$E(\pi_{\tau}(c))\xi_{\tau}=P\pi_{\tau}(c)\xi_{\tau}$ for $c\in\mathcal{M}$,
where $\xi_{\tau}\in\mathcal{H}_{\tau}$ is the canonical cyclic vector and $P$
is the projection of $\mathcal{H}_{\tau}$ onto the closure of
$\pi_{\tau}(a)\mathcal{H}_{\tau}$ – see [15, Lemma 1.5.11], for example, for
more details. Set $a_{1}\coloneqq\pi_{\tau}(a)$, and note that
$E(a_{1})=a_{1}$. Therefore, for $b_{1}\coloneqq E(\pi_{\tau}(b))$,

(3.33) $\displaystyle\|a_{1}-b_{1}\|_{2,\tau}$
$\displaystyle\leq\|\pi_{\tau}(a)-\pi_{\tau}(b)\|_{2,\tau}<\epsilon,$ and
Lemma 3.26 applied to $\\{\pi_{\tau}(a)\\}^{\prime\prime}$ implies (3.34)
$\displaystyle\|f(a_{1})-f(b_{1})\|_{2,\tau}$
$\displaystyle\leq\|a_{1}-b_{1}\|_{2,\tau}<\epsilon.$

Since $b_{1}$ is a self-adjoint contraction, $f(b_{1})=b_{1}$.

Hence, in the norm $\|\cdot\|_{2,\tau}$, we have strict approximations

(3.35)
$\pi_{\tau}(f(a))=f(a_{1})\approx_{\epsilon}f(b_{1})=b_{1}\approx_{\epsilon}a_{1}=\pi_{\tau}(a)\approx_{\epsilon}\pi_{\tau}(b).$

Therefore, $\|\pi_{\tau}(f(a))-\pi_{\tau}(b)\|_{2,\tau}<3\epsilon$. This
completes the proof for the case when $b$ is self-adjoint.

In the general case, view $\mathcal{M}\otimes M_{2}$ as a tracially complete
$C^{*}$-algebra as in Proposition 3.18. By the first part of the proof, there
is a self-adjoint contraction $a^{\prime\prime}=(a_{ij}^{\prime\prime})\in
A\otimes M_{2}$ such that

(3.36) $\Big{\|}a^{\prime\prime}-\begin{pmatrix}0&b\\\
b^{*}&0\end{pmatrix}\Big{\|}_{2,X\otimes\\{\mathrm{tr}_{2}\\}}<3\epsilon.$

Since $a^{\prime\prime}$ is self-adjoint,
$a_{21}^{\prime\prime}=a_{12}^{\prime\prime*}$. For any $\tau\in X$, we
compute that

(3.37)
$\begin{split}\|a_{12}^{\prime\prime}-b\|_{2,\tau}^{2}&\leq\frac{1}{2}\tau\big{(}|a_{11}^{\prime\prime}|^{2}+2|a_{12}^{\prime\prime}-b|^{2}+|a_{22}^{\prime\prime}|^{2}\big{)}\\\
&=\Big{\|}a^{\prime\prime}-\begin{pmatrix}0&b\\\
b^{*}&0\end{pmatrix}\Big{\|}_{2,\tau\otimes\\{\mathrm{tr}_{2}\\}}^{2}\\\
&<(3\epsilon)^{2}.\end{split}$

Hence, taking $a^{\prime}\coloneqq a_{12}^{\prime\prime}$, we have
$\|a^{\prime}-b\|_{2,X}<3\epsilon$. ∎

The following version of Kaplansky’s density theorem follows immediately from
the previous lemma.

######  Proposition 3.28 (cf. [80, Theorem 4.3.3]).

Let $(\mathcal{M},X)$ be a tracially complete $C^{*}$-algebra and let
$A\subseteq\mathcal{M}$ be a $\|\cdot\|_{2,X}$-dense $C^{*}$-subalgebra. Then
the unit ball of $A$ is $\|\cdot\|_{2,X}$-dense in the unit ball of
$\mathcal{M}$.

As an application, a tracially complete $C^{*}$-algebra is the tracial
completion of any of its $\|\cdot\|_{2,X}$-dense subalgebras.

######  Corollary 3.29.

Suppose $(\mathcal{M},X)$ is a tracially complete $C^{*}$-algebra and let
$A\subseteq\mathcal{M}$ be a $\|\cdot\|_{2,X}$-dense $C^{*}$-subalgebra.

  1. (i)

$X_{A}\coloneqq\\{\tau|_{A}:\tau\in X\\}\subseteq T(A)$ is a compact convex
set and the inclusion $A\hookrightarrow\mathcal{M}$ induces an isomorphism
$(\overline{A}^{X_{A}},X_{A})\rightarrow(\mathcal{M},X)$.

  2. (ii)

If $(\mathcal{M},X)$ is factorial, then $X_{A}$ is a closed face in $T(A)$.

  3. (iii)

If $(\mathcal{N},Y)$ is another tracially complete $C^{*}$-algebra and
$\phi\colon A\to\mathcal{N}$ is a ∗-homomorphism with $\phi^{*}(Y)\subseteq
X_{A}$, then $\phi$ has a unique extension to
$\tilde{\phi}\colon(\mathcal{M},X)\to(\mathcal{N},Y)$.

###### Proof.

(i): Proposition 3.28 implies the unit ball of $A$ is $\|\cdot\|_{2,X}$-dense
in the unit ball of $\mathcal{M}$. Therefore, if $\tau\in X$, then $\tau|_{A}$
has norm 1, and hence $X_{A}\subseteq T(A)$. As $X$ is compact and convex so
too is $X_{A}$. By Proposition 3.25, the inclusion
$A\hookrightarrow\mathcal{M}$ extends to a morphism
$\theta\colon(\overline{A}^{X_{A}},X_{A})\rightarrow(\mathcal{M},X)$ of
tracially complete $C^{*}$-algebras.

Note that $\theta$ is an embedding in the sense of Definition 3.11, and hence
is isometric by Proposition 3.12. Further, the range of $\theta$ contains the
$\|\cdot\|_{2,X}$-dense subalgebra $A\subseteq\mathcal{M}$. Since the unit
ball of $\overline{A}^{X_{A}}$ is $\|\cdot\|_{2,X_{A}}$-complete, Kaplansky’s
density theorem (Proposition 3.28) implies $\theta$ maps the unit ball of
$\overline{A}^{X_{A}}$ onto the unit ball of $\mathcal{M}$, and in particular
$\theta$ is surjective. Finally, for $\tau\in X$, $\tau\circ\theta^{-1}$ is a
$\|\cdot\|_{2,X}$-continuous trace on $\overline{A}^{X_{A}}$ extending
$\tau|_{A}$, so $\tau\in X_{A}$. This shows $\theta^{-1}$ is also a morphism
of tracially complete $C^{*}$-algebras.

(ii): In light of (i), this follows from Proposition 3.23(iv).

(iii): This follows from (i) and Proposition 3.25. ∎

###  3.5. Constructions

There is a recipe for producing constructions on tracially complete
$C^{*}$-algebras.

  1. (i)

Perform the corresponding (spatial) $C^{*}$-construction on the underlying
$C^{*}$-algebras.

  2. (ii)

Identify the suitable collection of traces on the newly constructed
$C^{*}$-algebra

  3. (iii)

Take the tracial completion with respect to these traces.

We illustrate this process here with direct sums, tensor products, and
sequential inductive limits. Products and reduced products (including
ultraproducts) are constructed in Section 5.1. In the case of direct sums,
step (iii) above is redundant as the $C^{*}$-direct sum is already tracially
complete.

######  Definition 3.30.

Let $(\mathcal{M},X)$ and $(\mathcal{N},Y)$ be tracially complete
$C^{*}$-algebras. Let $X\oplus Y$ denote the set of traces of the form
$\lambda\tau_{X}+(1-\lambda)\tau_{Y}$ for $\tau_{X}\in X$, $\tau_{Y}\in Y$ and
$0\leq\lambda\leq 1$. The direct sum of $(\mathcal{M},X)$ and
$(\mathcal{N},Y)$ is the tracially complete $C^{*}$-algebra
$(\mathcal{M}\oplus\mathcal{N},X\oplus Y)$.

Note that the tracially complete direct sum of two finite von Neumann algebras
$(\mathcal{M},T(\mathcal{M})$ and $(\mathcal{N},T(\mathcal{N}))$ is the finite
von Neumann algebra $(\mathcal{M}\oplus\mathcal{N},T(\mathcal{M}\oplus N))$.
As the extreme traces on $X\oplus Y$ are precisely the union of the extreme
traces on $X$ and the extreme traces on $Y$, the direct sum
$(\mathcal{M},X)\oplus(\mathcal{N},Y)$ is factorial if and only if both
$(\mathcal{M},X)$ and $(\mathcal{N},Y)$ are factorial.

Next up are tensor products.

######  Definition 3.31.

Let $(\mathcal{M},X)$ and $(\mathcal{N},Y)$ be tracially complete
$C^{*}$-algebras. Let $\mathcal{M}\otimes\mathcal{N}$ denote the minimal
$C^{*}$-tensor product of $\mathcal{M}$ and $\mathcal{N}$ and let $X\otimes
Y\subseteq T(\mathcal{M}\otimes\mathcal{N})$ be the closed convex hull of the
traces $\sigma\otimes\rho$ for $\sigma\in X$ and $\rho\in Y$. Define
$\mathcal{M}\bar{\otimes}\mathcal{N}$ to be the tracial completion of
$\mathcal{M}\otimes\mathcal{N}$ with respect to $X\otimes Y$. The tensor
product $(\mathcal{M},X)\bar{\otimes}(\mathcal{N},Y)$ is the pair
$(\mathcal{M}\bar{\otimes}\mathcal{N},X\otimes Y)$.

######  Proposition 3.32.

If $(\mathcal{M},X)$ and $(\mathcal{N},Y)$ are factorial tracially complete
$C^{*}$-algebras, then $(\mathcal{M},X)\bar{\otimes}(\mathcal{N},Y)$ is also
factorial.

###### Proof.

Let $F\subseteq T(\mathcal{M}\otimes\mathcal{N})$ be the closed face generated
by $X\otimes Y$. By Proposition 3.23(iv), it is enough to show that
$F=X\otimes Y$. In fact, by the Krein–Milman theorem, it is enough to show
$\partial_{e}F\subseteq X\otimes Y$. To this end, let
$\tau_{0}\in\partial_{e}F$ be given. Since $F$ is a closed face in
$T(\mathcal{M}\otimes\mathcal{N})$, we have
$\tau_{0}\in\partial_{e}T(\mathcal{M}\otimes\mathcal{N})$. By [10, Proposition
3.5], there are traces $\sigma_{0}\in T(\mathcal{M})$ and $\rho_{0}\in
T(\mathcal{N})$ such that $\tau_{0}=\sigma_{0}\otimes\rho_{0}$. Now, it is
enough to show $\sigma_{0}\in X$ and $\rho_{0}\in Y$ – we will only show the
former as the latter follows by symmetry.

By Theorem 2.4(ii), to show $\sigma_{0}\in X$, it is enough to show
$f(\sigma_{0})=0$ for every continuous affine function $f\colon
T(\mathcal{M})\rightarrow[0,1]$ with $f|_{X}=0$. Let $f$ be such a function
and apply Proposition 2.7 to produce a self-adjoint $a\in\mathcal{M}$ with
$\sigma(a)=f(\sigma)$ for all $\sigma\in T(\mathcal{M})$. Define

(3.38) $G\coloneqq\\{\tau\in T(\mathcal{M}\otimes\mathcal{N}):\tau(a\otimes
1_{\mathcal{N}})=0\\}.$

For any $\sigma\in T(\mathcal{M})$ and $\rho\in T(\mathcal{N})$, we have
$(\sigma\otimes\rho)(a\otimes 1_{\mathcal{N}})=f(\sigma)\geq 0$. By [10,
Proposition 3.5] and the Krein–Milman theorem,

(3.39)
$T(\mathcal{M}\otimes\mathcal{N})=\overline{\mathrm{co}}\,\big{\\{}\sigma\otimes\rho:\sigma\in
T(\mathcal{M}),\ \rho\in T(\mathcal{N})\big{\\}}.$

Hence $\tau(a\otimes 1_{\mathcal{N}})\geq 0$ for all $\tau\in
T(\mathcal{M}\otimes\mathcal{N})$. It follows that $G$ is a closed face in
$T(\mathcal{M}\otimes\mathcal{N})$. For all $\sigma\in X$ and $\rho\in Y$,

(3.40) $(\sigma\otimes\rho)(a\otimes 1_{\mathcal{N}})=\sigma(a)=f(\sigma)=0,$

and hence $X\otimes Y\subseteq G$. Then, as $G$ is a closed face containing
$X\otimes Y$, we have $F\subseteq G$, and so $\tau_{0}\in G$. Since
$\tau_{0}=\sigma_{0}\otimes\rho_{0}$, this shows that
$f(\sigma_{0})=\sigma_{0}(a)=0$. ∎

Finally, we construct sequential inductive limits in the category of tracially
complete $C^{*}$-algebras.

######  Definition 3.33.

Let

(3.41)
$\cdots\rightarrow(\mathcal{M}_{n},X_{n})\xrightarrow{\phi_{n}^{n+1}}(\mathcal{M}_{n+1},X_{n+1})\xrightarrow{\phi_{n+1}^{n+2}}(\mathcal{M}_{n+2},X_{n+2})\rightarrow\cdots$

be a sequential inductive system of tracially complete $C^{*}$-algebras. Form
the $C^{*}$-inductive limit
$A\coloneqq\varinjlim(\mathcal{M}_{n},\phi_{n}^{n+1})$ and let
$\hat{\phi}_{n}^{\infty}\colon\mathcal{M}_{n}\rightarrow A$ be the canonical
unital ∗-homomorphism. The inductive system (3.41) induces a projective system

(3.42) $\cdots\longleftarrow
X_{n}\xleftarrow{(\phi_{n}^{n+1})^{*}}X_{n+1}\xleftarrow{(\phi_{n+1}^{n+2})^{*}}X_{n+2}\longleftarrow\cdots$

of compact convex sets. Set

(3.43) $X\coloneqq\\{\tau\in T(A):\tau\circ\hat{\phi}_{n}^{\infty}\in
X_{n}\text{ for all }n\geq 1\\}\subseteq T(A)$

and note that $X$ is a compact convex subset of $T(A)$. Further, the maps
$(\hat{\phi}_{n}^{\infty})^{*}\colon X\rightarrow X_{n}$ induce an affine
homeomorphism

(3.44)
$X\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\varprojlim\big{(}X_{n},(\phi_{n}^{n+1})^{*}\big{)}$

(and in particular, $X$ is non-empty whenever all of the $X_{n}$ are non-
empty; see [35, Theorem VIII.3.5], for example). Define
$\mathcal{M}\coloneqq\overline{A}^{X}$ and

(3.45)
$\varinjlim\big{(}(\mathcal{M}_{n},X_{n}),\phi_{n}^{n+1}\big{)}\coloneqq(\mathcal{M},X)$

Finally, write $\phi_{n}^{\infty}\colon\mathcal{M}_{n}\rightarrow\mathcal{M}$
for the unital ∗-homomorphism obtained by composing $\hat{\phi}_{n}^{\infty}$
with the canonical map $\alpha_{X}\colon A\to\mathcal{M}$.

We now show the definition of inductive limit in Definition 3.33 satisfies the
required universal property and hence is an inductive limit in the category of
tracially complete $C^{*}$-algebras.

######  Proposition 3.34.

Let $\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ be a sequence of
tracially complete $C^{*}$-algebras and
$\big{(}\phi_{n}^{n+1}\colon(\mathcal{M}_{n},X_{n})\to(\mathcal{M}_{n+1},X_{n+1})\big{)}_{n=1}^{\infty}$
be a sequence of morphisms. Set

(3.46)
$(\mathcal{M},X)\coloneqq\varinjlim\big{(}(\mathcal{M}_{n},X_{n}),\phi_{n}^{n+1}\big{)}$

as in Definition 3.33 and write
$\phi_{n}^{\infty}\colon\mathcal{M}_{n}\rightarrow\mathcal{M}$ for the
canonical ∗-homomorphisms as above. Then $(\mathcal{M},X)$ is a tracially
complete $C^{*}$-algebra, $\phi_{n}^{\infty}$ is a morphism for $n\geq 1$, and
together, they define the inductive limit of
$\big{(}(\mathcal{M}_{n},X_{n}),\phi_{n}^{n+1}\big{)}_{n=1}^{\infty}$ in the
category of tracially complete $C^{*}$-algebras. If each $X_{n}$ is
metrisable, then so is $X$. Further, if each $(\mathcal{M}_{n},X_{n})$ is
factorial, then so is $(\mathcal{M},X)$.

###### Proof.

Define $A$, $X$, and $\hat{\phi}_{n}^{\infty}$ as in Definition 3.33, so that
$(\mathcal{M},X)=(\overline{A}^{X},X)$. By Proposition 3.23(iii),
$(\mathcal{M},X)$ is a tracially complete $C^{*}$-algebra. By construction,
$(\hat{\phi}_{n}^{\infty})^{*}(X)\subseteq X_{n}$, so
$(\phi_{n}^{\infty})^{*}(X)\subseteq X_{n}$ and each $\phi_{n}^{\infty}$ is a
morphism. As sequential projective limits of metrisable spaces are metrisable,
$X$ is metrisable if each $X_{n}$ is metrisable. If each
$(\mathcal{M}_{n},X_{n})$ is factorial, then $X_{n}$ is a face in
$T(\mathcal{M}_{n})$ for all $n\geq 1$. It follows from (3.43) that $X$ is a
face in $T(A)$. Proposition 3.23(iv) then implies that $(\mathcal{M},X)$ is
factorial.

To show that $(\mathcal{M},X)$ is the inductive limit of
$\big{(}(\mathcal{M}_{n},X_{n}),\phi_{n}^{n+1}\big{)}$, suppose we have a
tracially complete $C^{*}$-algebra $(\mathcal{N},Y)$ and morphisms

(3.47) $\psi_{n}\colon(\mathcal{M}_{n},X_{n})\to(\mathcal{N},Y),\qquad
n\in\mathbb{N},$

such that $\psi_{n+1}\circ\phi_{n}^{n+1}=\psi_{n}$. Then, working in the
category of unital $C^{*}$-algebras, we obtain a unique unital ∗-homomorphism
$\hat{\psi}\colon A\to\mathcal{N}$ such that
$\psi_{n}=\hat{\psi}\circ\hat{\phi}_{n}^{\infty}$ for each $n\in\mathbb{N}$.
For $\tau\in Y$, we have

(3.48) $\tau\circ\hat{\psi}\circ\hat{\phi}_{n}^{\infty}=\tau\circ\psi_{n}\in
X_{n}$

for all $n\geq 1$, so by (3.43), $\tau\circ\hat{\psi}\in X$. This shows
$\hat{\psi}^{*}(Y)\subseteq X$. Accordingly, there is a unique morphism
$\psi\colon(\mathcal{M},X)\rightarrow(\mathcal{N},Y)$ with
$\hat{\psi}=\psi\circ\alpha_{X}$ by Proposition 3.25. Moreover

(3.49)
$\psi_{n}=\hat{\psi}\circ\hat{\phi}_{n}^{\infty}=\psi\circ\alpha_{X}\circ\hat{\phi}_{n}^{\infty}=\psi\circ\phi_{n}^{\infty},$

as required. Uniqueness of $\psi$ follows from uniqueness in Proposition 3.25
as any morphism $\psi\colon(\mathcal{M},X)\to(\mathcal{N},Y)$ with
$\psi\circ\phi_{n}=\psi_{n}$ for all $n$ satisfies
$\psi\circ\alpha_{X}=\hat{\psi}$. ∎

For each metrisable Choquet simplex $X$, we now use an inductive limit
construction to produce the concrete model $(\mathcal{R}_{X},X)$ of a
tracially complete $C^{*}$-algebra covered by Theorem C as discussed in the
overview of results. This is achieved by mimicking construction of a simple AF
algebra $A$ for which $T(A)=X$, found in [6, 54], and can be deduced from
these results by considering the tracial completion of such an AF algebras.
The details are slightly easier in the tracially complete setting since
$K_{0}(\mathcal{R})\cong\mathbb{R}$ and one need not worry about simplicity.

######  Example 3.35.

Let $X$ be a metrisable Choquet simplex. By [55, Theorem 11.6], we can write
$X$ as the inverse limit of a system of finite dimensional simplices

(3.50)
$X_{1}\stackrel{{\scriptstyle\alpha_{2}^{1}}}{{\longleftarrow}}X_{2}\stackrel{{\scriptstyle\alpha_{3}^{2}}}{{\longleftarrow}}\cdots,$

where the connecting maps are continuous and affine. We will construct an
inductive system of tracially complete $C^{*}$-algebras realising this data.

Set $\mathcal{M}_{n}\coloneqq\mathcal{R}^{\oplus\,\partial_{e}X_{n}}$, where
$\mathcal{R}$ is the hyperfinite II1 factor. Note that $T(\mathcal{M}_{n})$
can be canonically identified with $X_{n}$ and
$\big{(}\mathcal{M}_{n},T(\mathcal{M}_{n})\big{)}$ is tracially complete and
factorial. Next, we can choose a ∗-homomorphism
$\phi_{n}^{n+1}\colon\mathcal{M}_{n}\to\mathcal{M}_{n+1}$ that induces the map
$\alpha_{n+1}^{n}$. To do this explicitly, write
$\partial_{e}X_{n+1}=\\{x_{1},\dots,x_{k}\\}$ and
$\partial_{e}X_{n}=\\{y_{1},\dots,y_{l}\\}$. For each $i=1,\dots,k$, write
$\alpha_{n+1}^{n}(x_{i})$ as the convex combination
$\sum_{j=1}^{l}\lambda_{i,j}y_{j}$. Fixing $i$, find a partition of unity
$p_{i,1},\dots,p_{i,l}\in\mathcal{R}$ of projections such that
$\tau(p_{i,j})=\lambda_{i,j}$ for each $j$. For each $i$ and $j$, choose any
unital ∗-homomorphism $\psi_{i,j}\colon\mathcal{R}\to
p_{i,j}\mathcal{R}p_{i,j}$ (which exists since $\mathcal{R}$ has full
fundamental group, a result which goes back to Murray and von Neumann in
[83]). Then define $\phi_{n}^{n+1}\colon\mathcal{R}^{\oplus
l}\to\mathcal{R}^{\oplus k}$ by

(3.51) $\phi_{n}^{n+1}(a_{1}\oplus\cdots\oplus
a_{l})\coloneqq\Big{(}\sum_{j=1}^{l}\psi_{1,j}(a_{j}),\dots,\sum_{j=1}^{l}\psi_{k,j}(a_{j})\Big{)},$

and note that $(\phi_{n}^{n+1})^{*}=\alpha_{n+1}^{n}$.

Define
$(\mathcal{R}_{X},X^{\prime})\coloneqq\varinjlim\big{(}(\mathcal{M}_{n},X_{n}),\phi_{n}^{n+1}\big{)}$.
As $(\phi_{n}^{n+1})^{*}=\alpha_{n+1}^{n}$, (3.44) provides an isomorphism
$X\rightarrow X^{\prime}$, so after identifying $X$ with $X^{\prime}$ via this
isomorphism, $(\mathcal{R}_{X},X)$ is the desired factorial tracially complete
$C^{*}$-algebra.

A priori, the construction outlined above depends not only on the Choquet
simplex $X$ but also on the choice of inverse limit and the choices made when
defining connecting maps. However, it will follow from Theorem 9.13 that the
tracially complete $C^{*}$-algebra $(\mathcal{R}_{X},X)$ depends only on $X$.
Moreover, as $X$ varies over all metrisable Choquet simplices, these will
provide models for the classifiable tracially complete $C^{*}$-algebras (see
Theorem 9.15, which contains Theorem C).

######  Remark 3.36.

The infinite tensor product $\bigotimes_{n=1}^{\infty}(\mathcal{M}_{n},X_{n})$
of a countable family $(\mathcal{M}_{n},X_{n})$ of tracially complete
$C^{*}$-algebras can now be constructed as the inductive limit of the finite
tensor products $\bigotimes_{i=1}^{n}(\mathcal{M}_{i},X_{i})$ with the obvious
connecting maps.

###  3.6. $W^{*}$-bundles

In Proposition 3.6, we showed Ozawa’s $W^{*}$-bundles can be viewed as
tracially complete $C^{*}$-algebras. We show here that by a reformulation of a
theorem due to Ozawa, any factorial tracially complete $C^{*}$-algebra
$(\mathcal{M},X)$ with $X$ a Bauer simplex can be given the structure of a
$W^{*}$-bundle whose fibres are factors. This is essentially due to Ozawa in
[86]. Recall from Section 2.1 that if $X$ is a Bauer simplex, then there is an
affine homeomorphism

(3.52)
$X\overset{\cong}{\longrightarrow}\mathrm{Prob}(\partial_{e}X)\colon\tau\mapsto\mu_{\tau},$

where $\mu_{\tau}$ is the measure with barycentre $\tau$.

######  Theorem 3.37 (cf. [86, Theorem 3]).

If $(\mathcal{M},X)$ is a factorial tracially complete $C^{*}$-algebra such
that $X$ is a Bauer simplex with $K\coloneqq\partial_{e}X$, then there is a
unique embedding $C(K)\subseteq Z(\mathcal{M})$ such that

(3.53)
$\tau(fa)=\int_{K}f(\sigma)\sigma(a)\,\mathrm{d}\mu_{\tau}(\sigma),\qquad f\in
C(K),\,a\in\mathcal{M},\,\tau\in X.$

Further, the map $E\colon\mathcal{M}\rightarrow C(K)$ given by
$E(a)(\tau)\coloneqq\tau(a)$ is a conditional expectation endowing
$\mathcal{M}$ with the structure of a $W^{*}$-bundle.

###### Proof.

We first show the embedding of $C(K)$ is unique, assuming it exists. Let
$\phi,\psi\colon C(K)\rightarrow Z(\mathcal{M})$ be two embeddings satisfying
(3.53). Fix $f\in C(K)$. If $\tau\in K$, then $\mu_{\tau}$ is the point mass
at $\tau$, so (3.53) gives

(3.54) $\tau(\phi(f)\psi(f))=f(\tau)\tau(\psi(f))=f(\tau)^{2}.$

Similarly, $\tau(\phi(f)^{2})=\tau(\psi(f)^{2})=f(\tau)^{2}$. Now, if $f\in
C(K)$ is self-adjoint, we have

(3.55)
$\|\phi(f)-\psi(f)\|_{2,\tau}^{2}=\tau\big{(}(\phi(f)-\psi(f))^{2}\big{)}=0$

for all $\tau\in\partial_{e}X$, and hence for all $\tau\in X$. Therefore,
$\phi(f)=\psi(f)$ for all self-adjoint $f\in C(K)$, and hence for all $f\in
C(K)$.

Note that once we know the embedding of $C(K)$ exists, then $E$ is necessarily
a tracial conditional expectation onto $C(K)$ such that
$\|\cdot\|_{2,X}=\|\cdot\|_{2,E}$. In particular, the unit ball of
$\mathcal{M}$ is $\|\cdot\|_{2,E}$-complete.

The hardest portion of the proof is the existence of the embedding of $C(K)$.
This is constructed in [86, Theorem 3]. Note that there is a standing
assumption of metrisability in [86]. The metrisability in this result is only
needed so that $\partial_{e}X$ is Borel, but in our case, this is clear since
$\partial_{e}X$ is closed. ∎

The use of factoriality in Theorem 3.37 is subtle – it enters in the proof
implicitly through the computation of the centre of
$\pi_{\tau}(\mathcal{M})^{\prime\prime}$ for $\tau\in X$ in [86, Lemma 10]. In
more detail, if $\tau\in X$, and $z\in
Z(\pi_{\tau}(\mathcal{M})^{\prime\prime})$, then we may define
$z\tau\in\mathcal{M}^{*}$ by
$(z\tau)(a)\coloneqq\langle\pi_{\tau}(a)z\xi_{\tau},\xi_{\tau}\rangle$. Then
$z\tau$ is a tracial linear functional with $|z\tau|\leq\|z\|\tau$. The
assumption that $X$ is a face implies $z\tau$ belongs to the span of $X$ –
this is necessary to define the representing measure $\mu_{z\tau}$ on
$\partial_{e}X$ as needed in the proof of [86, Lemma 10]. As the following
example demonstrates, Theorem 3.37 can fail without factoriality even for the
one-dimensional simplex. The salient point is that in a $W^{*}$-bundle the
fibres are fairly independent of each other, whereas in a non-factorial
tracially complete algebra $(\mathcal{M},X)$, the GNS representation
$\pi_{\tau}$ with respect to the extreme traces $\tau\in\partial_{e}X$ may not
be.

######  Example 3.38.

Let $\mathcal{M}\coloneqq\mathcal{R}\oplus\mathcal{R}\oplus\mathcal{R}$, and
let $\tau_{i}\in T(\mathcal{M})$ be given by
$\tau_{i}(a_{1},a_{2},a_{3})\coloneqq\tau_{\mathcal{R}}(a_{i})$ for $i=1,2,3$.
Define $\sigma_{1},\sigma_{2}\in T(\mathcal{M})$ by

(3.56)
$\sigma_{1}\coloneqq\frac{1}{2}(\tau_{1}+\tau_{2})\qquad\text{and}\qquad\sigma_{2}\coloneqq\frac{1}{2}(\tau_{1}+\tau_{3})$

Let $Y\coloneqq\mathrm{co}\\{\sigma_{1},\sigma_{2}\\}\subseteq
T(\mathcal{M})$. Then $(\mathcal{M},Y)$ is a tracially complete
$C^{*}$-algebra and $Y$ is a Bauer simplex. Note that $(\mathcal{M},Y)$ is not
factorial – in fact, neither extreme point of $Y$ is an extreme point of
$T(\mathcal{M})$. Let $K\coloneqq\partial_{e}Y=\\{\sigma_{1},\sigma_{2}\\}$.

We will show there is no embedding of $C(K)\cong\mathbb{C}^{2}$ into
$Z(\mathcal{M})\cong\mathbb{C}^{3}$ satisfying (3.53). Note that any embedding
satisfying (3.53) is unital. Let $f_{1},f_{2}\in C(K)$ be given by
$f_{1}(\sigma_{1})\coloneqq 1$ and $f_{1}(\sigma_{2})\coloneqq 0$, and
$f_{2}\coloneqq 1_{C(K)}-f_{1}$. Exploiting the symmetry of $\tau_{2}$ and
$\tau_{3}$, it suffices to show that the three unital embeddings
$j_{1},j_{2},j_{3}\colon C(K)\rightarrow Z(\mathcal{M})$ determined by

(3.57) $j_{1}(f_{1})=(1_{\mathcal{R}},0,0),\
j_{2}(f_{1})=(1_{\mathcal{R}},1_{\mathcal{R}},0),\ \text{and}\
j_{3}(f_{1})=(0,1_{\mathcal{R}},0)$

fail (3.53).

In the first case, for $a\coloneqq(0,1_{\mathcal{R}},0)$,
$\sigma_{1}(j_{1}(f_{1})a)=0$, but

(3.58)
$\int_{K}f_{1}(\sigma)\sigma(a)\,\mathrm{d}\mu_{\sigma_{1}}=f_{1}(\sigma_{1})\sigma_{1}(a)=\frac{1}{2}.$

In the second case, for $a\coloneqq(1_{\mathcal{R}},0,0)$,
$\sigma_{2}(j_{2}(f_{2})a)=0$, but

(3.59)
$\int_{K}f_{2}(\sigma)\sigma(a)\,\mathrm{d}\mu_{\sigma_{2}}=f_{2}(\sigma_{2})\sigma_{2}(a)=\frac{1}{2}.$

In the third case, for $a\coloneqq(1_{\mathcal{R}},0,0)$,
$\sigma_{1}(j_{3}(f_{1})a)=0$, but

(3.60)
$\int_{K}f_{1}(\sigma)\sigma(a)\,\mathrm{d}\mu_{\sigma_{1}}=f_{1}(\sigma_{1})\sigma_{1}(a)=\frac{1}{2}.$

So (3.53) fails in all three cases.

##  4\. Amenability for tracially complete $C^{*}$-algebras

In this section, we show how to combine fibrewise amenability to obtain a
global uniform 2-norm completely positive approximation property. The main
result is Theorem 4.9, characterising morphisms into tracially complete
$C^{*}$-algebras which are _tracially nuclear_ – the appropriate notion of
amenability which feeds into classification. Theorem 1.2 will follow as a
special case of Theorem 4.9.

###  4.1. Definition and basic properties

Our notion of amenability for tracially complete $C^{*}$-algebras is given by
the following version of the completely positive approximation property.

######  Definition 4.1.

Let $A$ be a $C^{*}$-algebra and let $(\mathcal{N},Y)$ be a tracially complete
$C^{*}$-algebra. We say that a c.p. map $\theta\colon A\rightarrow\mathcal{N}$
is _tracially nuclear_ if there are nets of finite dimensional
$C^{*}$-algebras $F_{\lambda}$ and c.p. maps

(4.1)
$A\xrightarrow{\psi_{\lambda}}F_{\lambda}\xrightarrow{\phi_{\lambda}}\mathcal{N}$

such that for all $a\in A$,

(4.2) $\lim_{\lambda}\|\phi_{\lambda}(\psi_{\lambda}(a))-\theta(a)\|_{2,Y}=0.$

Further, we say $(\mathcal{N},Y)$ is _amenable_ if $\mathrm{id}_{\mathcal{N}}$
is tracially nuclear.

As usual, we may restrict to sequences in Definition 4.1 when $A$ is
separable.343434This requires having a uniform bound on the norms of
$\phi_{\lambda}$ and $\psi_{\lambda}$, which is always possible by Proposition
4.4.

Using the canonical inclusion and projection maps, it immediate that
amenability passes to direct sums.

######  Proposition 4.2.

For tracially complete $C^{*}$-algebras $(\mathcal{M},X)$ and
$(\mathcal{N},Y)$ the direct sum $(\mathcal{M}\oplus\mathcal{N},X\oplus Y)$ is
amenable if and only if both $(\mathcal{M},X)$ and $(\mathcal{N},Y)$ are
amenable.

Recall that if $\mathcal{N}$ is a von Neumann algebra, then a c.p. map
$\theta\colon A\to\mathcal{N}$ is _weakly nuclear_ if there are c.p. maps as
in (4.1) such that for all $a\in\mathcal{N}$,
$\phi_{\lambda}(\psi_{\lambda}(a))\rightarrow a$ in the weak∗-topology on
$\mathcal{N}$. When $\tau$ is a faithful normal trace,
$(\mathcal{N},\\{\tau\\})$ is a tracially complete $C^{*}$-algebra, tracial
nuclearity and weak nuclearity agree (and likewise, semidiscreteness of
$\mathcal{N}$ agrees with amenability of $(\mathcal{N},\\{\tau\\}$). We defer
the proof to Proposition 4.5 as we first need some prerequisite results.

The following standard lemma allows us to reduce problems about tracial
nuclearity to the unital case. We use $A^{\dagger}$ for the forced unitisation
of $A$; i.e. when $A$ is unital, we add a new unit to form $A^{\dagger}\cong
A\oplus\mathbb{C}$.

######  Lemma 4.3.

Suppose $A$ is a $C^{*}$-algebra, $(\mathcal{N},Y)$ is a tracially complete
$C^{*}$-algebra, and $\theta\colon A\rightarrow\mathcal{N}$ is a c.p.c. map.
Then $\theta$ is tracially nuclear if and only if the unitisation
$\theta^{\dagger}\colon A^{\dagger}\rightarrow\mathcal{N}$ is also tracially
nuclear.

###### Proof.

First note that $\theta$ factorises as the inclusion $A\to A^{\dagger}$
followed by $\theta^{\dagger}$, so that tracial nuclearity of
$\theta^{\dagger}$ implies that of $\theta$.

Conversely, suppose that $\theta$ is tracially nuclear. Let $(e_{\lambda})$ be
an approximate unit for $A$ consisting of positive contractions and define
$\theta_{\lambda}\colon A^{\dagger}\rightarrow\mathcal{N}$ by

(4.3) $\theta_{\lambda}(a+\alpha
1_{A^{\dagger}})\coloneqq\theta(e_{\lambda}ae_{\lambda})+\alpha
1_{\mathcal{N}}.$

Then $\theta_{\lambda}$ is unital. Further, $\theta_{\lambda}$ is tracially
nuclear as it is the sum of the tracially nuclear maps

(4.4) $\displaystyle a+\alpha 1_{A^{\dagger}}$
$\displaystyle\mapsto\theta(e_{\lambda}(a+\alpha 1_{A^{\dagger}})e_{\lambda})$
and (4.5) $\displaystyle a+\alpha 1_{A^{\dagger}}$
$\displaystyle\mapsto\alpha(1_{\mathcal{N}}-\theta(e_{\lambda}^{2}))$

Further, since each $\theta_{\lambda}$ is unital and
$\|\theta_{\lambda}(a)-\theta(a)\|\rightarrow 0$ for all $a\in A$, we have
$\|\theta_{\lambda}(a)-\theta^{\dagger}(a)\|\rightarrow 0$ for all $a\in
A^{\dagger}$. As each $\theta_{\lambda}$ is tracially nuclear, so is $\theta$.
∎

As with the analogous versions of the completely positive approximation
property for both $C^{*}$-algebras and von Neumann algebras, we can arrange
for a uniform bound on the norms of the $\phi_{\lambda}$ and $\psi_{\lambda}$
in Definition 4.1. The proof here follows the proof of the von Neumann
algebraic version in [15, Proposition 3.8.2] taking care to avoid the use of
Borel functional calculus, which, in general, does not exist in tracially
complete $C^{*}$-algebras.

######  Proposition 4.4.

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

(4.6)
$A\xrightarrow{\psi_{\lambda}}F_{\lambda}\xrightarrow{\phi_{\lambda}}\mathcal{N}$

as in Definition 4.1 with $\|\psi_{\lambda}\|\leq\|\theta\|$ and
$\phi_{\lambda}(1_{F_{\lambda}})=1_{\mathcal{N}}$. Further, if $A$ and
$\theta$ are unital, we may arrange each $\psi_{\lambda}$ to be unital.

###### Proof.

Rescaling $\theta$, we may assume that $\|\theta\|\leq 1$. Then by adding a
unit to $A$ and using Lemma 4.3, we may assume $A$ and $\theta$ are unital. We
will construct u.c.p. maps $\phi_{\lambda}$ and $\psi_{\lambda}$ as in (4.6)
approximately factorising $\theta$.

Since $\theta$ is assumed to be tracially nuclear, there are nets of finite
dimensional $C^{*}$-algebras $F_{\lambda}$ and c.p. maps

(4.7)
$A\xrightarrow{\psi_{\lambda}^{\prime\prime}}F_{\lambda}\xrightarrow{\phi_{\lambda}^{\prime\prime\prime}}\mathcal{N}$

such that for all $a\in A$,

(4.8)
$\lim_{\lambda}\|\phi^{\prime\prime\prime}_{\lambda}(\psi^{\prime\prime}_{\lambda}(a))-\theta(a)\|_{2,Y}=0.$

By [15, Lemma 2.2.5], there is a u.c.p. map $\psi_{\lambda}^{\prime}\colon
A\rightarrow F_{\lambda}$ such that

(4.9)
$\psi_{\lambda}^{\prime\prime}(a)=\psi_{\lambda}^{\prime\prime}(1_{A})^{1/2}\psi_{\lambda}^{\prime}(a)\psi_{\lambda}^{\prime\prime}(1_{A})^{1/2},\qquad
a\in A.$

Define $\phi_{\lambda}^{\prime\prime}\colon F_{\lambda}\rightarrow\mathcal{N}$
by

(4.10)
$\phi_{\lambda}^{\prime\prime}(x)\coloneqq\phi_{\lambda}^{\prime\prime\prime}\big{(}\psi_{\lambda}^{\prime\prime}(1_{A})^{1/2}x\psi_{\lambda}^{\prime\prime}(1_{A})^{1/2}\big{)},\qquad
x\in F_{\lambda},$

and note that
$\phi_{\lambda}^{\prime\prime}\circ\psi_{\lambda}^{\prime}=\phi_{\lambda}^{\prime\prime\prime}\circ\psi_{\lambda}^{\prime\prime}$.
Therefore, we have

(4.11)
$\lim_{\lambda}\|\phi_{\lambda}^{\prime\prime}(\psi_{\lambda}^{\prime}(a))-\theta(a)\|_{2,Y}=0,\qquad
a\in A.$

Define a continuous function $f\colon\mathbb{R}\rightarrow[0,1]$ by

(4.12) $f(t)\coloneqq\begin{cases}1,&t\leq 1;\\\ 2-t,&1<t<2;\\\ 0,&t\geq
2.\end{cases}$

Let $b_{\lambda}\coloneqq f(\phi_{\lambda}^{\prime\prime}(1_{F_{\lambda}}))$
and define $\phi_{\lambda}^{\prime}\colon F_{\lambda}\rightarrow\mathcal{N}$
by

(4.13) $\phi_{\lambda}^{\prime}(x)\coloneqq
b_{\lambda}\phi^{\prime\prime}_{\lambda}(x)b_{\lambda},\qquad x\in
F_{\lambda}.$

By elementary calculus, $0\leq tf(t)^{2}\leq 1$ for all $t\in[0,\infty)$.
Therefore, $\|\phi_{\lambda}^{\prime}(1_{F_{\lambda}})\|\leq 1$, and
$\phi_{\lambda}^{\prime}$ is a c.p.c. map. Also, for all $t\in\mathbb{R}$, we
have $0\leq 1-f(t)\leq|t-1|$, and so

(4.14) $0\leq
1_{\mathcal{N}}-b_{\lambda}\leq|\phi^{\prime\prime}_{\lambda}(1_{F_{\lambda}})-1_{\mathcal{N}}|.$

Since $\theta$ and $\psi_{\lambda}^{\prime}$ are unital, (4.11) and (4.14)
imply

(4.15) $\lim_{\lambda}\|1_{\mathcal{N}}-b_{\lambda}\|_{2,Y}=0.$

Fix $a\in A$. Since the elements $b_{\lambda}$ are contractions, we have

(4.16)
$\|\phi_{\lambda}^{\prime}(\psi_{\lambda}^{\prime}(a))-b_{\lambda}\theta(a)b_{\lambda}\|_{2,Y}\leq\|\phi^{\prime\prime}_{\lambda}(\psi^{\prime}_{\lambda}(a))-\theta(a)\|_{2,Y},$

so
$\lim_{\lambda}\|\phi^{\prime}_{\lambda}(\psi^{\prime}_{\lambda}(a))-b_{\lambda}\theta(a)b_{\lambda}\|_{2,Y}=0$
by (4.11). Moreover, as multiplication in $\mathcal{N}$ is
$\|\cdot\|_{2,Y}$-continuous on $\|\cdot\|$-bounded sets, we have

(4.17)
$\lim_{\lambda}\|b_{\lambda}\theta(a)b_{\lambda}-\theta(a)\|_{2,Y}=0,\qquad
a\in A,$

by (4.15). Therefore,

(4.18)
$\lim_{\lambda}\|\phi_{\lambda}^{\prime}(\psi_{\lambda}^{\prime}(a))-\theta(a)\|_{2,Y}=0,\qquad
a\in A.$

While $\psi^{\prime}_{\lambda}$ was arranged to be unital, the map
$\phi^{\prime}_{\lambda}$ need not be. We modify the maps once more to correct
this. Let $\rho$ be a state on $A$ and define $\psi_{\lambda}\colon
A\rightarrow F_{\lambda}\oplus\mathbb{C}$ by
$\psi_{\lambda}(a)\coloneqq(\psi^{\prime}_{\lambda}(a),\rho(a))$. Then
$\psi_{\lambda}$ is u.c.p. as both $\psi^{\prime}_{\lambda}$ and $\rho$ are.
Also, define $\phi_{\lambda}\colon
F_{\lambda}\oplus\mathbb{C}\rightarrow\mathcal{N}$ by

(4.19)
$\phi_{\lambda}(x,\alpha)\coloneqq\phi_{\lambda}^{\prime}(x)+\alpha(1_{\mathcal{N}}-\phi_{\lambda}^{\prime}(1_{F_{\lambda}}))$

Then $\phi_{\lambda}$ is u.c.p. as $\phi^{\prime}_{\lambda}$ is c.p.c. By
(4.18), to see that

(4.20)
$\lim_{\lambda}\|\phi_{\lambda}(\psi_{\lambda}(a))-\theta(a)\|_{2,Y}=0,\qquad
a\in A,$

it suffices to show that
$\lim_{\lambda}\|1_{\mathcal{N}}-\phi_{\lambda}(1_{F_{\lambda}})\|_{2,Y}=0$.
However, this follows immediately from taking $a\coloneqq 1_{A}$ in (4.18)
using that the maps $\psi_{\lambda}^{\prime}$ and $\theta$ are both unital. ∎

We now return to the promised connection between tracial and weak nuclearity.

######  Proposition 4.5.

Let $A$ be a $C^{*}$-algebra and $(\mathcal{N},\tau)$ be a tracial von Neumann
algebra. A c.p. map $\theta\colon A\to(\mathcal{N},\\{\tau\\})$ is tracially
nuclear if and only if it is weakly nuclear. In particular,
$(\mathcal{N},\tau)$ is semidiscrete as a tracial von Neumann algebra if and
only if $(\mathcal{N},\\{\tau\\})$ is amenable as a tracially complete
$C^{*}$-algebra.

###### Proof.

The key observation is that the strong operator topology on the operator norm
unit ball of $\mathcal{N}$ coincides with the topology induced by
$\|\cdot\|_{2,\tau}$ (see [7, Proposition III.2.2.17], for example). From
here, assume that $\theta$ is weakly nuclear. By scaling $\theta$, we may
assume $\|\theta\|\leq 1$. Then, as in the proof of Lemma 4.3, the unitisation
$\theta^{\dagger}\colon A^{\dagger}\rightarrow\mathcal{N}$ is weakly nuclear.
Hence we may assume that $A$ and $\theta$ are unital. By [15, Proposition
3.8.2], there are nets of finite dimensional $C^{*}$-algebras and c.p.c. maps

(4.21)
$A\xrightarrow{\psi_{\lambda}}F_{\lambda}\xrightarrow{\phi_{\lambda}}\mathcal{N}$

such that $\phi_{\lambda}(\psi_{\lambda}(a))\rightarrow a$ weak∗, and in
particular in the weak operator topology for all $a\in A$. Since the set of
c.p.c. maps $A\rightarrow\mathcal{N}$ which admit c.p.c. factorisations
through a finite dimensional $C^{*}$-algebra form a convex set ([15, Lemma
2.3.6 and Remark 2.3.7]), the Hahn–Banach theorem, in the form of [15, Lemma
3.8.1], shows that there is a net of c.p.c. maps $\theta_{\lambda}\colon
A\rightarrow\mathcal{N}$ such that $\theta_{\lambda}(a)\rightarrow\theta(a)$
in the strong-operator topology for all $a\in A$, and each $\theta_{\lambda}$
admits a c.p.c. factorisation through a finite dimensional $C^{*}$-algebra. As
each $\theta_{\lambda}$ is contractive, this implies
$\|\theta(a)-\theta_{\lambda}(a)\|_{2,\tau}\rightarrow 0$ for all $a\in A$,
and hence $\theta$ is tracially nuclear.

Conversely, suppose that $\theta$ is tracially nuclear. Again, after scaling
$\theta$, we may assume that $\theta$ is contractive. By Proposition 4.4,
there are nets of finite dimensional $C^{*}$-algebras and c.p.c. maps

(4.22)
$A\xrightarrow{\psi_{\lambda}}F_{\lambda}\xrightarrow{\phi_{\lambda}}\mathcal{N}$

such that $\|\theta(a)-\phi_{\lambda}(\psi_{\lambda}(a))\|_{2,\tau}\rightarrow
0$ for all $a\in A$. Then $\phi_{\lambda}(\psi_{\lambda}(a))\rightarrow a$ in
the strong operator topology, and hence in the weak operator topology. As the
weak operator topology and weak∗ topology on $\mathcal{N}$ agree on bounded
sets, the convergence holds weak∗, and this implies that $\theta$ is weakly
nuclear. ∎

It is known that if $A$ is an exact $C^{*}$-algebra and $\mathcal{N}$ is a von
Neumann algebra, then any weakly nuclear map $A\rightarrow\mathcal{N}$ is
nuclear – see [59, Remark 3.4] (as observed there, this characterises
exactness of $A$). This suggests the following question, which we have been
unable to answer even with the additional assumptions that $(\mathcal{N},Y)$
is factorial and has CPoU. The difficulty is that the passage from weak
nuclearity to nuclearity involves taking point-weak∗-limit points.353535The
proof in [59, Remark 3.4] uses Kirchberg’s $\mathcal{O}_{2}$-embedding
theorem, but this can be avoided using the following slight variation of their
proof. Let $\theta\colon A\rightarrow\mathcal{N}$ be weakly nuclear. Fix nets
of finite dimensional $C^{*}$-algebras $F_{\lambda}$ and c.p.c. maps
$\psi_{\lambda}\colon A\rightarrow F_{\lambda}$ and $\phi_{\lambda}\colon
F_{\lambda}\rightarrow\mathcal{N}$ such that
$\phi_{\lambda}\circ\psi_{\lambda}$ converges to $\theta$ point-weak∗. Let
$\pi\colon A\rightarrow\mathcal{B}(\mathcal{H})$ be a faithful representation
of $A$ and use Arveson’s extension theorem to find c.p.c. maps
$\tilde{\psi}_{\lambda}\colon\mathcal{B}(\mathcal{H})\rightarrow F_{\lambda}$
with $\tilde{\psi}_{\lambda}\circ\pi=\psi_{\lambda}$. Let
$\rho\colon\mathcal{B}(\mathcal{H})\rightarrow\mathcal{N}$ be a point-weak∗
limit point of $\phi\circ\tilde{\psi}_{\lambda}$ and note that
$\theta=\rho\circ\pi$. As $A$ is exact, $\pi$ is nuclear, and hence so is
$\theta$. We do not have the required compactness for this in the setting of
tracially complete $C^{*}$-algebras.

######  Question 4.6.

Suppose $A$ is an exact $C^{*}$-algebra and $(\mathcal{N},Y)$ is a tracially
complete $C^{*}$-algebra. Is every tracially nuclear map
$A\rightarrow\mathcal{N}$ nuclear?

###  4.2. Fibrewise amenability

We work towards showing that amenability of tracially complete
$C^{*}$-algebras can be detected in its tracial von Neumann algebra
completions. The basic strategy is to use a convexity argument similar to the
one in the fundamental result from [34, 24, 25] stating that if $A$ is a
$C^{*}$-algebra such that $A^{**}$ is semidiscrete, then $A$ is nuclear (see
[15, Proposition 2.3.8], for example).

We set up the convexity argument in a somewhat general setting for later use
in Section 6.4. Both the statement and proof are abstracted from an argument
in the proof of Ozawa’s [86, Theorem 3] computing the centre of tracial
completions of $C^{*}$-algebras. A similar Hahn–Banach argument appears in the
proof of (8)$\Rightarrow$(1) in [48, Theorem 2.5].

######  Lemma 4.7.

Let $\mathcal{C}$ be a convex subset of a real vector space and let
$(\mathcal{M},X)$ be a tracially complete $C^{*}$-algebra. Suppose we are
given $m\in\mathbb{N}$ and a finite collection
$f_{1},\dots,f_{m}\colon\mathcal{C}\rightarrow\mathcal{M}$ of affine functions
such that for all $\tau\in X$ and $\epsilon>0$, there is
$c_{\tau}\in\mathcal{C}$ such that

(4.23) $\displaystyle\max_{1\leq i\leq m}\|f_{i}(c_{\tau})\|_{2,\tau}$
$\displaystyle<\epsilon.$ Then for all $\epsilon>0$, there is
$c\in\mathcal{C}$ such that (4.24) $\displaystyle\max_{1\leq i\leq
m}\|f_{i}(c)\|_{2,X}$ $\displaystyle<\epsilon.$

###### Proof.

Let $\Lambda$ be the directed set of pairs $(\mathcal{T},\epsilon)$ where
$\mathcal{T}\subseteq X$ is a non-empty finite set and $\epsilon>0$, equipped
with the ordering
$(\mathcal{T}_{1},\epsilon_{1})\leq(\mathcal{T}_{2},\epsilon_{2})$ if and only
if $\mathcal{T}_{1}\subseteq\mathcal{T}_{2}$ and
$\epsilon_{1}\geq\epsilon_{2}$.

Fix $\lambda\coloneqq(\mathcal{T}_{\lambda},\epsilon_{\lambda})\in\Lambda$ and
let
$\sigma_{\lambda}\coloneqq|\mathcal{T}_{\lambda}|^{-1}\sum_{\tau\in\mathcal{T}_{\lambda}}\tau\in
X$ denote the average of the traces in $\mathcal{T}_{\lambda}$. By assumption,
there is $c_{\lambda}\in\mathcal{C}$ such that

(4.25) $\displaystyle|\mathcal{T}_{\lambda}|^{1/2}\max_{1\leq i\leq
m}\|f_{i}(c_{\lambda})\|_{2,\sigma_{\lambda}}$
$\displaystyle<\epsilon_{\lambda}.$ It follows from the definition of
$\sigma_{\lambda}$ that (4.26)
$\displaystyle\max_{\tau\in\mathcal{T}_{\lambda}}\max_{1\leq i\leq
m}\|f_{i}(c_{\lambda})\|_{2,\tau}$ $\displaystyle<\epsilon_{\lambda}.$

For $\lambda\in\Lambda$ and $i=1,\ldots,m$, define $h_{i,\lambda}\colon
X\to\mathbb{R}$ by

(4.27) $h_{i,\lambda}(\tau)\coloneqq\|f_{i}(c_{\lambda})\|_{2,\tau}^{2}$

and note that $h_{i,\lambda}\in{\rm Aff}(X)$. For all $i=1,\ldots,m$,
$h_{i,\lambda}\rightarrow 0$ pointwise on $X$ by (4.26), and hence Proposition
2.1 implies $h_{i,\lambda}\rightarrow 0$ weakly. View
$h_{\lambda}\coloneqq(h_{i,\lambda},\ldots,h_{m,\lambda})$ as a net in the
Banach space ${\rm Aff}(X)^{\oplus m}$, and note that $h_{\lambda}\rightarrow
0$ weakly.

Fix $\epsilon>0$. By the Hahn–Banach theorem, there are $l\in\mathbb{N}$,
$\lambda_{1},\dots,\lambda_{l}\in\Lambda$, and real numbers
$\alpha_{1},\ldots,\alpha_{l}\geq 0$ such that $\sum_{k=1}^{l}\alpha_{k}=1$
and

(4.28) $\max_{1\leq i\leq
m}\Big{\|}\sum_{k=1}^{l}\alpha_{k}h_{i,\lambda_{k}}\Big{\|}_{\infty}<\epsilon^{2}.$

Define $c\coloneqq\sum_{k=1}^{l}\alpha_{k}c_{\lambda_{k}}\in\mathcal{C}$. For
$i=1,\ldots,m$, using the triangle inequality and the Cauchy–Schwarz
inequality, we have

(4.29)
$\begin{split}\|f_{i}(c)\|_{2,X}&=\Big{\|}\sum_{k=1}^{l}\alpha_{k}f_{i}(c_{\lambda_{k}})\Big{\|}_{2,X}\\\
&\leq\sum_{k=1}^{l}\alpha_{k}\|f_{i}(c_{\lambda_{k}})\|_{2,X}\\\
&=\sum_{k=1}^{l}\big{(}\alpha_{k}^{1/2})\big{(}\alpha_{k}^{1/2}\|f_{i}(c_{\lambda_{k}})\|_{2,X}\big{)}\\\
&\leq\Big{(}\sum_{k=1}^{l}\alpha_{k}\Big{)}^{1/2}\Big{(}\sum_{k=1}^{l}\alpha_{k}\|f_{i}(c_{\lambda_{k}})\|_{2,X}^{2}\Big{)}^{1/2}\\\
&=\Big{(}\sum_{k=1}^{l}\alpha_{k}\|f_{i}(c_{\lambda_{k}})\|_{2,X}^{2}\Big{)}^{1/2}\end{split}$

Then for $i=1,\ldots,m$ and $\tau\in X$, we have

(4.30)
$\|f_{i}(c)\|_{2,\tau}^{2}\stackrel{{\scriptstyle\eqref{eq:hilambdaDef},\eqref{eq:convexity-
bound}}}{{\leq}}\sum_{k=1}^{l}\alpha_{k}h_{i,\lambda_{k}}(\tau)\stackrel{{\scriptstyle\eqref{eq:affine-
selection1}}}{{<}}\epsilon^{2}.$

Hence $\|f_{i}(c)\|_{2,X}<\epsilon$ for all $i=1,\ldots,m$. ∎

The following lemma is standard. For example, it follows from the proof of
[59, Lemma 1.1] by quoting the Choi–Effros lifting theorem in place of the
projectivity of order zero maps in the last paragraph.

######  Lemma 4.8.

If $F$ and $B$ are $C^{*}$-algebras with $F$ finite dimensional, $\pi\colon
B\rightarrow\mathcal{B}(H)$ is a ∗-homomorphism, and $\phi\colon
F\rightarrow\mathcal{\pi}(B)^{\prime\prime}$ is a c.p. map, then there is a
net of c.p. maps $\phi_{\lambda}\colon F\rightarrow B$ such that
$\|\phi_{\lambda}\|\leq\|\phi\|$ for all $\lambda$ and

(4.31) $\phi(b)=\text{\rm
strong${}^{*}$-}\lim_{\lambda}\pi(\phi_{i}(b)),\qquad b\in B.$

Further, if $\phi$ is unital, we may arrange for each $\phi_{\lambda}$ to be
unital.

We are now in a position to give the ‘fibrewise’ characterisation of tracially
nuclear ∗-homomorphisms (Theorem 4.9) from which Theorem 1.2 follows
immediately (by taking $A\coloneqq\mathcal{M}$ and
$\theta\coloneqq\mathrm{id}_{\mathcal{M}}$). Note that the following theorem
also implies the first part of Theorem D from the overview.

The equivalence of (ii) and (iii) in Theorem 4.9 is essentially obtained by
Brown in [12, Theorem 3.2.2] (which is a variation on Connes’ theorem), so the
main implication we need to show is (ii) implies (i). This will follow from
the Hahn–Banach argument of Lemma 4.7.

######  Theorem 4.9.

Let $A$ be a $C^{*}$-algebra, let $(\mathcal{M},X)$ be a tracially complete
$C^{*}$-algebra, and let $\theta\colon A\rightarrow\mathcal{M}$ be a c.p. map.
The following are equivalent:

  1. (i)

$\theta$ is tracially nuclear;

  2. (ii)

for all $\tau\in X$, $\pi_{\tau}\circ\theta\colon
A\rightarrow\pi_{\tau}(\mathcal{M})^{\prime\prime}$ is weakly nuclear.

If $\theta$ is a ∗-homomorphism then these are also equivalent to:

  1. (iii)

for every $\tau\in X$ with $\tau\circ\theta\neq 0$, the trace
$\|\tau\circ\theta\|^{-1}\cdot\tau\circ\theta\in T(A)$ is uniformly amenable
in the sense of [12, Definition 3.2.1].363636Brown implicitly only defines
uniformly amenable for traces on unital $C^{*}$-algebras in [15] (the
definition there requires u.c.p. maps out of $A$). We extend this definition
to the non-unital case, by saying that $\tau\in T(A)$ is uniformly amenable if
its unitisation, $\tau^{\dagger}\in T(A^{\dagger})$, is uniformly amenable.

###### Proof.

To see (i) implies (ii), note that if $\theta$ is tracially nuclear, then for
all $\tau\in X$, $\pi_{\tau}\circ\theta$ is tracially nuclear as a map into
the tracial von Neumann algebra
$(\pi_{\tau}(\mathcal{M})^{\prime\prime},\tau)$. As noted in the remarks
following Definition 4.1, for maps into tracial von Neumann algebras, weak and
tracial nuclearity are equivalent, so (ii) follows.

When $\theta$ is a ∗-homomorphism, the equivalence of (ii) and (iii) can be
reduced to the case that $A$ and $\theta$ are unital by adding a unit to $A$
and using Lemma 4.3. For $\tau\in X$, since there is a normal trace-preserving
conditional expectation
$\pi_{\tau}(\mathcal{M})^{\prime\prime}\rightarrow\pi_{\tau}(\theta(A))^{\prime\prime}$
(see [15, Lemma 1.5.10], for example),
$\pi_{\tau\circ\theta}=\pi_{\tau}\circ\theta$ is weakly nuclear when viewed as
a map into $\pi_{\tau}(\mathcal{M})^{\prime\prime}$ if and only if it is
weakly nuclear when viewed as a map into
$\pi_{\tau\circ\theta}(A)^{\prime\prime}=\pi_{\tau}(\theta(A))^{\prime\prime}$.
The equivalence of (ii) and (iii) then follows from the equivalence of (1) and
(6) in [12, Theorem 3.2.2].

It remains to show that (ii) implies (i). Assume that $\theta$ is contractive.
Fix a finite set $\mathcal{F}\subseteq A$ and $\epsilon>0$. Let $\mathcal{C}$
denote the set of all c.p. maps $A\rightarrow\mathcal{M}$ which factor by c.p.
maps through a finite dimensional $C^{*}$-algebra and note that $\mathcal{C}$
is convex (cf. [15, Lemma 2.3.6]). We will apply Lemma 4.7 to the affine
functions

(4.32)
$f_{a}\colon\mathcal{C}\rightarrow\mathcal{M}\colon\eta\mapsto\eta(a)-\theta(a),\qquad
a\in\mathcal{F}.$

For $\tau\in X$, $\pi_{\tau}\circ\theta$ is weakly nuclear by (ii), so there
are a finite dimensional $C^{*}$-algebra $F_{\tau}$ and c.p.c. maps

(4.33)
$A\xrightarrow{\psi_{\tau}}F_{\tau}\xrightarrow{\bar{\phi}_{\tau}}\pi_{\tau}(\mathcal{M})^{\prime\prime}$

such that

(4.34)
$\|\bar{\phi}_{\tau}(\psi_{\tau}(a))-\pi_{\tau}(\theta(a))\|_{2,\tau}<\epsilon,\qquad
a\in\mathcal{F}.$

By Lemma 4.8, there is then a c.p.c. map $\phi_{\tau}\colon
F_{\tau}\rightarrow\mathcal{M}$ such that

(4.35) $\|\phi_{\tau}(\psi_{\tau}(a))-\theta(a)\|_{2,\tau}<\epsilon,\qquad
a\in\mathcal{F}.$

Then $\eta_{\tau}\coloneqq\phi_{\tau}\circ\psi_{\tau}\in\mathcal{C}$ and
$\|f_{a}(\eta_{\tau})\|_{2,\tau}<\epsilon$ for all $a\in\mathcal{F}$. By Lemma
4.7, there is $\eta\in\mathcal{C}$ such that $\|f_{a}(\eta)\|_{2,X}<\epsilon$
for all $a\in\mathcal{F}$. Unpacking notation, $\eta$ is a c.p. map factoring
through a finite dimensional $C^{*}$-algebra, and

(4.36) $\|\eta(a)-\theta(a)\|_{2,X}<\epsilon,\qquad a\in\mathcal{F},$

so $\theta$ is tracially nuclear. ∎

Combining the previous result with Connes’ theorem gives a tracially complete
analogue of the fact that a von Neumann algebra completion of a nuclear
$C^{*}$-algebra is semidiscrete. Let $T_{\rm am}(A)\subseteq T(A)$ denote the
set of amenable traces on $A$.

######  Corollary 4.10.

If $A$ is an exact $C^{*}$-algebra with $T_{\rm am}(A)$ compact,373737Note
that $T_{\rm am}(A)$ is closed in $T(A)$ by [12, Proposition 3.5.1], so the
compactness is automatic if $A$ is unital. then the tracial completion of $A$
with respect to $T_{\mathrm{am}}(A)$ is an amenable factorial tracially
complete $C^{*}$-algebra. In particular, if $A$ is a nuclear $C^{*}$-algebra
with $T(A)$ compact, then the tracial completion of $A$ with respect to $T(A)$
is an amenable factorial tracially complete $C^{*}$-algebra.

###### Proof.

By [66, Lemma 3.4], $T_{\rm am}(A)$ is a closed face in $T(A)$. Therefore, the
tracial completion of $A$ with respect to $T_{\mathrm{am}}(A)$ is a factorial
tracially complete $C^{*}$-algebra by Proposition 3.23(iv).

As $A$ is exact, $A$ is locally reflexive by [65, Remark (11)], and hence by
[12, Theorem 4.3.3], all amenable traces on $A$ are uniformly amenable. By the
equivalence of (1) and (5) in [12, Theorem 3.2.2], it follows that
$\pi_{\tau}(A)^{\prime\prime}$ is semidiscrete for $\tau\in T_{\rm am}(A)$.
Using Proposition 3.23(vi), we also have that $\pi_{\tau}(\overline{A}^{T_{\rm
am}(A)})^{\prime\prime}$ is semidiscrete for all $\tau\in T_{\rm am}(A)$. By
Theorem 4.9, this implies that $\big{(}\overline{A}^{T_{\rm am}(A)},T_{\rm
am}(A)\big{)}$ is amenable. This proves the first sentence of the theorem. The
second sentence follows since $T(A)=T_{\rm am}(A)$ when $A$ is nuclear (see
[12, Theorem 4.2.1], for example). ∎

Without exactness, it need not be the case that all amenable traces are
uniformly amenable. Indeed, given a sequence $(k_{n})$ of natural numbers
converging to $\infty$, let $A\coloneqq\prod M_{k_{n}}$. For a free
ultrafilter $\omega$ on $\mathbb{N}$, the trace
$\tau_{\omega}((x_{n}))\coloneqq\lim_{n\to\omega}\mathrm{tr}_{k_{n}}(x_{n})$
is an amenable trace which is not uniformly amenable.383838This is because the
tracial von Neumann ultraproduct $\prod^{\omega}M_{n_{k}}$ is not amenable, as
it contains a copy of $L(F_{2})$.. This observation leads to the following
characterisation of those finite von Neumann algebras which are amenable as
tracially complete $C^{*}$-algebras over their trace space.

######  Proposition 4.11.

Let $\mathcal{M}$ be a semidiscrete finite von Neumann algebra. Then
$\big{(}\mathcal{M},T(\mathcal{M})\big{)}$ is amenable as a tracially complete
$C^{*}$-algebra if and only if its type II1 summand has only finitely many
extremal traces and it has no type ${\rm I}_{n}$ summand for sufficiently
large $n$.

###### Proof.

Suppose $\mathcal{M}$ satisfies the stated condition. Then the type I part
$\mathcal{M}_{\mathrm{I}}$ of $\mathcal{M}$ is a finite direct sum of matrices
over abelian $C^{*}$-algebras (see [104, Theorem V.1.27], for example) and
hence nuclear so has the completely positive approximation property in norm.
Hence $\big{(}\mathcal{M}_{\mathrm{I}},T(\mathcal{M}_{\mathrm{I}})\big{)}$ is
amenable as a tracially complete $C^{*}$-algebra. Let $e_{1},\dots,e_{m}$
denote the minimal central projections of the type II1 part $\mathcal{M}_{\rm
II}$ of $\mathcal{M}$ so that each
$\mathcal{M}e_{i}=\mathcal{M}_{\mathrm{II}}e_{i}$ is a semidiscrete factor,
whence $\big{(}\mathcal{M}e_{i},T(\mathcal{M}e_{i})\big{)}$ is an amenable
tracially complete $C^{*}$-algebra (by Proposition 4.5). Since the finite
direct sum of amenable tracially complete $C^{*}$-algebras is amenable
(Proposition 4.2), it follows that $\big{(}\mathcal{M},T(\mathcal{M})\big{)}$
is amenable.

Conversely, supposing the condition does not hold, we can find a sequence
$(n_{k})$ converging to $\infty$ and orthogonal central projections $(e_{k})$
in $\mathcal{M}$ such that there is a unital embedding
$M_{n_{k}}\to\mathcal{\mathcal{M}}e_{k}$.393939This uses the fact that for any
II1 von Neumann algebra $\mathcal{N}$, and any $n\in\mathbb{N}$, one can find
a unital embedding $M_{n}\to\mathcal{N}$. This goes back to Murray and von
Neumann. The proof for $n=2$ from [104, Proposition V.1.35] can be readily
modified to cover general $n$. In this way, we have an embedding of the
infinite product $C^{*}$-algebra $\prod M_{n_{k}}$ in $\mathcal{M}$. For each
$k$, let $\tau_{k}$ be a trace on $\mathcal{M}$ with $\tau_{k}(e_{k})=1$ and
for a free ultrafilter $\omega$ on $\mathbb{N}$, let
$\tau_{\omega}\coloneqq\lim_{k\to\omega}\tau_{k}$, which exists by compactness
of the trace space of $\prod M_{n_{k}}$. Then $\tau_{\omega}$ restricts to the
trace $(x_{n})\mapsto\lim_{n\to\omega}\mathrm{tr}_{n_{k}}(x_{n})$ on $\prod
M_{n_{k}}$. This restricted trace is not uniformly amenable (see Footnote 38).
As uniform amenability of traces passes to subalgebras (this is immediate from
the approximation form of the definition in [12, Definition 3.2.1]),
$\tau_{\omega}$ is not uniformly amenable, and hence
$\big{(}\mathcal{M},T(\mathcal{M})\big{)}$ is not amenable.

∎

In particular, the tracially complete $C^{*}$-algebra
$\big{(}\ell^{\infty}(\mathcal{R}),T(\ell^{\infty}(\mathcal{R}))\big{)}$ is
not amenable even though $\ell^{\infty}(\mathcal{R})$ is semidiscrete as a von
Neumann algebra.

We end this section by applying the characterisation of tracial nuclearity, to
show that it can be tested on $C^{*}$-subalgebras which are dense in the
uniform 2-norm. A naive argument (showing that the witnesses of tracial
nuclearity for $\phi|_{A}$ extend to witnesses of tracial nuclearity for
$\phi$) is not possible, as one cannot a priori assume uniform
$\|\cdot\|_{2,X}$-boundedness of the c.p.c. approximations in the definition
of tracial nuclearity.

######  Lemma 4.12.

Suppose $(\mathcal{M},X)$ and $(\mathcal{N},Y)$ are tracially complete
$C^{*}$-algebras and $\phi\colon(\mathcal{M},X)\rightarrow(\mathcal{N},Y)$ is
a morphism. If $A\subseteq\mathcal{\mathcal{M}}$ is a $\|\cdot\|_{2,X}$-dense
$C^{*}$-subalgebra, then $\phi$ is tracially nuclear if and only if
$\phi|_{A}$ is tracially nuclear.

###### Proof.

For all $\tau\in X$, the inclusion $A\hookrightarrow\mathcal{M}$ induces an
isomorphism
$\pi_{\tau}(A)^{\prime\prime}\rightarrow\pi_{\tau}(\mathcal{M})^{\prime\prime}$
(by combining Corollary 3.29(i) and Proposition 3.23(vi), for example). Since
a trace on a $C^{*}$-algebra is uniformly amenable if and only if it generates
a semidiscrete von Neumann algebra (see [12, Theorem
3.2.2(2)$\Leftrightarrow$(3)]), we have that for each $\tau\in Y$,
$\tau\circ\phi$ is uniformly amenable if and only if $\tau\circ\phi|_{A}$ is
uniformly amenable. The result follows from Theorem 4.9. ∎

##  5\. Reduced products and central sequences

In this section, we define the reduced product
$\prod^{\omega}(\mathcal{M}_{n},X_{n})$ associated with a sequence of
tracially complete $C^{*}$-algebras
$\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ and a free filter
$\omega$ on $\mathbb{N}$.

Aided by the language of reduced products and motivated by the analogous
conditions in von Neumann algebra theory, we introduce two properties of
(factorial) tracially complete $C^{*}$-algebras: the McDuff property and
property $\Gamma$. Both of these properties have been studied in the setting
of $C^{*}$-algebras with their uniform 2-norm in [22, 23], where they are
referred to as the _uniform McDuff property_ and _uniform property $\Gamma$_,
respectively. We view tracially complete $C^{*}$-algebras as the natural
framework to develop these properties, and the definitions given here extend
the definitions from [22, 23] when restricted to tracial completions of
$C^{*}$-algebras with respect to the full trace simplex. Moreover, in the
unique trace case, these properties reduce to the homonymous properties for
II1 factors.

###  5.1. Reduced products

Reduced products provide an algebraic setting for manipulating properties
involving approximations. The most common constructions are ultrapowers with
respect to a free ultrafilter on the natural numbers $\mathbb{N}$ and sequence
algebras consisting of the algebra of bounded sequences modulo the ideal of
$c_{0}$-sequences.

For many basic applications, ultrapowers and sequence algebras can be used
interchangeably, but each has its technical advantages. In settings where
traces are considered, ultrapowers are often more natural, as the resulting
set of limit traces is already convex; on the other hand, working with
sequence algebras allows the reparameterisation argument of [46, Theorem 4.3]
(see Theorem 5.11), which will be used in our classification result in Section
9. In order to allow for both constructions simultaneously, we will work with
reduced products defined with respect to a free filter.

For the remainder of the paper, $\omega$ will denote a free filter on the
natural numbers $\mathbb{N}$. We recall that a filter $\omega$ on the natural
numbers is free if and only if it contains all cofinite sets (see [15,
Appendix A], for example, for a general discussion of filters). The following
selection theorem of Kirchberg will be used frequently. It is most often
stated for ultrafilters, but the result (and proof) is equally valid for
general filters. For the readers convenience, we include the details.

######  Lemma 5.1 (Kirchberg’s $\epsilon$-test, cf. [67, Lemma A.1]).

Let $\omega$ be a free filter on $\mathbb{N}$. Let $(X_{n})_{n=1}^{\infty}$ be
a sequence of non-empty sets and for $k,n\in\mathbb{N}$, let
$f^{(k)}_{n}\colon X_{n}\rightarrow[0,\infty]$ be a function. Define functions
$f^{(k)}\colon X_{1}\times X_{2}\times\cdots\to[0,\infty]$ by

(5.1)
$f^{(k)}(x_{1},x_{2},\dots)\coloneqq\limsup_{n\to\omega}f^{(k)}_{n}(x_{n}).$

If for every $\epsilon>0$ and $k_{0}\in\mathbb{N}$, there exists $x\in
X_{1}\times X_{2}\times\cdots$ such that $f^{(k)}(x)<\epsilon$ for
$k=1,\dots,k_{0}$, then there exists $y\in X_{1}\times X_{2}\times\cdots$ such
that $f^{(k)}(y)=0$ for all $k\in\mathbb{N}$.

###### Proof.

For each $r\in\mathbb{N}$, there exists
$x^{(r)}=(x_{1}^{(r)},x_{2}^{(r)},\dots)\in X_{1}\times X_{2}\times\cdots$
such that $f^{(k)}(x^{(r)})<\frac{1}{r}$ for $k=1,\dots,r$. By (5.1), there
exists $I_{r}\in\omega$ such that $f_{n}^{(k)}(x_{n}^{(r)})<\frac{1}{r}$ for
all $n\in I_{r}$ and $k=1,\dots,r$. As $\omega$ is a free filter we may assume
that $I_{r}\subseteq\\{r,r+1,\dots\\}$.

For each $n\in\mathbb{N}$, if $n$ lies in $\bigcup_{r=1}^{\infty}I_{r}$, then
let $r_{n}\in\mathbb{N}$ be maximal such that $n\in I_{r_{n}}$ (noting that
$n\notin I_{r}$ for $r>n$) and set $y_{n}\coloneqq x_{n}^{(r_{n})}$.
Otherwise, define $y_{n}\in X_{n}$ arbitrarily. Fix $k,r\in\mathbb{N}$ with
$k\leq r$. Then for all $n\in I_{r}$, it follows that $r_{n}\geq r$, and hence

(5.2) $f^{(k)}_{n}(y_{n})<\frac{1}{r_{n}}\leq\frac{1}{r}.$

Thus

(5.3) $f^{(k)}(y)=\limsup_{n\to\omega}f_{n}(y_{n})\leq\sup_{n\in
I_{r}}f_{n}(y_{n})\leq\frac{1}{r}.$

Since this holds for all $r\geq k$, we obtain $f^{(k)}(y)=0$, as required. ∎

We now formally define reduced products.

######  Definition 5.2.

For a sequence $\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ of
tracially complete $C^{*}$-algebras, define a $C^{*}$-algebra404040 Here,
$\prod_{n=1}^{\infty}\mathcal{M}_{n}$ denotes the $\ell^{\infty}$-product.
Using (3.2), it is easy to see that
$\big{\\{}(a_{n})_{n=1}^{\infty}:\lim_{n\to\omega}\|a_{n}\|_{2,X_{n}}=0\big{\\}}$
is an ideal of $\prod_{n=1}^{\infty}\mathcal{M}_{n}$, so this quotient is a
$C^{*}$-algebra.

(5.4)
$\prod^{\omega}\mathcal{M}_{n}\coloneqq\prod_{n=1}^{\infty}\mathcal{M}_{n}\big{/}\big{\\{}(a_{n})_{n=1}^{\infty}:\lim_{n\to\omega}\|a_{n}\|_{2,X_{n}}=0\big{\\}}.$

For every sequence of traces
$(\tau_{n})_{n=1}^{\infty}\in\prod_{n=1}^{\infty}X_{n}$ and every ultrafilter
$\omega^{\prime}$ containing $\omega$, there is a trace defined on
$\prod^{\omega}\mathcal{M}_{n}$ by
$a\mapsto\lim_{n\rightarrow\omega^{\prime}}\tau_{n}(a_{n})$, where
$(a_{n})_{n=1}^{\infty}\in\prod_{n=1}^{\infty}\mathcal{M}_{n}$ is any sequence
representing $a$ – such traces are called _limit traces_. Let
$\sum^{\omega}X_{n}$ be the closed convex hull of the set of limit traces on
$\prod^{\omega}\mathcal{M}_{n}$.

Then the pair

(5.5)
$\prod^{\omega}(\mathcal{M}_{n},X_{n})\coloneqq\Big{(}\prod^{\omega}\mathcal{M}_{n},\sum^{\omega}X_{n}\Big{)}$

is called the _reduced product_ of the sequence
$\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ with respect to
$\omega$ (and the _ultraproduct_ when $\omega$ is an ultrafilter). In the case
when $\omega$ is the Fréchet filter, we write the reduced product as

(5.6)
$\prod^{\infty}(\mathcal{M}_{n},X_{n})\coloneqq\Big{(}\prod^{\infty}\mathcal{M}_{n},\sum^{\infty}X_{n}\Big{)}.$

Our first goal is to prove that the reduced product of a sequence of tracially
complete $C^{*}$-algebras (with respect to a given free filter $\omega$) is
itself a tracially complete $C^{*}$-algebra. Before doing that, we isolate a
useful lemma that will be used frequently in our analysis of reduced products.

######  Lemma 5.3.

Let $\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ be a sequence of
tracially complete $C^{*}$-algebras. If
$a\in\prod^{\omega}(\mathcal{M}_{n},X_{n})$ be represented by the sequence
$(a_{n})_{n=1}^{\infty}\in\prod_{n=1}^{\infty}\mathcal{M}_{n}$, then

(5.7) $\|a\|_{2,\sum^{\omega}X_{n}}=\limsup_{n\to\omega}\|a_{n}\|_{2,X_{n}}.$

###### Proof.

Given a sequence of traces $(\tau_{n})_{n=1}^{\infty}$ and an ultrafilter
$\omega^{\prime}\supseteq\omega$, let $\tau$ be the associated limit trace.
Since $\|a_{n}\|_{2,\tau_{n}}\leq\|a_{n}\|_{2,X_{n}}$ for all
$n\in\mathbb{N}$, we have

(5.8)
$\|a\|_{2,\tau}=\lim_{n\to\omega^{\prime}}\|a_{n}\|_{2,\tau_{n}}\leq\lim_{n\to\omega^{\prime}}\|a_{n}\|_{2,X_{n}}\leq\limsup_{n\to\omega}\|a_{n}\|_{2,X_{n}}.$

Hence
$\|a\|_{2,\sum^{\omega}X_{n}}\leq\limsup_{n\to\omega}\|a_{n}\|_{2,X_{n}}$.

Conversely, let $\omega^{\prime}\supseteq\omega$ be an ultrafilter with

(5.9)
$\lim_{n\to\omega^{\prime}}\|a_{n}\|_{2,X_{n}}=\limsup_{n\to\omega}\|a_{n}\|_{2,X_{n}}.$

For every $n\in\mathbb{N}$, there exists $\tau_{n}\in X_{n}$ with
$\|a_{n}\|_{2,\tau_{n}}>\|a_{n}\|_{2,X_{n}}-2^{-n}$. Let $\tau$ be the limit
trace corresponding to the sequence $(\tau_{n})_{n=1}^{\infty}$ and the
ultrafilter $\omega^{\prime}$. Then

(5.10)
$\|a\|_{2,\sum^{\omega}X_{n}}\geq\|a\|_{2,\tau}=\lim_{n\to\omega^{\prime}}\|a_{n}\|_{2,\tau_{n}}\geq\limsup_{n\to\omega}\|a_{n}\|_{2,X_{n}}.\qed$

We now prove that a reduced product of a sequence of tracially complete
$C^{*}$-algebras is a tracially complete $C^{*}$-algebra.

######  Proposition 5.4 (cf. [23, Lemma 1.6]).

Let $\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ be a sequence of
tracially complete $C^{*}$-algebras. Then
$\prod^{\omega}(\mathcal{M}_{n},X_{n})$ is a tracially complete
$C^{*}$-algebra.

###### Proof.

By Lemma 5.3 and the definition of $\prod^{\omega}\mathcal{M}_{n}$, it is
clear that $\|\cdot\|_{2,\sum^{\omega}X_{n}}$ is a norm. It remains to show
that the $\|\cdot\|$-closed unit ball is
$\|\cdot\|_{2,\sum^{\omega}X_{n}}$-complete.

Let $(a^{(k)})_{k=1}^{\infty}$ be a $\|\cdot\|_{2,\sum^{\omega}X_{n}}$-Cauchy
sequence in the unit ball of $\prod^{\omega}\mathcal{M}_{n}$, and for each
$k\in\mathbb{N}$, fix a sequence $(a_{n}^{(k)})_{n=1}^{\infty}$ of
contractions which represents $a^{(k)}$. Set

(5.11)
$\epsilon^{(k)}\coloneqq\sup\big{\\{}\|a^{(l)}-a^{(l^{\prime})}\|_{2,\sum^{\omega}X_{n}}:l,l^{\prime}\geq
k\big{\\}},$

noting that $\epsilon^{(k)}\rightarrow 0$ as $k\rightarrow\infty$ since
$(a^{(k)})_{k=1}^{\infty}$ is $\|\cdot\|_{2,\sum^{\omega}X_{n}}$-Cauchy.
Define functions $f^{(k)}_{n}\colon\\{b\in\mathcal{M}:\|b\|\leq
1\\}\rightarrow[0,2]$ by

(5.12)
$f^{(k)}_{n}(b)\coloneqq\max\big{\\{}\|b-a^{(k)}_{n}\|_{2,X_{n}}-\epsilon^{(k)},0\big{\\}}.$

For $k_{0}\in\mathbb{N}$ and $k=1,\dots,k_{0}$, since
$\|a^{(k_{0})}-a^{(k)}\|_{2,\sum^{\omega}X_{n}}\leq\epsilon^{(k)}$, we have
$\limsup_{n\rightarrow\omega}f^{(k)}_{n}(a_{n}^{(k_{0})})=0$ by Lemma 5.3.
Therefore, Kirchberg’s $\epsilon$-test (Lemma 5.1) gives a sequence
$(a_{n})_{n=1}^{\infty}$ of contractions representing an element $a$ in the
unit ball of $\prod^{\omega}\mathcal{M}_{n}$ such that
$\limsup_{n\rightarrow\omega}f_{n}^{(k)}(a_{n})=0$ for all $k\in\mathbb{N}$.
By Lemma 5.3 and (5.12), this means that

(5.13) $\|a-a^{(k)}\|_{2,\sum^{\omega}X_{n}}\leq\epsilon^{(k)}\rightarrow 0$

as $k\rightarrow\infty$. Hence the unit ball of
$\prod^{\omega}\mathcal{M}_{n}$ is
$\|\cdot\|_{2,\sum^{\omega}X_{n}}$-complete. ∎

A particularly relevant case of a reduced product is when
$\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ is a constant sequence;
i.e. for some tracially complete $C^{*}$-algebra $(\mathcal{M},X)$, we have
$(\mathcal{M}_{n},X_{n})=(\mathcal{M},X)$ for all $n\in\mathbb{N}$. In this
case, we write

(5.14)
$(\mathcal{M}^{\omega},X^{\omega})\coloneqq\prod^{\omega}(\mathcal{M}_{n},X_{n})$

and call $(\mathcal{M}^{\omega},X^{\omega})$ the _reduced power_ of
$(\mathcal{M},X)$ with respect to the free filter $\omega$, or the
_ultrapower_ if $\omega$ is an ultrafilter. Again, when $\omega$ is the
Fréchet filter, we write $(\mathcal{M}^{\infty},X^{\infty})$ in place of
$(\mathcal{M}^{\omega},X^{\omega})$.

Note that there is a natural embedding
$\iota_{(\mathcal{M},X)}\colon(\mathcal{M},X)\rightarrow(\mathcal{M}^{\omega},X^{\omega})$
of tracially complete $C^{*}$-algebras given by identifying $\mathcal{M}$ with
constant sequences in $\mathcal{M}^{\omega}$. Typically, the map
$\iota_{(\mathcal{M},X)}$ will be suppressed and we will view $\mathcal{M}$ as
a subalgebra of $\mathcal{M}^{\omega}$ – this is the case for example when
considering the central sequence algebra
$\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$.

For a $C^{*}$-algebra $A$ with $T(A)$ compact, we write $A^{\omega}$ for the
reduced power of $A$ in the norm $\|\cdot\|_{2,T(A)}$, which is defined in a
way analogous to Definition 5.2, and we write $T_{\omega}(A)$ for the limit
traces on $A^{\omega}$ induced by $T(A)$. These uniform tracial reduced powers
of $C^{*}$-algebras appear explicitly in connection with the Toms–Winter
conjecture in [23] (working with ultrafilters $\omega$) and in the abstract
approach to classification [16] (working with the Fréchet
filter).414141Various related constructions appeared earlier. A suitable
quotient of the norm ultrapower, which is isomorphic to the tracial
ultrapower, appeared in [68, 110], and all these ideas have their spiritual
origins in Matui and Sato’s work [75, 76]. Uniform tracial ultrapowers of
$W^{*}$-bundles also were used in [10]. In our formalism, the pair
$\big{(}A^{\omega},\overline{\rm co}(T_{\omega}(A))\big{)}$ is a tracially
complete $C^{*}$-algebra and, in fact, if $(\mathcal{M},X)$ is the tracial
completion of $A$ with respect to $T(A)$, then the canonical map
$\alpha_{X}\colon A\rightarrow\mathcal{M}$ induces an isomorphism

(5.15) $\big{(}A^{\omega},\overline{\rm
co}(T_{\omega}(A))\big{)}\rightarrow(\mathcal{M}^{\omega},X^{\omega}),$

defined on representative sequences by
$(a_{n})_{n=1}^{\infty}\mapsto(\alpha_{X}(a_{n}))_{n=1}^{\infty}$. This
follows from a more general result on the compatibility of tracial
completions: Proposition 5.6) below. We first introduce some more notation.

######  Definition 5.5.

Let $(A_{n})_{n=1}^{\infty}$ be a sequence of $C^{*}$-algebras and let
$X_{n}\subseteq T(A_{n})$ be a compact convex set for each $n\geq 1$. We write

(5.16)
$\prod^{\omega}A_{n}\coloneqq\prod_{n=1}^{\infty}A_{n}\big{/}\big{\\{}(a_{n})_{n=1}^{\infty}:\lim_{n\to\omega}\|a_{n}\|_{2,X_{n}}=0\big{\\}}.$

Let $\sum^{\omega}X_{n}$ be the closed convex hull of the limit traces on
$\prod^{\omega}A_{n}$ defined by sequences
$(\tau_{n})_{n=1}^{\infty}\in\prod_{n=1}^{\infty}X_{n}$. In the case of a
constant sequence, say $A_{n}=A$ and $X_{n}=X$, we write
$A^{\omega}=\prod^{\omega}A_{n}$ and $X^{\omega}=\sum^{\omega}X_{n}$.

######  Proposition 5.6.

Let $(A_{n})_{n=1}^{\infty}$ be a sequence of $C^{*}$-algebras, let
$X_{n}\subseteq T(A_{n})$ be a compact convex set for each $n\geq 1$, let
$(\mathcal{M}_{n},X_{n})$ be the tracial completion of $A_{n}$ with respect to
$X_{n}$ as in Definition 3.19, and let $\alpha_{n}\colon
A_{n}\rightarrow\mathcal{M}_{n}$ be the canonical map for $n\geq 1$. Then
$\big{(}\prod^{\omega}A_{n},\sum^{\omega}X_{n}\big{)}$ is a tracially complete
$C^{*}$-algebra and there is an isomorphism of tracially complete
$C^{*}$-algebras

(5.17)
$\prod^{\omega}\alpha_{n}\colon\Big{(}\prod^{\omega}A_{n},\sum^{\omega}X_{n}\Big{)}\xrightarrow{\cong}\Big{(}\prod^{\omega}\mathcal{M}_{n},\sum^{\omega}X_{n}\Big{)}$

defined at the level of representative sequences by
$(a_{n})_{n=1}^{\infty}\mapsto(\alpha_{n}(a_{n}))_{n=1}^{\infty}$.

###### Proof.

It is easy to see that $\alpha\coloneqq\prod^{\omega}\alpha_{n}$ is a
∗-homomorphism and is isometric with respect to the uniform 2-norms, so it
suffices to show that $\alpha$ is surjective. Fix
$b\in\prod^{\omega}\mathcal{M}_{n}$ and represent $b$ by a bounded sequence
$(b_{n})_{n=1}^{\infty}$. By the construction of the tracial completion
$(\mathcal{M}_{n},X_{n})$, there is, for each $n\geq 1$, $a_{n}\in A_{n}$ such
that $\|\alpha_{n}(a_{n})-b_{n}\|_{2,X_{n}}<\frac{1}{n}$ and
$\|a_{n}\|\leq\|b_{n}\|$. Then the sequence $(a_{n})_{n=1}^{\infty}$ defines
an element $a\in\prod^{\omega}A_{n}$ such that $\alpha(a)=b$. ∎

In the case of reduced powers, an important additional observation is that the
isomorphism defined in Proposition 5.6 is well behaved on the central
sequences.

######  Proposition 5.7.

Let $A$ be a $C^{*}$-algebra, let $X\subseteq T(A)$ be compact and convex, let
$(A^{\omega},X^{\omega})$ be the uniform tracial reduced power, let
$\iota:A\rightarrow A^{\omega}$ be given by constant sequences, let
$(\mathcal{M},X)$ be the tracial completion of $A$ with respect to $X$, and
let $\alpha_{X}:A\rightarrow\mathcal{M}$ be the canonical map. The isomorphism
$\alpha^{\omega}:(A^{\omega},X^{\omega})\rightarrow(\mathcal{M}^{\omega},X^{\omega})$
defined at the level of representative sequences by
$(a_{n})_{n=1}^{\infty}\mapsto(\alpha_{X}(a_{n}))_{n=1}^{\infty}$ satisfies

(5.18)
$\alpha^{\omega}(A^{\omega}\cap\iota(A)^{\prime})=\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}.$

More generally, for any $\|\cdot\|_{2,X}$-separable subset
$S\subseteq\mathcal{M}$, there is a $\|\cdot\|$-separable subset
$S_{0}\subseteq A$ such that
$\alpha(A^{\omega}\cap\iota(S_{0})^{\prime})\subseteq\mathcal{M}^{\omega}\cap
S^{\prime}$.

###### Proof.

By Proposition 5.6, $\alpha^{\omega}$ is an isomorphism. Since

(5.19) $\alpha^{\omega}(\iota(A))=\alpha_{X}(A)\subseteq\mathcal{M}^{\omega},$

we have

(5.20)
$\alpha^{\omega}(A^{\omega}\cap\iota(A)^{\prime})=\mathcal{M}^{\omega}\cap\alpha_{X}(A)^{\prime}.$

By Lemma 5.3, $\iota_{(\mathcal{M},X)}$ is an isometry for the respective
uniform 2-norms. Hence $\alpha_{X}(A)$ is $\|\cdot\|_{2,X^{\omega}}$-dense in
$\mathcal{M}$. In a tracially complete $C^{*}$-algebra, left and right
multiplication by a fixed element are continuous with respect to the uniform
2-norm by (3.2). Therefore, any element of $\mathcal{M}^{\omega}$ that
commutes with $\alpha_{X}(A)$ must also commute with $\mathcal{M}$. Hence
$\mathcal{M}^{\omega}\cap\alpha_{X}(A)^{\prime}=\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$.

Let $S\subseteq\mathcal{M}$ be $\|\cdot\|_{2,X}$-separable. Since
$\alpha_{X}(A)$ is $\|\cdot\|_{2,X^{\omega}}$-dense in $\mathcal{M}$, there is
a countable subset $S_{0}\subseteq A$ such that
$\overline{\alpha_{X}(S_{0})}^{\|\cdot\|_{2,X}}\supseteq S$. Then

(5.21)
$\alpha^{\omega}(A^{\omega}\cap\iota(S_{0})^{\prime})=\mathcal{M}^{\omega}\cap\alpha_{X}(S_{0})^{\prime}=\mathcal{M}^{\omega}\cap(\overline{\alpha_{X}(S_{0})}^{\|\cdot\|_{2,X}})^{\prime}\subseteq\mathcal{M}^{\omega}\cap
S^{\prime},$

as claimed. ∎

As with $C^{*}$-norm reduced products (see [15, Lemma 3.9.5]), matrix
amplifications commute with reduced products in the sense of the following
theorem. The proof is essentially the same as in the $C^{*}$-algebra setting.

######  Proposition 5.8.

For any sequence $\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ of
tracially complete $C^{*}$-algebras with reduced product $(\mathcal{M},X)$ and
$d\in\mathbb{N}$, there is a natural isomorphism

(5.22) $\big{(}\mathcal{M}\otimes
M_{d},X\otimes\\{\mathrm{tr_{d}}\\}\big{)}\xrightarrow{\cong}\prod^{\omega}\big{(}\mathcal{M}_{n}\otimes
M_{d},X_{n}\otimes\\{\mathrm{tr_{d}}\\}\big{)}.$

defined on representing sequences by $(a_{n})_{n=1}^{\infty}\otimes
b\mapsto(a_{n}\otimes b)_{n=1}^{\infty}$.

###### Proof.

The natural map

(5.23) $\phi\colon\Big{(}\prod_{n=1}^{\infty}\mathcal{M}_{n}\Big{)}\otimes
M_{d}\rightarrow\prod_{n=1}^{\infty}(\mathcal{M}_{n}\otimes M_{d})$

is an isomorphism of $C^{*}$-algebras (see the proof of [15, Lemma 3.9.4], for
example). Now, suppose
$(\tau_{n})_{n=1}^{\infty}\in\prod_{n=1}^{\infty}T(\mathcal{M}_{n})$ and
$\omega^{\prime}$ is an ultrafilter containing $\omega$, and let $\tau$ denote
the trace on $\prod_{n=1}^{\infty}\mathcal{M}_{n}$ given by

(5.24)
$\tau(a)\coloneqq\lim_{n\rightarrow\omega^{\prime}}\tau_{n}(a_{n}),\qquad
a=(a_{n})_{n=1}^{\infty}\in\mathcal{\prod}_{n=1}^{\infty}\mathcal{M}_{n}.$

Then for
$a=(a_{n})_{n=1}^{\infty}\in\prod_{n=1}^{\infty}(\mathcal{M}_{n}\otimes
M_{d})$, we have

(5.25)
$\lim_{n\rightarrow\omega^{\prime}}(\tau_{n}\otimes\mathrm{tr}_{d})(a_{n})=(\tau\otimes\mathrm{tr}_{d})(\phi^{-1}(a)).$

Therefore, $\phi$ descends to an isomorphism as in (5.22). ∎

We now turn to the question of traces on reduced products. It is well known
that the tracial ultrapower of II1 factor is also a II1 factor. We have been
unable to answer the general question for tracially complete $C^{*}$-algebras.
Under the additional hypothesis of CPoU (see Section 6), a positive answer to
the following question is given in Theorem 7.5. In particular, the answer is
affirmative in the presence of property $\Gamma$ (see Section 5.3) by Theorem
1.4.

######  Question 5.9.

If $\big{(}(\mathcal{M}_{n},X_{n})\big{)}_{n=1}^{\infty}$ is a sequence of
type II1 factorial tracially complete $C^{*}$-algebras, is the reduced power
$\big{(}\prod^{\omega}\mathcal{M}_{n},\sum^{\omega}X_{n}\big{)}$ also
factorial?

######  Remark 5.10.

In an earlier draft of this paper, we asked this question without the type II1
assumption. A counterexample in the type I setting was given by Vaccaro in
[113]. It is based on a family of $C^{*}$-algebras introduced in [89]:
$\mathcal{M}_{n}$ is the continuous sections of a bundle over the complex
projective space $\mathbb{C}P^{n}$ with fibre $M_{2}$, and
$X_{n}=T(\mathcal{M}_{n})$.

We end this subsection with a reparameterisation theorem which will allow us
to prove existence results for morphisms given both existence and uniqueness
results for approximate morphisms. This should be regarded as an abstract
version of the standard intertwining arguments commonly used in
$C^{*}$-algebra theory. An operator norm version of this result appears in
[46, Theorem 4.3], which, in turn, is a sequential version of [90, Proposition
1.3.7], attributed to Kirchberg.

If $(\mathcal{N},Y)$ is a tracially complete $C^{*}$-algebra and
$r\colon\mathbb{N}\rightarrow\mathbb{N}$ is a function such that
$\lim_{n\rightarrow\infty}r(n)=\infty$, then there is an induced endomorphism
$r^{*}$ of $(\mathcal{N}^{\infty},Y^{\infty})$ given on representing sequences
by

(5.26)
$r^{*}\big{(}(b_{n})_{n=1}^{\infty})\coloneqq(b_{r(n)})_{n=1}^{\infty},\qquad(b_{n})_{n=1}^{\infty}\in\ell^{\infty}(\mathcal{N}).$

Equivalently, viewing $\ell^{\infty}(\mathcal{N})$ as bounded functions
$\mathbb{N}\rightarrow\mathcal{N}$, $r^{*}$ is the map induced by

(5.27) $\ell^{\infty}(\mathcal{N})\rightarrow\ell^{\infty}(\mathcal{N})\colon
f\mapsto f\circ r.$

Note that the map $r^{*}$ would typically not be well-defined if
$(\mathcal{N}^{\infty},Y^{\infty})$ were replaced with an ultrapower – this is
the main reason for working with general reduced products of tracially
complete $C^{*}$-algebras.

In applications of the following theorem, the metric space $S$ will typically
be either a separable $C^{*}$-algebra with the operator norm or a tracially
complete $C^{*}$-algebra which is separable in its uniform 2-norm.

######  Theorem 5.11 (Intertwining via reparameterisation).

Let $(\mathcal{N},Y)$ be a tracially complete $C^{*}$-algebra, let $S$ be a
separable metric space, and let $\phi\colon S\rightarrow\mathcal{N}^{\infty}$
be a $\|\cdot\|_{2,Y^{\infty}}$-continuous function. Suppose also that every
unitary in $\mathcal{N}^{\infty}$ lifts to a unitary in
$\ell^{\infty}(\mathcal{N})$. If for every function
$r\colon\mathbb{N}\rightarrow\mathbb{N}$ with
$\lim_{n\rightarrow\infty}r(n)=\infty$, we have $r^{*}\circ\phi$ is
approximately unitarily equivalent to $\phi$, then there is a
$\|\cdot\|_{2,Y^{\infty}}$-continuous function $\psi\colon
S\rightarrow(\mathcal{N},Y)$ such that $\iota_{(\mathcal{N},Y)}\circ\psi$ is
unitarily equivalent to $\phi$.

The proof is a minor modification of the $C^{*}$-algebra version obtained in
[46, Theorem 4.3], where the reduced power is taken in the $C^{*}$-norm. In
the $C^{*}$-norm setting, the condition on unitaries is automatic since every
approximate unitary is close to a genuine unitary. We do not know if the
condition on unitaries in $(\mathcal{N}^{\infty},Y^{\infty})$ is necessary. In
all of our applications of Theorem 5.11, $(\mathcal{N},Y)$ will satisfy CPoU
(see Section 6), and hence the condition on unitaries will follow from
Corollary 7.11.

###### Proof of Theorem 5.11.

Define a sequence of functions $\phi_{n}\colon S\rightarrow Y$ by choosing a
$\|\cdot\|$-preserving lift
$(\phi_{n}(b))_{n=1}^{\infty}\in\ell^{\infty}(\mathcal{N})$ of
$\phi(b)\in\mathcal{N}^{\infty}$ for every $b\in S$. We claim that for every
finite subset $\mathcal{F}\subseteq S$, every $\epsilon>0$, and every
$m\in\mathbb{N}$, there exists $k\geq m$ such that for every $n\geq k$, there
exists a unitary $u\in\mathcal{N}$ such that

(5.28) $\max_{a\in F}\|u^{*}\phi_{n}(a)u-\phi_{k}(a)\|_{2,Y}<\epsilon,\quad
a\in\mathcal{F}.$

Suppose for a contradiction that the claim is false. Then there exist a finite
set $\mathcal{F}_{0}\subseteq S$, $\epsilon_{0}>0$, $m_{0}\in\mathbb{N}$, and
a sequence of natural numbers $(n_{k})_{k=m_{0}}^{\infty}$ with $n_{k}\geq k$
such that

(5.29)
$\max_{a\in\mathcal{F}_{0}}\|u^{*}\phi_{n_{k}}(a)u-\phi_{k}(a)\|_{2,Y}\geq\epsilon_{0}$

for all unitaries $u\in\mathcal{N}$ and all $k\geq m_{0}$. Let
$r:\mathbb{N}\rightarrow\mathbb{N}$ be given by $r(k)\coloneqq n_{k}$ for
$k\geq m_{0}$ and define $r(k)$ arbitrarily for $k<m_{0}$. Then
$\lim_{k\rightarrow\infty}r(k)=\infty$. By our hypothesis, $\phi$ and
$r^{*}\circ\phi$ are approximately unitarily equivalent. Therefore, there
exists a unitary $u\in\mathcal{N}^{\infty}$ such that

(5.30) $\|u^{*}(r^{*}\circ\phi)(a)u-\phi(a)\|_{2,Y^{\infty}}<\epsilon_{0}$

for all $a\in\mathcal{F}_{0}$. By our hypothesis, we may lift $u$ to a
sequence of unitaries $(u_{k})_{k=1}^{\infty}$ in $\mathcal{N}$. By Lemma 5.3,
we have

(5.31)
$\begin{split}\limsup_{k\rightarrow\infty}\|u_{k}^{*}\phi_{n_{k}}(a)u_{k}-\phi_{k}(a)\|_{2,Y}<\epsilon_{0}\end{split}$

for all $a\in\mathcal{F}_{0}$. Then (5.31) contradicts (5.29) for some
sufficiently large $k\in\mathbb{N}$. This proves the claim. We now use this to
construct $\psi$.

Let $(\mathcal{F}_{i})_{i=1}^{\infty}$ be an increasing sequence of finite
subsets whose union is dense in $S$. Applying the claim recursively, we obtain
an increasing sequence of natural numbers $(k_{n})_{n=1}^{\infty}$ and a
sequence of unitaries $(u_{n})_{n=1}^{\infty}$ in $\mathcal{N}$ such that

(5.32) $\|u_{n}^{*}\phi_{k_{n}}(a)u_{n}-\phi_{k_{n-1}}(a)\|_{2,Y}<2^{-n}$

for all $a\in\mathcal{F}_{n}$. Set $v_{n}\coloneqq u_{n}u_{n-1}\cdots u_{1}$
(and put $v_{0}\coloneqq 1_{\mathcal{N}}$). Then, since $v_{n}=u_{n}v_{n-1}$
and $\|v_{n-1}\|\leq 1$, we have

(5.33)
$\begin{split}\|v_{n}^{*}\phi_{k_{n}}(a)v_{n}-v_{n-1}^{*}\phi_{k_{n-1}}(a)v_{n-1}\|_{2,Y}<2^{-n}\end{split}$

for all $n\in\mathbb{N}$ and $a\in\mathcal{F}_{n}$.

By construction, for every $a\in\bigcup_{i=1}^{\infty}\mathcal{F}_{i}$, the
sequence $(v_{n}^{*}\phi_{k_{n}}(a)v_{n})_{n=1}^{\infty}$ is
$\|\cdot\|_{2,Y}$-Cauchy. Indeed, if $a\in\mathcal{F}_{i_{0}}$, then for
$n>m>i_{0}$, we have

(5.34)
$\|v_{m}^{*}\phi_{k_{m}}(a)v_{m}-v_{n}^{*}\phi_{k_{n}}(a)v_{n}\|_{2,Y}<\sum_{i=n+1}^{m}2^{-i}<2^{-n}.$

Let $b\in S$ and $\epsilon>0$. Since $\bigcup_{i=1}^{\infty}\mathcal{F}_{i}$
is dense in $S$ and $\phi:S\rightarrow(\mathcal{N}^{\infty},Y^{\infty})$ is
$\|\cdot\|_{2,Y^{\infty}}$-continuous, there exists $i_{0}\in\mathbb{N}$ and
some $a\in F_{i_{0}}$ with $\|\phi(b)-\phi(a)\|_{2,Y^{\infty}}<\epsilon$.
Hence, by Lemma 5.3, we have

(5.35) $\limsup_{k\to\infty}\|\phi_{k}(b)-\phi_{k}(a)\|_{2,Y}<\epsilon.$

Choose $N_{1}\in\mathbb{N}$ with $\|\phi_{k}(b)-\phi_{k}(a)\|_{2,Y}<\epsilon$
for all $k>N_{1}$, and choose $N_{2}\in\mathbb{N}$ with
$\|v_{m}^{*}\phi_{k_{m}}(a)v_{m}-v_{n}^{*}\phi_{k_{n}}(a)v_{n}\|_{2,Y}<\epsilon$
for all $n,m>N_{2}$. Then a simple $3\epsilon$-argument gives that

(5.36)
$\|v_{m}^{*}\phi_{k_{m}}(b)v_{m}-v_{n}^{*}\phi_{k_{n}}(b)v_{n}\|_{2,Y}<3\epsilon$

whenever $n,m>\max(N_{1},N_{2})$. Hence
$(v_{n}^{*}\phi_{k_{n}}(b)v_{n})_{n=1}^{\infty}$ is $\|\cdot\|_{2,Y}$-Cauchy
for all $b\in S$. Moreover, we have
$\|v_{n}^{*}\phi_{k_{n}}(b)v_{n}\|\leq\|\phi_{n}(b)\|\leq\|\phi(b)\|$ for all
$n\in N$. Since $(\mathcal{N},Y)$ is a tracially complete $C^{*}$-algebra,
$(v_{n}^{*}\phi_{k_{n}}(b)v_{n})_{n=1}^{\infty}$ converges in the uniform
2-norm for all $b\in S$. Hence we may define
$\psi:S\rightarrow(\mathcal{N},Y)$ by
$\psi(b)\coloneqq\lim_{n\to\infty}v_{n}^{*}\phi_{k_{n}}(b)v_{n}$.

By construction, $\iota_{(\mathcal{N},Y)}\circ\psi$ is unitarily equivalent to
$r^{*}\circ\phi$, where $r:\mathbb{N}\rightarrow\mathbb{N}$ is given by
$r(n)\coloneqq k_{n}$, via the unitary represented by the sequence
$(v_{k_{n}})_{n=1}^{\infty}$. Since $r^{*}\circ\phi$ is unitarily equivalent
to $\phi$ by hypothesis (using Kirchberg’s $\epsilon$-test to replace
approximate unitary equivalence with unitary equivalence), we conclude that
$\iota_{(\mathcal{N},Y)}\circ\psi$ is unitarily equivalent to $\phi$. ∎

###  5.2. The McDuff property

Central sequences in II1 factors have been studied beginning with the
foundational work of Murray and von Neumann ([83]) when they used them to
distinguish the hyperfinite II1 factor from any free group factor. The
systematic study of central sequences was later instigated by two breakthrough
results of McDuff: the existence of infinitely many (and in fact, uncountably
many) non-isomorphic II1 factors ([77, 78]) and, of particular relevance to
us, her characterisation those II1 factors $\mathcal{M}$ with separable
predual which tensorially absorb the hyperfinite II1 factor (i.e.
$\mathcal{M}\cong\mathcal{M}\bar{\otimes}\mathcal{R}$) as those $\mathcal{M}$
with non-abelian central sequence algebras. Motivated by this last result, a
II1 factor $\mathcal{M}$ is said to have the _McDuff property_ if there are
approximately central unital embeddings of matrix algebras into
$\mathcal{M}$.424242From a ultrapower viewpoint, McDuff’s work shows that the
central sequence algebra $\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$ is
non-abelian if and only if it is type II1 (and so admits unital embeddings of
all matrix algebras). For II1 factors with non-separable predual, the
equivalence of tensorial absorption of $\mathcal{R}$ and the McDuff property
no longer holds ([44, Theorem 1.5]). Experience has shown that the formulation
in terms of approximately central matrix embeddings is the correct way to
extend the McDuff property to the non-separable predual situation (where one
should of course work with ultrapowers over uncountable sets). Indeed, the
ultrapower $\mathcal{M}^{\omega}$ of a McDuff II1 factor $\mathcal{M}$ with
separable predual has the McDuff property but is not stable under tensoring by
$\mathcal{R}$ – see [16, Footnote 65], which uses [50].

The McDuff property of II1 factors has been of considerable interest to
$C^{*}$-algebraists working in the classification programme because of its
relation to $\mathcal{Z}$-stability at both conceptual and technical levels
([111, 75, 68, 110, 97, 22, 16]).

In this section, we generalise the McDuff property to the setting of tracially
complete $C^{*}$-algebras and show that the McDuff property is equivalent to
$\mathcal{R}$-stability in the separable case (Theorem 5.18). We begin with a
formal definition of the McDuff property before establishing some useful
technical reformulations and permanence properties.

######  Definition 5.12 (cf. [22, Definition 4.2]).

Let $(\mathcal{M},X)$ be a tracially complete $C^{*}$-algebra. We say that
$(\mathcal{M},X)$ is _McDuff_ if for any $\|\cdot\|_{2,X}$-separable set
$S\subseteq\mathcal{M}$ and $k\geq 1$, there is a unital embedding
$M_{k}\rightarrow\mathcal{M}^{\omega}\cap S^{\prime}$.434343Of course, when
$\mathcal{M}$ is $\|\cdot\|_{2,X}$-separable, we can just take
$S=\mathcal{M}$.

The _uniform McDuff property_ was defined for a separable $C^{*}$-algebra $A$
with $T(A)$ compact in [22, Definition 4.2] as the existence of unital
embeddings $M_{k}\rightarrow A^{\omega}\cap A^{\prime}$ for all $k\geq 1$. By
Proposition 5.7, this is consistent with our definition.

######  Proposition 5.13.

Let $A$ be a separable $C^{*}$-algebra with $T(A)$ compact. Then $A$ is
uniformly McDuff in the sense of [22] if and only if its tracial completion
with respect to $T(A)$ is McDuff in the sense of Definition 5.12.

We now establish some equivalent reformulations of the McDuff property. Note
that the equivalence of (i) and (ii) in the following theorem shows that the
McDuff property is independent of choice of free filter $\omega$. The result
is standard in the setting of $\mathrm{II}_{1}$ factors, and the same
techniques work in this context.

######  Proposition 5.14 (cf. [10, Proposition 3.11]).

Let $(\mathcal{M},X)$ be a tracially complete $C^{*}$-algebra. The following
are equivalent:

  1. (i)

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

  2. (ii)

Given a finite set $\mathcal{F}\subseteq\mathcal{M}$ and $\epsilon>0$, there
is a contraction $v\in\mathcal{M}$ such that

(5.37) $\max_{a\in\mathcal{F}}\|[v,a]\|_{2,X}<\epsilon,\
\|v^{*}v+vv^{*}-1_{\mathcal{M}}\|_{2,X}<\epsilon,\ \text{and}\
\|v^{2}\|_{2,X}<\epsilon.$

  3. (iii)

For each $\|\cdot\|_{2,X}$-separable subset $S\subseteq\mathcal{M}^{\omega}$,
there is a contraction $v\in\mathcal{M}^{\omega}\cap S^{\prime}$ such
$v^{2}=0$ and $vv^{*}+vv^{*}=1_{\mathcal{M}^{\omega}}$.

  4. (iv)

For each $\|\cdot\|_{2,X}$-separable subset $S\subseteq\mathcal{M}^{\omega}$,
there is $k\geq 2$ and a unital embedding
$M_{k}\rightarrow\mathcal{M}^{\omega}\cap S^{\prime}$.

  5. (v)

For each $\|\cdot\|_{2,X}$-separable subset $S\subseteq\mathcal{M}^{\omega}$,
there is a unital embedding $\mathcal{R}\rightarrow\mathcal{M}^{\omega}\cap
S^{\prime}$.

###### Proof.

(i)$\Rightarrow$(ii): Given a finite set $\mathcal{F}\subseteq\mathcal{M}$,
fix a unital embedding $\phi\colon
M_{2}\rightarrow\mathcal{M}^{\omega}\cap\mathcal{F}^{\prime}$. Let
$(v_{k})_{k=1}^{\infty}\subseteq\mathcal{M}$ be a sequence of contractions
representing $\phi(e_{1,2})$ and set $v\coloneqq v_{k}$ for some suitable
index $k$.

(ii)$\Rightarrow$(iii): Let $\\{s^{(i)}:i\in\mathbb{N}\\}$ be a countable
dense set in $S$. For each $i\in\mathbb{N}$, let
$(s^{(i)}_{n})_{n=1}^{\infty}$ be a sequence representing $s^{(i)}$. For each
$n\in\mathbb{N}$, set $\epsilon_{n}\coloneqq 2^{-n}$ and
$\mathcal{F}_{n}\coloneqq\\{s^{(i)}_{n}:i\leq n\\}$. Let $v_{n}\in\mathcal{M}$
be given as in (ii) with $(\mathcal{F}_{n},\epsilon_{n})$ in place of
$(\mathcal{F},\epsilon)$. Then the sequence $(v_{n})_{n=1}^{\infty}$ induces a
contraction $v\in\mathcal{M}^{\omega}\cap S^{\prime}$ with $v^{2}=0$ and
$v^{*}v+vv^{*}=1_{\mathcal{M}^{\omega}}$.

(iii)$\Rightarrow$(iv): Let $v\in\mathcal{M}^{\omega}\cap S^{\prime}$ be as in
(iii). Since $v$ is a contraction and $v^{2}=0$, we have $v^{*}v$ and $vv^{*}$
are orthogonal positive contractions. As $v^{*}v+vv^{*}=1_{\mathcal{M}}$,
$v^{*}v$ and $vv^{*}$ are projections, and in particular, $v$ is a partial
isometry. It follows that the $C^{*}$-subalgebra generated by $v$ is spanned
by $1_{\mathcal{M}}$, $v$, $v^{*}$, and $v^{*}v$. As this subalgebra is non-
commutative and has dimension at most four, it is isomorphic to $M_{2}$. This
verifies (iv) with $k=2$.

(iv)$\Rightarrow$(v): Fix the subset $S$ from (v). Using (iv), let $k_{1}\geq
2$ and let $\phi_{1}\colon M_{k_{1}}\rightarrow\mathcal{M}^{\omega}\cap
S^{\prime}$ be a unital embedding. Let $S_{1}\coloneqq
C^{*}(\phi_{1}(M_{k_{1}})\cup S)$ and use (iv) again to produce an integer
$k_{2}\geq 2$ and a unital embedding $\phi_{2}\colon
M_{k_{2}}\rightarrow\mathcal{M}^{\omega}\cap S_{1}^{\prime}$. Continuing
inductively, there are integers $k_{n}\geq 2$ and unital embeddings
$\phi_{n}\colon M_{k_{n}}\rightarrow\mathcal{M}^{\omega}\cap S^{\prime}$ with
commuting ranges. The $\phi_{n}$ induce a unital embedding

(5.38)
$\mathcal{R}\cong\overline{\bigotimes}_{n=1}^{\,\infty}(M_{k_{n}},\mathrm{tr}_{k_{n}})\rightarrow\mathcal{M}^{\omega}\cap
S^{\prime}.$

(v)$\Rightarrow$(i): This follows as there is a unital embedding
$M_{k}\rightarrow\mathcal{R}$ for all $k\geq 1$. ∎

######  Remark 5.15.

Other equivalent properties to McDuffness can be given by strengthening
Definition 5.12 to allow $S$ to be any $\|\cdot\|_{2,X^{\omega}}$-separable
subset of $\mathcal{M}^{\omega}$ or weakening any of (iii), (iv) or (v) to
only have $S\subseteq\mathcal{M}$. In all cases the argument (for the non-
trivial direction) is by reindexing.

For example, to go from the weakening of (iv) to (iv), given a
$\|\cdot\|_{2,X}$-separable subset $S\subseteq\mathcal{M}$, let
$T\subseteq\mathcal{M}$ be a countable set consisting of the sequence entries
of lifts of a countable dense subset of $S$. Then, given a unital embedding
$\phi:M_{k}\to\mathcal{M}^{\omega}\cap T^{\prime}$, we may lift $\phi$ to a
sequence of c.p.c. maps $\phi_{n}:M_{k}\to\mathcal{M}$ which are
asymptotically multiplicative, unital, and commute with $T$. An appropriate
reindexing $(\phi_{m_{n}})$ will provide a unital embedding
$\phi:M_{k}\to\mathcal{M}^{\omega}\cap S^{\prime}$.

As a corollary of the equivalence between (i) and (ii) above, we obtain the
following permanence properties.

######  Corollary 5.16.

Inductive limits and reduced products of McDuff tracially complete
$C^{*}$-algebras are McDuff.

The following result gives a large supply of examples of McDuff tracially
complete $C^{*}$-algebras. The result extends [23, Proposition 2.3], and the
proof is very similar.

######  Proposition 5.17 (cf. [23, Proposition 2.3]).

If $A$ is a separable $\mathcal{Z}$-stable $C^{*}$-algebra and $X\subseteq
T(A)$ is a compact convex set, then $(\overline{A}^{X},X)$ is a McDuff
tracially complete $C^{*}$-algebra.

###### Proof.

Let $\omega$ be a free ultrafilter on $\mathbb{N}$, let
$A_{\omega}\coloneqq\ell^{\infty}(A)/c_{\omega}(A)$ be the operator norm
ultrapower of $A$, and let $\mathcal{M}\coloneqq\overline{A}^{X}$. By [67,
Proposition 4.4(4)], there is a unital embedding
$\phi\colon\mathcal{Z}\rightarrow(A_{\omega}\cap A^{\prime})/A^{\perp}$, where

(5.39) $A^{\perp}\coloneqq\\{a\in A_{\omega}:ab=ba=0\text{ for all }b\in
A\\}.$

The natural map $\alpha_{X}\colon A\rightarrow\mathcal{M}$ induces a
∗-homomorphism $q\colon A_{\omega}\rightarrow\mathcal{M}^{\omega}$. Since
$\alpha_{X}$ maps the unit ball of $A$ onto a $\|\cdot\|_{2,X}$-dense subset
of the unit ball of $\mathcal{M}$, we have $q$ is surjective and
$q(A_{\omega}\cap
A^{\prime})\subseteq\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$.

Following the proof of [23, Lemma 1.10], we show $q(A^{\perp})=0$. Assume
$b\in A^{\perp}$ with $0\leq b\leq 1$. Let $\tau\in X^{\omega}\subseteq
T(\mathcal{M}^{\omega})$ be given and note that

(5.40) $\tau\circ q|_{A}=\tau|_{\mathcal{M}}\circ\alpha_{X}.$

Combining Corollary 3.29(i) with (5.40) shows that $\tau\circ q|_{A}$ has norm
1. Fix $\epsilon>0$ and a positive contraction $e\in A$ with $\tau(q(e))\geq
1-\epsilon$. Since $b\in A^{\perp}$ and $e\in A$ are orthogonal positive
contractions, their sum is also a positive contraction. Therefore,

(5.41) $0\leq\tau(q(b))=\tau(q(b+e))-\tau(q(e))\leq 1-(1-\epsilon)=\epsilon.$

Since $\epsilon>0$ was arbitrary, $\tau(q(b))=0$ for all $\tau\in X^{\omega}$,
and since $q(b)\geq 0$, this implies $q(b)=0$. Hence $q(A^{\perp})=0$.

Let $\bar{q}\colon(A_{\omega}\cap
A^{\prime})/A^{\perp}\rightarrow\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$
be the ∗-homomorphism determined by $q$. Then
$\bar{q}\circ\phi\colon\mathcal{Z}\rightarrow\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$
is a unital ∗-homomorphism. As $\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$
is a $\|\cdot\|_{2,X^{\omega}}$-closed, unital $C^{*}$-subalgebra of
$\mathcal{M}^{\omega}$, we may view
$\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$ as a tracially complete
$C^{*}$-algebra as in the comments preceding Definition 3.11. Since
$\mathcal{Z}$ has a unique trace $\tau$ and
$\pi_{\tau}(\mathcal{Z})^{\prime\prime}\cong\mathcal{R}$, Proposition 3.25
allows us to extend $\bar{q}\circ\phi$ to a unital embedding
$\mathcal{R}\rightarrow\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$. Since
$M_{k}$ embeds unitally in $\mathcal{R}$ for all $k\in\mathbb{N}$, we see that
$\mathcal{M}$ has the McDuff property. ∎

The following result gives a tensor product characterisation of McDuff
tracially complete $C^{*}$-algebras in the separable setting, analogous to
McDuff’s original result for II1 factors from [79]. Our proof is an adaptation
of the argument found in [10, Proposition 3.11] in the setting of
$W^{*}$-bundles. These follow the framework of the analogous results for
absorption of a strongly self-absorbing $C^{*}$-algebra (see [111, Theorem
2.2] or [93, Theorem 7.2.2]), which are powered by an Elliott intertwining
argument.

######  Theorem 5.18.

Suppose $(\mathcal{M},X)$ is a tracially complete $C^{*}$-algebra such that
$\mathcal{M}$ is $\|\cdot\|_{2,X}$-separable. Then $(\mathcal{M},X)$ is
uniformly McDuff if and only if
$(\mathcal{M},X)\bar{\otimes}(\mathcal{R},\mathrm{tr}_{\mathcal{R}})\cong(\mathcal{M},X)$.

###### Proof.

Since $\mathcal{M}$ is $\|\cdot\|_{2,X}$-separable and McDuff, there exists a
unital embedding
$\phi:\mathcal{R}\rightarrow\mathcal{M}^{\omega}\cap\mathcal{M}^{\prime}$.
Take a set-theoretic lift of $\phi$ to a sequence
$(\phi_{n}\colon\mathcal{R}\to\mathcal{M})_{n=1}^{\infty}$ of maps. We define
commuting unital embeddings

(5.42)
$\alpha,\beta:\mathcal{R}\rightarrow(\mathcal{M}\bar{\otimes}\mathcal{R})^{\omega}\cap(\mathcal{M}\otimes
1_{\mathcal{R}})^{\prime}$

at the level of representative sequences by

(5.43) $\alpha(x)\coloneqq(1_{\mathcal{M}}\otimes
x)_{n=1}^{\infty}\quad\text{and}\quad\beta(x)\coloneqq(\phi_{n}(x)\otimes
1_{\mathcal{R}})_{n=1}^{\infty},$

respectively.

Let $M_{2^{\infty}}$ be the UHF algebra with supernatural number $2^{\infty}$,
which we view as a $\|\cdot\|_{2,\mathrm{tr}_{\mathcal{R}}}$-dense subalgebra
of $\mathcal{R}$. As $M_{2^{\infty}}$ is nuclear, there exists an embedding

(5.44) $\gamma:\mathcal{M}\otimes M_{2^{\infty}}\otimes
M_{2^{\infty}}\rightarrow(\mathcal{M}\bar{\otimes}\mathcal{R})^{\omega}$

such that

(5.45) $\gamma(a\otimes b\otimes c)=(a\otimes
1_{\mathcal{R}})\alpha(b)\beta(c)=(a\otimes b)\beta(c)$

for all $a\in\mathcal{M}$ and $b,c\in M_{2^{\infty}}$. Since $M_{2^{\infty}}$
has a unique trace, it follows that $\gamma$ is trace-preserving444444To ease
notation, set $A\coloneqq\mathcal{M}\otimes\mathcal{M}_{2^{\infty}}$ and
$(\mathcal{N},Y)=(\mathcal{M},X)\bar{\otimes}(\mathcal{R},\mathrm{tr}_{\mathcal{R}})$,
and view $A$ as a $\|\cdot\|_{2,Y}$-dense subalgebra of $\mathcal{N}$. Then
$\gamma$ is a map
$A\otimes\mathcal{M}_{2^{\infty}}\rightarrow(\mathcal{N}\otimes\mathcal{R})^{\omega}$
given by $\gamma(a\otimes c)=a\beta(c)$. It suffices to show
$\gamma^{*}(\tau)=\tau|_{A}\otimes\mathrm{tr}_{2^{\infty}}$ for all $\tau\in
Y^{\infty}$. Fix $\tau\in Y^{\infty}$ and positive $a\in A$. If $\tau(a)=0$,
then $\tau(a\beta(c))=0=\tau(a)\mathrm{tr}_{2^{\infty}}(c)$ by the
Cauchy–Schwarz inequality. If $\tau(a)\neq 0$, then
$c\mapsto\tau(\gamma(a\otimes c))/\tau(a)$ is a trace on $M_{2^{\infty}}$ and
hence is the unique trace $\mathrm{tr}_{2^{\infty}}$. Therefore, for all $a\in
A$ and $c\in M_{2^{\infty}}$, we have $\tau(\gamma(a\otimes
c))=\tau(a)\mathrm{tr}_{2^{\infty}}(c)$. and, accordingly, extends to an
embedding of the tracial completion

(5.46)
$\bar{\gamma}:\mathcal{M}\bar{\otimes}\mathcal{R}\bar{\otimes}\mathcal{R}\rightarrow(\mathcal{M}\bar{\otimes}\mathcal{R})^{\omega}$

with

(5.47) $\bar{\gamma}(a\otimes b\otimes c)=(a\otimes
1_{\mathcal{R}})\alpha(b)\beta(c)=(a\otimes b)\beta(c)$

for all $a\in\mathcal{M}$ and $b,c\in\mathcal{R}$.

Recall that $\mathcal{R}$ has an approximately inner half-flip, in the sense
that there exists unitaries $(\bar{u}_{m})_{m=1}^{\infty}$ in
$\mathcal{R}\bar{\otimes}\mathcal{R}$ such that for all $b\in\mathcal{R}$

(5.48) $\lim_{m\to\infty}\|\bar{u}_{m}^{*}(b\otimes
1_{\mathcal{R}})\bar{u}_{m}-1_{\mathcal{R}}\otimes
b\|_{2,\mathrm{tr}_{\mathcal{R}}\otimes\mathrm{tr}_{\mathcal{R}}}=0.$

Moreover, since $\mathcal{R}\bar{\otimes}\mathcal{R}$ is a von Neumann
algebra, each unitary $\bar{u}_{m}$ is of the form $\exp(2\pi i\bar{h}_{m})$
for some self-adjoint $\bar{h}_{m}\in\mathcal{R}\bar{\otimes}\mathcal{R}$.

Set
$u_{m}\coloneqq\bar{\gamma}(1_{\mathcal{M}}\otimes\bar{u}_{m})\in(\mathcal{M}\bar{\otimes}\mathcal{R})^{\omega}\cap(\mathcal{M}\otimes
1_{\mathcal{R}})^{\prime}$ (it is in this commutant because
$\bar{\gamma}(\mathcal{M}\otimes 1_{\mathcal{R}}\otimes
1_{\mathcal{R}})=\mathcal{M}\otimes 1_{\mathcal{R}}$). Then the two embeddings
$\mathcal{M}\bar{\otimes}\mathcal{R}\rightarrow(\mathcal{M}\bar{\otimes}\mathcal{R})^{\omega}$
given by

(5.49) $a\otimes b\mapsto\bar{\gamma}(a\otimes b\otimes
1_{\mathcal{R}})=a\otimes b\ \ \text{and}\ \ a\otimes
b\mapsto\bar{\gamma}(a\otimes 1_{\mathcal{R}}\otimes b)=(a\otimes 1)\beta(b)$

are approximately unitary equivalent in
$\|\cdot\|_{2,(X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\})^{\omega}}$, via the
sequence $(u_{m})_{m=1}^{\infty}$. Since $\beta(b)\in(\mathcal{M}\otimes
1_{\mathcal{R}})^{\omega}$, it follows that

(5.50)
$\lim_{m\to\infty}\inf\Big{\\{}\|u_{m}^{*}xu_{m}-z\|_{2,(X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\})^{\omega}}:\begin{matrix}z\in(\mathcal{M}\otimes
1_{\mathcal{R}})^{\omega}\\\ \|z\|\leq 1\end{matrix}\Big{\\}}=0$

for all contractions $x\in\mathcal{M}\otimes\mathcal{R}$. We also have
$u_{m}=\exp(2\pi ih_{m})$, where
$h_{m}=\bar{\gamma}(1_{\mathcal{M}}\otimes\bar{h}_{m})$. Since we can lift the
self adjoint $h_{m}\in(\mathcal{M}\bar{\otimes}\mathcal{R})^{\omega}$ to a
sequence of self-adjoint elements $(h_{m,n})_{n=1}^{\infty}$ in
$\ell^{\infty}(\mathcal{M}\bar{\otimes}\mathcal{R})$, we can find a sequence
of unitaries $(u_{m,n})_{n=1}^{\infty}$ in $\mathcal{M}\otimes\mathcal{R}$
representing $u_{m}$.

We are now ready to construct an isomorphism
$\psi:\mathcal{M}\rightarrow\mathcal{M}\otimes\mathcal{R}$. Let
$(x_{k})_{k=1}^{\infty}$ be a $\|\cdot\|_{2,X}$-dense sequence in the unit
ball of $\mathcal{M}$ and $(y_{k})_{k=1}^{\infty}$ be a
$\|\cdot\|_{2,X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\}}$-dense sequence in the
unit ball of $\mathcal{M}\otimes\mathcal{R}$. We shall iteratively produce
unitaries $w_{k}$ in $\mathcal{M}\otimes\mathcal{R}$ and contractions
$z_{k}^{(j)}\in\mathcal{M}$ for $k\in\mathbb{N}$ and $1\leq j\leq k$. The
construction begins with $w_{0}\coloneqq 1$. Fix $k\geq 1$ and suppose that
$w_{s},z^{(r)}_{s}\in\mathcal{M}$ for $1\leq r\leq s<k$ have already been
constructed.

By (5.50), there exists contractions
$\bar{z}^{(1)},\ldots,\bar{z}^{(k)}\in(\mathcal{M}\otimes
1_{\mathcal{R}})^{\omega}$ and $m_{k}\in\mathbb{N}$ such that

(5.51)
$\begin{split}\|u_{m_{k}}^{*}w_{k-1}^{*}y_{j}w_{k-1}u_{m_{k}}-\bar{z}_{k}^{(j)}\otimes
1_{\mathcal{R}}\|_{2,(X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\})^{\omega}}&<\frac{1}{k}.\end{split}$

for $j=1,\ldots,k-1$. Let $(u_{m_{k},n})_{n=1}^{\infty}$ be a sequence of
unitaries in $\mathcal{M}\otimes\mathcal{R}$ representing $u_{m_{k}}$ and let
$(z^{(j)}_{k,n})_{n=1}^{\infty}$ be sequences of contractions representing
each $\bar{z}_{k}^{(j)}$. Since
$u_{m_{k}}\in(\mathcal{M}\bar{\otimes}\mathcal{R})^{\omega}\cap(\mathcal{M}\otimes
1_{\mathcal{R}})^{\prime}$, we can choose $n_{k}\in\mathbb{N}$ such that

(5.52) $\displaystyle\
\|u_{m_{k},n_{k}}^{*}w_{k-1}^{*}y_{j}w_{k-1}u_{m_{k},n_{k}}-z^{(j)}_{k,n_{k}}\otimes
1_{\mathcal{R}}\|_{2,X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\}}$
$\displaystyle<\frac{1}{k},$ (5.53)
$\displaystyle\|[u_{m_{k},n_{k}},x_{j}\otimes
1_{\mathcal{R}}]\|_{2,X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\}}$
$\displaystyle<2^{-k},$

for all $1\leq j\leq k$, and

(5.54) $\begin{split}\|[u_{m_{k},n_{k}},z^{(r)}_{s}\otimes
1_{\mathcal{R}}]\|_{2,X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\}}&<2^{-k},\end{split}$

for all $1\leq r\leq s<k$. Set $w_{k}\coloneqq w_{k-1}u_{m_{k},n_{k}}$ and set
$z^{(j)}_{k}\coloneqq z^{(j)}_{k,n_{k}}$ for $j=1,\ldots,k$.

For $j\in\mathbb{N}$, the sequence $(w_{k}(x_{j}\otimes
1_{\mathcal{R}})w_{k}^{*})_{k=1}^{\infty}$ is
$\|\cdot\|_{2,X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\}}$-Cauchy by (5.53) as
$\sum_{k=1}^{\infty}2^{-k}$ converges. Since $(x_{j})_{j=1}^{\infty}$ is a
$\|\cdot\|_{2,X}$-dense sequence in the unit ball of $\mathcal{M}$, a
$3\epsilon$-argument gives that $(w_{k}(a\otimes
1_{\mathcal{R}})w_{k}^{*})_{n=1}^{\infty}$ is
$\|\cdot\|_{2,X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\}}$-Cauchy for all
$a\in\mathcal{M}$. As $(w_{k}(a\otimes
1_{\mathcal{R}})w_{k}^{*})_{n=1}^{\infty}$ is $\|\cdot\|$-bounded, we may
define a map $\psi:\mathcal{M}\rightarrow\mathcal{M}\otimes\mathcal{R}$ by
$\psi(a)\coloneqq\lim_{k\to\infty}w_{k}(a\otimes 1_{\mathcal{R}})w_{k}^{*}$.
Since the $w_{k}$ are unitaries, we see that $\psi$ is an injective
∗-homomorphism and we have
$(\tau\otimes\mathrm{tr}_{\mathcal{R}})\circ\psi=\tau$ for all $\tau\in X$.
Surjectivity follows from the
$\|\cdot\|_{2,X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\}}$-density of
$(y_{j})_{j=1}^{\infty}$ in the unit ball of $\mathcal{M}$, because for $k>j$
we have

(5.55)
$\begin{array}[]{rcl}\|y_{j}-\psi(z^{(j)}_{k})\|_{2,X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\}}\\!\\!&\stackrel{{\scriptstyle\eqref{eqn:inductive-
McDuff3}}}{{\leq}}&\\!\displaystyle\sum_{r=k+1}^{\infty}2^{-r}+\|w_{k}(z^{(j)}_{k,n_{k}}\otimes
1_{\mathcal{R}})w_{k}^{*}\|\\\ &\stackrel{{\scriptstyle\eqref{eqn:inductive-
McDuff1}}}{{<}}&\\!\displaystyle\sum_{r=k+1}^{\infty}2^{-r}+\frac{1}{k},\end{array}$

which converges to zero as $k\to\infty$. Indeed, as the unit ball of
$\mathcal{M}$ is $\|\cdot\|_{2,X}$-complete and $\psi$ preserves the uniform
2-norm, its image must be
$\|\cdot\|_{2,X\otimes\\{\mathrm{tr}_{\mathcal{R}}\\}}$-closed. ∎

###  5.3. Property $\Gamma$

In its original formulation, a II1 factor $\mathcal{M}$ has property $\Gamma$
if there is an approximately central net of trace-zero unitaries in
$\mathcal{M}$. Since
$\mathcal{R}\cong\overline{\bigotimes}_{n=1}^{\infty}(\mathcal{R},\mathrm{tr}_{\mathcal{R}})$,
the hyperfinite II1 factor has property $\Gamma$. Consequently, all McDuff
factors have property $\Gamma$. On the other hand, Murray and von Neumann’s
$14\epsilon$-argument shows that the factors associated to free groups do not
have property $\Gamma$ ([83, Lemma 6.2.1]).

Dixmier extended the work of Murray and von Neumann, proving that property
$\Gamma$ was equivalent to the existence of systems of approximately central
projections that sum to the unit ([32]). It is through this reformulation that
most structural consequence of property $\Gamma$ are obtained for II1 factors
(see [26, 27, 49, 91]), and so it was the basis for the definition of _uniform
property $\Gamma$_ for $C^{*}$-algebras introduced in [23] and studied further
in [22].

Here, we define property $\Gamma$ for tracially complete $C^{*}$-algebras. It
is an immediate consequence of Proposition 5.7 and a reindexing argument (see
Remark 5.24 below) that a $C^{*}$-algebra $A$ with $T(A)$ compact has uniform
property $\Gamma$ in the sense of [23, Definition 2.1] if and only if its
tracial completion with respect to $T(A)$ has property $\Gamma$ as defined
here (see Proposition 5.20).

######  Definition 5.19 (cf. [23, Definition 2.1]).

Let $(\mathcal{M},X)$ be a factorial tracially complete $C^{*}$-algebra. We
say that $(\mathcal{M},X)$ has _property $\Gamma$_ if for any
$\|\cdot\|_{2,X}$-separable subset $S\subseteq\mathcal{M}$ and any
$k\in\mathbb{N}$ there exist projections
$p_{1},\ldots,p_{k}\in\mathcal{M}^{\omega}\cap S^{\prime}$ summing to
$1_{\mathcal{M}^{\omega}}$ such that

(5.56) $\tau(ap_{i})=\frac{1}{k}\tau(a),\qquad a\in S,\ \tau\in X^{\omega},\
i=1,\dots,k.$

######  Proposition 5.20.

Let $A$ be a separable $C^{*}$-algebra with $T(A)$ compact. Then $A$ has
uniform property $\Gamma$ as in [23, Definition 2.1] if and only if its
tracial completion with respect to $T(A)$ has property $\Gamma$ in the sense
of Definition 5.19.

######  Remark 5.21.

We have chosen to restrict our definition of property $\Gamma$ to the case of
factorial tracially complete $C^{*}$-algebras. Although Definition 5.19 makes
sense for non-factorial tracially complete $C^{*}$-algebras, in the absence of
results, we are not confident that this would be appropriate definition
outside of the factorial setting.

The following simple observation provides many examples of factorial tracially
complete $C^{*}$-algebras with property $\Gamma$. Combining this with
Proposition 5.17 shows that the tracial completion of a separable
$\mathcal{Z}$-stable $C^{*}$-algebra with respect to a compact face of traces
has property $\Gamma$ (cf. [23, Proposition 2.3]).

######  Proposition 5.22.

If $(\mathcal{M},X)$ is a factorial McDuff tracially complete $C^{*}$-algebra,
then $(\mathcal{M},X)$ satisfies property $\Gamma$.

###### Proof.

Let $S\subseteq\mathcal{M}$ be a $\|\cdot\|_{2,X}$-separable set and fix
$k\geq 1$. Fix a unital embedding $\phi\colon
M_{k}\rightarrow\mathcal{M}^{\omega}\cap S^{\prime}$. If $a\in S$, then the
function $\tau(a\phi(\,\cdot\,))$ is a tracial functional on $M_{k}$. Hence
$\tau(a\phi(e_{ii}))=\tau(a\phi(e_{11}))$ for all $i=1,\ldots,k$, and since
$\phi$ is unital and $\sum_{i=1}^{n}e_{ii}=1_{M_{k}}$, we have
$\tau(a\phi(e_{ii}))=\frac{1}{k}\tau(a)$ for all $a\in S$. Thus the
projections $p_{i}\coloneqq\phi(e_{ii})$, $i=1,\ldots,k$, satisfy the
conditions in Definition 5.19. ∎

The following proposition records several equivalent formulation of property
$\Gamma$.

######  Proposition 5.23 (cf. [22, Proposition 2.3]).

For a factorial tracially complete $C^{*}$-algebra $(\mathcal{M},X)$, the
following are equivalent:

  1. (i)

$(\mathcal{M},X)$ satisfies property $\Gamma$.

  2. (ii)

For every finite set $\mathcal{F}\subseteq\mathcal{M}$ and $\epsilon>0$, there
is a self-adjoint contraction $p\in\mathcal{M}$ such that for all
$a\in\mathcal{F}$ and all $\tau\in X$,

(5.57) $\|[p,a]\|_{2,X}<\epsilon,\ \|p-p^{2}\|_{2,X}<\epsilon,\ \text{and}\ \
\big{|}\tau(ap)-\frac{1}{2}\tau(a)\big{|}<\epsilon.$

  3. (iii)

There is $c\in(0,1)$ such that for every $\|\cdot\|_{2,X}$-separable set
$S\subseteq\mathcal{M}^{\omega}$, there is a projection
$p\in\mathcal{M}^{\omega}\cap S^{\prime}$ such that $\tau(ap)=c\tau(a)$ for
all $a\in S$ and $\tau\in X^{\omega}$.

  4. (iv)

There exists a faithful trace $\sigma\in T(\mathbb{C}^{2})$ such that for
every $\|\cdot\|_{2,X}$-separable set $S\subseteq\mathcal{M}^{\omega}$, there
is a unital embedding $\phi:\mathbb{C}^{2}\rightarrow\mathcal{M}^{\omega}\cap
S^{\prime}$ such that $\tau(a\phi(x))=\tau(a)\sigma(x)$ for all $a\in S$,
$x\in\mathbb{C}^{2}$, and $\tau\in X^{\omega}$.

  5. (v)

For every $\|\cdot\|_{2,X}$-separable set $S\subseteq\mathcal{M}^{\omega}$,
there is a ∗-homomorphism $\phi\colon
L^{\infty}[0,1]\rightarrow\mathcal{M}^{\omega}\cap S^{\prime}$ such that

(5.58) $\tau(a\phi(f))=\tau(a)\int_{0}^{1}f(t)dt$

for all $a\in S$, $f\in L^{\infty}[0,1]$, and $\tau\in X^{\omega}$.

###### Proof.

(i)$\Rightarrow$(ii): Apply the definition of property $\Gamma$ with
$k\coloneqq 2$ and $S\coloneqq\mathcal{F}$ and take a suitable element of a
representative sequence for one of the projections witnessing property
$\Gamma$ to give (ii).

(ii)$\Rightarrow$(iii): Let $\\{s^{(i)}:i\in\mathbb{N}\\}$ be a countable
dense set in $S$. For each $i\in\mathbb{N}$, let
$(s^{(i)}_{n})_{n=1}^{\infty}$ be a sequence representing $s^{(i)}$. For each
$n\in\mathbb{N}$, set $\epsilon_{n}\coloneqq 2^{-n}$ and
$\mathcal{F}_{n}\coloneqq\\{s^{(i)}_{n}:i\leq n\\}$. Let $p_{n}\in\mathcal{M}$
be given as in (ii) with $(\mathcal{F}_{n},\epsilon_{n})$ in place of
$(\mathcal{F},\epsilon)$. Then the sequence $(p_{n})_{n=1}^{\infty}$ induces a
projection $p\in\mathcal{M}^{\omega}\cap S^{\prime}$ such that
$\tau(ap)=\frac{1}{2}\tau(a)$ for all $a\in S$ and $\tau\in X^{\omega}$.

(iii)$\Rightarrow$(iv): Let $e_{1},e_{2}\in\mathbb{C}^{2}$ be the standard
basis vectors and let $p\in\mathcal{M}^{\omega}\cap S^{\prime}$ be the
projection constructed in (iii). Define
$\phi:\mathbb{C}^{2}\rightarrow\mathcal{M}^{\omega}\cap S^{\prime}$ by
$\phi(e_{1})\coloneqq p$ and $\phi(e_{2})\coloneqq
1_{\mathcal{M}^{\omega}}-p$. Define $\sigma(x_{1},x_{2})\coloneqq
cx_{1}+(1-c)x_{2}$. Then $\tau(a\phi(x))=\tau(a)\sigma(x)$ for all $a\in S$,
$x\in\mathbb{C}^{2}$, and $\tau\in X^{\omega}$.

(iv)$\Rightarrow$(v): Let
$\phi_{1}\colon\mathbb{C}^{2}\rightarrow\mathcal{M}^{\omega}\cap S^{\prime}$
be a unital embedding such that $\tau(a\phi_{1}(x))=\tau(a)\sigma(x)$ for all
$a\in S$, $x\in\mathbb{C}^{2}$, and $\tau\in X^{\omega}$. Then define
$S_{1}\coloneqq C^{*}(\phi_{1}(\mathbb{C}^{2})\cup S)$ and use (iv) again to
produce a unital embedding
$\phi_{2}\colon\mathbb{C}^{2}\rightarrow\mathcal{M}^{\omega}\cap
S_{1}^{\prime}$ such that $\tau(a\phi_{2}(x))=\tau(a)\sigma(x)$ for all $a\in
S_{1}$, $x\in\mathbb{C}^{2}$, and $\tau\in X^{\omega}$.

Continuing inductively, there are unital embeddings
$\phi_{n}\colon\mathbb{C}^{2}\rightarrow\mathcal{M}^{\omega}\cap S^{\prime}$
with commuting ranges such that the induced embedding

(5.59)
$\phi\colon\bigotimes_{n=1}^{\infty}\mathbb{C}^{2}\rightarrow\mathcal{M}^{\omega}\cap
S^{\prime}$

satisfies $\tau(a\phi(x))=\tau(a)\sigma^{\otimes\infty}(x)$ for all $a\in S$,
$x\in\bigotimes_{n=1}^{\infty}\mathbb{C}^{2}$, and $\tau\in X^{\omega}$.

Since $\mathcal{M}^{\omega}\cap S^{\prime}$ is tracially complete, $\phi$
extends to a ∗-homomorphism

(5.60)
$\bar{\phi}\colon\pi_{\sigma^{\otimes\infty}}\Big{(}\bigotimes_{n=1}^{\infty}\mathbb{C}^{2}\Big{)}^{\prime\prime}\rightarrow\mathcal{M}^{\omega}\cap
S^{\prime}.$

Writing, $\bar{\sigma}$ for the trace on
$\pi_{\sigma^{\otimes\infty}}\Big{(}\bigotimes_{n=1}^{\infty}\mathbb{C}^{2}\Big{)}^{\prime\prime}$
induced by $\sigma^{\otimes\infty}$, we have
$\tau(a\bar{\phi}(x))=\tau(a)\bar{\sigma}(x)$ for all $a\in S$,
$x\in\pi_{\sigma^{\otimes\infty}}\big{(}\bigotimes_{n=1}^{\infty}\mathbb{C}^{2}\big{)}^{\prime\prime}$,
and $\tau\in X^{\omega}$. The uniqueness of the standard probability space
then provides an isomorphism

(5.61)
$\pi_{\tau_{0}}\Big{(}\bigotimes_{n=1}^{\infty}\mathbb{C}^{2}\Big{)}^{\prime\prime}\cong
L^{\infty}[0,1]$

which carries $\bar{\sigma}$ to the trace on $L^{\infty}[0,1]$ given by
integration with respect to the Lebesgue measure.

(v)$\Rightarrow$(i): For $k\geq 1$ and $1\leq i\leq k$, set
$p_{i}\coloneqq\phi(\chi_{[(i-1)/k,i/k)})$. ∎

######  Remark 5.24.

Using reindexing, as in Remark 5.15, we can see that Definition 5.12 is
equivalent to an _a priori_ stronger definition where $S$ is allowed to be any
$\|\cdot\|_{2,X^{\omega}}$-separable subset of $\mathcal{M}^{\omega}$.
Similarly, each of (iii), (iv) and (v) is equivalent to an _a priori_ weaker
statement with $S\subseteq\mathcal{M}$.

The local characterisation of property $\Gamma$ given in Proposition 5.23(ii)
provides the following permanence property. Recall that factoriality is
preserved by inductive limits by Proposition 3.34.

######  Proposition 5.25.

Property $\Gamma$ is preserved by inductive limits of factorial tracial
complete $C^{*}$-algebras.

######  Remark 5.26.

It is also true that the reduced product of a sequence of factorial tracially
complete $C^{*}$-algebras with property $\Gamma$ is both factorial and has
property $\Gamma$, but we do not yet have the machinery to prove the factorial
part of this claim.454545Recall that in general it is not true that a reduced
product of factorial tracially complete $C^{*}$-algebras is factorial (see
Remark 5.10). Once the factorial issue is sorted, it is clear that the
condition in Proposition 5.23(ii) is preserved by reduced products. See
Corollary 7.7(ii).

The definition of property $\Gamma$ requires that one can tracially divide all
elements of a tracially complete $C^{*}$-algebra in an approximately central
fashion, whereas for a II1 factor $\mathcal{M}$ it suffices just to divide the
unit (i.e. (5.56) only needs to hold for $a=1_{\mathcal{M}^{\omega}}$). In
[22, Proposition 3.2], several of the present authors observed that such a
result is also true for separable $C^{*}$-algebras with Bauer tracial
simplices. The same holds in the tracially complete setting with a very
similar proof.

######  Proposition 5.27 (cf. [22, Corollary 3.2]).

Let $(\mathcal{M},X)$ be a factorial tracially complete $C^{*}$-algebra such
that $X$ is a Bauer simplex. Suppose that for any $\|\cdot\|_{2,X}$-separable
subset $S\subseteq\mathcal{M}$ and $k\in\mathbb{N}$, there exists pairwise
orthogonal projections $p_{1},\dots,p_{k}\in\mathcal{M}^{\omega}\cap
S^{\prime}$ with $\tau(p_{i})=1/k$ for all $i=1,\dots,k$. Then
$(\mathcal{M},X)$ has property $\Gamma$.

###### Proof.

First assume that $\mathcal{M}$ is $\|\cdot\|_{2,X}$-separable, so we can take
$\mathcal{S}\coloneqq\mathcal{M}$ in the statement. Using Remark 5.24, it will
suffice to show that we automatically have

(5.62) $\tau(ap_{i})=\frac{1}{k}\tau(a),\qquad a\in\mathcal{M},\ \tau\in
X^{\omega},\ i=1,\ldots k.$

By [10, Proposition 3.9], $\mathcal{M}^{\omega}$ is a $W^{*}$-bundle over
$(\partial_{e}X)^{\omega}$ (in the notation defined before [10, Proposition
3.9]),464646In [10], $\omega$ is taken to be an ultrafilter, but the same
considerations apply to any free filter on $\mathbb{N}$. so it suffices to
prove (5.62) for $\tau\in(\partial_{e}X)^{\omega}$, which means it suffices to
assume that $\tau$ is a limit trace coming from a sequence $(\tau_{n})$ of
traces in $\partial_{e}X$ and an ultrafilter $\omega^{\prime}\supseteq\omega$.
In this case, since $\partial_{e}X$ is compact,
$\tau|_{\mathcal{M}}\in\partial_{e}X$ (as it is the $\omega^{\prime}$-limit of
the $\tau_{n}$). Define $\sigma\colon\mathcal{M}\to\mathbb{C}$ by
$\sigma(x)\coloneqq\tau(xp_{i})$. This is a positive tracial functional such
that $\sigma(1_{\mathcal{M}})=\frac{1}{k}$; since $\tau$ is an extreme point
in $T(\mathcal{M})$, we have $\sigma=\frac{1}{k}\tau$, which amounts to
(5.62).

In the non-separable case, we can use the fact that factoriality is separably
inheritable (Theorem A.3(i)) to see that for any $\|\cdot\|_{2,X}$-separable
subset $S$ of $\mathcal{M}$, there exists a $\|\cdot\|_{2,X}$-separable
factorial tracially complete subalgebra $\mathcal{M}_{0}$ of $\mathcal{M}$
which contains $S$ and satisfies the hypotheses of the proposition and,
therefore, has property $\Gamma$. Consequently, $\mathcal{M}$ has property
$\Gamma$ as the projections witnessing property $\Gamma$ for $\mathcal{M}_{0}$
also witness property $\Gamma$ for $\mathcal{M}$. ∎

In the setting of $W^{*}$-bundles, abstracting the results in [110, 97, 68],
an amenable factorial type II1 tracially complete $C^{*}$-algebra
$(\mathcal{M},X)$ with $\partial_{e}X$ compact and finite dimensional turns
out to have property $\Gamma$. This will be contained in the forthcoming work
by the third- and fifth-named authors, which will show that a factorial
$W^{*}$-bundle over a finite dimensional space has property $\Gamma$ when
viewed as a tracially complete $C^{*}$-algebra if and only if each fibre has
property $\Gamma$ as a von Neumann algebra. We show here that this holds in
the zero dimensional setting.

######  Proposition 5.28.

If $(\mathcal{M},X)$ is a factorial tracially complete $C^{*}$-algebra such
that $\partial_{e}X$ is compact and totally disconnected and
$\pi_{\tau}(\mathcal{M})^{\prime\prime}$ has property $\Gamma$ for each
$\tau\in\partial_{e}X$, then $(\mathcal{M},X)$ has property $\Gamma$. In
particular, if $(\mathcal{M},X)$ is an amenable type $\mathrm{II}_{1}$
factorial tracially complete $C^{*}$-algebra such that $\partial_{e}X$ is
compact and totally disconnected, then $(\mathcal{M},X)$ has property
$\Gamma$.

###### Proof.

Note that the second statement follows from the first. Indeed, if
$(\mathcal{M},X)$ is as in the second statement, then for all
$\tau\in\partial_{e}X$, $\pi_{\tau}(\mathcal{M})^{\prime\prime}$ is
semidiscrete by Theorem 1.2 and is a II1 factor by Proposition 3.14. It then
follows from [29, Corollary 2.2] that $\pi_{\tau}(\mathcal{M})^{\prime\prime}$
has property $\Gamma$.

We now prove the first statement. As $\partial_{e}X$ is compact, $\mathcal{M}$
can be viewed as a $W^{*}$-bundle over $K\coloneqq\partial_{e}X$ by Theorem
3.37. Let $C(K)\subseteq\mathcal{M}$ be the corresponding inclusion. Fix a
finite set $\mathcal{F}\subset\mathcal{M}$, $\epsilon>0$, and
$k\in\mathbb{N}$. By Proposition 5.27, it suffices to show that there are
positive contractions $p_{1},\ldots,p_{k}\in\mathcal{M}$ such that

  1. (i)

$\|[p_{i},a]\|_{2,X}<\epsilon$ for all $a\in\mathcal{F}$ and $i=1,\ldots,k$,

  2. (ii)

$\|p_{i}-p_{i}^{2}\|_{2,X}<\epsilon$ for all $i,\ldots,k$,

  3. (iii)

$\|p_{i}p_{i^{\prime}}\|_{2,X}<\epsilon$ for $i,i^{\prime}=1,\ldots,k$ with
$i\neq i^{\prime}$, and

  4. (iv)

$|\tau(p_{i})-1/k|<\epsilon$ for all $\tau\in X$ and $i=1,\ldots,k$.

For each $\tau\in K$, as the II1 factor
$\pi_{\tau}(\mathcal{M})^{\prime\prime}$ satisfies property $\Gamma$, there
are mutually orthogonal projections
$\bar{p}_{1,\tau},\ldots,\bar{p}_{k,,\tau}\in\pi_{\tau}(\mathcal{M})^{\prime\prime}$
with trace $1/k$ such that

(5.63) $\|[\bar{p}_{i,\tau},\pi_{\tau}(a)]\|_{2,\tau}<\epsilon,\qquad
a\in\mathcal{F},\ i=1,\ldots,k.$

Using Kaplansky’s density theorem to approximate each $\bar{p}_{\tau,i}$ by a
positive contraction in $\mathcal{M}$ in the norm $\|\cdot\|_{2,\tau}$
together with the compactness of $K$, there is a finite open cover
$U_{1},\ldots,U_{n}$ of $K$ and positive contractions $p_{i,j}\in\mathcal{M}$
for $i=1,\ldots,k$ and $j=1,\ldots,n$ such that

  1. (i′)

$\|[p_{i,j},a]\|_{2,U_{j}}<\epsilon$ for all $a\in\mathcal{F}$,
$i=1,\ldots,k$, and $j=1,\ldots,n$,

  2. (ii′)

$\|p_{i,j}-p_{i,j}^{2}\|_{2,U_{j}}<\epsilon$ for all $i,\ldots,k$ and
$j=1,\ldots,n$,

  3. (iii′)

$\|p_{i,j}p_{i^{\prime},j}\|_{2,U_{j}}<\epsilon$ for $i,i^{\prime}=1,\ldots,k$
with $i\neq i^{\prime}$ and $j=1,\ldots,n$, and

  4. (iv′)

$|\tau(p_{i,j})-1/k|<\epsilon$ for all $\tau\in U_{j}$ and $i=1,\ldots,k$.

As $K$ is totally disconnected, after refining the open cover, we may assume
that the sets $U_{1},\ldots,U_{n}$ form a clopen partition of $K$. Then the
elements

(5.64) $p_{i}\coloneqq\sum_{j=1}^{n}\chi_{U_{j}}p_{i,j}\in\mathcal{M},\qquad
j=1,\ldots n,$

are positive contractions satisfying (i)–(iv). ∎

It remains open whether the approximately central division of the unit as in
Proposition 5.27 implies property $\Gamma$ outside the setting of Bauer
simplices ([22, Question 3.5]). This question is just as valid in the
tracially complete framework.

######  Question 5.29.

Does Proposition 5.27 hold for all factorial tracially complete
$C^{*}$-algebras (i.e. without assuming that $X$ is a Bauer simplex)?

Immediately following [22, Proposition 5.10], the second-, third-, and last
two named authors asserted that all the Villadsen algebras of the first type –
namely the examples from [114], which were used by Toms to give striking
counterexamples to Elliott’s original classification conjecture ([109]) – have
property $\Gamma$. These algebras are so-called diagonal AH algebras (i.e.
inductive limits of homogeneous algebras of a particularly nice form), and
they prove a fertile testing ground for comparing regularity properties of
very different natures, with the analysis in [112] leading to a positive
solution to the Toms–Winter conjecture on this class of algebras. Uniform
tracial completions of diagonal AH algebras satisfy the central divisibility-
of-the-unit hypothesis of Proposition 5.27 (this is [22, Proposition 5.10])
and the assertion that Villadsen algebras of the first type have property
$\Gamma$ was based on [112, Section 8] (where it is suggested, based on
computations in [108, Theorem 4.1], that these have Bauer simplices of
traces).474747In [22], the second-, third-, and last two named authors
incorrectly referenced the non-existent [109, Theorem 4.1] rather than [108,
Theorem 4.1]. However recent results of Elliott, Li, and Niu ([36, Theorem
4.5]) show that, in fact, the trace space of Villadsen algebras of the first
type is the Poulsen simplex (whenever the algebra is not $\mathcal{Z}$-stable)
– maximally far from being Bauer! Accordingly, the argument in [22] is not
valid, and it does not seem straightforward to determine whether the Villadsen
type I algebras have property $\Gamma$.

######  Question 5.30.

Do the uniform tracial completions of the (non-$\mathcal{Z}$-stable) Villadsen
algebras of the first type have property $\Gamma$?

The connecting maps in the inductive limit construction of a Villadsen type I
algebra involve a combination of coordinate projections and point evaluations,
the latter of which ensure simplicity of the inductive limit. In the case of a
non-$\mathcal{Z}$-stable Villadsen type I algebra $A$, the point evaluations
are relatively sparse (see [112, Theorem 3.4 and Lemma 5.1]). As such they do
not affect the structure of the tracial completion of $A$. For this reason,
the heart of Question 5.30 lies in determining whether $\Gamma$ holds for the
tracially complete inductive limit of the trivial $W^{*}$-bundles

(5.65) $C([0,1])\to C([0,1]^{2},M_{2})\to C([0,1]^{4},M_{4})\to\cdots,$

where each stage is equipped with all its traces, and the connecting maps are
given as the direct sum of the two coordinate projections.

##  6\. Complemented partitions of unity

This section formally introduces the notation of CPoU for factorial tracially
complete $C^{*}$-algebras as discussed in Section 1.4. The definition of CPoU
is given in Section 6.1. Some examples of tracially complete $C^{*}$-algebras
with CPoU can be found in Section 6.2, and permanence properties are discussed
in Section 6.3. The proof Theorem 1.4 (property $\Gamma$ implies CPoU) will be
given in Section 6.4.

###  6.1. Formal definition and its reformulations

The following definition extends the notion of CPoU introduced in [23] in the
$C^{*}$-setting to factorial tracial complete $C^{*}$-algebras. More
precisely, a separable $C^{*}$-algebra $A$ with $T(A)$ compact has CPoU in the
sense of [23] if and only its tracial completion with respect to $T(A)$ has
CPoU in the sense of Definition 6.1 below; this is immediate from the
definition and Proposition 5.7, which identifies the uniform 2-norm central
sequence algebra of $A$ with that of its uniform tracial completion.484848In
[23], CPoU was only defined in the separable setting, but we will not make
this restriction here. Recall our convention that (unless specified otherwise)
$\omega$ denotes a free filter on $\mathbb{N}$. It will follow from the local
characterisation in Proposition 6.2 that Definition 6.1 does not depend on the
choice of filter $\omega$.

######  Definition 6.1 (cf. [23, Definition 3.1]).

Let $(\mathcal{M},X)$ be a factorial tracially complete $C^{*}$-algebra. We
say that $(\mathcal{M},X)$ has _complemented partitions of unity_ (CPoU) if
for any $\|\cdot\|_{2,X}$-separable subset $S\subseteq\mathcal{M}$, any family
$a_{1},\dots,a_{k}$ of positive elements in $\mathcal{M}$, and any scalar

(6.1) $\delta>\sup_{\tau\in X}\min_{1\leq i\leq k}\tau(a_{i}),$

there exist orthogonal projections
$p_{1},\dots,p_{k}\in\mathcal{M}^{\omega}\cap S^{\prime}$ summing to
$1_{\mathcal{M}^{\omega}}$ such that

(6.2) $\tau(a_{i}p_{i})\leq\delta\tau(p_{i}),\qquad\tau\in X^{\omega},\
i=1,\dots,k.$

We note that when $\mathcal{M}$ is $\|\cdot\|_{2,X}$-separable, it suffices to
take $S\coloneqq\mathcal{M}$. As with property $\Gamma$, we have have chosen
to restrict the definition to the setting of factorial tracially complete
$C^{*}$-algebras since it is not clear if this definition is appropriate
outside of the factorial setting.

The following proposition provides an equivalent formulation of CPoU that
avoids the language of reduced products. Moreover, a reindexing argument shows
that Definition 6.1 is equivalent to an _a priori_ stronger definition where
$a_{1},\ldots,a_{k}\in\mathcal{M}^{\omega}$ and $S$ can be any
$\|\cdot\|_{2,X}$-separable subspace of $\mathcal{M}^{\omega}$.

######  Proposition 6.2.

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

  1. (i)

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

  2. (ii)

for every finite set $\mathcal{F}\subseteq\mathcal{M}$, $\epsilon>0$,
$a_{1},\dots,a_{k}\in\mathcal{M}_{+}$, and

(6.3) $\delta>\sup_{\tau\in X}\min_{1\leq i\leq k}\tau(a_{i}),$

there exist orthogonal positive contractions $e_{1},\dots,e_{k}\in\mathcal{M}$
such that

(6.4) $\displaystyle\Big{\|}\sum_{j=1}^{k}e_{j}-1_{\mathcal{M}}\Big{\|}_{2,X}$
$\displaystyle<\epsilon,$
$\displaystyle\max_{x\in\mathcal{F}}\|[e_{i},x]\|_{2,X}$
$\displaystyle<\epsilon,$ $\displaystyle i=1,\dots,k,$
$\displaystyle\tau(a_{i}e_{i})-\delta\tau(e_{i})$ $\displaystyle<\epsilon,$
$\displaystyle\tau\in X,\ i=1,\dots,k;$

  3. (iii)

for every $\|\cdot\|_{2,X}$-separable subset $S\subseteq\mathcal{M}$,
$a_{1},\dots,a_{k}\in\mathcal{M}^{\omega}$, and

(6.5) $\delta>\sup_{\tau\in X^{\omega}}\min_{1\leq i\leq k}\tau(a_{i}),$

there exist orthogonal projections
$p_{1},\dots,p_{k}\in\mathcal{M}^{\omega}\cap S^{\prime}$ summing to
$1_{\mathcal{M}^{\omega}}$ such that

(6.6) $\tau(a_{i}p_{i})\leq\delta\tau(p_{i}),\qquad\tau\in X^{\omega},\
i=1,\dots,k.$

###### Proof.

(i)$\Rightarrow$(ii): Let $\epsilon>0$, let
$a_{1},\dots,a_{k}\in\mathcal{M}_{+}$, let $\mathcal{F}\subseteq\mathcal{M}$
be a finite subset, and let $\delta>0$ be such that (6.1) holds. Since
$(\mathcal{M},X)$ has CPoU, we obtain orthogonal projections
$p_{1},\ldots,p_{k}\in\mathcal{M}^{\omega}\cap\mathcal{F}^{\prime}$ summing to
$1_{\mathcal{M}^{\omega}}$ such that (6.2) hold. We can lift
$p_{1},\ldots,p_{k}\in\mathcal{M}^{\omega}$ to orthogonal positive
contractions $e^{(1)},\ldots,e^{(k)}\in\ell^{\infty}(\mathcal{M})$ by [1,
Proposition 2.6] (this is a special case of projectivity of cones over finite
dimensional $C^{*}$-algebras; see [72, Corollary 3.8]). Write
$(e^{(i)}_{j})_{j=1}^{\infty}$ for a sequence of positive contractions
representing $e^{(i)}$ for $i=1,\ldots,k$. Note that for each
$j\in\mathbb{N}$, $e^{(1)}_{j},\ldots,e^{(k)}_{j}$ are orthogonal positive
contractions. Since the $p_{i}$ sum to $1_{\mathcal{M}^{\omega}}$ and commute
with $\mathcal{F}$, we have

(6.7)
$\limsup_{j\to\omega}\left\|e^{(1)}_{j}+\cdots+e^{(k)}_{j}-1_{\mathcal{M}}\right\|_{2,X}=0,$

and

(6.8) $\limsup_{j\to\omega}\|[e^{(i)}_{j},x]\|_{2,X}=0,\quad x\in\mathcal{F},\
i=1,\ldots,k.$

Moreover, we must have

(6.9) $\limsup_{j\to\omega}\sup_{\tau\in
X}\Big{(}\tau\big{(}a_{i}e^{(i)}_{j}\big{)}-\delta\tau\big{(}e^{(i)}_{j}\big{)}\Big{)}\leq
0,\quad i=1,\ldots,k,$

as otherwise we could choose a sequence of traces $(\tau_{j})_{j=1}^{\infty}$
in $X$ and an ultrafilter $\omega^{\prime}\supseteq\omega$ such that the
corresponding limit trace $\tau$ does not satisfy
$\tau(a_{i}p_{i})\leq\delta\tau(p_{i})$. Therefore, taking $e_{i}\coloneqq
e^{(i)}_{j}$ for suitable choice of $j$, the orthogonal positive contractions
$e_{1},\ldots,e_{k}$ will satisfy (6.4).

(ii)$\Rightarrow$(iii): Let $S\subseteq\mathcal{M}^{\omega}$ be a
$\|\cdot\|_{2,X^{\omega}}$-separable subset and fix a countable dense subset
$\\{s^{(i)}:i\in\mathbb{N}\\}$ of $S$. For each $i\in\mathbb{N}$, let
$(s^{(i)}_{j})_{j=1}^{\infty}$ be a sequence representing $s^{(i)}$. Let
$a_{1},\ldots,a_{k}\in\mathcal{M}^{\omega}$ be positive contractions in
$\mathcal{M}^{\omega}$. Let $(a^{(i)}_{j})_{j=1}^{\infty}$ be a sequence of
positive contractions representing $a_{i}$ for $i=1,\ldots,k$. Let $\delta>0$