Proof. Let $\tau _0$ be a normal trace on N. Replacing N with $\pi _{\tau _0}(N)$ (where $\pi _\tau $ is the Gelfand–Naimark–Segal representation corresponding to $\tau _0$ ) if necessary, we may assume that $\tau _0$ is faithful. We need to show that $\theta $ and $\pi $ are unitarily equivalent as maps into the tracial ultrapower $N^\omega _\tau $ of N with respect to $\tau _0$ .

Define $\mu :A\rightarrow \mathbb {M}_2(N^\omega _\tau )$ by

$$ \begin{align*}\mu(a)= \begin{bmatrix}\theta(a)&0\\0&\pi(a)\end{bmatrix}.\end{align*} $$

Then $\mu $ is a weakly nuclear *-homomorphism. We claim that $M:=\pi (A)''$ is hyperfinite.

Indeed, let $\psi _n:A\rightarrow F_n$ and $\varphi _n:F_n\rightarrow M$ be contractive completely positive maps for finite-dimensional C*-algebras $F_n$ such that $(\varphi _n\circ \psi _n)$ converges in the point-weak* topology to $\mu $ . Fix a unital normal representation $M\subseteq \mathbb {B}(\mathcal {H})$ and define maps $\eta ,\eta _n:A\otimes _{\mathrm {alg}} M'\rightarrow \mathbb {B}(\mathcal {H})$ by $\eta (a\otimes b)= \mu (a)b$ and ${\eta _n(a\otimes b)}= {\varphi _n\circ \psi _n(a)b}$ , so that $(\eta _n)$ converges to $\eta $ in the point-weak* topology. The $\eta _n$ are continuous with respect to the minimal tensor product since they factor through $F_k\otimes M'$ and hence so is $\eta $ . A conditional expectation of $\mathbb {B}(\mathcal {H})$ onto $M'$ is then provided by [Reference Brown and Ozawa4, Proposition 3.6.5], confirming that M is injective and hence hyperfinite by Connes’ theorem [Reference Connes9, Theorem 6].

Define projections

$$ \begin{align*}p_1= \begin{bmatrix}1&0\\0&0\end{bmatrix},\quad p_2= \begin{bmatrix}0&0\\0&1\end{bmatrix}\end{align*} $$

in $\mathbb {M}_2(N^\omega _\tau )\cap \pi (A)'$ . Then $\tau (p_1x)= \tau (p_2x)$ for every normal trace $\tau $ and every $x\in \pi (A)$ and hence also for every $x\in \pi (A)''$ . By [Reference Castillejos, Evington, Tikuisis, White and Winter8, Lemma 4.5], $\tau (p_1)= \tau (p_2)$ for every trace $\tau $ on $\mathbb {M}_2(N^\omega _\tau )\cap \pi (A)'$ . Thus, there is a unitary $u=[u_{i,j}]\in \mathbb {M}_2(N^\omega _\tau )\cap \pi (A)'$ such that $u^*p_1u=p_2$ and hence $u_{1,1}= 0$ and $u_{1,2}^*u_{1,2}= 1_{N^\omega _\tau }$ . Therefore, $u_{1,2}$ is unitary since $N^\omega _\tau $ is finite, and $\theta (a)u_{1,2}=u_{1,2}\pi (a)$ since $u\in \pi (A)'$ .

Proof of Lemma 2.3.

Let $\mathcal {F}\subset A$ be a finite set of contractions, $\mathcal {S}$ a finite set of normal states on N and $\epsilon>0$ . Define a normal state $\rho = \sum _{\rho '\in \mathcal {S}}\rho '/|\mathcal {S}|$ , so that $\|b\|_{\rho '}^2\leq |\mathcal {S}|\|b\|_\rho ^2$ for every $\rho '\in \mathcal {S}$ and $b\in N$ . Thus, we need only make reference to $\rho $ in the proof, rather than the entire set $\mathcal {S}$ .

Note that N may be decomposed into $N_{1}\oplus N_\infty $ for von Neumann algebras $N_{1},N_\infty $ that are respectively finite and properly infinite. Similarly, $\pi = \pi _{1}\oplus \pi _\infty $ for weakly nuclear *-homomorphisms $\pi _{1}:A\rightarrow N_{1}$ and $\pi _{\infty }:A\rightarrow N_{\infty }$ . We shall deal with each summand separately by assuming that it composes all of N.

Case 1. N is properly infinite. We begin by following the proof of [Reference Carrión and Schafhauser6, Proposition 3]. By the assumption of weak nuclearity, there are $k\in \mathbb {Z}^+$ and unitary completely positive maps $A\stackrel {\psi '}{\longrightarrow }\mathbb {M}_{k}\stackrel {\dot {\varphi }'}{\longrightarrow }N$ such that $\|\dot {\varphi }'\circ \psi '(x)-\pi (x)\|_{\rho }\leq \epsilon $ for every $x\in \mathcal {F}$ . Since A is separable, we may fix a faithful unital representation $A\subseteq \mathbb {B}(\mathcal {H})$ such that $\mathcal {H}$ is separable and A contains no nonzero compact operators. Voiculescu’s theorem [Reference Davidson11, Theorem II.5.3] provides an isometry $v:\mathbb {C}^{k}\rightarrow \mathcal {H}$ such that $\|v^*xv-\psi '(x)\|\leq \epsilon $ for every $x\in \mathcal {F}$ . Likewise, quasidiagonality provides a finite-rank projection $p\in \mathbb {B}(\mathcal {H})$ such that $\|pv-p\|\leq \epsilon $ and $\|px-xp\|\leq \epsilon $ for every $x\in \mathcal {F}$ . Define unitary completely positive maps $A\stackrel {\psi }{\longrightarrow }\mathbb {B}(p\mathcal {H})\stackrel {\dot {\varphi }}{\longrightarrow }N$ by $\psi (a)= pap$ and $\dot {\varphi }(T)= \dot {\varphi }'(v^*Tv)$ . Thus, for all $x,y\in \mathcal {F}$ ,

$$ \begin{align*} \|\psi(x)\psi(y)-\psi(xy)\|&\leq \|p\|\,\|xp-px\|\,\|yp\|\leq\epsilon,\\ \|\dot{\varphi}\circ\psi(x)-\pi(x)\|_{\rho}& \leq \|\dot{\varphi}'\|\,\|v^*\psi(x)v-\psi'(x)\|+\epsilon \\&\leq \|v^*p-v^*\|\,\|xpv\|+\|v^*x\|\,\|pv-v\|+2\epsilon\leq 4\epsilon. \end{align*} $$

We now follow the proof of [Reference Brown, Carrión and White3, Lemma 2.4], beginning on page 50 with the explanation that the reference itself is following the proof of [Reference Haagerup12, Proposition 2.2]. Our properly infinite assumption finally kicks in, allowing us to find a unital embedding $\iota :\mathbb {B}(p\mathcal {H})\rightarrow N$ . By [Reference Haagerup12, Proposition 2.1], there is an isometry $w\in N$ such that $\dot {\varphi }(T)= w^*\iota (T)w$ for all $T\in \mathbb {B}(p\mathcal {H})$ . Then [Reference Haagerup12, page 167] shows how w may be approximated by a unitary u so that

$$ \begin{align*}\|\!\operatorname{\mathrm{Ad}}(u^*)\circ\iota(T)-\dot{\varphi}(T)\|_\rho\leq \|T\|\epsilon\leq \epsilon\end{align*} $$

for every $T\in \psi (\mathcal {F})\subset \mathbb {B}(p\mathcal {H})$ . Therefore, this case is proven by defining the *-homomorphism $\varphi = \operatorname {\mathrm {Ad}}(u^*)\circ \iota $ .

Case 2. N is finite. This proof follows exactly in the footsteps of that of [Reference Brown, Carrión and White3, Lemma 2.5], until its last two paragraphs. At that point we note that $\theta $ is nuclear by definition: we have finite-dimensional unitary completely positive maps $\widetilde {\psi }_n:=\bigoplus _{k=1}^n\psi _k$ mapping $A\rightarrow \bigoplus _{k=1}^nF_k$ and *-homomorphisms $\widetilde {\phi }_n:\bigoplus _{k=1}^nF_k\rightarrow N^\omega $ given by

$$ \begin{align*} \widetilde{\phi}_n((T_1,\ldots,T_n))= q((\phi_1(T_1),\ldots, \phi_{n-1}(T_{n-1}),\phi_{n}(T_{n}),\phi_{n}(T_n),\ldots )_n). \end{align*} $$

Thus, $\theta = \lim _\omega \phi _n\circ \psi _n= \lim _n\widetilde {\phi }_n\circ \widetilde {\psi }_n$ . This allows us to use Theorem 2.1 to deduce that $\theta $ is strong* approximately unitarily equivalent to $\pi ^\omega $ , and the proof is finished using Kirchberg’s $\epsilon $ -test as in the original.

Proof of Lemma 1.2.

The composition of a trace with an order-zero map is again (rescalable into) a trace [Reference Winter and Zacharias15, Corollary 3.4], so each $\tau \circ \varphi _n$ is a trace on the finite-dimensional C*-algebra $F_n$ . By assumption, the $\psi _n$ are approximately multiplicative and $\tau \circ \varphi _n\circ \psi _n\rightarrow \tau \circ \pi $ in the weak* topology. Therefore, $\tau \circ \pi $ is quasidiagonal.

Proof of Theorem 1.3.

Let $A\stackrel {\psi _n}{\longrightarrow }F_n\stackrel {\varphi _n}{\longrightarrow }B^{**}$ be an approximately multiplicative $\sigma $ -strong*-decomposition. By Arveson’s extension theorem, we have conditional expectations $E_n:B^{**}\rightarrow \varphi _n(F_n)$ . Moreover, by Alaoglu’s theorem, we may pass $(E_n)$ to a subsequence that converges to a linear map $E:B^{**}\rightarrow B^{**}$ in the point- $\sigma $ -weak topology (see [Reference Brown and Ozawa4, Theorem 1.3.7]).

Fix $\epsilon>0$ , $a\in A$ and a normal functional $\eta \in B^*$ . Then $\eta \circ E$ is also a normal functional, so by the previous paragraph there exists $m\in \mathbb {N}$ such that $n>m$ implies that each of the following hold:

$$ \begin{align*} |\eta(E_n(\pi(a))-E(\pi(a)))|&\leq\epsilon/4,\\ |\eta\circ E(\pi(a)-\varphi_n\circ\psi_n(a))|&\leq\epsilon/4,\\ |\eta(E(\varphi_n\circ\psi_n(a))-E_n(\varphi_n\circ\psi_n(a)))|&\leq\epsilon/4,\\ |\eta(\varphi_n\circ\psi_n(a)-\pi(a))|&\leq\epsilon/4. \end{align*} $$

Combined,

$$ \begin{align*} |\eta(E_n\circ\pi(a)-\pi(a))|&\leq\epsilon \end{align*} $$

and therefore the inclusion $\pi (A)\subseteq B^{**}$ is weakly nuclear.

We now use a Hahn–Banach argument akin to the proof (as in [Reference Brown and Ozawa4, Proposition 2.3.6]) that if $A^{**}$ is semidiscrete, then A is nuclear. Let X be the set of all maps from $\pi (A)$ to $B^{**}$ of the form $\varphi \circ \psi $ for two unitary completely positive maps $\psi :\pi (A)\rightarrow F$ and $\varphi :F\rightarrow B^{**}$ and finite-dimensional F. Then X is convex.

Indeed, let $w\in (0,1)$ and $\varphi _0\circ \psi _0,\varphi _1\circ \psi _{1}\in X$ . Then it follows that ${\psi :=\psi _0\oplus \psi _1:\pi (A)\rightarrow F_0\oplus F_1}$ is unitary completely positive, as is the map $\varphi :F_0\oplus F_1\rightarrow B^{**}$ given by $\varphi ((M_0,M_1))= w\varphi _0(M_0)+(1-w)\varphi _1(M_1)$ . Thus, $w\varphi _0\circ \psi _0+(1-w)\varphi _1\circ \psi _1= \varphi \circ \psi \in ~X$ .

Fix $\mathcal {F}=\{b_i~|~i\in [1,k]\}\subset \pi (A)$ . By weak nuclearity, the k-tuple $(b_i)_{i=1}^k\in \bigoplus _{i=1}^kB^{**}$ is in the weak-closure of the set $\{(\varphi \circ \psi (b_i))_{i=1}^k~|~\varphi \circ \psi \in X\}$ and hence also in its norm-closure by the Hahn–Banach theorem. Therefore, there is a map $\varphi \circ \psi \in X$ such that

$$ \begin{align*} \max_i\|b_i-\varphi\circ\psi(b_i)\|= \|(b_i)_{i=1}^k-(\varphi\circ\psi(b_i))_{i=1}^k\|<\epsilon.\\ \end{align*} $$

Proof. $\boldsymbol {\Rightarrow }$ : We retread the proof of Lemma 2.3, beginning by fixing normal state $\rho $ , finite $\mathcal {F}\subset A$ and $\epsilon>0$ .

We may skip the first paragraph of the properly infinite case, instead using the assumed approximately multiplicative norm-decomposition to provide $A\stackrel {\psi }{\longrightarrow }F\stackrel {\dot {\varphi }}{\longrightarrow }B$ that satisfy the necessary inequalities. We now need only use [Reference Haagerup12] to find a unitary $u\in N$ so that $\varphi = \operatorname {\mathrm {Ad}}(u^*)\circ \iota $ approximates $\dot {\varphi }$ for a unital embedding $\iota :F\rightarrow B^{**}$ .

The finite case does not require that A be quasidiagonal. Of course, property (i) of our norm-decomposition witnesses the nuclearity of $\pi $ and hence $\pi :A\rightarrow B^{**}$ is weakly nuclear. Therefore, Lemma 1.2 allows us finish this case, and this direction.

$\boldsymbol {\Leftarrow }$ : This is a perturbation of [Reference Hirshberg, Kirchberg and White13, Theorem 1.4]. Let $A\stackrel {\psi _n}{\longrightarrow }F_n\stackrel {\dot {\varphi }_n}{\longrightarrow }B^{**}$ be an approximately multiplicative $\sigma $ -strong*-decomposition. Using [Reference Hirshberg, Kirchberg and White13, Lemma 1.1], we can find order-zero unitary completely positive maps $\varphi _{n}:F_n\rightarrow B$ (note the range) such that $\varphi _n\circ \psi _n(x)\rightarrow \pi (x)$ in the $\sigma $ -weak topology for every $x\in A$ . We once again conclude with a Hahn–Banach argument.

Proof. Fix $\epsilon>0$ and a finite subset of contractions $\mathcal {G}\subset D$ . Let $\lambda :\mathrm {span}(\mathcal {G})\rightarrow A$ be a local lifting. By nuclearity, there are $i\in \mathbb {N}$ and unitary completely positive maps $A\stackrel {\psi }{\longrightarrow }\mathbb {M}_i\stackrel {\dot {\varphi }}{\longrightarrow }B$ such that $\|\dot {\varphi }\circ \psi (a)-\pi (a)\|<\epsilon $ for every $a\in \lambda (\mathcal {G})$ . Arveson’s extension theorem then provides a unitary completely positive $\psi ':D\rightarrow \mathbb {M}_i$ that agrees with $\psi \circ \lambda $ on $\mathrm {span}(\mathcal {G})$ . Thus, for any $d\in \mathcal {G}$ ,

$$ \begin{align*}\|\dot{\varphi}\circ\psi'(d)-\pi_2(d)\|= \|\dot{\varphi}\circ\psi(\lambda(d))-\pi(\lambda(d))\|<\epsilon.\end{align*} $$

From this, we conclude that $\pi _2$ is nuclear. This allows us to apply the properly infinite case of Lemma 2.3 to get approximately multiplicative decompositions of $\pi _2:D\rightarrow B^{**}$ . This case may then be extended to work for A itself by replacing the resultant $\psi _n$ with $\psi _n\circ \pi _1$ . The finite case goes through for A unmodified, resulting in approximately multiplicative decompositions of $\pi :A\rightarrow B^{**}$ .

Proof of Theorem 1.4.

As mentioned, Proposition 2.9 already provides the backward direction, so we need now only address the forward one. Let $A\stackrel {\psi _n}{\longrightarrow }F_n\stackrel {\varphi _n}{\longrightarrow }B$ be the approximately multiplicative norm-decomposition. The decomposition itself witnesses the nuclearity of $\pi $ . The quasidiagonality of every $\tau \circ \pi $ was shown in Lemma 1.2. The quasidiagonality of $\pi $ itself is a consequence of [Reference Dadarlat10, Theorem 4.8]; for the convenience of the reader, we present a distillation of the proof.

Since A is exact, we may treat it as a C*-subalgebra of some C*-algebra C such that the inclusion is nuclear. Using Arveson’s extension theorem, we may treat the domains of the $\psi _n$ as C. They induce a unitary completely positive map $\Psi $ from C to an ultraproduct $\prod _\omega F_n$ , where $(\psi _n(a))_n$ is a representative sequence of $\Psi (a)$ . Note that the approximate multiplicativity of $(\psi _n)$ makes $\Psi |_A$ a *-homomorphism. Likewise, we have a contractive completely positive map $\Phi $ from $\prod _\omega F_n$ to the ultrapower $B^\omega $ given by $\Phi ((T_n)_n)= (\varphi _n(T_n))_n$ . Thus, $\Phi \circ \Psi (a)= (\varphi _n\circ \psi _n(a))_n= (\pi (a))_n$ , which we may identify with $\pi (a)$ itself by treating B as a C*-subalgebra of $B^\omega $ through constant sequences. Thus, $\Phi |_{\Psi (A)}$ must also be a *-homomorphism.

Fix $\epsilon>0$ and a finite subset of contractions $\mathcal {G}\subset \Psi (A)$ . Also, let $\lambda :\mathrm {span}(\mathcal {G})\rightarrow A$ be a local lifting of $\Psi $ . By nuclearity of $A\subseteq C$ , there exist a finite-dimensional C*-algebra G and contractive completely positive maps $A\stackrel {\theta }{\longrightarrow }G\stackrel {\xi }{\longrightarrow }C$ such that, for every $d\in \mathcal {G}$ ,

$$ \begin{align*}\|(\Psi\circ\xi)\circ(\theta\circ\lambda)(d)-d\|= \|\Psi\circ\xi\circ\theta\circ\lambda(d)-\Psi\circ\lambda(d)\|\leq \|\xi\circ\theta\circ\lambda(d)-\lambda(d)\|<\epsilon.\end{align*} $$

Another use of Arveson’s extension theorem yields $\theta ':\Psi (A)\rightarrow G$ with restriction $\theta '|_{\mathcal {G}}= \theta \circ \lambda $ .

Thus, the inclusion $\Psi (A)\subseteq \prod _\omega F_n$ is nuclear. By the Choi–Effros lifting theorem [Reference Brown and Ozawa4, Theorem C.3], this inclusion lifts to a contractive completely positive map to $\prod _n F_n$ . Therefore, $\Psi (A)$ is quasidiagonal (see, for example, [Reference Brown and Ozawa4, Exercise 7.1.3]).