II1 factor embeds into $\mathcal{R}^{\omega}$ (see the paragraph above [29,
Notation 5.6]). Note that since every II1 factor contains $\mathcal{R}$, we
have that every II1 factor which embeds into $\mathcal{R}^{\omega}$ also
embeds into $\mathcal{N}^{\omega}$ for each II1 factor $\mathcal{N}$.

Our embedding result will be along these lines showing that if $A$ is
separable $C^{*}$-algebra, $(\mathcal{N},Y)$ is a type II1 factorial tracially
complete $C^{*}$-algebra with CPoU, then there is a ∗-homomorphism
$A\rightarrow\mathcal{N}^{\omega}$ with prescribed tracial data in the case
that all the relevant traces on $A$ factorise through $\mathcal{R}^{\omega}$.
We first provide the following local characterisation of traces factorising
through $\mathcal{R}^{\omega}$.

######  Definition 9.6.

A trace $\tau$ on a $C^{*}$-algebra $A$ is _hyperlinear_ if there is a net of
self-adjoint linear maps $(\psi_{\lambda}\colon A\rightarrow M_{d(\lambda)})$
such that for all $a,b\in A$,

  1. (i)

$\|\psi_{\lambda}(ab)-\psi_{\lambda}(a)\psi_{\lambda}(b)\|_{2}\rightarrow 0$,

  2. (ii)

$\mathrm{tr}_{d(\lambda)}(\psi_{\lambda}(a))\rightarrow\tau(a)$, and

  3. (iii)

$\displaystyle\limsup_{\lambda}\|\psi_{\lambda}(a)\|<\infty$.

For applications to classification, the hyperlinear traces of interest will be
amenable, so that the maps $\psi_{i}$ can be taken to be c.p.c. However, we
will prove the existence theorem in terms of hyperlinear traces since it takes
little extra effort.

######  Remark 9.7.

We collect here a few general observations about hyperlinear traces.

  1. (i)

As usual, when $A$ is unital, we may arrange for each $\psi_{\lambda}$ to be
unital, and when $A$ is separable, we can arrange for the net
$(\psi_{\lambda})$ to be a sequence.

  2. (ii)

If $A$ is separable, we choose a sequence $(\psi_{n})_{n=1}^{\infty}$ as above
and view $M_{d(n)}$ as a unital subalgebra of $\mathcal{R}$. In this way, the
$\psi_{n}$ induce a ∗-homomorphism $\psi\colon
A\rightarrow\mathcal{R}^{\omega}$ with
$\mathrm{tr}_{\mathcal{R}^{\omega}}\circ\psi=\tau$. Note that part (iii) of
Definition 9.6 is needed to guarantee that $\psi_{n}$ is well-defined.

  3. (iii)

If $G$ is a discrete group, then $G$ is hyperlinear if and only if the
canonical trace on the reduced group $C^{*}$-algebra $C^{*}_{\lambda}(G)$ is
hyperlinear. This is the reason for the terminology.

  4. (iv)

Definition 9.6(iii) is equivalent to

(9.16) $\limsup_{\lambda}\|\psi_{\lambda}(a)\|\leq\|a\|,\qquad a\in A.$

Indeed, if $\Lambda$ is the index set of the net and we view each
$M_{d(\lambda)}$ as a subalgebra of $\mathcal{R}$, , then the $\psi_{\lambda}$
induce a ∗-homomorphism $\psi\colon
A\rightarrow\ell^{\infty}(\Lambda,\mathcal{R})/c_{0}(\Lambda,\mathcal{R})$,
where $c_{0}(\Lambda,\mathcal{R})$ consists of all bounded strongly null nets.
As ∗-homomorphisms are contractive, (9.16) follows.

The following is our main existence result for ∗-homomorphisms, although in
the classification theorem, it will be accessed through Corollary 9.9 which
gives a restatement in terms of reduced products when the domain is separable.
We keep track of the additional details regarding factorisations through
finite dimensional $C^{*}$-algebras as it is conceivable that these will play
a role, for example, in subsequent nuclear dimension calculations.

######  Theorem 9.8.

Suppose $A$ is a $C^{*}$-algebra, $(\mathcal{N},Y)$ is a type II1 factorial
tracially complete $C^{*}$-algebra with CPoU, and $\gamma\colon Y\rightarrow
T(A)$ is a continuous affine map such that $\gamma(\tau)$ is hyperlinear for
all $\tau\in Y$. For every finite set $\mathcal{F}\subseteq A$ and
$\epsilon>0$, there are a finite dimensional $C^{*}$-algebra $F$, a self-
adjoint linear map $\psi\colon A\rightarrow F$, and a unital ∗-homomorphism
$\phi\colon F\rightarrow\mathcal{N}$ such that for all $a,b\in\mathcal{F}$:

  1. (i)

$\|\psi(ab)-\psi(a)\psi(b)\|_{2,T(F)}<\epsilon$,

  2. (ii)

$|\tau(\phi(\psi(a)))-\gamma(\tau)(a)|<\epsilon$ for all $\tau\in Y$, and

  3. (iii)

$\|\psi(a)\|<\|a\|+\epsilon$.

Moreover, if $A$ is unital, we may arrange for $\psi$ to be unital, if
$\gamma(\tau)$ is amenable for all $\tau\in Y$, we may arrange for $\psi$ to
be c.p.c., and if $\gamma(\tau)$ is quasidiagonal for all $\tau\in T(A)$, we
may arrange for

  1. (i′)

$\|\psi(ab)-\psi(a)\psi(b)\|<\epsilon$

for all $a,b\in\mathcal{F}$. Finally, the last three statements can be
performed simultaneously.

###### Proof.

Since $Y$ is a closed face in $T(\mathcal{N})$, we have that $Y$ is a Choquet
simplex (Theorem 2.6). Applying Theorem 2.2 to the continuous affine maps
$\mathrm{ev}_{a}\circ\gamma$ for $a\in\mathcal{F}$, there is a finite
dimensional Choquet simplex $Z$ together with continuous affine maps
$\alpha^{\prime}\colon Z\rightarrow Y$ and $\beta\colon Y\rightarrow Z$ so
that

(9.17)
$|\gamma(\alpha^{\prime}(\beta(\tau)))(a)-\gamma(\tau)(a)|<\epsilon,\qquad
a\in\mathcal{F},\ \tau\in Y.$

Define $\alpha\coloneqq\gamma\circ\alpha^{\prime}\colon Z\rightarrow T(A)$ so
that $\alpha(\tau)$ is hyperlinear for all $\tau\in Z$ and

(9.18) $|\alpha(\beta(\tau))(a)-\gamma(\tau)(a)|<\epsilon,\qquad
a\in\mathcal{F},\ \tau\in Y.$

Let $\rho_{1},\ldots,\rho_{n}\in Z$ denote the extreme points of $Z$. For
$i=1,\ldots,n$, since $\alpha(\rho_{i})\in T(A)$ is hyperlinear (by the
assumption on $\gamma$), there are $d_{i}\in\mathbb{N}$ and ∗-linear maps
$\psi_{i}\colon A\rightarrow M_{d_{i}}$ such that for all $a,b\in\mathcal{F}$,

(9.19)
$\begin{split}\|\psi_{i}(ab)-\psi_{i}(a)\psi_{i}(b)\|_{2,\mathrm{tr}_{d_{i}}}&<\epsilon,\\\
|\mathrm{tr}_{d_{i}}(\psi_{i}(a))-\alpha(\rho_{i})(a)|&<\epsilon,\quad\text{and}\\\
\|\psi_{i}(a)\|-\|a\|&<\epsilon.\end{split}$

Define $F\coloneqq\bigoplus_{i=1}^{n}M_{d_{i}}$ and let
$\psi\coloneqq\bigoplus_{i=1}^{n}\phi_{i}\colon A\to F$. We identify $Z$ with
$T(F)$ via the affine map given by

(9.20) $Z\rightarrow
T(F)\colon\rho_{i}\mapsto\mathrm{tr}_{d_{i}}\circ\pi_{i},$

where $\pi_{i}\colon F\rightarrow M_{d_{i}}$ denotes the projection map for
$i=1,\ldots,n$. By Lemma 9.5, there is a unital ∗-homomorphism $\phi\colon
F\rightarrow\mathcal{N}$ such that $\tau\circ\phi=\beta(\tau)$ for all
$\tau\in Y$. Then all three conditions in the theorem follow from the
corresponding conditions in (9.19).

The additional claims in the theorem follow by choosing the $\psi_{i}$ with
the appropriate properties in the second paragraph of the proof. ∎

######  Corollary 9.9.

Suppose $A$ is a separable $C^{*}$-algebra, $(\mathcal{N},Y)$ is a type II1
factorial tracially complete $C^{*}$-algebra with CPoU. Given a continuous
affine map $\gamma\colon Y\rightarrow T(A)$ such that $\gamma(\tau)$ is
hyperlinear for all $\tau\in Y$, there is a unital ∗-homomorphism
$\theta\colon A\rightarrow\mathcal{N}^{\omega}$ such that
$\tau\circ\theta=\gamma(\tau|_{\mathcal{N}})$ for all $\tau\in Y^{\omega}$.

###### Proof.

Since $A$ is separable, we may choose a countable dense
$\mathbb{Q}[i]$-subalgebra $A_{0}\subseteq A$. By Theorem 9.8, there are
sequences $(F_{n})_{n=1}^{\infty}$ of finite dimensional $C^{*}$-algebras,
$(\psi_{n}\colon A_{0}\rightarrow F_{n})_{n=1}^{\infty}$ of self-adjoint
linear maps, and $(\phi_{n}\colon F_{n}\rightarrow\mathcal{N})_{n=1}^{\infty}$
of unital ∗-homomorphisms such that for all $a,b\in A_{0}$ and $\tau\in Y$,

  1. (i)

$\|\psi_{n}(ab)-\psi_{n}(a)\psi_{n}(b)\|_{2,T(F_{n})}\rightarrow 0$,

  2. (ii)

$|\tau(\phi(\psi(a)))-\gamma(\tau)(a)|\rightarrow 0$, and

  3. (iii)

$\displaystyle\limsup_{n\rightarrow\infty}\|\psi(a_{n})\|\leq\|a\|$.

The sequences $(\phi_{n})_{n=1}^{\infty}$ and $(\psi_{n})_{n=1}^{\infty}$
induce ∗-homomorphisms

(9.21)
$A_{0}\xrightarrow{\psi}\prod^{\omega}\big{(}F_{n},T(F_{n})\big{)}\xrightarrow{\phi}\mathcal{N}^{\omega}.$

Define $\theta_{0}\coloneqq\phi\circ\psi$. By (iii), $\theta_{0}$ is
contractive, and hence extends to a ∗-homomorphism $\theta\colon
A\rightarrow\mathcal{N}^{\omega}$ with the required property. ∎

If the range of $\gamma$ is contained in the uniformly amenable traces on $A$,
then $\theta$ will necessarily be tracially nuclear by Theorem 4.9. In the
next subsection, we will use this observation together with the uniqueness
result in Corollary 9.4 to strengthen this existence theorem – namely,
$\theta$ can be chosen to take values in $\mathcal{N}$ (Theorem 9.12).

As a first application of Theorem 9.8, we establish a $W^{*}$-bundle version
of the Connes’ embedding problem assuming a positive solution in each fibre.
In the following result, we assume $\omega$ is a free ultrafilter on
$\mathbb{N}$ (as opposed to a free filter on $\mathbb{N}$ as above).

######  Corollary 9.10.

Suppose $\mathcal{M}$ is a $W^{*}$-bundle over a compact metrisable space $K$
with separably acting fibres. If each fibre of $\mathcal{M}$ admits a trace-
preserving embedding into $\mathcal{R}^{\omega}$, then there is an embedding
$\mathcal{M}\hookrightarrow C_{\sigma}(K,\mathcal{R})^{\omega}$ which
restricts to the identity on $C(K)$.

###### Proof.

As in Proposition 3.6, we may view $\mathcal{N}\coloneqq
C_{\sigma}(K,\mathcal{R})$ as a factorial tracially complete $C^{*}$-algebra
$(\mathcal{N},Y)$ where $Y\cong\mathrm{Prob}(K)$ is the set of traces given by
integrating the trace on $\mathcal{R}$ over $K$. It is easy to see that
$\mathcal{N}$ is McDuff, and hence $(\mathcal{N},Y)$ has CPoU by Theorem 1.4.
Similarly, we may view $\mathcal{M}$ as a tracially complete $C^{*}$-algebra
$(\mathcal{M},X)$ where $X\cong\mathrm{Prob}(K)$ where again, a Radon
probability measure on $K$ induces on trace on $\mathcal{M}$ by integrating
the traces on the fibres of $\mathcal{M}$.

Let $\gamma\colon Y\rightarrow X$ be the continuous affine map induced by the
identity map on $\mathrm{Prob}(K)$. By hypothesis, $\gamma(\tau)\in X$ is
hyperlinear for all $\tau\in K$. It is easy to show that the hyperlinear
traces on a unital $C^{*}$-algebra form a closed convex set, and hence
$\gamma(\tau)\in X$ is hyperlinear for all $\tau\in Y$. Since $K$ is
metrisable and each fibre of $\mathcal{M}$ is separably acting, we have that
$\mathcal{M}$ is $\|\cdot\|_{2,X}$-separable. Let $A\subseteq\mathcal{M}$ be a
$\|\cdot\|$-separable unital $C^{*}$-subalgebra.

By Corollary 9.9, there is a ∗-homomorphism $\theta_{0}\colon
A\rightarrow\mathcal{N}^{\omega}$ such that
$\gamma(\tau)|_{A}=\tau\circ\theta_{0}$ for all $\tau\in Y^{\omega}$. Then
$\theta_{0}$ extends by continuity to a morphism
$(\mathcal{M},X)\rightarrow(\mathcal{N}^{\omega},Y^{\omega})$. By
construction, the inclusion map $C(K)\hookrightarrow\mathcal{N}^{\omega}$ and
$\theta|_{C(K)}\colon C(K)\hookrightarrow\mathcal{N}^{\omega}$ agree on
traces, and hence are unitarily equivalent by Corollary 9.4. If
$u\in\mathcal{N}^{\omega}$ is a unitary conjugating $\theta|_{C(K)}$ to the
inclusion map, then $\mathrm{ad}(u)\circ\theta$ is the desired embedding. ∎

######  Question 9.11.

Can Corollary 9.10 be improved to provide an embedding $\mathcal{M}\to
C_{\sigma}(K,\mathcal{R}^{\omega})$ (which restricts to the identity on
$C(K)$)?

###  9.3. Classification and consequences

The existence and uniqueness results for morphisms in the last sections
together can be combined with tracially complete versions of standard
intertwining arguments to produce our main classification theorems. First, we
use the intertwining via reparameterisation technique (Theorem 5.11) to obtain
a classification result for tracially nuclear morphisms.

######  Theorem 9.12.

Let $(\mathcal{N},Y)$ be a a type II1 factorial tracially complete
$C^{*}$-algebra with CPoU.

  1. (i)

Let $A$ be a separable $C^{*}$-algebra. If $\gamma\colon Y\rightarrow T(A)$ is
a continuous affine map such that $\gamma(\tau)$ is uniformly amenable for all
$\tau\in Y$, then there is a tracially nuclear ∗-homomorphism $\phi\colon
A\rightarrow\mathcal{N}$ such that $\tau\circ\phi=\gamma(\tau)$ for all
$\tau\in Y$, and this $\phi$ is unique up to approximate unitary equivalence.

  2. (ii)

Let $(\mathcal{M},X)$ be a $\|\cdot\|_{2,X}$-separable tracially complete
$C^{*}$-algebra. If $\gamma\colon Y\rightarrow X$ is a continuous affine
function such that $\gamma(Y)$ is contained in the uniformly amenable traces
on $\mathcal{M}$ then there is a tracially nuclear ∗-homomorphism
$\phi\colon(\mathcal{M},X)\rightarrow(\mathcal{N},Y)$ such that
$\tau\circ\phi=\gamma(\tau)$ for all $\tau\in Y$, and this $\phi$ is unique up
to approximate unitary equivalence.

  3. (iii)

Let $(\mathcal{M},X)$ be a $\|\cdot\|_{2,X}$-separable amenable tracially
complete $C^{*}$-algebra. If $\gamma\colon Y\rightarrow X$ is a continuous
affine function, then there is a tracially nuclear ∗-homomorphism
$\phi\colon(\mathcal{M},X)\rightarrow(\mathcal{N},Y)$ such that
$\tau\circ\phi=\gamma(\tau)$ for all $\tau\in Y$, and this $\phi$ is unique up
to approximate unitary equivalence.

###### Proof.

(i). The uniqueness part of (i) is Theorem 9.3. For the existence part of (i),
since uniformly amenable traces are hyperlinear, Corollary 9.9 implies there
is a ∗-homomorphism $\theta_{\infty}\colon A\rightarrow\mathcal{N}^{\infty}$
with $\tau\circ\theta_{\infty}=\gamma(\tau|_{\mathcal{N}})$ for all $\tau\in
Y^{\infty}$. By Theorem 4.9, $\theta$ is tracially nuclear. Note that if
$r\colon\mathbb{N}\rightarrow\mathbb{N}$ is a function with
$\lim_{n\rightarrow\infty}r(n)=\infty$ and
$r^{*}\colon\mathcal{N}^{\infty}\rightarrow\mathcal{N}^{\infty}$ is the
reparameterisation map as in Theorem 5.11, then

(9.22) $\tau\circ\theta_{\infty}=\gamma(\tau|_{\mathcal{N}})=\gamma((\tau\circ
r^{*})|_{\mathcal{N}})=\tau\circ r^{*}\circ\theta_{\infty},\qquad\tau\in
Y^{\infty}.$

By Corollary 9.4, $r^{*}\circ\theta$ and $\theta$ are unitarily equivalent. By
Corollary 7.11, unitaries in $\mathcal{N}^{\infty}$ lift to unitaries in
$\ell^{\infty}(\mathcal{N})$, and hence Theorem 5.11 provides a ∗-homomorphism
$\theta\colon A\rightarrow\mathcal{N}$ such that
$\iota_{\mathcal{N}}\circ\theta$ and $\theta_{\infty}$ are unitarily
equivalent. Then $\theta$ satisfies $\tau\circ\theta=\gamma(\tau)$ for all
$\tau\in\mathcal{N}$, and Theorem 4.9 implies that $\theta$ is tracially
nuclear.

(ii). Let $A\subseteq\mathcal{M}$ be a separable $\|\cdot\|_{2,X}$-dense
$C^{*}$-subalgebra. Then uniqueness follows from uniqueness in (i) as all
morphisms between tracially complete $C^{*}$-algebras are automatically
contractive between the uniform 2-norms. Part (i) gives a (tracially nuclear)
∗-homomorphism $\tilde{\phi}\colon A\to\mathcal{N}$ with
$\tau\circ\tilde{\phi}=\tilde{\gamma}(\tau)$ for $\tau\in Y$, which extends
uniquely to $\phi\colon(\mathcal{M},X)\to(\mathcal{N},Y)$ by Corollary
3.29(iii). The extension is tracially nuclear by Lemma 4.12. As $A$ is
$\|\cdot\|_{2,X}$-dense in $\mathcal{M}$, $\tau\circ\phi=\gamma(\tau)$ for
$\tau\in Y$.

(iii). This is immediate from (ii) as amenability of $\mathcal{M}$ forces all
traces in $X$ to be uniformly amenable (by Theorem 4.9(iii), taking $\theta$
to be $\mathrm{id}_{\mathcal{M}}$). ∎

The classification theorem for regular amenable tracially complete
$C^{*}$-algebras is obtained from a tracially complete version of the two-
sided Elliott intertwining argument (the $C^{*}$-version of which can be found
as [93, Corollary 2.3.4], for example; note that [38, Theorem 3] abstracts
this $C^{*}$-version to an abstract intertwining result for categories with
notions of inner automorphisms and metric structure on the morphism sets, and
one could apply it to the category of separable tracially complete
$C^{*}$-algebras). As with $C^{*}$-classification results, any isomorphism at
the level of the invariants – in this case the designated traces – lifts to an
isomorphism of tracially complete $C^{*}$-algebras. The second part of Theorem
D from the overiew follows from part (ii) below, with the first part of
Theorem D having already been proven in Theorem 4.9.

######  Theorem 9.13.

Suppose that $(\mathcal{M},X)$ and $(\mathcal{N},Y)$ are type II1 semidiscrete
factorial tracially complete $C^{*}$-algebras with property $\Gamma$ such that
$\mathcal{M}$ is $\|\cdot\|_{2,X}$-separable and $\mathcal{N}$ is
$\|\cdot\|_{2,Y}$-separable.

  1. (i)

$(\mathcal{M},X)\cong(\mathcal{N},Y)$ if and only if $X\cong Y$, and moreover,

  2. (ii)

For any affine homeomorphism $\gamma\colon Y\rightarrow X$, there is an
isomorphism $\theta\colon(\mathcal{M},X)\rightarrow\mathcal{(}\mathcal{N},Y)$
such that $\tau\circ\theta=\gamma(\tau)$ for all $\tau\in Y$.

###### Proof.

The first part follows from the second, so fix an isomorphism $\gamma$ as in
(ii). Both $(\mathcal{M},X)$ and $(\mathcal{N},Y)$ have CPoU by Theorem 1.4.
So, by two applications of the existence portion of Theorem 9.12(ii), there
are tracially nuclear morphisms
$\phi_{0}\colon(\mathcal{M},X)\rightarrow(\mathcal{N},Y)$ and
$\psi_{0}\colon(\mathcal{N},Y)\rightarrow(\mathcal{M},X)$ such that

(9.23)
$\tau\circ\phi_{0}=\gamma(\tau)\qquad\text{and}\qquad\sigma\circ\psi_{0}=\gamma^{-1}(\sigma)$

for all $\tau\in Y$ and $\sigma\in X$. In particular,

(9.24)
$\tau\circ\phi_{0}\circ\psi_{0}=\tau\qquad\text{and}\qquad\sigma\circ\psi_{0}\circ\phi_{0}=\sigma$

for all $\tau\in Y$ and $\sigma\in X$. Using the uniqueness portion of Theorem
9.12(ii) twice, we have $\phi_{0}\circ\psi_{0}$ is approximately unitarily
equivalent to $\mathrm{id}_{\mathcal{N}}$ and $\psi_{0}\circ\phi_{0}$ is
approximately unitarily equivalent to $\mathrm{id}_{\mathcal{M}}$.

Let $(\mathcal{F}_{n}^{\prime})_{n=1}^{\infty}$ and
$(\mathcal{G}_{n}^{\prime})_{n=1}^{\infty}$ be increasing sequences of finite
subsets of $\mathcal{M}$ and $\mathcal{N}$, respectively, whose unions are
dense in the respective uniform 2-norms. We will inductively construct
increasing sequences $(\mathcal{F}_{n})_{n=1}^{\infty}$ and
$(\mathcal{G}_{n})_{n=1}^{\infty}$ of finite subsets of $\mathcal{M}$ and
$\mathcal{N}$, respectively, and sequences of ∗-homomorphisms
$(\phi_{n}\colon\mathcal{M}\rightarrow\mathcal{N})_{n=1}^{\infty}$ and
$(\psi_{n}\colon\mathcal{N}\rightarrow\mathcal{M})_{n=1}^{\infty}$ such that
for all $n\geq 1$,

  1. (i)

$\mathcal{F}_{n}^{\prime}\subseteq\mathcal{F}_{n}$ and
$\mathcal{G}_{n}^{\prime}\subseteq\mathcal{G}_{n}$,

  2. (ii)

$\phi_{n}$ and $\psi_{n}$ are unitarily equivalent to $\phi_{0}$ and
$\psi_{0}$, respectively,

  3. (iii)

$\phi_{n}(\mathcal{F}_{n})\subseteq\mathcal{\mathcal{missing}}G_{n}$ and
$\psi_{n}(\mathcal{G}_{n})\subseteq\mathcal{F}_{n+1}$, and

  4. (iv)

for all $a\in\mathcal{F}_{n}$ and $b\in\mathcal{G}_{n}$, we have

(9.25) $\displaystyle\|\psi_{n}(\phi_{n}(a))-a\|_{2,X}$ $\displaystyle<2^{-n}$
and (9.26) $\displaystyle\|\phi_{n}(\psi_{n-1}(b))-b\|_{2,Y}$
$\displaystyle<2^{-n}.$

Assuming that $\mathcal{F}_{n-1}$, $\mathcal{G}_{n-1}$, $\phi_{n-1}$, and
$\psi_{n-1}$ have been constructed, let
$\mathcal{F}_{n}\coloneqq\mathcal{F}_{n}^{\prime}\cup\mathcal{F}_{n-1}\cup\psi_{n-1}(\mathcal{G}_{n-1})$.
Since $\phi_{n-1}$ and $\psi_{n-1}$ are unitarily equivalent to $\phi_{0}$ and
$\psi_{0}$, respectively, and $\phi_{0}\circ\psi_{0}$ is approximately
unitarily equivalent to $\mathrm{id}_{\mathcal{M}}$, we have
$\phi_{n-1}\circ\psi_{n-1}$ approximately unitarily equivalent to
$\mathrm{id}_{\mathcal{M}}$. Therefore, there is a ∗-homomorphism
$\phi_{n}\colon\mathcal{M}\rightarrow\mathcal{N}$ which satisfies (9.26), and
is unitarily equivalent to $\phi_{n-1}$, and hence also $\phi_{0}$. The
construction of $\mathcal{G}_{n}$ and $\psi_{n}$ is similar.

If $m\geq n\geq 1$ and $a\in\mathcal{F}_{n}$, then

(9.27)
$\begin{split}\|\phi_{m+1}(a)-\phi_{m}(a)\|_{2,Y}&\leq\|\phi_{m+1}(a-\psi_{m}(\phi_{m}(a)))\|_{2,Y}\\\
&\hphantom{==}+\|\phi_{m+1}(\psi_{m}(\phi_{m}(a)))-\phi_{m}(a)\|_{2,Y}\\\
&<2^{-m}+2^{-m-1}.\end{split}$

In particular, $(\phi_{m}(a))_{m=1}^{\infty}$ is norm-bounded and
$\|\cdot\|_{2,Y}$-Cauchy for all $a\in\mathcal{F}_{n}$. So
$(\phi_{n}(a))_{n=1}^{\infty}$ is norm-bounded and $\|\cdot\|_{2,Y}$-Cauchy
for all $a\in\bigcup_{n=1}^{\infty}\mathcal{F}_{n}$, and hence also for all
$a\in\mathcal{M}$. Similarly, $(\psi_{n}(b))_{n=1}^{\infty}$ is norm-bounded
and $\|\cdot\|_{2,X}$-Cauchy for all $b\in\mathcal{N}$.

Define $\phi\colon\mathcal{M}\rightarrow\mathcal{N}$ and
$\psi\colon\mathcal{N}\rightarrow\mathcal{M}$ by

(9.28)
$\phi(a)\coloneqq\lim_{n\rightarrow\infty}\phi_{n}(a)\qquad\text{and}\qquad\psi(a)\coloneqq\lim_{n\rightarrow\infty}\psi_{n}(a).$

By construction, $\phi$ and $\psi$ are approximately unitarily equivalent to
$\phi_{0}$ and $\psi_{0}$, respectively, and in particular,
$\tau\circ\phi=\tau\circ\phi_{0}$ for all $\tau\in Y$ and
$\sigma\circ\psi=\sigma\circ\psi_{0}$ for all $\sigma\in X$. To complete the
proof, note that $\phi$ and $\psi$ are mutual inverses using (9.25) and
(9.26). ∎

Theorem A from the overview is an immediate consequence of Theorem 9.13,
provided (as set out in Footnote 1) we interpret it to mean that for unital
separable nuclear $\mathcal{Z}$-stable $C^{*}$-algebras $A$ and $B$, one has
$\big{(}\overline{A}^{T(A)},T(A)\big{)}\cong\big{(}\overline{B}^{T(B)}\big{)}$
if and only if $T(A)$ and $T(B)$ are affinely homeomorphic. The point is that
the ‘easy direction’ – recovering the trace space from an isomorphism – is a
tautology when the isomorphism is in the category of tracially complete
$C^{*}$-algebras.606060To recover $T(A)$ from $\overline{A}^{T(A)}$ as a
$C^{*}$-algebra requires the forthcoming solution to the trace problem
(Question 1.1) for tracially complete $C^{*}$-algebras with CPoU due to the
third-named author. To see the ‘hard direction’, note that if $A$ is a unital
separable nuclear $\mathcal{Z}$-stable $C^{*}$-algebra, then
$\big{(}\overline{A}^{T(A)},T(A)\big{)}$ is $\|\cdot\|_{2,T(A)}$-separable,
semidiscrete by Corollary 4.10, factorial by Proposition 3.23(iv), and
satisfies property $\Gamma$ by Propositions 5.17 and 5.22. The result then
follows from Theorem 9.13(i).

The classification result for morphisms provides the following strengthened
version of completely positive approximation property in which the ‘upward
maps’ can be taken to be ∗-homomorphisms (and the downward maps are
approximately multiplicative). Such a result holds for injective von Neumann
algebras, and this was the starting point to obtaining strengthened forms of
the completely positive approximation property involving _decomposable
approximations_ in the $C^{*}$-algebraic setting ([14, 18, 59]).616161In [59,
Section 1] a local reflexivity argument is given which shows how to use
hyperfiniteness of a von Neumann algebra $\mathcal{M}$ to obtain nets of
finite dimensional algebras $F_{i}$ together with u.c.p. maps
$\psi_{i}\colon\mathcal{M}\to F_{i}$, $\psi_{i}\colon F_{i}\to\mathcal{M}$
such that $\phi_{i}$ a ∗-homomorphism and $\phi_{i}(\psi_{i}(x))\to x$ in the
weak∗-topology for all $x\in\mathcal{M}$. In fact all of Connes’, Popa’s, and
Haagerup’s proofs of injectivity implies hyperfiniteness ([29, 92, 57]) output
such approximations in the case of factors.

######  Theorem 9.14.

Suppose $A$ is a $C^{*}$-algebra, $(\mathcal{N},Y)$ is a type II1 factorial
tracially complete $C^{*}$-algebra, and $\theta\colon A\rightarrow\mathcal{N}$
is a tracially nuclear ∗-homomorphism. Then there are nets

(9.29)
$A\xrightarrow{\psi_{\lambda}}F_{\lambda}\xrightarrow{\phi_{\lambda}}\mathcal{N}$

of finite dimensional $C^{*}$-algebras $F_{\lambda}$ and c.p.c. maps
$\phi_{\lambda}$ and $\psi_{\lambda}$ such that

  1. (i)

$\|\phi_{\lambda}(\psi_{\lambda}(a))-\theta(a)\|_{2,Y}\rightarrow 0$ for all
$a\in A$,

  2. (ii)

$\|\psi_{\lambda}(ab)-\psi_{\lambda}(a)\psi_{\lambda}(b)\|_{2,T(F_{\lambda})}\rightarrow
0$ for all $a,b\in A$,

  3. (iii)

each $\phi_{\lambda}$ is a unital ∗-homomorphism.

If $\|\tau\circ\theta\|^{-1}(\tau\circ\theta)$ is quasidiagonal whenever
$\tau\in Y$ with $\tau\circ\theta\neq 0$, we may further arrange for

  1. (ii′)

$\|\psi_{\lambda}(ab)-\psi_{\lambda}(a)\psi_{\lambda}(b)\|\rightarrow 0$ for
all $a,b\in A$.

In either case, when $A$ and $\theta$ are unital, we may arrange for each
$\psi_{\lambda}$ to be unital.

###### Proof.

Adding a unit to $A$, we may assume that $A$ is unital (Lemma 4.3). Further,
as the conclusion of Theorem 9.14 is a local condition, it suffices to prove
it when $A$ is separable. By Theorem 4.9, $\tau\circ\theta$ is uniformly
amenable for each $\tau\in T(A)$. By Theorem 9.8 (applied to the affine map
$\tau\mapsto\tau\circ\theta$), there are finite dimensional $C^{*}$-algebras
$F_{n}$ and u.c.p. maps

(9.30)
$A\xrightarrow{\psi_{n}}F_{n}\xrightarrow{\phi^{\prime}_{n}}\mathcal{N}$

so that for all $a,b\in A$ and $\tau\in Y$, we have

(9.31) $\|\psi_{n}(ab)-\psi_{n}(a)\psi_{n}(b)\|_{2,T(F_{n})}\rightarrow 0,$

and

(9.32) $|\tau(\phi^{\prime}_{n}(\psi_{n}(a)))-\tau(\theta(a))|\rightarrow 0,$

and such that each $\phi^{\prime}_{n}$ is a unital ∗-homomorphism. Further, if
$\tau\circ\theta$ is quasidiagonal for all $\tau\in T(A)$, we may replace
(9.31) with

(9.33) $\|\psi_{n}(ab)-\psi_{n}(a)\psi_{n}(b)\|\rightarrow 0$

for all $a,b\in A$.

The maps $(\phi^{\prime}_{n}\circ\psi_{n})_{n=1}^{\infty}$ induce a
∗-homomorphism $\theta^{\prime}\colon A\rightarrow\mathcal{N}^{\infty}$ with
$\tau\circ\theta^{\prime}=\tau\circ\theta$ for all $\tau\in Y^{\infty}$. By
Theorem 4.9, $\theta^{\prime}$ is tracially nuclear, and hence by Corollary
9.4, $\theta$ and $\theta^{\prime}$ are unitarily equivalent. If
$u\in\mathcal{N}^{\omega}$ is a unitary with
$\mathrm{ad}(u)\circ\theta^{\prime}=\theta$, then by Corollary 7.11, there is
a sequence of unitaries $(u_{n})_{n=1}^{\infty}\subseteq\mathcal{N}$ lifting
$u$. The result follows by setting
$\phi_{n}\coloneqq\mathrm{ad}(u_{n})\circ\phi_{n}^{\prime}$. ∎

We end by proving the structure and classification theorems from the overview
(Theorem B and C). Most of the implications involved are already in place, and
it remains to use the classification theorem to pass back from hyperfiniteness
to the McDuff property via CPoU.

######  Theorem 9.15.

Let $(\mathcal{M},X)$ be a type II1 factorial tracially complete
$C^{*}$-algebra. Then the following conditions are equivalent:

  1. (i)

$(\mathcal{M},X)$ hyperfinite;

  2. (ii)

$(\mathcal{M},X)$ is amenable and has CPoU;

  3. (iii)

$(\mathcal{M},X)$ is amenable and satisfies property $\Gamma$;

  4. (iv)

$(\mathcal{M},X)$ is amenable and McDuff.

In this setting, if $\mathcal{M}$ is also assumed to be
$\|\cdot\|_{2,X}$-separable, then $(\mathcal{M},X)\cong(\mathcal{R}_{X},X)$
(see Example 3.35).

###### Proof.

The implications (iv)$\implies$(iii), (iii)$\implies$(ii), and
(i)$\implies$(ii) hold by Proposition 5.22, Theorem 1.4, and Theorem 8.3,
respectively. Further, Theorem 9.14 applied to
$\theta=\mathrm{id}_{\mathcal{M}}$ shows (ii)$\implies$ (i). Suppose (ii)
holds and $(\mathcal{M},X)$ is $\|\cdot\|_{2,X}$-separable. Both
$(\mathcal{M},X)$ and $(\mathcal{R}_{X},X)$ satisfy the conditions of Theorem
9.13, and hence $(\mathcal{M},X)\cong(\mathcal{R}_{X},X)$. As
$(\mathcal{R}_{X},X)$ is McDuff, this also shows $(\mathcal{M},X)$ is McDuff
and finishes the proof in the separable setting.

It remains to show (ii) implies (iv) without separability. Assume
$(\mathcal{M},X)$ satisfies CPoU. By the local characterisation of McDuff’s
property (Proposition 5.14(ii)), it suffices to show that every finite set
$\mathcal{F}\subseteq\mathcal{M}$ is contained in a unital
$\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebra of $\mathcal{M}$ which is
factorial and McDuff as a tracially complete $C^{*}$-algebra. Using the
separable inheritability of the conditions in (ii) (Theorem A.3(ii) and (vi),
together with Proposition A.2), this follows from the fact that (ii) implies
(iv) in the separable setting. ∎

##  Appendix A Separabilisation

In this appendix we collect the machinery needed to reduce our main structural
results – Theorems B and 1.4 – to the case of separable tracially complete
$C^{*}$-algebras, and prove the non-metrisable version of Theorem 2.2.

###  A.1. Separabilising tracially complete $C^{*}$-algebras

The following definition is modelled on Blackadar’s notion of separable
inheritability for $C^{*}$-algebras ([7, Section II.8.5]). Recall from Section
3.1 (before Definition 3.11) that if $(\mathcal{M},X)$ is a tracially complete
$C^{*}$-algebra then a tracially complete $C^{*}$-subalgebra of
$(\mathcal{M},X)$ is $(\mathcal{M}_{0},X_{0})$ where
$\mathcal{M}_{0}\subseteq\mathcal{M}$ is a unital $\|\cdot\|_{2,X}$-closed
$C^{*}$-subalgebra and $X_{0}\subseteq T(\mathcal{M})$ is the set of traces
arising as restrictions of traces in $X$ to $\mathcal{M}_{0}$.

######  Definition A.1 (cf. [7, Section II.8.5]).

We say that a property $(P)$ of tracially complete $C^{*}$-algebras is
_separably inheritable_ if

  1. (i)

whenever $(\mathcal{M},X)$ is a tracially complete $C^{*}$-algebra satisfying
$(P)$ and $S\subseteq\mathcal{M}$ is a $\|\cdot\|_{2,X}$-separable subset of
$\mathcal{M}$, there is a tracially complete $C^{*}$-subalgebra
$(\mathcal{M}_{0},X_{0})$ which is $\|\cdot\|_{2,X_{0}}$-separable, satisfies
$(P)$ and such that $S\subseteq\mathcal{M}_{0}$,

  2. (ii)

if $\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ is a sequence of
tracially complete $C^{*}$-algebras satisfying $(P)$ and
$\phi_{n}^{n+1}\colon(\mathcal{M}_{n},X_{n})\rightarrow(\mathcal{M}_{n+1},X_{n+1})$
is an embedding for each $n\geq 1$, then
$\varinjlim\,\big{(}(\mathcal{M}_{n},X_{n}),\phi_{n}^{n+1}\big{)}$ also
satisfies $(P)$.

We say that $(P)$ is _strongly separably inheritable_ if, in addition,

  1. (iii)

if $(\mathcal{M},X)$ is a tracially complete $C^{*}$-algebra such that for
every $\|\cdot\|_{2,X}$-separable subset $S$ of $\mathcal{M}$, there exists a
tracially complete $C^{*}$-subalgebra $(\mathcal{M}_{0},X_{0})$ satisfying
$(P)$ and containing $S$, then $(\mathcal{M},X)$ satisfies $(P)$.

Adapting the language of [99, Definition 1.4] to tracially complete
$C^{*}$-algebras, condition (iii) asks that $(\mathcal{M},X)$ satisfies $(P)$
whenever it separably satisfies $(P)$.

If $(P)$ is a separably inheritable property and $(Q)$ satisfies condition
(iii), then in order to prove $(P)\implies(Q)$ for all tracially complete
$C^{*}$-algebras, it is enough to prove it for $\|\cdot\|_{2,X}$-separable
tracially complete $C^{*}$-algebras.

The following will allow us to construct separable subalgebras that satisfy
several separably inheritable properties simultaneously. The proof is an easy
modification of the corresponding result for $C^{*}$-algebras given in [7,
Proposition II.8.5.3].

######  Proposition A.2.

The conjunction of countably many (strongly) separably inheritable properties
is (strongly) separably inheritable.

###### Proof.

Let $\big{(}(P_{\lambda})\big{)}_{\lambda\in\Lambda}$ be a collection of
separably inheritable properties of tracially complete $C^{*}$-algebras
indexed by some countable set $\Lambda$, and let $(P)$ be the conjunction of
the $(P_{\lambda})$.

It is clear that Definition A.1(ii) holds for $(P)$ as it holds for each
$(P_{\lambda})$. To see Definition A.1(i), let $(\mathcal{M},X)$ be a
tracially complete $C^{*}$-algebras satisfying $(P)$ and let
$S\subseteq\mathcal{M}$ be a $\|\cdot\|_{2,X}$-separable subset. Fix a
surjective map $f\colon\mathbb{N}\rightarrow\Lambda$ such that each
$\lambda\in\Lambda$ has infinitely many preimages. Using Definition A.1(i) for
the properties $(P_{\lambda})$, inductively construct an increasing sequence
of $\|\cdot\|_{2,X}$-closed and $\|\cdot\|_{2,X}$-separable
$C^{*}$-subalgebras $\mathcal{M}_{n}\subseteq\mathcal{M}$ such that
$S\subseteq\mathcal{M}_{1}$ and $\mathcal{M}_{n}$ satisfies $(P_{f(n)})$. As
each $(P_{\lambda})$ satisfies Definition A.1(ii), the
$\|\cdot\|_{2,X}$-closed union of the $\mathcal{M}_{n}$ satisfies $(P)$.

For strong separable inheritability, it is clear that Definition A.1(iii) is
closed under (arbitrary) conjucntions. ∎

We will show that many of the properties of tracially compete $C^{*}$-algebras
defined in this paper are strongly separably inheritable.

######  Theorem A.3.

The following properties are strongly separably inheritable:

  1. (i)

factoriality,

  2. (ii)

amenability,

  3. (iii)

hyperfiniteness,

  4. (iv)

McDuff’s property

  5. (v)

factoriality and property $\Gamma$, and

  6. (vi)

factoriality and CPoU.

In the case of the last two conditions, we include the factoriality condition
as we have not defined property $\Gamma$ or CPoU in the non-factorial setting.

The rest of this subsection is devoted to the proof. We will show each
condition separately. For factoriality, most of the work is contained in Lemma
6.8.

###### Proof of Theorem A.3(i).

A sequential direct limit of factorial tracially complete $C^{*}$-algebras is
factorial by Proposition 3.34. Now suppose $(\mathcal{M},X)$ is a factorial
tracially complete $C^{*}$-algebra and $S\subseteq\mathcal{M}$ is a
$\|\cdot\|_{2,X}$-separable subset of $\mathcal{M}$.

By Lemma 6.8, there is a $\|\cdot\|$-separable unital $C^{*}$-algebra
$A\subseteq\mathcal{M}$ containing a $\|\cdot\|_{2,X}$-dense subset of $S$
such that

(A.1) $X_{A}\coloneqq\\{\tau|_{A}:\tau\in X\\}\subseteq T(A)$

is a closed face. Let $(\mathcal{N},Y)$ denote the tracial completion of $A$
with respect to $X_{A}$ and note that $(\mathcal{N},Y)$ is factorial by
Proposition 3.23(iv). The inclusion $A\rightarrow\mathcal{M}$ extends to a
morphism $\phi\colon(\mathcal{N},Y)\rightarrow(\mathcal{M},X)$ by Proposition
3.25. Note that $\phi$ is an embedding whose range contains $S$. Also, the
unit ball of $\phi(\mathcal{N})$ is $\|\cdot\|_{2,X}$-closed in the unit ball
of $\mathcal{M}$, and it follows from Lemma 3.27 that $\phi(\mathcal{N})$ is
$\|\cdot\|_{2,X}$-closed in $\mathcal{M}$.

Finally, suppose that $(\mathcal{M},X)$ is a tracially complete
$C^{*}$-algebra such that every $\|\cdot\|_{2,X}$-separable subset $S$ is
contained in a factorial tracially complete $C^{*}$-subalgebra of
$(\mathcal{M},X)$. To show that $(\mathcal{M},X)$ is factorial, let
$\tau_{1},\tau_{2},\tau\in T(\mathcal{M})$ be such that $\tau\in X$ and $\tau$
is a non-trivial convex combination of $\tau_{1}$ and $\tau_{2}$, and we will
check that $\tau_{1}\in X$. For any finite subset $\mathcal{F}$ of
$\mathcal{M}$, we can find a factorial tracially complete $C^{*}$-subalgebra
$(\mathcal{M}_{0},X_{0})$ such that $\mathcal{F}\subseteq\mathcal{M}_{0}$.
Since $\tau|_{\mathcal{M}_{0}}$ is a non-trivial convex combination of
$\tau_{1}|_{\mathcal{M}_{0}}$ and $\tau_{2}|_{\mathcal{M}_{0}}$, and $X_{0}$
is a face, it follows that $\tau_{1}|_{\mathcal{M}_{0}}\in X_{0}$. By
definition of $X_{0}$, this means that there exists $\sigma_{\mathcal{F}}\in
X$ such that
$\tau_{1}|_{\mathcal{M}_{0}}=\sigma_{\mathcal{F}}|_{\mathcal{M}_{0}}$, so in
particular, $\tau_{1}|_{\mathcal{F}}=\sigma_{\mathcal{F}}|_{\mathcal{F}}$. By
doing this over all finite subsets $\mathcal{F}$, we see that $\tau_{1}$ is a
weak∗-limit of traces in $X$, and therefore $\tau_{1}\in X$ as required. ∎

It is somewhat subtle to show that amenability is preserved by inductive
limits. Even in the tracial von Neumann algebra setting, the only known proof
that the weak closure of an increasing union of semidiscrete von Neumann
algebras is semidiscrete relies on the equivalence of semidiscreteness and
injectivity from Connes’ work ([29]). Our argument for parts (i) and (ii) of
Definition A.1 goes via the extension result for tracially nuclear morphisms
(Lemma 4.12) proved using the local-to-global characterisations of amenability
in Section 4.

###### Proof of Theorem A.3(ii).

For condition (iii) of Definition A.1, fix a finite subset $\mathcal{F}$ of
$\mathcal{M}$ and $\epsilon>0$. Use the hypothesis to find an amenable
tracially complete $C^{*}$-subalgebra $(\mathcal{M}_{0},X_{0})$ of
$\mathcal{M}$ containing $\mathcal{F}$. By Arveson’s extension theorem, the
maps witnessing amenability of $(\mathcal{M}_{0},X_{0})$ can be extended to
give a finite dimensional $C^{*}$-algebra $F$ and c.p.c. maps
$\mathcal{M}\stackrel{{\scriptstyle\psi}}{{\rightarrow}}F\stackrel{{\scriptstyle\phi}}{{\rightarrow}}\mathcal{M}$
with $\|\phi(\psi(x))-x\|<\epsilon$ for $x\in\mathcal{F}$. Working with a net
indexed over finite sets $\mathcal{F}$ and $\epsilon>0$ gives amenability of
$\mathcal{M}$.

Suppose $\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ is a sequence
of amenable tracially complete $C^{*}$-algebras and
$\phi_{n}^{n+1}\colon(\mathcal{M}_{n},X_{n})\rightarrow(\mathcal{M}_{n+1},X_{n+1})$
is an embedding for each $n\geq 1$ and consider
$(\mathcal{M},X)\coloneqq\varinjlim\,\big{(}(\mathcal{M}_{n},X_{n}),\phi_{n}^{n+1}\big{)}$.
If $A\subseteq\mathcal{M}$ is the $C^{*}$-algebraic direct limit of the
$\mathcal{M}_{n}$, then the inclusion $A\hookrightarrow\mathcal{M}$ is
tracially nuclear (using Arveson’s extension theorem as above). By Lemma 4.12,
$(\mathcal{M},X)$ is amenable, and this shows Definition A.1(ii).

Suppose now that $(\mathcal{M},X)$ is an amenable tracially complete
$C^{*}$-algebra. We’ll first show that for every unital
$\|\cdot\|_{2,X}$-separable, $\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebra
$\mathcal{M}_{0}\subseteq\mathcal{M}$, there is a $\|\cdot\|_{2,X}$-separable,
$\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebra $\mathcal{N}\subseteq\mathcal{M}$
containing $\mathcal{M}_{0}$ so that the inclusion
$\mathcal{M}_{0}\hookrightarrow\mathcal{N}$ is tracially nuclear.

Fix a $\|\cdot\|$-separable, $\|\cdot\|_{2,X}$-dense $C^{*}$-algebra
$A\subseteq\mathcal{M}_{0}$ and an increasing sequence $(\mathcal{F}_{n})$ of
finite subsets of $A$ with $\|\cdot\|$-dense union, and for each $n\geq 1$,
use Proposition 4.4 to construct a finite dimensional $C^{*}$-algebra $F_{n}$
and c.p.c. maps

(A.2) $A\xrightarrow{\psi_{n}}F_{n}\xrightarrow{\phi_{n}}\mathcal{M}$

so that

(A.3) $\|\phi_{n}(\psi_{n}(a))-a\|_{2,X}<\frac{1}{n},\qquad
a\in\mathcal{F}_{n}.$

Let $\mathcal{N}$ denote the tracially complete $C^{*}$-subalgebra of
$\mathcal{M}$ generated by
$\mathcal{M}_{0}\cup\bigcup_{n=1}^{\infty}\phi_{n}(\mathcal{F}_{n})$. By
construction, $\mathcal{N}$ is $\|\cdot\|_{2,X}$-separable and the inclusion
$A\hookrightarrow\mathcal{N}$ is tracially nuclear, and by Lemma 4.12, so is
the inclusion $\mathcal{M}_{0}\hookrightarrow\mathcal{N}$.

Applying this result inductively, given a $\|\cdot\|_{2,X}$-separable subset
$S\subseteq\mathcal{M}$, we may construct an increasing sequence
$(\mathcal{M}_{n})_{n=1}^{\infty}$ of $\|\cdot\|_{2,X}$-separable,
$\|\cdot\|_{2,X}$-closed subalgebras of $\mathcal{M}$ such that
$S\subseteq\mathcal{M}_{1}$ and the inclusions
$\mathcal{M}_{n}\hookrightarrow\mathcal{M}_{n+1}$ are tracially nuclear. If
$\mathcal{N}$ is the $\|\cdot\|_{2,X}$-closure of the union of the
$\mathcal{M}_{n}$, then $\mathcal{N}$ is $\|\cdot\|_{2,X}$-separable. If
$A\subseteq\mathcal{N}$ is the $\|\cdot\|$-closure of the union of the
$\mathcal{M}_{n}$, then the inclusion $A\hookrightarrow\mathcal{N}$ is
tracially nuclear (by the same application of Arveson’s extension theorem used
in the previous parts), and hence $\mathcal{N}$ is amenable by Lemma 4.12.
This shows Definition A.1(i). ∎

The proof that hyperfiniteness is separably inheritable is standard.

###### Proof of Theorem A.3(iii).

Verifying Definition A.1(ii) and (iii) is a standard $\epsilon/3$ argument. To
see Definition A.1(i) suppose $(\mathcal{M},X)$ is a hyperfinite tracially
complete $C^{*}$-algebra and let $S$ be a $\|\cdot\|_{2,X}$-separable subset
of $\mathcal{M}$, and let $\mathcal{M}_{1}$ denote the tracially complete
$C^{*}$-subalgebra of $\mathcal{M}$ generated by $S$.

As $\mathcal{M}_{1}$ is $\|\cdot\|_{2,X}$-separable, fix an increasing
sequence $(\mathcal{F}_{k})$ of finite sets with dense union in
$\mathcal{M}_{1}$. For each $k\geq 1$, let $F_{k}\subseteq\mathcal{M}$ be a
finite dimensional $C^{*}$-algebra such that for every $a\in\mathcal{F}_{k}$,
there is a $b\in F_{k}$ with $\|a-b\|_{2,X}<\frac{1}{k}$. Let
$\mathcal{M}_{2}$ denote the tracially complete $C^{*}$-subalgebra generated
by $\mathcal{M}_{1}$ and the $F_{k}$.

Iterating this construction, there is an increasing sequence of
$\|\cdot\|_{2,X}$-closed $C^{*}$-algebras
$\mathcal{M}_{n}\subseteq\mathcal{M}$ such that for every $n\geq 1$, finite
set $\mathcal{F}\subseteq\mathcal{M}_{n}$, and $\epsilon>0$, there is a finite
dimensional $C^{*}$-algebra $F\subseteq\mathcal{M}_{n+1}$ such that for all
$a\in\mathcal{F}$ there is a $b\in F$ with $\|a-b\|_{2,X}<\epsilon$. Then the
$\|\cdot\|_{2,X}$-closure of the union of the $\mathcal{M}_{n}$ is a tracially
complete hyperfinite subalgebra of $\mathcal{M}$ containing $S$. ∎

The separable inheritability of McDuff’s property and property $\Gamma$ are
easy consequences of the local characterisation of these properties in
Proposition 5.14(ii) and Proposition 5.25 respectively.

###### Proof of Theorem A.3(iv) and (v).

McDuff’s property is preserved by sequential inductive limits by Corollary
5.16, and the combination of factoriality and property $\Gamma$ is preserved
under sequential inductive limits by Propositions 3.34 and 5.25, respectively.
In particular, Definition A.1(ii) holds for both conditions. Likewise,
Proposition 5.14(ii) shows that Definition A.1(iii) holds for McDuff’s
property, while Proposition 5.23(ii) (together with the fact that Definition
A.1(iii) holds for factoriality) handles this condition for factorial
tracially complete $C^{*}$-algebras with property $\Gamma$.

For Definition A.1(i), let $(\mathcal{M},X)$ be a McDuff tracially complete
$C^{*}$-algebra and let $S\subseteq\mathcal{M}$ be a
$\|\cdot\|_{2,X}$-separable set; write $\mathcal{M}_{1}$ for the
$\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebra of $\mathcal{M}$ generated by $S$
which is necessarily $\|\cdot\|_{2,X}$-separable. Inductively, given a
$\|\cdot\|_{2,X}$-separable, $\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebra
$\mathcal{M}_{n}$ of $\mathcal{M}$, Proposition 5.14(ii) provides a sequence
$(v_{m})_{m=1}^{\infty}\subseteq\mathcal{M}$ such that for any finite subset
$\mathcal{F}\subseteq\mathcal{M}_{n}$ and $\epsilon>0$, there is $m\geq 1$
such that for all $a\in\mathcal{F}$,

(A.4) $\|[v_{m},a]\|_{2,X}<\epsilon,\
\|v_{m}^{*}v_{m}+v_{m}v_{m}^{*}-1_{\mathcal{M}}\|_{2,X}<\epsilon,\ \text{and}\
\|v_{m}^{2}\|_{2,X}<\epsilon.$

Let $\mathcal{M}_{n+1}$ be the $\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebra of
$\mathcal{M}$ generated by $\mathcal{M}_{n}\cup\\{v_{m}:m\in\mathbb{N}\\}$.
Let $\mathcal{N}$ denote the $\|\cdot\|_{2,X}$-closure of the union of the
$\mathcal{M}_{n}$. Then $\mathcal{N}$ is a $\|\cdot\|_{2,X}$-separable,
$\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebra of $\mathcal{M}$ containing $S$
and satisfying the approximation property in Proposition 5.14(ii). So
$\mathcal{N}$ is McDuff, and Definition A.1(i) holds.

The argument for Definition A.1(i) works very similarly for the combination of
factoriality and property $\Gamma$. For the inductive step, given a
$\|\cdot\|_{2,X}$-separable, $\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebra of
$\mathcal{M}$ which is factorial with the tracially complete structure induced
from $\mathcal{M}$, we replace the sequence $(v_{m})_{m=1}^{\infty}$ with a
sequence $(p_{m})_{m=1}^{\infty}$ of contractions such that for any finite
subset $\mathcal{F}\subset\mathcal{M}_{n}$ and $\epsilon>0$, there is $m$ with

(A.5) $\|p_{m}-p_{m}^{2}\|_{2,X}<\epsilon,\ \|[p_{m},a]\|_{2,X}<\epsilon\
\text{and}\ \sup_{\tau\in
X}\big{|}\tau(ap_{m})-\frac{1}{2}\tau(a)\big{|}<\epsilon$

for all $a\in\mathcal{F}$ which is given by the characterisation of property
$\Gamma$ in Proposition 5.23(ii). Now use the corresponding part of Theorem
A.3(i) to obtain a $\|\cdot\|_{2,X}$-separable, $\|\cdot\|_{2,X}$-closed
$C^{*}$-subalgebra $\mathcal{M}_{n+1}$ of $\mathcal{M}$ containing
$\mathcal{M}_{n}\cup\\{p_{m}:m\in\mathbb{N}\\}$ which is factorial in the
induced tracially complete structure. Just as in the previous paragraph, the
$\|\cdot\|_{2,X}$-closure of the union of the $\mathcal{M}_{n}$ is factorial
(by Theorem A.3(i)) with property $\Gamma$ (via Proposition 5.23(ii)). ∎

Finally we establish that CPoU is strongly separably inheritable for factorial
tracially complete $C^{*}$-algebras. This works in a very similar fashion to
separable inheritability of the McDuff property and property $\Gamma$.

###### Proof of Theorem A.3(vi).

The combination of factoriality and CPoU is preserved under inductive limits
by Propositions 3.34 and 6.12. Similarly, Proposition 6.2(ii) (together with
the fact that Definition A.1(iii) holds for factoriality) shows that
factoriality and CPoU satisfies Definition A.1(iii).

Suppose now that $(\mathcal{M},X)$ is a factorial tracially complete
$C^{*}$-algebra satisfying CPoU. Let $S\subseteq\mathcal{M}$ be a
$\|\cdot\|_{2,X}$-separable subset of $\mathcal{M}$. We start by constructing
an increasing sequence $(\mathcal{N}_{n})_{n=1}^{\infty}$ of
$\|\cdot\|_{2,X}$-separable, $\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebras of
$\mathcal{M}$ such that

  * •

$S\subseteq\mathcal{N}_{1}$,

  * •

each $\mathcal{N}_{n}$ is factorial as a tracially complete $C^{*}$-algebra,
and

  * •

for each $k,n\geq 1$, if $a_{1},\ldots,a_{k}\in(\mathcal{N}_{n})_{+}$ and
$\delta>0$ with

(A.6) $\sup_{\tau\in X}\min_{1\leq i\leq k}\tau(a_{i})<\delta,$

then there are projections
$p_{1},\ldots,p_{k}\in(\mathcal{N}_{n+1})^{\omega}\cap\mathcal{N}_{n}^{\prime}$
such that

(A.7)
$\sum_{j=1}^{k}p_{j}=1_{\mathcal{M}^{\omega}}\qquad\text{and}\qquad\tau(a_{i}p_{i})\leq\delta\tau(p_{i})$

for all $i=1,\ldots,k$ and $\tau\in X^{\omega}$.

We will construct the algebras $\mathcal{N}_{n}$ by induction starting with
$\mathcal{N}_{1}$ being any unital factorial $\|\cdot\|_{2,X}$-closed
$C^{*}$-subalgebra of $\mathcal{M}$ containing $S$, which exists as
factoriality is separably inheritable (part (i) of Theorem A.3).

Suppose $n\geq 1$ and $\mathcal{N}_{n}$ has been constructed. Let $S_{n}$ be a
countable, $\|\cdot\|_{2,X}$-dense subset of $(\mathcal{N}_{n})_{+}$, and let
$K_{n}$ be the set of all pairs
$\kappa=(\mathcal{G}_{\kappa},\delta_{\kappa})$ where
$\mathcal{G}_{\kappa}\subseteq S_{n}$ is a finite set and $\delta_{\kappa}>0$
is rational with

(A.8) $\sup_{\tau\in
X}\min_{a\in\mathcal{G}_{\kappa}}\tau(a)<\delta_{\kappa}.$

Note that $K_{n}$ is a countable set. Since $(\mathcal{M},X)$ satisfies CPoU,
for each $\kappa\in K_{n}$ and $a\in\mathcal{G}_{\kappa}$, there are
projections $p_{\kappa,a}\in\mathcal{M}^{\omega}\cap\mathcal{N}_{n}^{\prime}$
such that

(A.9)
$\sum_{b\in\mathcal{G}_{\kappa}}p_{\kappa,b}=1_{\mathcal{M}^{\omega}}\qquad\text{and}\qquad\tau(ap_{\kappa,a})\leq\delta\tau(p_{\kappa,a})$

for all $a\in\mathcal{G}_{\kappa}$ and $\tau\in X^{\omega}$. For each
$\kappa\in K_{n}$ and $a\in\mathcal{G}_{\kappa}$, let
$(p_{\kappa,a,m})_{m=1}^{\infty}\subseteq\mathcal{M}$ be a sequence in
$\mathcal{M}$ representing $p_{\kappa,a}$. As factoriality is separably
inheritable (part (i) of the theorem), there is a unital
$\|\cdot\|_{2,X}$-separable, $\|\cdot\|_{2,X}$-closed $C^{*}$-subalgebra
$\mathcal{N}_{n+1}\subseteq\mathcal{M}$ which contains $\mathcal{N}_{n}$ and
each $p_{\kappa,a,m}$ for all $m\geq 1$, $\kappa\in K_{n}$ and
$a\in\mathcal{G}_{\kappa}$, and is factorial as a tracially complete
$C^{*}$-algebra. Then $\mathcal{N}_{n+1}$ satisfies the required properties.

Let $\mathcal{N}\subseteq\mathcal{M}$ be the $\|\cdot\|_{2,X}$-closure of the
union of the algebras $\mathcal{N}_{n}$. Then $\mathcal{N}$ is a
$\|\cdot\|_{2,X}$-closed, $\|\cdot\|_{2,X}$-separable $C^{*}$-subalgebra of
$\mathcal{M}$ which contains $S$. As each $\mathcal{N}_{n}$ is factorial, so
too is $\mathcal{N}$ by Proposition 3.34. The third condition on the
$\mathcal{N}_{n}$ ensures that $\mathcal{N}$ satisfies CPoU, and this
completes the proof. ∎

###  A.2. Proof of Theorem 2.2

The final section of the appendix is devoted to finite dimensional
approximations of non-metrisable Choquet simplices. We show how reduce Theorem
2.2 to the metrisable case ([40, Lemma 2.8], which is a corollary of the
fundamental work of Lazar and Lindenstrauss in [70]). The strategy is another
variation of Blackadar’s separable inheritability, this time for compact
convex sets; roughly, we want to know the property of being a Choquet simplex
is “metrisably inheritable” among compact convex sets. We find it conceptually
easier to do this in the dual picture and consider separably inheritable
properties of Archimedean order unit spaces.

Choquet simplices can be characterised in terms of Archimedean order unit
spaces using Kadison duality as follows. An ordered vector space $V$ satisfies
_Riesz interpolation_ if for all $f_{1},f_{2},g_{1},g_{2}\in V$ with
$f_{i}\leq g_{j}$ for $i,j=1,2$, there is an $h\in V$ with $f_{i}\leq h\leq
g_{j}$ for $i,j=1,2$.

######  Theorem A.4 ([3, Corollary II.3.11]).

Suppose $X$ is a compact convex set and $V$ is an Archimedean order unit
space.

  1. (i)

$X$ is a Choquet simplex if and only if $\mathrm{Aff}(X)$ has Riesz
interpolation.

  2. (ii)

$V$ has Riesz interpolation if and only if $S(V)$ is a Choquet simplex.

###### Proof.

The first statement is [3, Corollary II.3.11], and the second statement
follows from the first by Kadison duality (see Section 2.1). ∎

We will show Riesz interpolation is a separably inheritable property in
Proposition A.6. To facilitate this, we show that Riesz interpolation can be
detected on a dense set.

######  Lemma A.5.

Let $V$ be an Archimedean order unit space. Then $V$ satisfies Riesz
interpolation if and only if there exists a unital dense rational subspace
$V_{0}\subseteq V$ with Riesz interpolation.

###### Proof.

The forward direction is clear by taking the subspace to be $V$. In the other
direction, suppose $V_{0}\subseteq V$ is a unital dense rational subspace of
$V$. Since all states (i.e. positive and unital functionals) on both $V_{0}$
and $V$ are bounded and $V_{0}$ is dense in $V$, the restriction map
$S(V)\rightarrow S(V_{0})$ is an affine homeomorphism. As $V_{0}$ has Riesz
interpolation, $S(V_{0})$ is a Choquet simplex by [55, Corollary 10.6], and
hence $S(V)$ is a Choquet simplex. So $V$ has Riesz interpolation. ∎

Finally, we arrive at the main separabilisation result needed in this
subsection. The proof is similar to the proofs in the previous subsection, and
also to the proofs of separabilisation results in other settings, such as
those in [7, Section II.8.5]

######  Proposition A.6.

Riesz interpolation is a separably inheritable property of Archimedean order
unit spaces in the following sense.

  1. (i)

If $V$ is an Archimedean order unit space with Riesz interpolation and $S$ is
a separable subset of $V$, then there is a separable unital closed subspace
$V_{0}\subseteq V$ that contains $S$ and has Riesz interpolation.

  2. (ii)

For a sequence $(V_{n})_{n=1}^{\infty}$ of separable Archimedean order unit
spaces with Riesz interpolation and unital order embeddings
$V_{n}\hookrightarrow V_{n+1}$, the inductive limit of the $V_{n}$ also has
Riesz interpolation.

###### Proof.

Part (ii) follows from Lemma A.5 since the non-closed union of the $V_{n}$
will have Riesz interpolation.

Assume in (i) that $V$ and $S$ are given, and without loss of generality,
assume $S$ is countable and $1\in S$. Let $V_{1}$ be the rational vector space
spanned by $S$ so that $V_{1}$ is countable.

We can inductively construct an increasing sequence of countable rational
subspaces $(V_{n})_{n=1}^{\infty}$ of $V$ such that for all $n\geq 1$ and
$f_{1},f_{2},g_{1},g_{2}\in V_{n}$ with $f_{i}\leq g_{j}$ for $1\leq i,j\leq
2$, there is an $h\in V_{n+1}$ such that $f_{i}\leq h\leq g_{j}$ for all
$1\leq i,j\leq 2$. Indeed, as $V$ has Riesz interpolation, for any such
quadruple $(f_{1},f_{2},g_{1},g_{2})$ from $V_{n}$, we may choose a
corresponding $h\in V$. As the set of all such quadruples in $V_{n}$ is
countable, we may define $V_{n+1}$ as the rational span of $V_{n}$ and the
corresponding functions $h$ corresponding to these quadruples.

Now, $\bigcup_{n=1}^{\infty}V_{n}$ has Riesz interpolation. Let $V_{0}$ be the
closure of this union and note that $V_{0}$ has Riesz interpolation by Lemma
A.5. ∎

With these results in hand, we prove Theorem 2.2 by reducing it to the
metrisable setting. We restate the theorem for the convenience of the reader.

See 2.2

###### Proof of Theorem 2.2.

View $\mathbb{R}^{d}$ as an ordered vector space with the coordinatewise order
and with order unit $(1,\ldots,1)$. By Kadison duality, it suffices to show
that for every finite set $\mathcal{F}\subseteq\mathrm{Aff}(X)$ and
$\epsilon>0$, there are an integer $d$ and unital positive linear maps

(A.10)
$\mathrm{Aff}(X)\xrightarrow{\psi}\mathbb{R}^{d}\xrightarrow{\phi}\mathrm{Aff}(X)$

such that $\|\phi(\psi(f))-f\|<\epsilon$ for all $f\in\mathcal{F}$.

By Proposition A.6, there is a separable unital closed subspace $V_{0}$ of
$\mathrm{Aff}(X)$ that contains $\mathcal{F}$ and satisfies Riesz
interpolation. By the Hahn–Banach theorem (applied in each coordinate), any
unital positive map $V_{0}\rightarrow\mathbb{R}^{d}$ extends to a unital
positive map $\mathrm{Aff}(X)\rightarrow\mathbb{R}^{d}$ (since unital
functionals are positive exactly when they have norm 1). Therefore, it
suffices to find an integer $d$ and unital positive linear maps

(A.11) $V_{0}\xrightarrow{\psi}\mathbb{R}^{d}\xrightarrow{\phi}V_{0}$

such that $\|\phi(\psi(f))-f\|<\epsilon$ for all $f\in\mathcal{F}$. Since
$V_{0}$ has Riesz interpolation, $S(V_{0})$ is a Choquet simplex by Theorem
A.4(ii). Using Kadison duality to identify $V_{0}$ with
$\mathrm{Aff}(S(V_{0}))$ the result follows from the separable case, [40,
Lemma 2.8]. ∎

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