2.1 Tracial von Neumann algebras

A tracial von Neumann algebra is a pair $(M,\tau)$ consisting of a von Neumann algebra M and a fixed faithful normal tracial state $\tau:M\rightarrow\mathbb C$ . We start by recalling some terminology concerning tracial von Neumann algebras; see [Reference Anantharaman and PopaAP] for more information.

Let $(M,\tau)$ be a tracial von Neumann algebra. Any unital embedding $M_0\subset M$ , where $(M_0,\tau_0)$ is a tracial von Neumann algebra, will be assumed to be trace preserving, i.e., such that $\tau_0=\tau_{|M_0}$ . We denote by $\mathrm{E}_{M_0}:M\rightarrow M_0$ the conditional expectation onto $M_0$ . For $x\in M$ , $\|x\|$ and $\|x\|_2:= \sqrt{\tau(x^*x)}$ denote the operator norm and 2-norm of x, respectively. We denote by $\mathrm{L}^2(M)$ the Hilbert space obtained by completing M with respect to the 2-norm and by $\mathcal U(M)$ the unitary group of M. A projection $p\in M$ is called minimal if $pMp\cong \mathbb{C} 1$ . We call M diffuse if it has no minimal projections.

The von Neumann algebra generated by a self-adjoint set $S\subset M$ with $1\in S$ is equal to its double commutant $S'':= (S')'$ , where for $S\subset \mathbb B(\mathcal H)$ we write $S':=\{x\in \mathbb B(\mathcal H)\mid xy=yx$ for all $y\in S\}$ . For von Neumann subalgebras $M_1,M_2\subset M$ , which we will always assume unital, we denote by $M_1\vee M_2:= (M_1\cup M_2)''$ the von Neumann algebra generated by $M_1$ and $M_2$ .

We denote by $\mathcal Z(M):= M'\cap M$ the center of M. Given a projection $p\in M$ , we denote by $z_M(p)$ the smallest projection $z\in\mathcal Z(M)$ with $z\geq p$ and call it the central support of p.

If $p,q\in M$ are projections, then $\tau(p\wedge q)=\tau(p)+\tau(q)-\tau(p\vee q)\geq \tau(p)+\tau(q)-1$ . Here $p\wedge q$ denotes the largest lower bound, i.e., the largest projection r satisfying $r\leq p$ and $r\leq q$ , and $p\vee q$ denotes the smallest upper bound, i.e., the smallest projection r satisfying $p\leq r$ and $q\leq r$ . We say that p and q are in general position if $\tau(p\wedge q)=\max\{\tau(p)+\tau(q)-1,0\}$ .

We continue with two elementary lemmas.

Proof. Let $S=\{1\leq j\leq n\mid p\wedge q_j\neq 0\}$ . If $S=\emptyset$ , the conclusion is immediate as $p\neq 0$ . If $|S|=1$ , let $1\leq k\leq n$ such that $S=\{k\}$ . Since $q_{k}\neq 1$ , we get that $\tau(p\wedge q_k)=\tau(p)+\tau(q_k)-1<\tau(p)$ , and hence $p-\sum_{1\leq j\leq n}p\wedge q_j=p-p\wedge q_k\neq 0$ . If $|S|\geq 2$ , then as $p\neq 1$ , $(|S|-1)(\tau(p)-1)<0$ , and thus

$$\tau\bigg(\sum_{1\leq j\leq n}p\wedge q_j\bigg)=\sum_{j\in S} (\tau(p)+\tau(q_j)-1)=\tau(p)+ \bigg(\sum_{j\in S}\tau(q_j)-1\bigg)+(|S|-1)(\tau(p)-1)<\tau(p).$$

This altogether implies that $p-\sum_{1\leq j\leq n}p\wedge q_j\neq 0$ .

This result is proved in [Reference DykemaDyk93, Theorem 1.2] when $M_1$ and $M_2$ are freely independent, but the same arguments work in the more general context here. We recall the proof for the reader’s convenience.

2.2 Minimal projections

Recall that for a tracial von Neumann algebra $(M,\tau)$ we denote

$$\mathrm{e}(M)=\max\{\tau(p)\mid\text{$p\in M$ a minimal projection}\}.$$

For instance, $\mathrm{e}(M)=0$ if M is diffuse, and $\mathrm{e}(M)=\max\{\tau(p_i)\mid 1\leq i\leq n\}$ if $M=\mathbb Cp_1\oplus\cdots\oplus\mathbb Cp_n$ is finite-dimensional and abelian. Moreover, we see that $\mathrm{e}(M)=\mathrm{e}(A)$ , if $A\subset M$ is a maximal abelian von Neumann subalgebra (MASA). The following lemma shows that whenever $\mathrm{e}(M)\neq 0$ , we can in fact find a finite-dimensional abelian von Neumann subalgebra $A\subset M$ with $\mathrm{e}(M)=\mathrm{e}(A)$ .

To prove the moreover assertion, assume that $\dim(A)\leq 2$ . Since $L\geq 3$ , and $K\geq 3$ if $N=\infty$ , we deduce that in the above construction, we necessarily have $p=0$ and $N\in\mathbb N$ . Thus, M is finite-dimensional and $A=\oplus_{i=1}^N\mathbb Cp_i$ is a MASA. Since $N=\dim(A)\leq 2$ , this forces M to be isomorphic to either $\mathbb C^2$ or $\mathbb M_2(\mathbb C)$ .

We also record the following easy observation.

2.3 Full free products of $C^*$ -algebras

In this section we briefly recall the definition of the (unital full) free product of two unital $C^*$ -algebras, see also, e.g., [Reference Voiculescu, Dykema and NicaVDN92]. This is the co-product in the category of unital $C^*$ -algebras.

2.4 The trace space of a $C^*$ -algebra

Let A be a unital, separable $C^*$ -algebra. We denote by $\mathrm{T}(A)$ the compact convex set of all traces on A endowed with the weak $^*$ -topology. We denote by $\partial_{\mathrm{e}}\mathrm{T}(A)$ the set of extreme points of $\mathrm{T}(A)$ . Then $\mathrm{T}(A)$ is a metrizable Choquet simplex, i.e., every $\varphi\in\mathrm{T}(A)$ is the barycenter of a unique Borel probability measure supported on $\partial_{\mathrm{e}}\mathrm{T}(A)$ .

A tracial representation of A is a triple $(M,\tau, \pi)$ where $(M,\tau)$ is a tracial von Neumann algebra and $\pi:A\to M$ is a $*$ -homomorphism with $\pi(A)''=M$ . In such cases, $\tau\circ\pi\in\mathrm{T}(A)$ . Two tracial representations $(M_1,\tau_1,\pi_1)$ and $(M_2,\tau_2,\pi_2)$ give rise to the same trace, namely $\tau_1\circ\pi_2 =\tau_2 \circ \pi_2$ , if and only if they are quasi-equivalent; namely, if there exists a $*$ -isomorphism $\Psi:M_1 \rightarrow M_2$ such that $\tau_1=\tau_2\circ\Psi$ and $\pi_2 = \Psi\circ \pi_1$ .

Conversely, given $\varphi\in\mathrm{T}(A)$ , there exists a unique (up to quasi-equivalence) tracial representation $(M,\tau,\pi)$ of A such that $\varphi=\tau\circ\pi$ . More precisely, $\pi$ is the GNS representation associated to $\varphi$ . We say that $\varphi$ is finite-dimensional if $\dim(M)<\infty$ . We denote by $\mathrm{T}_{\mathrm{fin}}(A)$ the set of all finite-dimensional traces $\varphi\in\mathrm{T}(A)$ and let $\mathrm{T}_{\infty}(A)=\mathrm{T}(A)\setminus\mathrm{T}_{\mathrm{fin}}(A)$ . Note that $\varphi\in\partial_{\mathrm{e}}\mathrm{T}(A)$ if and only if M is a factor (i.e., $\pi(A)''$ is either a II $_1$ factor or isomorphic to $\mathbb M_n(\mathbb C)$ , for some $n\in\mathbb N$ ). We say that $\varphi\in \partial_{\mathrm{e}}\mathrm{T}(A)$ is n-dimensional, for some $n\in\mathbb N$ , if $M\cong\mathbb M_n(\mathbb C)$ . Finally, we say that $\varphi$ is von Neumann amenable if M is an amenable von Neumann algebra.

For future reference, we record the following useful known lemma concerning convergence of traces in the weak $^*$ -topology versus the Wasserstein topology.

The main content of this lemma is the moreover assertion. This assertion can be deduced from [Reference Gangbo, Jekel, Nam and ShlyakhtenkoGJNS22, Proposition 5.26] when A is a finitely generated $C^*$ -algebra. For the reader’s convenience we provide a short self-contained proof. Note that the moreover assertion in particular applies if $\varphi\in\mathrm{T}_{\mathrm{fin}}(A)$ . Note also that condition (2) is equivalent to $\varphi_n\rightarrow\varphi$ in the Wasserstein topology.

For the moreover assertion, assume that $\varphi$ is von Neumann amenable and that (1) holds. Let $\pi:A\rightarrow N$ and $\rho_n:A\rightarrow N_n$ be tracial representations associated to $\varphi$ and $\varphi_n$ for every $n\in\mathbb N$ . Define $M=N*(*_{n\in\mathbb N}N_n)*\mathrm{L}(\mathbb F_2)$ and let $\tau$ be its canonical trace. Then M is a II $_1$ factor [Reference Anantharaman and PopaAP, Corollary 5.3.8]. Viewing N and $N_n$ as subalgebras of M, we moreover have $\varphi=\tau\circ\pi$ and $\varphi_n=\tau\circ\rho_n$ for every $n\in\mathbb N$ .

We proceed with the following claim.

Given the claim, it is clear that the maps $\pi_n:A\rightarrow M$ given by $\pi_n(a)=u_n\rho_n(a)u_n^*$ satisfy condition (2). Hence proving the claim will finish the proof of the lemma.

Proof of the claim. Let $(a_k)\subset (A)_1$ be a $\|\cdot\|$ -dense sequence. For every $n\in\mathbb N$ , let

$$\varepsilon_n:= \inf\bigg\{\sum_{k=1}^\infty2^{-k}\|u\rho_n(a_k)u^*-\pi(a_k)\|_2\mid u\in\mathcal U(M)\bigg\}.$$

The claim is equivalent to the statement that $\varepsilon_n\rightarrow 0$ . Assume by contradiction that this is false. Then we can find $\delta>0$ and a subsequence $(\varepsilon_{n_k})\subset(\varepsilon_n)$ such that $\varepsilon_{n_k}\geq \delta$ for every $k\in\mathbb N$ .

Let $\omega$ be a free ultrafilter on $\mathbb N$ and $M^{\omega}$ be the ultrapower von Neumann algebra together with its canonical trace given by $\tau^{\omega}((x_k)_{k\in\omega})=\lim_{k\rightarrow\omega}\tau(x_k)$ . Define $*$ -homomorphisms $\widetilde\pi,\rho:A\rightarrow M^{\omega}$ by setting $\widetilde\pi(a)=(\pi(a))_{k\in\omega}$ and $\rho(a)=(\rho_{n_k}(a))_{k\in\omega}$ for every $a\in A$ . Then for every $a\in A$ , we have

$$\tau^{\omega}(\rho(a))=\lim_{k\rightarrow\omega}\tau(\rho_{n_k}(a))=\lim_{k\rightarrow\omega}\varphi_{n_k}(a)=\varphi(a)=\tau(\pi(a))=\tau^{\omega}(\widetilde\pi(a)).$$

Thus, there is a trace-preserving $*$ -isomorphism $\theta:\widetilde\pi(A)''\rightarrow\rho(A)''$ such that $\theta(\widetilde\pi(a))=\rho(a)$ for every $a\in A$ . Since $\varphi$ is von Neumann amenable, $\widetilde\pi(A)''$ is amenable. By Connes’ theorem [Reference ConnesCon76], $\widetilde\pi(A)''$ is approximately finite-dimensional. Since M is a II $_1$ factor, a well-known fact (see, e.g., [Reference Gangbo, Jekel, Nam and ShlyakhtenkoGJNS22, Lemma 5.23]) implies the existence of $v=(v_k)_{k\in\omega}\in \mathcal U(M^\omega)$ , where $v_k\in \mathcal U(M)$ for every $k\in\mathbb N$ , such that $\widetilde\pi(a)=v\rho(a)v^*$ for every $a\in A$ . In other words, $\lim_{k\rightarrow\omega}\|\pi(a)-v_k\rho_{n_k}(a)v_k^*\|_2=0$ for every $a\in A$ . This implies that $\lim_{k\rightarrow\omega}\varepsilon_{n_k}=0$ , contradicting that $\varepsilon_{n_k}\geq\delta$ , for every $k\in\mathbb N$ . This finishes the proof of the claim.

Thus, this also completes the proof of the lemma.

2.5 The tracial quotient of a $C^*$ -algebra

Before continuing, we note that studying the trace space of a $C^*$ -algebra amounts to studying its tracial quotient.

Definition 2.8. Let A be a unital, separable $C^*$ -algebra with $\mathrm{T}(A)\neq\emptyset$ . We define $A_{\mathrm{tr}}$ to be the quotient

\[ A_{\mathrm{tr}} := A/I_{\mathrm{tr}}, \]

where $I_{\mathrm{tr}} = \{a\in A\mid \varphi(a^*a)=0 \text{ for all }\varphi\in\mathrm{T}(A)\}$ . We call $A_{\mathrm{tr}}$ the tracial quotient of A.

We also note that the condition from Theorem A that $\mathrm{T}(A_i)$ does not consist of a single one-dimensional trace can be reformulated as $A_{i,\mathrm{tr}}\not\cong \mathbb{C}$ (i.e., $A_i$ is not ‘tracially equivalent’ to $\mathbb C$ ).

We record the following easy observation.

Let $\rho:A\to M$ be a tracial representation. Since $\rho_{|A_i}$ factors through $\pi_i$ , we can find unital $*$ -homomorphisms $\delta_i: A_{i,\mathrm{tr}}\to M$ such that $\rho_{|A_i} = \delta_i\circ \pi_i$ for $i=1,2$ . Defining the unital $*$ -homomorphism $\delta:A_{1,\mathrm{tr}}\ast A_{2,\mathrm{tr}}\to M$ by $\delta_{|A_{i,\mathrm{tr}}} = \delta_i$ for $i=1,2$ , we get that $\rho = \delta\circ \pi$ .

This implies that the continuous affine injective map $\mathrm{T}(A_{1,\mathrm{tr}}\ast A_{2,\mathrm{tr}})\ni \varphi\mapsto \varphi\circ \pi\in \mathrm{T}(A)$ is surjective, and is therefore an affine homeomorphism. This finishes the proof of the lemma.

3.1 An obstruction to diffuseness

Let $M_1,M_2$ be von Neumann subalgebras of a tracial von Neumann algebra $(M,\tau)$ . Assume that there exist projections $p\in \mathcal Z(M_1),q\in \mathcal Z(M_2)$ such that $M_1p=\mathbb Cp,M_2q=\mathbb Cq$ and $\tau(p)+\tau(q)>1$ . Then $M_1(p\wedge q)=(M_1p)(p\wedge q)=\mathbb C(p\wedge q)$ , and similarly $M_2(p\wedge q)=\mathbb C(p\wedge q)$ . These facts imply that $p\wedge q\in \mathcal Z(M_1\vee M_2)$ and $(M_1\vee M_2)(p\wedge q)=\mathbb C(p\wedge q)$ . Since $\tau(p)+\tau(q)>1$ , it follows that $M_1\vee M_2$ has a one-dimensional direct summand.

We next generalize this fact by assuming that $M_2q$ is finite-(but not necessarily one-) dimensional.

Following the formula immediately before [Reference DykemaDyk93, Definition 3.3] we define $p'=\sum_{i=1}^kp_i$ , where

$$p_i=\bigwedge\limits_{1\leq j\leq k}v_{i,j}(p\wedge q_j)v_{i,j}^*=(p\wedge q_i)\wedge\bigwedge\limits_{1\leq j\leq k, j\neq i}v_{i,j}(p\wedge q_j)v_{i,j}^*\quad \text{for every}\ 1\leq i\leq k.$$

Then $p_i=v_{i,j}p_jv_{i,j}^*$ , and hence $p'v_{i,j}=p'q_iv_{i,j}=p_iv_{i,j}=v_{i,j}p_j=v_{i,j}q_jp'=v_{i,j}p'$ for every $1\leq i,j\leq k$ . This implies that p' commutes with $M_2$ . Since $p'\leq p$ , p' also commutes with $M_1$ . Since $p'\in M_1\vee M_2$ , we derive that $p'\in \mathcal Z(M_1\vee M_2)$ . Since $p_i\leq p\wedge q$ , for every $1\leq i\leq k$ , we also get that $p'\leq p\wedge q$ .

To prove the lower bound for $\tau(p')$ , recall that for any projections $r\in M$ , $e,f\in rMr$ we have

$$\tau(e\wedge f)=\tau(e)+\tau(f)-\tau(e\vee f)\geq \tau(e)+\tau(f)-\tau(r).$$

Let $1\leq i\leq k$ . By using repeatedly the last inequality, we get that

\begin{align*}\tau(p_i)&\geq \sum_{j=1}^k\tau(p\wedge q_j)-(k-1)\tau(q_i)\\&\geq \sum_{j=1}^k(\tau(p)+\tau(q_j)-1)-(k-1)\tau(q_i)\\&=k\left(\tau(p)+\frac{\tau(q)}{k^2}-1\right)\!.\end{align*}

Thus, $\tau(p')=\sum_{i=1}^k\tau(p_i)\geq k^2(\tau(p)+({\tau(q)}/{k^2})-1)$ , proving that $p'\neq 0$ and the desired lower bound.

Since $M_1p'=\mathbb Cp'$ , we have $Mp'=M_2p'$ . As $p'\leq q$ is non-zero and $M_2q\cong\mathbb M_k(\mathbb C)$ is a factor, we get that $Mp'\cong\mathbb M_k(\mathbb C)$ , which finishes the proof.

3.2 Finite-dimensional extreme traces

In this subsection, we use the foregoing results to establish an obstruction to the trace simplex of a free product $C^*$ -algebra being a Poulsen simplex. We also refer to Lemma 6.5 for a converse.

Let $(\psi_n)_{n\in\mathbb{N}}\subset\partial_{\mathrm{e}}\mathrm{T}(A)$ be a sequence such that $\psi_n\rightarrow\varphi$ . For $i\in\{1,2\}$ , let $\mu_{i,n}$ be the Borel probability measure on $\partial_{\mathrm{e}}\mathrm{T}(A_i)$ whose barycenter is ${\psi_n}_{|A_i}$ . Since $\varphi_i=\varphi_{|A_i}\in\partial_{\mathrm{e}}\mathrm{T}(A_i)$ is isolated, Corollary 3.2 gives that $\lambda_{i,n}:= \mu_{i,n}(\{\varphi_i\})\rightarrow 1$ for every $i\in\{1,2\}$ .

For $n\in\mathbb N$ , let $\pi_n:A\rightarrow M_n$ be a tracial representation, where $(M_n,\tau_n)$ is a tracial factor, such that $\psi_n=\tau_n\circ\pi_n$ and $\pi_n(A)''=M_n$ . For $i\in\{1,2\}$ , let $M_{i,n}=\pi_n(A_i)''$ . Then there exist projections $p_{1,n}\in \mathcal Z(M_{1,n})$ , $p_{2,n}\in \mathcal Z(M_{2,n})$ and a $*$ -isomorphism $\Psi:\mathbb M_k(\mathbb C)\rightarrow M_{2,n}p_{2,n}$ such that $\tau_n(p_{1,n})=\lambda_{1,n}$ , $\tau_n(p_{2,n})=\lambda_{2,n}$ ,

(3.1) \begin{equation}\pi_n(a_1)p_{1,n}=\rho_1(a_1)p_{1,n}\ \text{and}\,\pi_n(a_2)p_{2,n}=\Psi(\rho_2(a_2))\quad \text{for every}\ a_1\in A_1\ \text{and}\ a_2\in A_2.\end{equation}

By Proposition 3.3, there exists a projection $p_n'\in \mathcal Z(M_n)$ such that $p_n'\leq p_{1,n}\wedge p_{2,n}$ and $\tau(p_n')\geq k^2(\tau(p_{1,n})+({\tau(p_{2,n})}/{k^2})-1)=k^2(\lambda_{1,n}+({\lambda_{2,n}}/{k^2})-1)$ . Then $\tau(p_n')\rightarrow 1$ . Since $M_n$ is a factor we conclude that $p_n'=1$ for n large enough. Hence $p_{1,n}=p_{2,n}=1$ for n large enough.

Let $a=b_1c_1\cdots b_mc_m$ , where $b_1,\ldots,b_m\in A_1$ and $c_1,\ldots,c_m\in A_2$ , for some $m\in\mathbb N$ . By using (3.1) we get that $\pi_n(a)=\rho_1(a_1)\Psi(\rho_2(a_2))=\Psi(\rho_1(a_1)\rho_2(a_2))=\Psi(\rho(a))$ , where $a_1=b_1\cdots b_m\in A_1$ and $a_2=c_1\cdots c_m\in A_2$ . As $\mathbb M_k(\mathbb C)$ is a factor, and thus has a unique trace, we get that $\tau_n(\Psi(y))=\mathrm{tr}_k(y)$ for every $y\in\mathbb M_k(\mathbb C)$ . These facts imply that

\begin{align*} \tau_n(\pi_n(a))= \tau_n(\Psi(\rho(a)))=\mathrm{tr}_k(\rho(a))=\varphi(a).\end{align*}

Since the linear span of such $a\in A$ is norm dense in A, we conclude that $\psi_n=\tau_n\circ\pi_n=\varphi$ for n large enough. This implies that $\varphi\in\partial_{\mathrm{e}}\mathrm{T}(A)$ is isolated.

3.3 Infinite free products of $C^*$ -algebras

In this section, we prove Proposition I, showing that the obstructions established in the previous sections for finite free products disappear when considering infinite free products.

Let $i\geq 1$ . We claim that there is a tracial representation $\rho_i:A_i\rightarrow N_i=\rho_i(A_i)''$ such that any minimal projection of $N_i$ has trace at most $\frac{1}{2}$ . If $A_i$ admits a factorial tracial representation $\pi_i$ of dimension at least 2, we can take $\rho_i=\pi_i$ . Otherwise, all factorial tracial representations of $A_i$ are one-dimensional. Since $\text{T}(A_i)$ does not consist of a single one-dimensional trace, there are two distinct one-dimensional tracial representations $\pi_i^1,\pi_i^2:A_i\rightarrow\mathbb C$ . Endow $\mathbb C^2$ with the trace assigning $\tfrac{1}{2}$ to (1, 0) and let $\rho_i=\pi_i^1\oplus\pi_i^2:A\rightarrow\mathbb C^2$ . Then any minimal projection in $\rho_i(A_i)''=\mathbb C^2$ has trace $\tfrac{1}{2}$ .

For $n\geq 1$ , let $M_n=M*(*_{i>n}N_i)$ and define a $*$ -homomorphism $\pi_n:A\rightarrow M_n$ by letting ${\pi_n}_{|A_i}=\pi_{|A_i}$ if $1\leq i\leq n$ and ${\pi_n}_{|A_i}=\rho_i$ if $i>n$ . Denote $\varphi_n:=\tau\circ\pi_n\in\text{T}(A)$ . Since $\pi_n(a)=\pi(a)$ , for every $a\in*_{i=1}^nA_i$ , we get that $\|\pi_n(a)-\pi(a)\|_2\rightarrow 0$ , for every $a\in A$ . Thus, $\varphi_n\rightarrow\varphi$ .

In order to finish the proof, it is enough to argue that $\varphi_n\in\partial_{\mathrm{e}}\text{T}(A)\cap \text{T}_\infty(A)$ or, equivalently, that $\pi_n(A)''$ is a II $_1$ factor for every $n\geq 1$ . Let $n\geq 1$ and put $P_n=\pi(*_{i=1}^nA_i)''$ . By the construction of $\pi_n$ we have that $\pi_n(A)''=P_n*(*_{i=n+1}^\infty N_i)$ . Thus, letting $Q_n=P_n*(*_{i=n+5}^\infty N_i)$ , we have

(3.2) \begin{equation}\pi_n(A)''=Q_n*(N_{n+1}*N_{n+2})*(N_{n+3}*N_{n+4}).\end{equation}

Since every minimal projection in $N_i$ has trace at most $\frac{1}{2}$ , [Reference DykemaDyk93] implies that $N_i*N_{i+1}$ is diffuse for every $i\geq 1$ . Indeed, each $N_i$ is either a II $_1$ factor, isomorphic to a matrix algebra $\mathbb{M}_n(\mathbb{C})$ for some $n\geq 2$ , or isomorphic to $\mathbb{C}^2$ with each minimal projection of trace $\frac{1}{2}$ . If either $N_i$ or $N_{i+1}$ is a II $_1$ factor, and hence diffuse, then $N_i*N_{i+1}$ is obviously diffuse. In the remaining cases, the formulas in [Reference DykemaDyk93, Theorem 1.1, Proposition 3.2, Theorem 3.6] show that the free product $N_i*N_{i+1}$ is equal to a diffuse von Neumann algebra plus additional finite-dimensional direct summands with weights determined by the traces of the minimal projections in $N_i$ and $N_{i+1}$ . From these formulas it is immediate that all the finite-dimensional summands disappear if all these minimal projections have trace at most $\frac{1}{2}$ .

Finally, using the above fact, the free product decomposition (3.2) of $\pi_n(A)''$ , and [Reference PopaPop83] (or [Reference Ioana, Peterson and PopaIPP08, Theorem 1.1]) we get that $\mathcal Z(\pi_n(A)'')\subset (N_{n+1}*N_{n+2})\cap (N_{n+3}*N_{n+4})=\mathbb C1$ . Indeed, [Reference Ioana, Peterson and PopaIPP08, Theorem 1.1] shows, in particular, that if $M_1$ and $M_2$ are diffuse von Neumann algebras, then $M_1'\cap M_1*M_2\subset M_1$ and $M_2'\cap M_1*M_2\subset M_2$ , which immediately yields the claim.

We conclude that $\pi_n(A)''$ is a II $_1$ factor, as desired, thus finishing the proof.

4.1 Perturbations in amalgamated free products

Definition 4.1 allows for perturbations within a larger von Neumann algebra $M_\varepsilon$ . We will construct this ambient von Neumann algebra $M_\varepsilon$ as an iterated amalgamated free product, utilizing results from [Reference Ioana, Peterson and PopaIPP08] concerning relative commutants of subalgebras, to establish factoriality. The following lemma provides a criterion for when a particular type of perturbation within an amalgamated free product generates a II $_1$ factor.

We recall that an element u in a tracial von Neumann algebra $(M,\tau)$ is called a Haar unitary if u is unitary and it satisfies $\tau(u^n)=0$ for all $n\in \mathbb{Z}\backslash \{0\}$ . Additionally, for von Neumann subalgebras $P,Q\subset M$ , we write $P\prec_M Q$ if a corner of P embeds into Q inside M in the sense of Popa, i.e., if there exist projections $p_0\in P$ , $q_0\in Q$ , a $^*$ -homomorphism $\psi:p_0Pp_0\to q_0Qq_0$ and a non-zero partial isometry $v\in q_0Mp_0$ such that $\psi(x)v=vx$ for all $x\in p_0Pp_0$ (see Theorem 2.1 and Corollary 2.3 in [Reference PopaPop06]). In particular, we point out that if P is diffuse, then $P\nprec_M \mathbb{C}1$ , and if P is type II $_1$ and Q is type I, then $P\nprec_M Q$ .

Assume by contradiction that $N_t$ is not a II $_1$ factor. Then there is a projection $p\in \mathcal Z(N_t)\setminus\{0,1\}$ ; put $y=p-\mathrm{E}_{A_2}(p)$ . By assumption (2), there exists a factor P with $A_2\subset P\subset N_t$ . This implies that $\mathrm{E}_{P}(p)\in\mathcal Z(P)=\mathbb{C} 1$ , and therefore $\mathrm{E}_{A_2}(p)=\mathrm{E}_{A_2}(\mathrm{E}_{P}(p)) =\tau(p)$ . It follows that $y=p-\tau(p)$ . From this we deduce that y commutes with $u_t M_2 u_t^*$ (in fact with $N_t$ ) and that y is invertible. We will show that the existence of an element $y\in M$ with these two properties, along with the condition $A_2\subsetneq M_2$ , leads to a contradiction.

Indeed, as $A_2\subsetneq M_2$ , we can find a non-zero element $x\in M_2$ with $\mathrm{E}_{A_2}(x)=0$ . Since y commutes with $u_tM_2u_t^*$ , we have

(4.1) \begin{equation}u_txu_t^*y=yu_txu_t^*.\end{equation}

Put $v_t=u_t-\tau(u_t)$ , and let Q denote the orthogonal projection from $\mathrm{L}^2(\widetilde M)$ onto the $\|\cdot\|_2$ -closure of the linear span of $\{z_1z_2z_3z_4\mid z_1,z_3\in A_2\overline{\otimes}\mathrm{L}(\mathbb Z), z_2,z_4\in M \text{ and for all} 1\leq i\leq 4: \mathrm{E}_{A_2}(z_i)=0\}$ . It is immediate that $Q(u_txu_t^*y)=v_txv_t^*y$ and $Q(yu_txu_t^*)=0$ , and (4.1) thus implies that $v_txv_t^*y=0$ . Using that $\mathrm{E}_{A_2}(x)=\mathrm{E}_{A_2}(y)=\mathrm{E}_{A_2}(v_t)=0$ , $\mathrm{E}_{A_2}(v_t^*v_t)=\mathrm{E}_{A_2}(v_tv_t^*)=\|v_t\|_2^2\cdot 1$ and $v_t\in A_2'\cap\widetilde M$ , we deduce that

\begin{align*} 0=\|v_txv_t^*y\|_2^2=\tau(y^*v_tx^*v_t^*v_txv_t^*y)=&\|v_t\|_2^2\tau(y^*v_tx^*xv_t^*y)\\=&\|v_t\|_2^2\tau(y^*v_t\mathrm{E}_{A_2}(x^*x)v_t^*y) \\=&\|v_t\|_2^2\tau(y^*\mathrm{E}_{A_2}(x^*x)v_tv_t^*y)\\=&\|v_t\|_2^4\tau(y^*\mathrm{E}_{A_2}(x^*x)y).\end{align*}

In the third equality we used that since $\mathrm{E}_{A_2}(x)=\mathrm{E}_{A_2}(y)=\mathrm{E}_{A_2}(v_t)=\mathrm{E}_{A_2}(v_t^*v_t-\|v_t\|_2^2\cdot 1)=0$ , $x,y\in M$ and $v_t,v_t^*v_t-\|v_t\|_2^2\cdot 1\in A_2\overline{\otimes}\text{L}(\mathbb Z)$ , the alternating word $y^*v_tx^*(v_t^*v_t-\|v_t\|_2^2\cdot 1)xv_t^*y$ has trace zero. Similarly, in the fourth and sixth equalities, we used that the alternating words $y^*v_t(x^*x-\text{E}_{A_2}(x^*x))v_t^*y$ and $(y^*\text{E}_{A_2}(x^*x))(v_tv_t^*-\|v_t\|_2^2\cdot 1)y$ have traces equal to zero.

Since $v_t\neq 0$ , we derive that $y^*\mathrm{E}_{A_2}(x^*x)y=0$ . Since y is invertible, it follows that $\mathrm{E}_{A_2}(x^*x)=0$ and hence $x=0$ , which is a contradiction. In conclusion, $N_t$ is a II $_1$ factor.

Note that the above proof still works if we replace $u_t$ with any unitary from $\text{L}(\mathbb Z)\setminus\mathbb C1$ . We nevertheless formulated Lemma 4.3 in this particular way which is more common in the free probability context.

We obtain the following immediate consequence of Lemma 4.3, establishing the special case of Theorem E when one of the algebras is diffuse.

Using Lemma 4.3, we show next that the property of being approximately factorial is inherited from abelian subalgebras.

Fix $\varepsilon>0$ and define $\widetilde{M}_\varepsilon=M_{\varepsilon}*_{v_{\varepsilon} A_2v_{\varepsilon}^*}(v_{\varepsilon} A_2v_{\varepsilon}^*\overline{\otimes}\mathrm{L}(\mathbb Z))$ . Let $u\in \mathrm{L}(\mathbb Z)$ be a Haar unitary and $h=h^*\in \mathrm{L}(\mathbb Z)$ with $u=\exp(ih)$ . Let $t>0$ such that $u_t=\exp(ith)$ satisfies $\|u_t-1\|_2<\varepsilon-\|v_\varepsilon-1\|_2$ .

Since $A_1 \vee v_\varepsilon A_2v_\varepsilon^*$ is a II $_1$ factor, we get that $M_1 \vee v_\varepsilon A_2v_\varepsilon^*$ is of type II $_1$ . Since $A_2$ is of type I, we derive that $M_1 \vee v_\varepsilon A_2v_\varepsilon^*\nprec_{ M_\varepsilon} v_\varepsilon A_2v_\varepsilon^*$ . Since we also assume that $A_2\subsetneq M_2$ , we derive from Lemma 4.3 that $M_1 \vee u_tv_{\varepsilon}M_2v_{\varepsilon}^*u_t^*$ is a II $_1$ factor. Since $\|u_tv_\varepsilon-1\|_2\leq \|u_t-1\|_2+\|v_\varepsilon-1\|_2<\varepsilon$ , we conclude that the triple $(M_1,M_2,M)$ is approximately factorial.

Last, we establish approximate factoriality in the presence of projections of trace $\frac{1}{2}$ and factoriality of one of the algebras.

By Proposition 5.2 (see also [Reference Collins and KempCK14, Theorem 1.9]), there exist a II $_1$ factor $M_\varepsilon$ and $v_\varepsilon\in \mathcal U(M_\varepsilon)$ such that $M\subset M_\varepsilon$ , $\|v_\varepsilon-1\|_2<\varepsilon$ and $A_1\vee v_\varepsilon A_2v_\varepsilon^*$ is diffuse. Here, we used the fact that if two projections p,q of trace $\frac{1}{2}$ are such that each of the pairs of projections $(p,q),(p,1-q),(1-p,q),(1-p,1-q)$ are in general position, then $p\wedge q=p\wedge (1-q)=(1-p)\wedge q=(1-p)\wedge (1-q)=0$ . This fact implies that $A_1\vee v_\varepsilon A_2v_\varepsilon^*$ has no one-dimensional summands and is thus diffuse by Proposition 5.2.

Define $\widetilde{M}_\varepsilon=M_{\varepsilon}*_{v_{\varepsilon} A_2v_{\varepsilon}^*}(v_{\varepsilon} A_2v_{\varepsilon}^*\overline{\otimes}\mathrm{L}(\mathbb Z))$ . Let $u\in \mathrm{L}(\mathbb Z)$ be a Haar unitary and $h=h^*\in \mathrm{L}(\mathbb Z)$ such that $u=\exp(ih)$ . Let $t>0$ such that $u_t=\exp(ith)$ satisfies $\|u_t-1\|_2<\varepsilon-\|v_\varepsilon-1\|_2$ .

Since $A_1\vee v_\varepsilon A_2v_\varepsilon^*$ is diffuse and $A_2$ is finite-dimensional, we get that $M_1\vee v_\varepsilon A_2v_\varepsilon^*\nprec_{M_\varepsilon}v_\varepsilon A_2v_\varepsilon^*$ . Since $M_2$ is a factor, $A_2\subsetneq M_2$ and thus $v_\varepsilon A_2v_\varepsilon^*\subsetneq v_\varepsilon M_2v_\varepsilon^*$ . Using that $M_2$ is a factor again, we may thus apply Lemma 4.3 and derive that $M_1\vee u_tv_{\varepsilon}M_2v_{\varepsilon}^*u_t^*$ is a II $_1$ factor. It is left to note that $\|u_tv_\varepsilon-1\|_2\leq \|u_t-1\|_2+\|v_\varepsilon-1\|_2<\varepsilon$ , so we conclude that the triple $(M_1,M_2,M)$ is approximately factorial.

4.2 General position of von Neumann subalgebras

We continue with a technical result, given as Theorem 4.9, pertaining to von Neumann algebras of a specific form, and which we will rely on in the proof of Theorem E. We postpone its proof to the next section. In order to state the theorem, we first introduce the following notation.

For algebras of class $\mathcal{C}$ , Theorem 4.9 establishes a more general version of Theorem E. To motivate its statement, note that, given von Neumann algebras $\mathbb{C}1\neq M_1,M_2\in\mathcal{C}$ with $\dim(M_1)+\dim(M_2)\geq 5$ , Theorem E would imply that we can find a unitary v close to 1 (in some II $_1$ factor containing $M_1$ and $M_2$ ) such that $M_1\vee vM_2v^*$ is a II $_1$ factor if the condition $\mathrm{e}(M_1)+\mathrm{e}(M_2)\leq 1$ is satisfied. If this condition fails, then that is never possible, since we will always have one-dimensional direct summands of the form $\mathbb{C}(p\wedge q)$ for central projections $p\in\mathcal{Z}(M_1)$ and $q\in\mathcal{Z}(vM_2v^*)$ satisfying $\tau(p)+\tau(q)>1$ (cf. the first paragraph of § 3.1). Nevertheless, Theorem 4.9 tells us that, even if $\mathrm{e}(M_1)+\mathrm{e}(M_2) > 1$ , we can find a unitary v close to 1 such that those are the only one-dimensional direct summands of $M_1\vee vM_2v^*$ , they have the minimal possible support projections, and the remaining direct summand is a II $_1$ factor. We formalize this in the next definition.

Recall that two projections p and q in a tracial von Neumann algebra $(M,\tau)$ are said to be in general position if $\tau(p\wedge q) = \max\{\tau(p) + \tau(q) - 1, 0\}$ .

For the proof of Theorem 4.9, we refer to § 5. In the remaining part of this section, we use Theorem 4.9 to establish Theorem E.

4.3 Proof of Theorem E

Before turning to the proof of Theorem E in full generality, we note that Theorem 4.9 immediately implies Theorem E when $M_1$ and $M_2$ are finite-dimensional abelian.

If $M_1$ or $M_2$ is diffuse, then Corollary 4.4 implies that $(M_1,M_2,M)$ is approximately factorial. Thus, we may assume that neither $M_1$ nor $M_2$ is diffuse. By Lemma 2.3, we can thus find a finite-dimensional abelian von Neumann subalgebra $A_i\subset M_i$ , for $i=1,2$ , such that $A_i\neq \mathbb C1$ , $\mathrm{e}(A_i)=\mathrm{e}(M_i)$ . Moreover, $\dim(A_i)\geq 3$ unless $M_i$ is isomorphic to $\mathbb C^2$ or $\mathbb M_2(\mathbb C)$ .

In particular, $\mathrm{e}(A_1)+\mathrm{e}(A_2)=\mathrm{e}(M_1)+\mathrm{e}(M_2)\leq 1$ , and since $A_1\neq \mathbb C1$ and $A_2\neq \mathbb C1$ , we get $\mathrm{dim}(A_1)+\mathrm{dim}(A_2)\geq 4$ . We finish the proof by considering the following two cases.

Case 1. $\mathrm{dim}(A_1)+\mathrm{dim}(A_2)\geq 5$ . In this case, Corollary 4.10 implies that the triple $(A_1,A_2,M)$ is approximately factorial. If $M_1=A_1$ and $M_2=A_2$ , then we get that $(M_1,M_2,M)$ is approximately factorial. Otherwise, we may assume, without loss of generality, that $A_2\subsetneq M_2$ . In this case, we obtain that the triple $(M_1,M_2,M)$ is approximately factorial by applying Proposition 4.5.

Case 2. $\mathrm{dim}(A_{1})+\mathrm{dim}(A_{2})=4$ . In this case, $\mathrm{dim}(A_1)=\mathrm{dim}(A_2)=2$ . Thus, $M_1$ and $M_2$ are isomorphic to either $\mathbb C^2$ or $\mathbb M_2(\mathbb C)$ . Since $\mathrm{dim}(M_1)+\mathrm{dim}(M_2)\geq 5$ , at least one of $M_1$ or $M_2$ is not isomorphic to $\mathbb C^2$ . Thus, without loss of generality, we may assume that $M_2 \cong \mathbb M_2(\mathbb C)$ .

Since $\mathrm{dim}(A_1)=\mathrm{dim}(A_2)=2$ , we have that $A_1=\mathbb Cp_1\oplus\mathbb C(1-p_1)$ and $A_2=\mathbb Cp_2\oplus\mathbb C(1-p_2)$ for some projections $p_1,p_2\in M$ with $\tau(p_1)\geq\frac{1}{2}$ and $\tau(p_2)\geq\frac{1}{2}$ . Since $1\geq \mathrm{e}(A_1)+\mathrm{e}(A_2)=\tau(p_1)+\tau(p_2)$ , we conclude that $\tau(p_1)=\tau(p_2)=\frac{1}{2}$ . We may thus proceed by applying Proposition 4.6 and conclude that the triple $(M_1,M_2,M)$ is approximately factorial.

5.1 Pairs of two-dimensional algebras

Before turning to the proof of Theorem 4.9, we note that the assumption that $\dim(M_1)+\dim(M_2)\geq 5$ from Theorem 4.9 is necessary. Indeed, if $M_1,M_2\neq\mathbb C1$ , having $\dim(M_1)+\dim(M_2)<5$ forces that $\dim(M_1)=\dim(M_2)=2$ , i.e., $M_1 \cong \mathbb{C}^2 \cong M_2$ . In this case, $M_1\vee vM_2v^*$ is a type I von Neumann algebra, and therefore has no type II $_1$ direct summand for any unitary v in any II $_1$ factor containing $M_1$ and $M_2$ .