}_{i}}\varphi(t)^{*}(I-P_{{\mathcal{H}}_{i}})\varphi(t)P_{{\mathcal{H}}_{i}}$
$\displaystyle=P_{{\mathcal{H}}_{i}}\varphi(t)^{*}\varphi(t)P_{{\mathcal{H}}_{i}}-P_{{\mathcal{H}}_{i}}\varphi(t)^{*}P_{{\mathcal{H}}_{i}}\varphi(t)P_{{\mathcal{H}}_{i}}$
$\displaystyle\leq
P_{{\mathcal{H}}_{i}}\varphi(t^{*}t)P_{{\mathcal{H}}_{i}}-P_{{\mathcal{H}}_{i}}\varphi(t)^{*}P_{{\mathcal{H}}_{i}}\varphi(t)P_{{\mathcal{H}}_{i}}$
$\displaystyle=\pi(t^{*}t)P_{{\mathcal{H}}_{i}}-\pi(t)^{*}\pi(t)P_{{\mathcal{H}}_{i}}=0.$

Therefore $(I-P_{{\mathcal{H}}_{i}})\varphi(t)P_{{\mathcal{H}}_{i}}=0$ and so

$\lim_{\eta}u_{\beta_{\eta}}^{*}\psi(t)u_{\beta_{\eta}}=\varphi(t)=\sum_{i\in
I}\varphi(t)P_{{\mathcal{H}}_{i}}=\sum_{i\in
I}P_{{\mathcal{H}}_{i}}\varphi(t)P_{{\mathcal{H}}_{i}}=\sum_{i\in
I}\pi(t)P_{{\mathcal{H}}_{i}}=\pi(t).$

∎

Recall that Lemma 2.2 states that the implication in Proposition 3.7 is known
to be an equivalence when the approximate unique extension property is
replaced by the usual unique extension property. On the contrary, the converse
will be shown to be far from true for $*$-representations that possess the
approximate unique extension property (Proposition 3.9). First, we require a
consequence to Proposition 3.7 that allows us to construct many more
$*$-representations that possess the approximate unique extension property.

######  Proposition 3.8.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space.
Suppose that $\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a $*$-representation possessing the approximate
unique extension property. Then, the following statements hold.

  1. (i)

If $\sigma:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\sigma})$
is a $*$-representation such that $\ker\sigma=\ker\pi$, then there is some
cardinal $\kappa$ such that $\sigma^{(\kappa)}$ possesses the approximate
unique extension property.

  2. (ii)

If $\rho:\mathrm{C}^{*}_{e}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\rho})$
is an isometric $*$-representation, then there is a cardinal $\mu$ such that
$\rho^{(\mu)}$ possesses the approximate unique extension property.

###### Proof.

(i): Define an infinite cardinal
$\kappa=\max\\{\dim{\mathcal{H}}_{\pi},\dim{\mathcal{H}}_{\sigma},\aleph_{0}\\}$.
For each $t\in\mathrm{C}^{*}({\mathcal{M}})\setminus\ker\sigma$, we have that

$\text{rank}(\sigma^{(\kappa)}(t))=\kappa\text{rank}(\sigma(t))=\kappa,$

and likewise, $\text{rank}(\pi^{(\kappa)}(t))=\kappa.$ Since
$\ker\pi=\ker\sigma$, Theorem 2.1 implies that $\sigma^{(\kappa)}$ is
approximately unitarily equivalent to $\pi^{(\kappa)}$. By Propositions 3.5
and 3.7, we find that $\sigma^{(\kappa)}$ possesses the approximate unique
extension property.

(ii): By Theorem 2.3 (i), there is an isometric $*$-representation
$\beta:\mathrm{C}^{*}_{e}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\beta})$
that possesses the unique extension property. Thus, it follows by (i) that
there is a cardinal $\mu$ such that $\rho^{(\mu)}$ possesses the approximate
unique extension property. ∎

Due to Proposition 3.8 (ii), we may confirm that the approximate unique
extension property is badly behaved with respect to sub-representations.

######  Proposition 3.9.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space. Assume
that whenever $\pi:\mathrm{C}^{*}_{e}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a $*$-representation possessing the approximate
unique extension property and ${\mathcal{F}}\subset{\mathcal{H}}_{\pi}$ is a
closed subspace that is reducing for $\text{Im}\pi$, then we have that the
$*$-representation

$t\mapsto\pi(t)\mid_{\mathcal{F}},\ \ \ \ \ \ \
t\in\mathrm{C}^{*}_{e}({\mathcal{M}}),$

also possesses the approximate unique extension property. Then, we have that
every $*$-representation of $\mathrm{C}^{*}_{e}({\mathcal{M}})$ has the
approximate unique extension property.

###### Proof.

Let $\beta$ be an isometric $*$-representation of
$\mathrm{C}^{*}_{e}({\mathcal{M}})$ on a separable Hilbert space that
possesses the approximate unique extension property (which exists by either
Theorem 2.3 or Proposition 3.8) and let
$\pi:\mathrm{C}^{*}_{e}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation. Then, there is a cardinal $\kappa$ such that
$\beta^{(\kappa)}$ has the approximate unique extension property by
Proposition 3.7. By Theorem 2.1, we have that $\beta^{(\kappa)}\oplus\pi$ is
approximately unitarily equivalent to $\beta^{(\kappa)}$. So
$\beta^{(\kappa)}\oplus\pi$ possesses the approximate unique extension
property by Propostion 3.5. By the assumption, we conclude that $\pi$ has the
approximate unique extension property. ∎

We conclude this section by connecting our work with Arveson’s hyperrigidity
conjecture. To start, we note the following.

######  Proposition 3.10.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a separable unital operator
space. Then, the following statements are equivalent.

  1. (i)

Every irreducible $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$
possesses the approximate unique extension property.

  2. (ii)

Every $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ possesses the
approximate unique extension property.

###### Proof.

For the non-trivial direction, assume that every irreducible
$*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ possesses the
approximate unique extension property and let
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation. As $\mathrm{C}^{*}({\mathcal{M}})$ is separable, we may
express $\pi=\bigoplus_{i\in I}\pi_{i}$ where
$\pi_{i}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{i})$ is a
$*$-representation on a separable Hilbert space ${\mathcal{H}}_{i}$. Since
${\mathcal{H}}_{i}$ is separable, we have that $\pi_{i}$ is approximately
unitarily equivalent to a direct sum of irreducible $*$-representations
[voiculescu1976non, Corollary 1.6]. By the assumption, Proposition 3.5, and
Proposition 3.7, we may conclude that each $\pi_{i}$ has the approximate
unique extension property. Thus, $\pi$ has the approximate unique extension
property by Proposition 3.7. ∎

We remain unsure if separability can be removed in Proposition 3.10. Indeed,
we required work of Voiculescu [voiculescu1976non], which states that every
$*$-representation is approximately unitarily equivalent to a direct sum of
irreducible $*$-representations provided that certain separability conditions
are met. However, we are unaware if there is an analogue to this fact in the
non-separable setting.

Also, note that Proposition 3.10 may be viewed as a positive answer to an
approximate version of Arveson’s hyperrigidity conjecture. Indeed, as
reflected in Proposition 3.10, to determine the class of $*$-representations
that possess the approximate unique extension property, it is sufficient to
study those _irreducible_ $*$-representations that possess the approximate
unique extension property.

We may now state the main result of this section.

######  Theorem 3.11.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a separable unital operator
space. Consider the following statements:

  1. (i)

The operator space ${\mathcal{M}}$ is hyperrigid.

  2. (ii)

Every irreducible $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ is a
boundary representation.

  3. (iii)

Every irreducible $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ is
approximately unitarily equivalent to a boundary representation.

  4. (iv)

Every $*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ has the
approximate unique extension property.

Then, we have that (i)$\Rightarrow$(ii)$\Rightarrow$(iii)$\Rightarrow$(iv) and
(iii)$\not\Rightarrow$(ii).

###### Proof.

The directions (i)$\Rightarrow$(ii)$\Rightarrow$(iii) are trivial and
(iii)$\Rightarrow$(iv) is a consequence to Theorem 3.6 and Proposition 3.10.
Finally, (iii)$\not\Rightarrow$(ii) by considering Example 2. ∎

Later, it will be revealed that the approximate unique extension property
coincides with the usual unique extension property for those irreducible
$*$-representations whose image contains a compact operator (Theorem 4.8).
Consequently, statements (ii), (iii), and (iv) will be equivalent whenever
$\mathrm{C}^{*}({\mathcal{M}})$ is postliminal.

##  4\. Split sequences for completely positive extensions

Suppose that $\pi$ has the approximate unique extension property with respect
to a unital operator space ${\mathcal{M}}$. We will now present a restriction
on the structure of the unital completely positive extensions of
$\pi\mid_{\mathcal{M}}$. This will be the driving force of our arguments
throughout this section and is the main result of our work.

For this recall that, given $\mathrm{C}^{*}$-algebras
${\mathfrak{A}}\subset{\mathfrak{B}}$, a _conditional expectation_ is a
contractive linear projection $E:{\mathfrak{B}}\rightarrow{\mathfrak{A}}$.
When $E$ is multiplicative, we say that the map is _homomorphic_. By
Tomiyama’s Theorem [brown2008textrm, Theorem 1.5.10], a conditional
expectation is automatically completely positive and satisfies
$E(aba^{\prime})=aE(b)a^{\prime}$ for every $a,a^{\prime}\in{\mathfrak{A}}$
and $b\in{\mathfrak{B}}$. Consequently, a homomorphic conditional expectation
is a $*$-homomorphism.

######  Theorem 4.1.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space.
Suppose that $\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a $*$-representation possessing the approximate
unique extension property and let
$\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$. Then, the following statements
hold.

  1. (i)

There is a homomorphic conditional expectation
$\Gamma_{\psi}:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\text{Im}\pi$
satisfying $\Gamma_{\psi}\circ\psi=\pi$.

  2. (ii)

If ${\mathcal{H}}_{\pi}$ is separable, then
$\text{rank}(\pi(t))\leq\text{rank}(\psi(t))$ for each
$t\in\mathrm{C}^{*}({\mathcal{M}})$.

  3. (iii)

We have that $\ker\Gamma_{\psi}$ is the smallest closed two-sided ideal of
$\mathrm{C}^{*}(\text{Im}\psi)$ that contains $\text{Im}(\psi-\pi)$.

  4. (iv)

We have that $\text{Im}(id-\Gamma_{\psi})=\ker\Gamma_{\psi}$ and
$\mathrm{C}^{*}(\text{Im}\psi)=\ker\Gamma_{\psi}+\text{Im}\pi$.

  5. (v)

We have that $\psi=\pi$ if and only if $\Gamma_{\psi}$ is isometric.

###### Proof.

(i): As $\pi$ possesses the approximate unique extension property, there is a
net of unitaries $u_{\beta}\in B({\mathcal{H}}_{\pi})$ such that
$u_{\beta}^{*}\psi(t)u_{\beta}\rightarrow\pi(t)$ in the weak operator topology
for each $t\in\mathrm{C}^{*}({\mathcal{M}})$. By [kleski2014korovkin, Lemma
2.1], we may conclude that convergence is in the strong operator topology.
Consequently, given a $*$-polynomial $p$ in non-commuting variables and
finitely many elements $t_{1},\ldots,t_{n}\in\mathrm{C}^{*}({\mathcal{M}})$,
we have that

$\lim_{\beta}u_{\beta}^{*}p(\psi(t_{1}),\ldots,\psi(t_{n}))u_{\beta}=p(\pi(t_{1}),\ldots,\pi(t_{n}))$

in the strong operator topology. Since the linear span of words in
$\text{Im}\psi$ is dense within $\mathrm{C}^{*}(\text{Im}\psi)$, we obtain a
$*$-homomorphism

$\Gamma_{\psi}:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\text{Im}\pi,\ \ \ \ \
\ x\mapsto\lim_{\beta}u_{\beta}^{*}xu_{\beta}.$

Evidently, $\Gamma_{\psi}\circ\psi=\pi$. Note that

$\text{Im}\pi=\mathrm{C}^{*}(\pi({\mathcal{M}}))=\mathrm{C}^{*}(\psi({\mathcal{M}}))\subset\mathrm{C}^{*}(\text{Im}\psi).$

Since

$(\Gamma_{\psi}\circ\pi)\mid_{\mathcal{M}}=(\Gamma_{\psi}\circ\psi)\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$

and $\Gamma_{\psi}$ is a $*$-representation, it follows that
$\Gamma_{\psi}\circ\pi=\pi$. Therefore, $\Gamma_{\psi}$ is a conditional
expectation as desired.

(ii): It is well-known that the rank function is lower semicontinous in the
weak operator topology provided that ${\mathcal{H}}_{\pi}$ is separable. Then
(ii) follows instantly since there is a net of unitaries $u_{\beta}\in
B({\mathcal{H}}_{\pi})$ such that
$u_{\beta}^{*}\psi(t)u_{\beta}\rightarrow\pi(t)$ in the weak operator
topology.

(iii): Since $\Gamma_{\psi}\circ\psi=\pi=\Gamma_{\psi}\circ\pi$, it follows
that

$\text{Im}(\psi-\pi)\subset\ker\Gamma_{\psi}.$

Let ${\mathfrak{J}}$ be the smallest closed two-sided ideal of
$\mathrm{C}^{*}(\text{Im}\psi)$ containing $\text{Im}(\psi-\pi)$. Necessarily,
we have that ${\mathfrak{J}}\subset\ker\Gamma_{\psi}$. We will show that each
element $\zeta\in\mathrm{C}^{*}(\text{Im}\psi)$ may expressed as $J+\pi(t)$
where $J\in{\mathfrak{J}}$ and $t\in\mathrm{C}^{*}({\mathcal{M}})$. Observe
that

${\mathfrak{D}}=\\{J+\pi(t)~{}:~{}J\in{\mathfrak{J}},\
t\in\mathrm{C}^{*}({\mathcal{M}})\\}$

is a closed subset of $\mathrm{C}^{*}(\text{Im}\psi)$ as it is the pre-image
of $\text{Im}\pi/{\mathfrak{J}}$ under the quotient map
$q:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\mathrm{C}^{*}(\text{Im}\psi)/{\mathfrak{J}}$.
Hence, by density, it suffices to consider when
$\zeta\in\mathrm{C}^{*}(\text{Im}\psi)$ lies within the linear span of words
in $\text{Im}\psi$. In this case, since $\psi$ and
$\mathrm{C}^{*}({\mathcal{M}})$ are unital, we may obtain a finite sum of
words

$\zeta=\sum_{i}\psi(t_{i_{1}})\ldots\psi(t_{i_{n}})$

where $n$ is a fixed integer and
$t_{k_{\ell}}\in\mathrm{C}^{*}({\mathcal{M}})$ for each $k,\ell$. For a fixed
$i$, we will show that $\psi(t_{i_{1}})\ldots\psi(t_{i_{n}})$ may be expressed
as $J+\pi(t)$ by an easy inductive argument. For the base step, observe that

$\psi(t_{i_{1}})=\psi(t_{i_{1}})-\pi(t_{i_{1}})+\pi(t_{i_{1}})$

and $(\psi(t_{i_{1}})-\pi(t_{i_{1}}))\in{\mathfrak{J}}$. Now, for the
inductive step, we may obtain $K\in{\mathfrak{J}}$ and
$s\in\mathrm{C}^{*}({\mathcal{M}})$ such that
$\psi(t_{i_{1}})\ldots\psi(t_{i_{n-1}})=K+\pi(s)$. Then we see that

$\displaystyle\psi(t_{i_{1}})\ldots\psi(t_{i_{n}})$
$\displaystyle=(K+\pi(s))\psi(t_{i_{n}})$
$\displaystyle=(K+\pi(s))(\psi(t_{i_{n}})-\pi(t_{i_{n}})+\pi(t_{i_{n}}))$
$\displaystyle=K(\psi(t_{i_{n}})-\pi(t_{i_{n}})+\pi(t_{i_{n}}))+\pi(s)(\psi(t_{i_{n}})-\pi(t_{i_{n}}))+\pi(st_{i_{n}}).$

Therefore, we have that the claim also holds for $\zeta$.

Now, let $k\in\ker\Gamma_{\psi}$ and express $k=J+\pi(t)$ where
$J\in{\mathfrak{J}}$ and $t\in\mathrm{C}^{*}({\mathcal{M}})$. Then we obtain
that

$k-J=\pi(t)=\Gamma_{\psi}(\pi(t))=\Gamma_{\psi}(k-J)=0.$

So $k=J$ and we have that ${\mathfrak{J}}=\ker\Gamma_{\psi}$.

(iv): This follows from a standard fact on linear idempotents, but we record
it for later arguments. Since $\Gamma_{\psi}$ is a linear idempotent, it is
easy to see that $\text{Im}(id-\Gamma_{\psi})=\ker\Gamma_{\psi}$. Then,

$\mathrm{C}^{*}(\text{Im}\psi)=\text{Im}(id-\Gamma_{\psi})+\text{Im}\Gamma_{\psi}=\ker\Gamma_{\psi}+\text{Im}\pi.$

(v): This is an immediate consequence of (iii). ∎

A statement similar to (i) was already known for faithful $*$-representations
of nuclear $\mathrm{C}^{*}$-algebras satisfying the property that all factor
representations possess the unique extension property [kleski2014korovkin,
Theorem 4.2]. The construction therein required non-trivial machinery on
direct integrals and disintegration of measures. In contrast, our methods in
Theorem 4.1 provide a non-trivial implication for all unital operator spaces
that fail to be hyperrigid. Indeed, whenever
${\mathcal{M}}\subset\mathrm{C}^{*}_{e}({\mathcal{M}})$ fails to be
hyperrigid, there is a $*$-representation of
$\mathrm{C}^{*}_{e}({\mathcal{M}})$ that has the approximate unique extension
property but not the usual unique extension property by Proposition 2.4 and
Theorem 3.6.

As a consequence to Theorem 4.1, we may derive a statement of independent
interest. As far as we are aware, the following has not been recorded
elsewhere.

######  Corollary 4.2.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space.
Suppose that $\pi:\mathrm{C}^{*}_{e}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is an isometric $*$-representation and let
$\psi:\mathrm{C}^{*}_{e}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be
a unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$. Then, the following statements
hold.

  1. (i)

The map $\psi$ is completely isometric.

  2. (ii)

We have that
$\mathrm{C}^{*}_{e}(\text{Im}\psi)\cong\mathrm{C}^{*}_{e}({\mathcal{M}})$.

###### Proof.

(i): By Proposition 3.8, there is a cardinal $\kappa$ such that
$\Pi:=\pi^{(\kappa)}$ has the approximate unique extension property. Note that
the map $\Psi:=\psi^{(\kappa)}$ agrees with $\Pi$ over ${\mathcal{M}}$. Thus,
by Theorem 4.1, there is a homomorphic conditional expectation
$\Gamma_{\Psi}:\mathrm{C}^{*}(\text{Im}\Psi)\rightarrow\text{Im}\Pi$
satisfying $\Gamma_{\Psi}\circ\Psi=\Pi$. Since $\Pi$ is isometric, it
immediately follows that $\Psi$ is completely isometric. Finally, this implies
that $\psi$ itself must be completely isometric.

(ii): By statement (i), there is a complete order isomorphism
$\iota:\text{Im}\psi\rightarrow\text{Im}\pi$ such that $\iota\circ\psi=\pi$.
Since $\text{Im}\pi$ is a $\mathrm{C}^{*}$-algebra, it then follows that
$\text{Im}\pi\cong\mathrm{C}^{*}_{e}({\mathcal{M}})$ is the
$\mathrm{C}^{*}$-envelope for $\text{Im}\psi$. ∎

The argument in Corollary 4.2 may be repurposed. If $\pi$ has the approximate
unique extension property and $\psi$ is a completely positive extension such
that $\ker\pi\subset\ker\psi$, then in fact we have equality of kernels and
hence, there is a complete order isomorphism
$\iota:\text{Im}\psi\rightarrow\text{Im}\pi$ given by
$\Gamma_{\psi}\mid_{\text{Im}\psi}$. In the language of
[kennedy2021noncommutative, Theorem 10.5], we may conclude that the nc state
space of the operator system $\text{Im}\psi$ is an “nc Bauer simplex.”

We briefly remark that the existence of the conditional expectation
$\Gamma_{\psi}$ provides a tool for determining when a completely positive
extension $\psi$ of $\pi\mid_{\mathcal{M}}$ will coincide with the
$*$-representation $\pi$.

######  Corollary 4.3.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space.
Suppose that $\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a $*$-representation possessing the approximate
unique extension property and let
$\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$. Then, the following statements
are equivalent.

  1. (i)

The equality $\psi=\pi$ holds.

  2. (ii)

The linear map $(id-\Gamma_{\psi})$ is contractive.

###### Proof.

It is clear from Theorem 4.1 that $\Gamma_{\pi}$ is the identity
representation and so, (i)$\Rightarrow$ (ii) is immediate.

(ii)$\Rightarrow$(i): The map $(id-\Gamma_{\psi})$ is a linear projection onto
$\ker\Gamma_{\psi}$ by Theorem 4.1 (iv). By the assumption, combined with
Tomiyama’s Theorem [brown2008textrm, Theorem 1.5.10], we conclude that
$(id-\Gamma_{\psi})$ is a conditional expectation. Thus, given
$k\in\ker\Gamma_{\psi}$ and $x\in\mathrm{C}^{*}(\text{Im}\psi)$, we have that

$xk=(id-\Gamma_{\psi})(xk)=(id-\Gamma_{\psi})(x)k=xk-\Gamma_{\psi}(x)k$

and so $\Gamma_{\psi}(x)k=0$. As $x\in\mathrm{C}^{*}(\text{Im}\psi)$ is
arbitrary, we must have that $k=0$. So $\Gamma_{\psi}$ is injective and thus,
$\psi=\pi$ by Theorem 4.1 (v). ∎

For our next development, we will require some terminology. Let
${\mathcal{M}}\subset{\mathcal{N}}$ be unital operator spaces and let
$\mathrm{U}\mathrm{C}\mathrm{P}({\mathcal{M}},{\mathcal{N}})$ denote the
collection of unital completely positive maps from ${\mathcal{M}}$ to
${\mathcal{N}}$. There is a natural ordering among the idempotent elements of
$\mathrm{U}\mathrm{C}\mathrm{P}({\mathcal{M}},{\mathcal{N}})$ defined by
$\varphi\prec\varphi^{\prime}$ if and only if
$\varphi\circ\varphi^{\prime}=\varphi^{\prime}\circ\varphi=\varphi$. Given a
subspace ${\mathcal{F}}$ of ${\mathcal{M}}$, an idempotent $\varphi$ is said
to be a _minimal ${\mathcal{F}}$-projection_ in
$\mathrm{U}\mathrm{C}\mathrm{P}({\mathcal{M}},{\mathcal{N}})$ if $\varphi$ is
minimal among the idempotent elements of
$\mathrm{U}\mathrm{C}\mathrm{P}({\mathcal{M}},{\mathcal{N}})$ that fix
${\mathcal{F}}$. Minimal ${\mathcal{F}}$-projections have found previous
success in helping to prove existence of the injective envelope of an operator
system [paulsen2002completely, Chapter 15]. We will find that the map
$\Gamma_{\psi}$ enjoys this minimality condition and is uniquely determined by
its behaviour over $\text{Im}\psi$.

######  Theorem 4.4.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space.
Suppose that $\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a $*$-representation possessing the approximate
unique extension property and let
$\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$. Then, the following statements
hold.

  1. (i)

Every $*$-representation $\sigma:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow
B({\mathcal{H}}_{\sigma})$ such that $\sigma\mid_{\ker\Gamma_{\psi}}=0$ has
the unique extension property with respect to $\text{Im}\psi$.

  2. (ii)

The map $\Gamma_{\psi}$ is a minimal $\text{Im}\pi$-projection in
$\mathrm{U}\mathrm{C}\mathrm{P}(\mathrm{C}^{*}(\text{Im}\psi),B({\mathcal{H}}_{\pi}))$.

###### Proof.

(i): Let $\varphi:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow
B({\mathcal{H}}_{\sigma})$ be a unital completely positive map satisfying
$\varphi\mid_{\text{Im}\psi}=\sigma\mid_{\text{Im}\psi}.$ As $\sigma$ vanishes
over $\ker\Gamma_{\psi}$, we may express
$\sigma=\widetilde{\sigma}\circ\Gamma_{\psi}$ where
$\widetilde{\sigma}:\text{Im}\pi\rightarrow B({\mathcal{H}}_{\sigma})$ is a
$*$-representation. Observe that, by Theorem 4.1,

$(\varphi\circ\psi)(t)=(\sigma\circ\psi)(t)=(\widetilde{\sigma}\circ\Gamma_{\psi}\circ\psi)(t)=(\widetilde{\sigma}\circ\pi)(t),\
\ \ \ \ t\in\mathrm{C}^{*}({\mathcal{M}}).$

By the Schwarz inequality, we have that for each
$t\in\mathrm{C}^{*}({\mathcal{M}})$,

$\varphi(\psi(t)^{*}\psi(t))\leq(\varphi\circ\psi)(t^{*}t)=(\widetilde{\sigma}\circ\pi)(t^{*}t)=\varphi(\psi(t))^{*}\varphi(\psi(t))\leq\varphi(\psi(t)^{*}\psi(t))$

and so $\varphi(\psi(t)^{*}\psi(t))=\varphi(\psi(t))^{*}\varphi(\psi(t))$.
Consequently, we have that $\text{Im}\psi$ is a subset of the multiplicative
domain for $\varphi$ [paulsen2002completely, Theorem 3.18]. Therefore, since
$\varphi\mid_{\text{Im}\psi}=\sigma\mid_{\text{Im}\psi}$, we must have that
$\varphi=\sigma$.

(ii): Let $\Omega:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow
B({\mathcal{H}}_{\pi})$ be a unital completely positive idempotent such that
$\Omega\circ\pi=\pi$ and $\Omega\prec\Gamma_{\psi}$. In this case, by Theorem
4.1,

$\Omega\circ\psi=\Omega\circ\Gamma_{\psi}\circ\psi=\Omega\circ\pi=\pi.$

As $\Omega\mid_{\text{Im}\psi}=\Gamma_{\psi}\mid_{\text{Im}\psi}$, it follows
by (i) that $\Omega=\Gamma_{\psi}$. ∎

As a result of Theorem 4.4, we show next that there is a one-to-one
correspondence between the idempotents $\Gamma_{\psi}$ and the domain of such
maps. In particular, a completely positive extension $\psi$ will be equal to a
$*$-representation $\pi$ with the approximate unique extension property
provided that the image of the completely positive extension is sufficiently
small. This generalizes [clouatre2018unperforated, Corollary 3.3].

######  Corollary 4.5.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space.
Suppose that $\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a $*$-representation possessing the approximate
unique extension property and let
$\psi,\psi^{\prime}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ be unital completely positive maps satisfying
$\psi\mid_{\mathcal{M}}=\psi^{\prime}\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$.
Then, the following statements are equivalent.

  1. (i)

We have that $\ker\Gamma_{\psi}\subset\ker\Gamma_{\psi^{\prime}}$.

  2. (ii)

We have that
$\mathrm{C}^{*}(\text{Im}\psi)\subset\mathrm{C}^{*}(\text{Im}\psi^{\prime})$.

  3. (iii)

We have that
$\Gamma_{\psi^{\prime}}\mid_{\mathrm{C}^{*}(\text{Im}\psi)}=\Gamma_{\psi}$.

In particular, we have that $\Gamma_{\psi}=\Gamma_{\psi^{\prime}}$ if and only
if $\mathrm{C}^{*}(\text{Im}\psi)=\mathrm{C}^{*}(\text{Im}\psi^{\prime}).$
Further, $\psi=\pi$ whenever $\text{Im}\psi\subset\text{Im}\pi$.

###### Proof.

The implication (iii)$\Rightarrow$(i) is trivial and (i)$\Rightarrow$(ii)
follows directly from Theorem 4.1 (iv) because

$\mathrm{C}^{*}(\text{Im}\psi)=\ker\Gamma_{\psi}+\text{Im}\pi\subset\ker\Gamma_{\psi^{\prime}}+\text{Im}\pi=\mathrm{C}^{*}(\text{Im}\psi^{\prime}).$

(ii)$\Rightarrow$(iii): In this case, we obtain a well-defined unital
completely positive map
$\varphi:=\Gamma_{\psi^{\prime}}\circ\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow\text{Im}\pi$
satisfying

$\varphi\mid_{\mathcal{M}}=(\Gamma_{\psi^{\prime}}\circ\psi)\mid_{\mathcal{M}}=(\Gamma_{\psi^{\prime}}\circ\pi)\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}.$

Thus, by Theorem 4.1 (i),

$\pi(t)=(\Gamma_{\varphi}\circ\varphi)(t)=\varphi(t),\ \ \ \ \ \ \
t\in\mathrm{C}^{*}({\mathcal{M}}),$

on account of $\varphi(t)$ lying in the image of the idempotent
$\Gamma_{\varphi}:\mathrm{C}^{*}(\text{Im}\varphi)\rightarrow\text{Im}\pi.$ It
follows that $\Gamma_{\psi^{\prime}}\circ\psi=\pi$. However, by Theorem 4.4
(i), the map $\Gamma_{\psi}$ has the unique extension property with respect to
$\text{Im}\psi$. So, since
$\Gamma_{\psi^{\prime}}\mid_{\text{Im}\psi}=\Gamma_{\psi}\mid_{\text{Im}\psi}$,
it follows that
$\Gamma_{\psi^{\prime}}\mid_{\mathrm{C}^{*}(\text{Im}\psi)}=\Gamma_{\psi}$.

For the final statement note that, whenever
$\text{Im}\psi\subset\text{Im}\pi$, it follows from condition (iii) that

$\Gamma_{\psi}=\Gamma_{\pi}\mid_{\mathrm{C}^{*}(\text{Im}\psi)}=id,$

where $id$ denotes the identity representation of
$\mathrm{C}^{*}(\text{Im}\psi)$. By Theorem 4.1 (v), we conclude that
$\psi=\pi$. ∎

Recall that, by Proposition 3.10, to determine the class of
$*$-representations that possess the approximate unique extension property it
suffices to determine the class of _irreducible_ $*$-representations that
possess the approximate unique extension property (provided that
${\mathcal{M}}$ is separable). Any other $*$-representation is the result of a
direct sum or an approximate unitary equivalence of $*$-representations of
this form. Here, we will mention how the conditional expectation
$\Gamma_{\psi}$ behaves with respect to direct sums and approximate unitary
equivalences of completely positive extensions. We start with direct sums.

######  Corollary 4.6.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and
$\\{\pi_{i}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{i}):i\in
I\\}$ be a collection of $*$-representations that possess the approximate
unique extension property. For each $i\in I$, let
$\psi_{i}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{i})$ be a
unital completely positive map satisfying
$\psi_{i}\mid_{\mathcal{M}}=\pi_{i}\mid_{\mathcal{M}}$. Then, defining
$\pi=\bigoplus_{i\in I}\pi_{i}$ and $\psi=\bigoplus_{i\in I}\psi_{i}$, we have
that the map
$\Gamma_{\psi}:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\text{Im}\pi$ satisfies

$\Gamma_{\psi}((x_{i})_{i\in I})=(\Gamma_{\psi_{i}}(x_{i}))_{i\in I},\ \ \ \ \
\ (x_{i})_{i\in I}\in\mathrm{C}^{*}(\text{Im}\psi).$

###### Proof.

Note that $\psi$ is a unital completely positive map satisfying
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$ and recall that $\pi$ has the
approximate unique extension property by Proposition 3.7. Thus, if we define

$\Lambda:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\text{Im}\pi,\ \ \ \ \ \
(x_{i})_{i\in I}\mapsto(\Gamma_{\psi_{i}}(x_{i}))_{i\in I},$

then observe that $\Lambda$ is a completely positive map satisfying
$\Lambda\circ\psi=\pi.$ So we have that $\Lambda=\Gamma_{\psi}$ by Theorem 4.4
(i). ∎

Next, we address how the conditional expectation $\Gamma_{\psi}$ is affected
by approximately unitarily equivalent completely positive extensions. We say
that unital completely positive maps
$\psi,\psi^{\prime}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ are _approximately unitarily equivalent_ whenever
there is a net of unitary operators $w_{\gamma}\in B({\mathcal{H}}_{\pi})$
satisfying

$\lim_{\gamma}\|w_{\gamma}^{*}\psi^{\prime}(t)w_{\gamma}-\psi(t)\|=0,\ \ \ \ \
\ \ t\in\mathrm{C}^{*}({\mathcal{M}}).$

In particular, when $\psi$ and $\psi^{\prime}$ are approximately unitarily
equivalent, there is a $*$-isomorphism
$\Omega:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\mathrm{C}^{*}(\text{Im}\psi^{\prime})$
satisfying $\Omega\circ\psi=\psi^{\prime}$.

######  Corollary 4.7.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space.
Suppose that $\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ is a $*$-representation possessing the approximate
unique extension property and let
$\psi,\psi^{\prime}:\mathrm{C}^{*}({\mathcal{M}})\rightarrow
B({\mathcal{H}}_{\pi})$ be unital completely positive maps satisfying
$\psi\mid_{\mathcal{M}}=\psi^{\prime}\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$.
Assume that there is a $*$-isomorphism
$\Omega:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\mathrm{C}^{*}(\text{Im}\psi^{\prime})$
satisfying $\Omega\circ\psi=\psi^{\prime}.$ Then, we have that
$\mathrm{C}^{*}(\text{Im}\psi)=\mathrm{C}^{*}(\text{Im}\psi^{\prime})$ and
$\Gamma_{\psi}=\Gamma_{\psi^{\prime}}$.

###### Proof.

Observe that, by Theorem 4.1 (i), we have that

$\pi=\Gamma_{\psi^{\prime}}\circ\psi^{\prime}=\Gamma_{\psi^{\prime}}\circ\Omega\circ\psi.$

By Theorem 4.4 (i), it follows that
$\Gamma_{\psi^{\prime}}\circ\Omega=\Gamma_{\psi}$. Whence, we obtain that
$\ker\Gamma_{\psi^{\prime}}=\ker\Gamma_{\psi}$ on account of $\Omega$ being a
$*$-isomorphism. By Corollary 4.5, we conclude that
$\mathrm{C}^{*}(\text{Im}\psi)=\mathrm{C}^{*}(\text{Im}\psi^{\prime})$ and
$\Gamma_{\psi}=\Gamma_{\psi^{\prime}}$. ∎

We close this section by providing an example of when the approximate unique
extension property is equivalent to the usual unique extension property. This
allows us to improve Theorem 3.11 in a special case.

######  Theorem 4.8.

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ be a unital operator space and let
$\pi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
$*$-representation such that
${\mathfrak{K}}({\mathcal{H}})\subset\text{Im}\pi$. Then, the following
statements are equivalent.

  1. (i)

$\pi$ possesses the unique extension property.

  2. (ii)

$\pi$ is approximately unitarily equivalent to a boundary representation.

  3. (iii)

$\pi$ possesses the approximate unique extension property.

In particular, if ${\mathcal{M}}$ is separable and
$\mathrm{C}^{*}({\mathcal{M}})$ is postliminal, then every irreducible
$*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ is a boundary
representation if and only if every $*$-representation of
$\mathrm{C}^{*}({\mathcal{M}})$ possesses the approximate unique extension
property.

###### Proof.

The direction (i) $\Rightarrow$(ii) is trivial and (ii)$\Rightarrow$(iii) is
Theorem 3.6. For (iii)$\Rightarrow$(i), let
$\psi:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\pi})$ be a
completely positive extension of $\pi\mid_{\mathcal{M}}$. Then, observe that
$\Gamma_{\psi}$ is a $*$-representation that fixes
${\mathfrak{K}}({\mathcal{H}})$ and consequently, must be the identity
representation (for example, see [paulsen2011weak, Proposition 3.5]). In which
case, by Theorem 4.1 (v), we have that $\psi=\pi$ and so $\pi$ has the unique
extension property.

For the last statement, if every irreducible $*$-representation of
$\mathrm{C}^{*}({\mathcal{M}})$ is a boundary representation, then every
$*$-representation of $\mathrm{C}^{*}({\mathcal{M}})$ possesses the
approximate unique extension property by Theorem 3.11. For the converse, note
that every irreducible $*$-representation
$\sigma:\mathrm{C}^{*}({\mathcal{M}})\rightarrow B({\mathcal{H}}_{\sigma})$
possesses the approximate unique extension property. Since
$\mathrm{C}^{*}({\mathcal{M}})$ is postliminal, we have that
${\mathfrak{K}}({\mathcal{H}}_{\sigma})\subset\text{Im}\sigma$. Therefore, the
previous paragraph guarantees that $\sigma$ has the unique extension
property.∎

We remark that the last statement within Theorem 4.8 allows us to rephrase
Arveson’s hyperrigidity conjecture in a particular case. Let
${\mathcal{M}}\subset B({\mathcal{H}})$ be a separable unital operator space
such that $\mathrm{C}^{*}_{e}({\mathcal{M}})$ is postliminal and suppose that
there is a $*$-representation of $\mathrm{C}^{*}_{e}({\mathcal{M}})$ that does
not possess the unique extension property. Then, for Arveson’s conjecture to
be true, there must be a $*$-representation of
$\mathrm{C}^{*}_{e}({\mathcal{M}})$ that does not possess the approximate
unique extension property.

Finally, note that Theorem 4.8 allows us to readily identify many instances
where $*$-representations with the approximate unique extension property will
necessarily factor through the $\mathrm{C}^{*}$-envelope.

######  Example 3.

Let $H^{2}$ denote the classical Hardy space on the open unit disc
${\mathbb{D}}\subset{\mathbb{C}}$ and let ${\mathbb{T}}\subset{\mathbb{C}}$
denote the unit circle. We let $\mathrm{A}({\mathbb{D}})$ denote the disc
algebra, which is the closed subalgebra of holomorphic functions in
$\mathrm{C}(\overline{{\mathbb{D}}})$.

Consider the subalgebra ${\mathcal{A}}\subset B(H^{2})$ of multiplication
operators $M_{\varphi}$ where $\varphi\in\mathrm{A}({\mathbb{D}})$. The reader
may consult [agler2002pick],[douglas2012banach],[hoffman2007banach] for more
details on this algebra. For our purposes, we note that there is a well-known
split exact sequence of $\mathrm{C}^{*}$-algebras

${0}$${{\mathfrak{K}}(H^{2})}$${\mathrm{C}^{*}({\mathcal{A}})}$${\mathrm{C}({\mathbb{T}})}$${0}$

and that $\mathrm{C}^{*}({\mathcal{A}})$ is the Toeplitz algebra. Moreover, we
have that $\mathrm{C}^{*}_{e}({\mathcal{A}})\cong\mathrm{C}({\mathbb{T}})$
(for example, this follows by [clouatre2018multiplier, Corollary 6.4]).

Note that the image of the identity representation
$\pi:\mathrm{C}^{*}({\mathcal{A}})\rightarrow B(H^{2})$ contains
${\mathfrak{K}}(H^{2})$. Since $\pi$ does not factor through the
$\mathrm{C}^{*}$-envelope of ${\mathcal{A}}$, we have that $\pi$ cannot be a
boundary representation for ${\mathcal{A}}$ by Theorem 2.3 (ii). So $\pi$ does
not possess the approximate unique extension property either by Theorem 4.8.

##  5\. Concrete applications

In this section, we apply our work to a couple classes of operator spaces.
This allows us to reinterpret the classical Šaškin Theorem (Proposition 5.1)
and obtain new formulations of Arveson’s essential normality conjecture
(Theorem 5.2).

###  5.1. Function spaces

Here, we study the approximate unique extension property in the presence of
commutativity. For this, we recall the classical theorem of Šaškin
[vsavskin1967mil]. Given a compact metric space $X$ and a unital function
space ${\mathcal{M}}\subset\mathrm{C}(X)$, Šaškin proved that every
irreducible $*$-representation of $\mathrm{C}(X)$ is a boundary representation
for ${\mathcal{M}}$ if and only if, whenever
$\psi_{n}:\mathrm{C}(X)\rightarrow\mathrm{C}(X)$ is a sequence of unital
completely positive maps satisfying $\|\psi_{n}(m)-m\|\rightarrow 0$ for every
$m\in{\mathcal{M}}$, then we have that $\|\psi_{n}(t)-t\|\rightarrow 0$ for
every $t\in\mathrm{C}(X)$. A non-separable extension also holds upon replacing
sequences with nets [davidson2021choquet, Theorem 5.3].

######  Proposition 5.1.

Let $X$ be a compact Hausdorff space and ${\mathcal{M}}\subset\mathrm{C}(X)$
be a unital function space. Assume that $\mathrm{C}(X)$ is the
$\mathrm{C}^{*}$-envelope of ${\mathcal{M}}$. Then, the following statements
are equivalent.

  1. (i)

Whenever $\pi:\mathrm{C}(X)\rightarrow B({\mathcal{H}}_{\pi})$ is a
$*$-representation possessing the approximate unique extension property and
$\psi:\mathrm{C}(X)\rightarrow B({\mathcal{H}}_{\pi})$ is a unital completely
positive map such that $\mathrm{C}^{*}(\text{Im}\psi)$ is commutative and
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$, we have that $\psi=\pi$.

  2. (ii)

Whenever $\psi_{\alpha}:\mathrm{C}(X)\rightarrow\mathrm{C}(X)$ is a net of
unital completely positive maps satisfying $\|\psi_{\alpha}(g)-g\|\rightarrow
0$ for every $g\in{\mathcal{M}}$, then we have that
$\|\psi_{\alpha}(f)-f\|\rightarrow 0$ for every $f\in\mathrm{C}(X)$.

  3. (iii)

Every irreducible $*$-representation for $\mathrm{C}(X)$ is a boundary
representation.

###### Proof.

(ii)$\Leftrightarrow$(iii): This is the non-separable version of Šaškin’s
Theorem [davidson2021choquet, Theorem 5.3].

(iii)$\Rightarrow$(i): Let $\pi:\mathrm{C}(X)\rightarrow
B({\mathcal{H}}_{\pi})$ be a $*$-representation possessing the approximate
unique extension property and let $\psi:\mathrm{C}(X)\rightarrow
B({\mathcal{H}}_{\pi})$ be a unital completely positive map such that
$\mathrm{C}^{*}(\text{Im}\psi)$ is commutative and
$\psi\mid_{\mathcal{M}}=\pi\mid_{\mathcal{M}}$. If $Y$ is a compact Hausdorff
space and $\Omega:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\mathrm{C}(Y)$ is
$*$-isomorphism, then $\Omega\circ\psi:\mathrm{C}(X)\rightarrow\mathrm{C}(Y)$
is a unital completely positive map satisfying
$(\Omega\circ\psi)\mid_{\mathcal{M}}=(\Omega\circ\pi)\mid_{\mathcal{M}}.$
Thus, $\Omega\circ\psi=\Omega\circ\pi$ by [davidson2021choquet, Theorem 5.1]
and so, $\psi=\pi$ as desired.

(i)$\Rightarrow$(ii): Let $\pi:\mathrm{C}(X)\rightarrow
B({\mathcal{H}}_{\pi})$ be an isometric $*$-representation and let
$\psi_{\alpha}:\mathrm{C}(X)\rightarrow\mathrm{C}(X)$ be a net of unital
completely positive maps satisfying $\|\psi_{\alpha}(g)-g\|\rightarrow 0$ for
every $g\in{\mathcal{M}}$. Define a $*$-homomorphism

$\pi_{\mathcal{U}}:\mathrm{C}(X)\rightarrow\left(\prod_{\alpha}\text{Im}\pi\right)/{\mathfrak{J}},\
\ \ \ \ \ \ f\mapsto(\pi(f))_{\alpha}+{\mathfrak{J}},$

where ${\mathfrak{J}}$ is the closed two-sided ideal of
$\prod_{\alpha}\text{Im}\pi$ consisting of those nets
$(\pi(f_{\alpha}))_{\alpha}$ such that
$\|\pi(f_{\alpha})\|=\|f_{\alpha}\|\rightarrow 0$. By definition of
${\mathfrak{J}}$, we have that $\ker\pi_{\mathcal{U}}=\ker\pi$ and so
$\pi_{\mathcal{U}}$ is isometric. Upon representing
$(\prod_{\alpha}\text{Im}\pi)/{\mathfrak{J}}$ on some Hilbert space via an
isometric $*$-homomorphism $\iota$, we may apply Proposition 3.8 (ii) to
obtain a cardinal $\kappa$ such that
$(\iota\circ\pi_{\mathcal{U}})^{(\kappa)}$ has the approximate unique
extension property.

Now, define a unital completely positive map

$\psi:\mathrm{C}(X)\rightarrow\left(\prod_{\alpha}\text{Im}\pi\right)/{\mathfrak{J}},\
\ \ \ \ \ \ f\mapsto((\pi\circ\psi_{\alpha})(f))_{\alpha}+{\mathfrak{J}}.$

By assumption, we have that $\psi$ and $\pi_{\mathcal{U}}$ agree on
${\mathcal{M}}$. Therefore, we have that $(\iota\circ\psi)^{(\kappa)}$ and
$(\iota\circ\pi_{\mathcal{U}})^{(\kappa)}$ agree on ${\mathcal{M}}$ as well.
However, since $\mathrm{C}^{*}(\text{Im}(\iota\circ\psi)^{(\kappa)})$ is a
commutative $\mathrm{C}^{*}$-algebra, the assumption allows us to conclude
that $(\iota\circ\psi)^{(\kappa)}=(\iota\circ\pi_{\mathcal{U}})^{(\kappa)}$.
As $\iota$ and $\pi_{\mathcal{U}}$ are isometric, and by definition of
$\pi_{\mathcal{U}}$, it then follows that $\|\psi_{\alpha}(f)-f\|\rightarrow
0$ for every $f\in\mathrm{C}(X).$ ∎

We remark that Proposition 5.1 is very similar to [davidson2021choquet,
Theorem 5.1], but in condition (i) we may make the additional assumption that
$\pi$ has the approximate unique extension property. Due to Theorem 4.8, we
see that (a priori) the only non-trivial implication in condition (i) is in
regards to those $*$-representations that fail to be irreducible. On the other
hand, Proposition 5.1 guarantees that this condition is in fact equivalent to
a statement about irreducible $*$-representations.

Finally, we would like to note that it also appears somewhat notable that
there are many examples of unital operator spaces where condition (i) fails
(Example 1).

###  5.2. Essential normality

Next, we apply our machinery to a different class of operator systems and
relate our work to another well-studied conjecture of Arveson.

For these purposes, let $d\geq 2$ be a fixed positive integer,
${\mathbb{B}}_{d}\subset{\mathbb{C}}^{d}$ denote the open unit ball and
${\mathbb{C}}[z]={\mathbb{C}}[z_{1},\ldots,z_{d}]$ denote the algebra of
complex $d$-variate polynomials. The Drury-Arveson space $H_{d}^{2}$ is the
reproducing kernel Hilbert space on ${\mathbb{B}}_{d}$ that is associated with
the kernel

$k(z,w)=\frac{1}{1-\langle z,w\rangle},\ \ \ \ \ \ z,w\in{\mathbb{B}}_{d}.$

Let $M_{p}\in B(H_{d}^{2})$ denote the polynomial multiplier associated with
$p\in{\mathbb{C}}[z]$. The row operator

$M_{z}=(M_{z_{1}},\ldots,M_{z_{d}}):(H_{d}^{2})^{(d)}\longrightarrow
H_{d}^{2}$

is called the $d$-shift of the Drury-Arveson space. Next, fix a homogeneous
ideal $I\vartriangleleft{\mathbb{C}}[z]$. It is clear that $IH_{d}^{2}$ is
invariant for $M_{z}$. Then, we define a $d$-tuple $S=(S_{1},\ldots,S_{d})$ of
operators on ${\mathcal{F}}_{I}=H_{d}^{2}\ominus IH_{d}^{2}$ given by

$S_{i}=P_{{\mathcal{F}}_{I}}M_{z_{i}}\mid_{{\mathcal{F}}_{I}},\ \ \ \ \ \
1\leq i\leq d.$

Define ${\mathfrak{T}}_{I}=\mathrm{C}^{*}({\mathcal{S}}_{I})$ to be the
Toeplitz algebra associated with the homogeneous ideal $I$. Then, it is known
that ${\mathfrak{K}}({\mathcal{F}}_{I})\subset{\mathfrak{T}}_{I}$
[popescu2006operator, Theorem 1.3] and so we define
${\mathcal{O}}_{I}={\mathfrak{T}}_{I}/{\mathfrak{K}}({\mathcal{F}}_{I})$ to be
the Cuntz algebra associated with the ideal. Throughout, we let
$q:{\mathfrak{T}}_{I}\rightarrow{\mathcal{O}}_{I}$ denote the quotient map. We
now present the main conjecture on this tuple of operators.

Arveson’s Essential Normality Conjecture: If
$I\vartriangleleft{\mathbb{C}}[z]$ is a homogeneous ideal, then
$S_{i}S_{j}^{*}-S_{j}^{*}S_{i}$ is a compact operator for every $1\leq i,j\leq
d.$

Tuples of operators that satisfy the above criterion are said to be
_essentially normal_. Among the main interests in Arveson’s essential
normality conjecture is that $S$ is the universal operator $d$-tuple for
commuting row contractions which satisfy $p(S)=0$ for each $p\in I$
[shalit2009subproduct, Theorem 8.4]. Accordingly, the essential normality
conjecture is connected with a theory of non-commutative varieties on the
ball.

Kennedy and Shalit revealed a connection between the essential normality
conjecture and Arveson’s hyperrigidity conjecture. Assume that
$I\vartriangleleft{\mathbb{C}}[z]$ is a non-zero homogeneous ideal that
contains no linear polynomials and is not of finite codimension in
${\mathbb{C}}[z]$. If we let ${\mathcal{S}}_{I}$ denote the operator system
generated by $S_{1},\ldots,S_{d}$, then essential normality of the tuple $S$
is equivalent to hyperrigidity of the operator system ${\mathcal{S}}_{I}$
[kennedy2015essential, Theorem 4.12],[kennedy2016corrigendum]. Now, we will
reframe essential normality of the tuple in terms of the approximate unique
extension property.

######  Theorem 5.2.

Let $I\vartriangleleft{\mathbb{C}}[z]$ be a non-zero homogeneous ideal that
contains no linear polynomials and such that $I$ is not of finite codimension
in ${\mathbb{C}}[z]$. Then, the following statements are equivalent.

  1. (i)

There is an isometric $*$-representation of ${\mathcal{O}}_{I}$ that has the
approximate unique extension property with respect to $q({\mathcal{S}}_{I})$.

  2. (ii)

The tuple $S=(S_{1},\ldots,S_{d})$ is essentially normal.

  3. (iii)

The operator system ${\mathcal{S}}_{I}$ is hyperrigid.

  4. (iv)

Every irreducible $*$-representation of ${\mathfrak{T}}_{I}$ is a boundary
representation for ${\mathcal{S}}_{I}$.

###### Proof.

The direction (iii)$\Rightarrow$ (iv) is trivial, (iv)$\Rightarrow$(i) is a
consequence of Lemma 3.2 and Theorem 3.11, and (ii)$\Leftrightarrow$(iii) is
[kennedy2015essential, Theorem 4.12],[kennedy2016corrigendum].

(i)$\Rightarrow$ (ii): Let $\pi:{\mathcal{O}}_{I}\rightarrow
B({\mathcal{H}}_{\pi})$ be an isometric $*$-representation that has the
approximate unique extension property with respect to $q({\mathcal{S}}_{I})$
and consider $\sigma:=\pi\circ q$. Observe that $\sigma$ vanishes over the
compact operators on ${\mathcal{F}}_{I}$ and so, by [kennedy2015essential,
Proposition 4.13], there exists a unital completely positive map
$\psi:{\mathfrak{T}}_{I}\rightarrow B({\mathcal{H}}_{\pi})$ such that
$\psi\mid_{{\mathcal{S}}_{I}}=\sigma\mid_{{\mathcal{S}}_{I}}$ and

$\psi(S_{i}^{*}S_{j})=\sigma(S_{j}S_{i}^{*}),\ \ \ \ \ \ 1\leq i,j\leq d.$

Moreover, by [kennedy2015essential, Lemma 4.6], we may express
$\psi=\varphi\circ q$ where $\varphi:{\mathcal{O}}_{I}\rightarrow
B({\mathcal{H}}_{\pi})$ is a unital completely positive map. Since
$\varphi\mid_{q({\mathcal{S}}_{I})}=\pi\mid_{q({\mathcal{S}}_{I})}$, we obtain
a net of unitaries $u_{\beta}\in B({\mathcal{H}}_{\pi})$ such that
$u_{\beta}^{*}\varphi(x)u_{\beta}\rightarrow\pi(x)$ in the weak operator
topology for every $x\in{\mathcal{O}}_{I}$. Therefore,
$u_{\beta}^{*}\psi(t)u_{\beta}\rightarrow\sigma(t)$ in the weak operator
topology for every $t\in{\mathfrak{T}}_{I}$. As in the proof of Theorem 4.1,
we obtain a homomorphic conditional expectation
$\Gamma_{\psi}:\mathrm{C}^{*}(\text{Im}\psi)\rightarrow\text{Im}\sigma$
satisfying $\Gamma_{\psi}\circ\psi=\sigma$. In particular, we see that

$\sigma(S_{i}^{*}S_{j})=(\Gamma_{\psi}\circ\psi)(S_{i}^{*}S_{j})=\sigma(S_{j}S_{i}^{*}),\
\ \ \ \ \ 1\leq i,j\leq d.$

Since $\ker\sigma={\mathfrak{K}}({\mathcal{F}}_{I})$, we obtain that
$S_{i}^{*}S_{j}-S_{j}S_{i}^{*}$ is compact for every $1\leq i,j\leq d$. In
other words, the tuple $S=(S_{1},\ldots,S_{d})$ is essentially normal. ∎

######  Remark 5.3.

Mimicing the above proof, one may replace condition (i) by the statement that
there is a $*$-representation of ${\mathfrak{T}}_{I}$ possessing the
approximate unique extension property with respect to ${\mathcal{S}}_{I}$ with
kernel equal to the compact operators on ${\mathcal{F}}_{I}$.

The fact that the operator system ${\mathcal{S}}_{I}$ satisfies the conditions
of Arveson’s hyperrigidity conjecture had been previously unrecorded (see
[kennedy2015essential, Discussion after Theorem 4.12]). However, one can
derive this from the results in [kennedy2015essential]. Indeed, by
[kennedy2015essential, Theorem 3.3], the $\mathrm{C}^{*}$-envelope of
$q({\mathcal{S}}_{I})$ is commutative and so the boundary representations
consist of some collection of characters. Recall that boundary representations
must factor through the $\mathrm{C}^{*}$-envelope by Theorem 2.3 (ii). It is
then easy to check that, if a boundary representation
$\pi:{\mathfrak{T}}_{I}\rightarrow B({\mathcal{H}}_{\pi})$ for
${\mathcal{S}}_{I}$ vanishes over the compact operators on
${\mathcal{F}}_{I}$, then $\pi$ must be a character. Thus, if every
irreducible $*$-representation of ${\mathfrak{T}}_{I}$ is a boundary
representation, we must have that ${\mathcal{O}}_{I}$ is a commutative
$\mathrm{C}^{*}$-algebra. Consequently, the tuple $S$ is essentially normal
and thus, ${\mathfrak{T}}_{I}$ is hyperrigid by [kennedy2015essential, Theorem
4.12],[kennedy2016corrigendum].

A priori, condition (i) appears to be a weak stipulation and thus, one may
expect that there are many choices of homogeneous ideals that will result in
essentially normal tuples. However we remind the reader that, since essential
normality of the tuple $S$ is equivalent to ${\mathcal{O}}_{I}$ being the
$\mathrm{C}^{*}$-envelope of $q({\mathcal{S}}_{I})$, it is unclear whether
${\mathcal{O}}_{I}$ will necessarily have an isometric $*$-representation with
the approximate unique extension property.

## References