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If a factor $R$ contains a matrix algebra $M\subset R$ (i.e., a $M$ is a type $I_n$ factor), then $R \cong M \otimes M^c$ where $M^c=R\cap M'$ is the relative commutant. Each state $\omega$ on $R$ restricts to states $\omega\rvert_M$ on $M$ and $\omega\rvert_{M^c}$ on $M^c$. The product state $\omega^\times$ defined on $R$ by linear continuation of $\omega^\times(xy) = \omega\rvert_M(x)\omega\rvert_{M^c}(y)$, $x\in M$, $y\in M^c$, is called the (normal) factorization of $\omega$ relative to $M$ [Kadison–Ringrose 2, 11.4.13].

Let $R$ be a hyperfinite factor with dense increasing sequence finite type I factors $M_1\subset M_2\subset \dotsb \subset R$ and let $\omega$ be a normal state on $R$. Denote by $\omega_n^\times$ the normal factorization of $\omega$ relative to $M_n$. Does it follow for all normal states that the $\omega_n^\times$ approximate $\omega$ in norm? I.e. does $\lVert \omega - \omega^\times_n \rVert \to 0$?

Some remarks: It is clear that the sequence converges weakly. Very similar statements are proved in section 11.4 of Kadison–Ringrose 2.

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Since the answer to this question depend on the nature of $R$ and the choice of the sequence $M_n$, let's say that $R$ satisfies property $(P)$ w.r.t. the filtration $M_n$ if $\|\omega - \omega_n^\times\| \to 0$ for every normal state $\omega$ on $R$ where $\omega_n^\times$ is defined as in the question.

Below I prove the following results.

Proposition 1. If $R$ is the hyperfinite II$_1$ factor, then property $(P)$ holds w.r.t. any filtration $M_n$.

Recall that a factor is said to be ITPFI if it can be written as the infinite tensor product of a sequence of matrix algebras equipped with faithful states. By a result of Krieger (see [A. Connes and E.J. Woods, Approximately transitive flows and ITPFI factors. Ergodic Theory Dynam. Systems 5 (1985), 203-236] for a systematic account), not all injective factors are ITPFI.

Proposition 2. If $R$ is an injective factor that is not ITPFI, there is no filtration for which property (P) holds.

Proposition 3. If $R$ is an ITPFI factor, then property (P) holds w.r.t. some filtrations, but not necessarily w.r.t. all filtrations.

Proof of Proposition 1. Note that for all normal states $\omega$ and $\mu$ on $R$, we have that $\|\omega_n^\times -\mu_n^\times\| \leq 2 \|\omega - \mu\|$ for all $n$. Denote by $E_n$ the unique trace preserving conditional expectation of $R$ onto $M_n$. Denote by $S_0$ the set of normal states on $R$ that are of the form $\eta \circ E_n$ for some $n$ and some state $\eta$ on $M_n$. Then $S_0$ is norm dense in the set of all normal states on $R$. When $\mu\in S_0$, we have that $\mu_n^\times = \mu$ for $n$ large enough. In combination with the inequality above, it follows that $\|\omega - \omega_n^{\times}\| \to 0$.

Proof of Proposition 2. Assume that $R$ satisfies property (P) w.r.t. some filtration $M_n$. I prove that $R$ must be an ITPFI factor. Choose a faithful normal state $\omega$ on $R$. Replacing $M_n$ by a subsequence, we may assume that $\|\omega_n^\times - \omega_{n-1}^\times\| < 2^{-n}$ for all $n$.

Denote $A_n = M_n \cap M_{n-1}'$, so that we can view $M_n$ as the tensor product of the matrix algebras $A_1 \otimes \cdots \otimes A_n$. Define the faithful state $\mu_n$ on $A_n$ by restricting $\omega$ to $A_n$. Define the faithful state $\gamma_n$ on $M_n$ by $\gamma_n = \mu_1 \otimes \cdots \otimes \mu_n$. Finally define the faithful normal states $\eta_n$ on $R$ by $\eta_n(xy) = \gamma_n(x) \omega(y)$ for all $x \in M_n$ and $y \in R \cap M_n'$.

Denoting by $\omega_n^c$ the restriction of $\omega$ to $M_n^c = R \cap M_n'$, we may view $\eta_n$ as the tensor product $\mu_1 \otimes \cdots \otimes \mu_n \otimes \omega_n^c$. So we get that $\|\eta_n - \eta_{n-1}\| = \|(\mu_n \otimes \omega_n^c - \omega_{n-1}^c)|_{M_{n-1}^c}\|$. On $M_{n-1}^c$ the state $\mu_n \otimes \omega_n^c$ coincides with $\omega_n^\times$ and on $M_{n-1}^c$, the state $\omega_{n-1}^c$ coincides with $\omega_{n-1}^\times$. We thus conclude that $\|\eta_n-\eta_{n-1}\| \leq \|\omega_n^\times - \omega_{n-1}^\times\| < 2^{-n}$. It follows that $(\eta_n)$ is a Cauchy sequence in $R_*$. Denote by $\eta$ the normal state on $R$ such that $\|\eta-\eta_n\| \to 0$.

For all $i \geq n$, the restriction of $\eta_i$ to $M_n$ equals $\gamma_n$. So the restriction of $\eta$ to $M_n$ equals $\gamma_n$. Denote by $(N,\mu)$ the infinite tensor product of the matrix algebras $(A_n,\mu_n)$. The identity map on all $M_n$ then extends to a unitary operator $U : L^2(N,\mu) \to L^2(M,\eta)$. We have the faithful normal GNS representations $\pi_\mu : N \to B(L^2(N,\mu))$ and $\pi_\eta : M \to B(L^2(M,\eta))$. By construction, $U \pi_\mu(x) U^* = \pi_\eta(x)$ for all $x \in M_n$ and all $n$. It follows that $U \pi_\mu(N) U^* = \pi_\eta(M)$, so that $N \cong M$. So $M$ is an ITPFI factor.

Proof of Proposition 3. When $R$ is the infinite tensor product of the sequence of matrix algebras $A_n$ with faithful states $\mu_n$, we denote by $\mu$ the infinite product state on $R$ and we can take the filtration $M_n = A_1 \otimes \cdots \otimes A_n$. For every $n$, there is a unique $\mu$-preserving conditional expectation $E_n : R \to M_n$. From here the proof is identical to the proof of proposition 1.

As a counterexample, let $\mu_0$ be a faithful state on $A_0 = M_2(\mathbb{C}) \otimes M_2(\mathbb{C})$ that cannot be written as the tensor product of two states. Then also the norm distance of $\mu_0$ to the set of tensor product states is strictly positive, say larger than $\delta > 0$. Define $R$ as the infinite tensor product of copies of $(A_0,\mu_0)$, with infinite product state $\mu$. Define the filtration $M_n$ by \begin{align} & M_{2n} = \underbrace{A_0 \otimes \cdots \otimes A_0}_{\text{$n$ times}} \; ,\\ & M_{2n+1} = \underbrace{A_0 \otimes \cdots \otimes A_0}_{\text{$n$ times}} \otimes M_2(\mathbb{C}) \; . \end{align} So by definition, $M_n$ is a tensor product of $n$ copies of $M_2(\mathbb{C})$. I prove that $R$ does not satisfy property (P) w.r.t. this filtration $M_n$. Indeed, the restriction of $\mu - \mu_{2n+1}^\times$ to the $n+1$ copy of $A_0$ equals $\mu_0 - \mu_1 \otimes \mu_2$ for some states $\mu_1$ and $\mu_2$ on $M_2(\mathbb{C})$. Thus, $\|\mu - \mu_{2n+1}^\times\| > \delta$ for all $n$, so that property (P) does not hold w.r.t. the filtration $M_n$.

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  • $\begingroup$ Thanks for your answer Stefaan! I'm a bit puzzled by the "unique trace preserving conditional expectation" that you mention. I only understand this if the factor $R$ is also finite. $\endgroup$
    – Lau
    Oct 19, 2023 at 4:54
  • $\begingroup$ @Lauritz. I see. In my conventions (but not generally agreed upon), hyperfinite means "finite and approximately finite dimensional (AFD)". I will update my answer, because for certain type III factors the property does not hold. $\endgroup$ Oct 19, 2023 at 9:08

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