Transition isomorphisms and the relative flow

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

Let $N\subset M$ be an inclusion with a faithful normal conditional expectation $E$, and suppose that $N$ is a factor of type III₁ with separable predual. Conjugation by unitaries of $N$ that move one faithful normal state $\varphi$ of $N$ closer and closer to another one $\psi$ carries the relative bicentralizer of $\varphi$ onto that of $\psi$; partial isometries of $N$ that scale $\varphi$ by a factor $\lambda$ define an action of $\mathbb R_+^*$ on $\mathrm B(N\subset M,\varphi)$, the relative bicentralizer flow of Ando, Haagerup, Houdayer and Marrakchi [AHHM, Theorem A]. This lesson constructs both, following the absolute case of The bicentralizer flow. Two facts specific to inclusions are added: the flow commutes with the modular group of $\bar\varphi=\varphi\circ E$ (Theorem 3.1(6)), and the von Neumann algebra $\mathrm B^\sharp=N\vee\mathrm B(N\subset M,\varphi)$ does not depend on $\varphi$ (Theorem 4.1). The last section computes the relative bicentralizer of a matrix amplification.

We use: The relative bicentralizer (all of Sections 1 and 2); Lemmas 1.1 (reparametrization) and 1.2 (products of eigen-sequences) of The bicentralizer flow, which concern bounded sequences in a single algebra and are applied below in $N$ and in $M$; Lemma 1.2 (asymptotic eigenoperators), Lemma 1.3 (unitaries conjugating states), Proposition 4.1 (rows of approximate eigenoperators) and Corollary 4.2 (Connes–Størmer unitaries) of Asymptotic centralizers of type III₁ factors, applied to $N$; Lemmas 1.1, 1.3 and 1.4 and (B8) of Ultraproducts and the asymptotic centralizer; the modular expectation theorem (ME.1)–(ME.3) of Conditional expectations from modular invariance; the uniqueness of the decomposition of a self-adjoint normal functional into positive parts of minimal total norm, Corollary 2.8 of the polar decomposition lesson; and the theorem that any two infinite projections of a countably decomposable factor are equivalent, Proposition 15.2(4) of the projections lesson.

1. Setting and a trace lemma

Throughout this lesson $N$ is a factor of type III₁ with separable predual, $N\subset M$ with the same unit, $M$ countably decomposable, $E\colon M\to N$ a faithful normal conditional expectation, and $\omega$ a free ultrafilter. For faithful normal states $\varphi,\psi,\chi$ on $N$ we write $\bar\varphi=\varphi\circ E$, and so on, and $\mathrm B(\varphi)=\mathrm B(N\subset M,\varphi)$. The notation $M^\omega$, $\pi$, $N_{\varphi,\omega}\subset M^\omega$, $\mathcal G_\varphi$, $\mathcal U_\delta$ is that of the previous lesson. For a unitary $u\in N$, $u\bar\varphi u^*=(u\varphi u^*)\circ E$ because $E$ is $N$-bimodular, so $\begin{aligned} &\|u\bar\varphi u^*-\bar\psi\|_{M_*}\\ &\quad =\|u\varphi u^*-\psi\|_{N_*} \end{aligned}\tag{1.1}$ by the argument of Lemma 1.1 of the previous lesson. Likewise $\|v\bar\varphi-\lambda\bar\varphi v\|_{M_*}=\|v\varphi-\lambda\varphi v\|_{N_*}$ for $v\in N$ and $\lambda>0$. In particular, by Lemma 1.2 of the asymptotic centralizer lesson applied in $M$ to $\bar\varphi$, a bounded sequence $(v_n)$ in $N$ with $\lim_\omega\|v_n\varphi-\lambda\varphi v_n\|=0$ lies in $N_\omega$ and its class $V$ satisfies $V\bar\varphi^\omega=\lambda\bar\varphi^\omega V$.

Write $\tau$ for the trace of the II₁ factor $N_{\varphi,\omega}$; it is the restriction of $\bar\varphi^\omega$ (Lemma 1.2 of the previous lesson).

Lemma 1.1. For $x\in\mathrm B(\varphi)$ and $Z\in N_{\varphi,\omega}$, $\bar\varphi^\omega(xZ)=\bar\varphi(x)\tau(Z)$.

Proof. The inclusion $N_{\varphi,\omega}\subset M^\omega$ is normal, so $\rho(Z)=\bar\varphi^\omega(xZ)$ is a normal functional on $N_{\varphi,\omega}$. For $Z_1,Z_2\in N_{\varphi,\omega}$, $Z_2$ commutes with $\bar\varphi^\omega$ and with $x$ (Lemma 1.2 and Proposition 2.2 of the previous lesson), so $\begin{aligned} &\rho(Z_1Z_2)=(Z_2\bar\varphi^\omega)(xZ_1)\\ &\quad =(\bar\varphi^\omega Z_2)(xZ_1)=\bar\varphi^\omega(Z_2xZ_1)\\ &\quad =\bar\varphi^\omega(xZ_2Z_1)=\rho(Z_2Z_1). \end{aligned}$ We may assume $x=x^*$, so $\rho$ is self-adjoint. Let $\rho=\rho_+-\rho_-$ be its decomposition into normal positive functionals with $\|\rho\|=\|\rho_+\|+\|\rho_-\|$. For a unitary $U\in N_{\varphi,\omega}$, $\rho_\pm\circ\operatorname{Ad}U$ is another such decomposition of $\rho\circ\operatorname{Ad}U=\rho$, so by uniqueness $\rho_\pm$ are invariant under $\operatorname{Ad}U$, hence tracial. By (B8) a normal tracial positive functional on a II₁ factor is a multiple of $\tau$, so $\rho=\rho(1)\tau=\bar\varphi(x)\tau$. $\square$

2. Transition isomorphisms

A sequence of unitaries $u_n\in N$ with $\|u_n\varphi u_n^*-\psi\|\to0$ is a $(\psi,\varphi)$-sequence; such sequences exist by Corollary 4.2 of the asymptotic centralizer lesson.

Theorem 2.1. Let $(u_n)$ be a $(\psi,\varphi)$-sequence. For every $x\in\mathrm B(\varphi)$, $u_nxu_n^*$ converges $*$-strongly to an element $\beta_{\psi,\varphi}(x)\in\mathrm B(\psi)$, which does not depend on the choice of $(u_n)$. The map $\beta_{\psi,\varphi}\colon\mathrm B(\varphi)\to\mathrm B(\psi)$ is a $*$-isomorphism with $\bar\psi\circ\beta_{\psi,\varphi}=\bar\varphi$, it is the identity on $N'\cap M$, and $\beta_{\varphi,\varphi}=\mathrm{id},\qquad\beta_{\chi,\psi}\circ\beta_{\psi,\varphi}=\beta_{\chi,\varphi}.$ For every bounded sequence $(a_n)$ in $N$ with $\|a_n\varphi-\psi a_n\|\to0$ and every $x\in\mathrm B(\varphi)$, $\begin{aligned} &a_nx-\beta_{\psi,\varphi}(x)a_n\\ &\quad \to0\quad*\text{-strongly}. \end{aligned}\tag{2.1}$

Proof. Fix $\omega$. By (1.1) and Lemma 1.3 of the asymptotic centralizer lesson (in $M$, for the states $\bar\varphi,\bar\psi$), $U=\pi(u_n)$ is a unitary of $M^\omega$ with $U\bar\varphi^\omega U^*=\bar\psi^\omega$. If $(w_n)$ is another $(\psi,\varphi)$-sequence with class $W$, then $(w_n^*u_n)\in A_{\varphi,\omega}(N)$ by Lemma 1.2(2) of the flow lesson applied in $N$, so $W^*U\in N_{\varphi,\omega}$ commutes with $x\in\mathrm B(\varphi)$ (Proposition 2.2 of the previous lesson), that is, $UxU^*=WxW^* . \tag{2.2}$ For $s(n)\to\infty$, $(u_{s(n)})$ is again a $(\psi,\varphi)$-sequence. Hence (2.2), for every $\omega$, is the hypothesis of Lemma 1.1 of the flow lesson for $y_n=u_nxu_n^*$ in $M$: this sequence converges $*$-strongly to some $\beta_{\psi,\varphi}(x)\in M$, equal to $UxU^*$ in every $M^\omega$, and by (2.2) independent of the sequence.

Values in $\mathrm B(\psi)$. For $Y=\pi(y_n)$ with $(y_n)\in A_{\psi,\omega}(N)$, Lemma 1.2(3) of the flow lesson in $N$ gives $(u_n^*y_nu_n)\in A_{\varphi,\omega}(N)$, so $U^*YU\in N_{\varphi,\omega}$ commutes with $x$; hence $Y$ commutes with $UxU^*=\beta_{\psi,\varphi}(x)$. By Proposition 2.2 of the previous lesson, $\beta_{\psi,\varphi}(x)\in\mathrm B(\psi)$.

Algebraic properties. $x\mapsto UxU^*$ is an injective $*$-homomorphism; with $u_n=1$, $\beta_{\varphi,\varphi}=\mathrm{id}$. If $(v_n)$ is a $(\chi,\psi)$-sequence, then $(v_nu_n)$ is a $(\chi,\varphi)$-sequence and, with $V=\pi(v_n)$, $\beta_{\chi,\varphi}(x)=VUxU^*V^*=\beta_{\chi,\psi}(\beta_{\psi,\varphi}(x))$. So $\beta_{\varphi,\psi}$ inverts $\beta_{\psi,\varphi}$. Since $U^*\bar\psi^\omega U=\bar\varphi^\omega$, $\bar\psi(\beta_{\psi,\varphi}(x))=\bar\psi^\omega(UxU^*)=\bar\varphi(x)$. If $x\in N'\cap M$, then $u_nxu_n^*=x$.

Intertwining. Put $c_n=u_n^*a_n$. Since $u_n^*\psi-\varphi u_n^*=u_n^*(\psi-u_n\varphi u_n^*)$, $\begin{aligned} &\|[c_n,\varphi]\|\le\|a_n\varphi-\psi a_n\|\\ &\quad +\|a_n\|\,\|\psi-u_n\varphi u_n^*\|\to0 , \end{aligned}$ so $(c_n)\in\mathrm{AC}(N,\varphi)$. Write $a_nx-\beta_{\psi,\varphi}(x)a_n=u_nz_n$ with $\begin{aligned} &z_n=(c_nx-xc_n)\\ &\quad +(x-u_n^*\beta_{\psi,\varphi}(x)u_n)c_n . \end{aligned}$ The first term tends to $0$ $*$-strongly, because $x,x^*\in\mathrm B(\varphi)$. For the second, $e_n=\beta_{\psi,\varphi}(x)-u_nxu_n^*\to0$ $*$-strongly, and by (1.1) $\begin{aligned} &\|u_n^*e_nu_n\|^2_{\bar\varphi}=(u_n\bar\varphi u_n^*)(e_n^*e_n)\\ &\quad \le\bar\psi(e_n^*e_n)+\|e_n\|^2\|u_n\varphi u_n^*-\psi\|\to0 , \end{aligned}$ likewise for the adjoint; so $u_n^*e_nu_n\to0$ $*$-strongly, and its product with $(c_n)\in A_{\bar\varphi,\omega}(M)$ tends to $0$ $*$-strongly along every $\omega$, because $I_\omega$ is an ideal of $A_{\bar\varphi,\omega}(M)$ (Theorem 5.1(5) of the ultraproduct lesson). Hence $z_n\to0$ $*$-strongly. Finally, $(u_n)\in N_\omega$ for every $\omega$ (Lemma 1.3 of the asymptotic centralizer lesson, with (1.1)), so $(u_nz_n)\in I_\omega$ for every $\omega$, that is, $u_nz_n\to0$ $*$-strongly. $\square$

Corollary 2.2. If $\psi=v\varphi v^*$ for a unitary $v\in N$, then $\beta_{\psi,\varphi}(x)=vxv^*$.

Proof. The constant sequence $v$ is a $(\psi,\varphi)$-sequence. $\square$

3. The flow

Let $\lambda>0$. A finite family of bounded sequences $(v_{k,n})_n$ in $N$, $1\le k\le m$, is an admissible $\lambda$-family along $\omega$ for $\varphi$ if $\begin{aligned} &\lim_\omega\|v_{k,n}\varphi-\lambda\varphi v_{k,n}\|\\ &\quad =0\quad(1\le k\le m),\\ &\lim_\omega\Big\|1-\sum_kv_{k,n}v_{k,n}^*\Big\|^\sharp_{\varphi}=0 . \end{aligned}$ The classes $V_k\in M^\omega$ satisfy $V_k\bar\varphi^\omega=\lambda\bar\varphi^\omega V_k$ and $\sum_kV_kV_k^*=1$ (Section 1 and Lemma 1.1 of the previous lesson). By Proposition 4.1 of the asymptotic centralizer lesson applied to $(N,\varphi)$, admissible $\lambda$-families exist with ordinary convergence; for $\lambda\le1$ one sequence suffices.

Theorem 3.1. For every $\lambda>0$ there is a unital $*$-automorphism $\beta^\varphi_\lambda$ of $\mathrm B(\varphi)$ such that:

for every $\omega$, every admissible $\lambda$-family along $\omega$ and every $x\in\mathrm B(\varphi)$, $\sum_kV_kxV_k^*=\beta^\varphi_\lambda(x)$ in $M^\omega$; for an admissible family with ordinary convergence, $\sum_kv_{k,n}xv_{k,n}^*\to\beta^\varphi_\lambda(x)$ $*$-strongly;
for every bounded sequence $(w_n)$ in $N$ with $\|w_n\varphi-\lambda\varphi w_n\|\to0$ and every $x\in\mathrm B(\varphi)$, $w_nx-\beta^\varphi_\lambda(x)w_n\to0$ $*$-strongly; in particular $wx=\beta^\varphi_\lambda(x)w$ for every $w\in N$ with $w\varphi=\lambda\varphi w$;
$\beta^\varphi_1=\mathrm{id}$, $\beta^\varphi_\lambda\beta^\varphi_\mu=\beta^\varphi_{\lambda\mu}$, and $\bar\varphi\circ\beta^\varphi_\lambda=\bar\varphi$ on $\mathrm B(\varphi)$;
$\beta^\varphi_\lambda(x)\to x$ strongly as $\lambda\to1$, for every $x\in\mathrm B(\varphi)$; the action is continuous for the $u$-topology;
$\beta^\psi_\lambda\circ\beta_{\psi,\varphi}=\beta_{\psi,\varphi}\circ\beta^\varphi_\lambda$;
$\beta^\varphi_\lambda\circ\sigma^{\bar\varphi}_t=\sigma^{\bar\varphi}_t\circ\beta^\varphi_\lambda$ on $\mathrm B(\varphi)$ for all $t\in\mathbb R$;
$\beta^\varphi_\lambda(x)=x$ for $x\in N'\cap M$.

Proof. (1) Let $(v_{k,n})$, $(w_{l,n})$ be admissible $\lambda$-families along $\omega$, with classes $V_k,W_l$. By Lemma 1.2(1) of the flow lesson in $N$, $(w_{l,n}^*v_{k,n})_n\in A_{\varphi,\omega}(N)$, so $W_l^*V_k\in N_{\varphi,\omega}$ commutes with $x\in\mathrm B(\varphi)$, and $\begin{aligned} &\sum_kV_kxV_k^*\\ &\quad =\sum_{k,l}W_lW_l^*V_kxV_k^*\\ &\quad =\sum_{k,l}W_lxW_l^*V_kV_k^*\\ &\quad =\sum_lW_lxW_l^* . \end{aligned}\tag{3.1}$ For a family with ordinary convergence and $s(n)\to\infty$, the reparametrized family is admissible along every $\omega$; by (3.1) and Lemma 1.1 of the flow lesson, $\sum_kv_{k,n}xv_{k,n}^*$ converges $*$-strongly to an element $\beta^\varphi_\lambda(x)\in M$, equal to $\sum_kV_kxV_k^*$ in every $M^\omega$; by (3.1) every admissible family gives the same element.

Values in $\mathrm B(\varphi)$, multiplicativity. Let $Y=\pi(y_n)$ with $(y_n)\in A_{\varphi,\omega}(N)$. The sequence $(v_{k,n}^*y_nv_{l,n})_n$ lies in $A_{\varphi,\omega}(N)$: writing $v_k,y,v_l$ for the $n$-th terms, $v_k^*yv_l\varphi\approx\lambda v_k^*y\varphi v_l\approx\lambda v_k^*\varphi yv_l\approx\varphi v_k^*yv_l ,$ with errors at most $\|v_k\|\|y\|\,\|v_l\varphi-\lambda\varphi v_l\|$, $\lambda\|v_k\|\|v_l\|\,\|[y,\varphi]\|$ and $\|y\|\|v_l\|\,\|\lambda v_k^*\varphi-\varphi v_k^*\|$, where $\|\lambda v_k^*\varphi-\varphi v_k^*\|=\|v_k\varphi-\lambda\varphi v_k\|$ by taking adjoints in $N_*$. So $x$ commutes with $V_k^*YV_l$, and $\begin{aligned} &\beta^\varphi_\lambda(x)Y=\sum_{k,l}V_kxV_k^*YV_lV_l^*\\ &\quad =\sum_{k,l}V_kV_k^*YV_lxV_l^*=Y\beta^\varphi_\lambda(x). \end{aligned}$ By Proposition 2.2 of the previous lesson, $\beta_\lambda^\varphi(x)\in\mathrm B(\varphi)$. With $Y=1$, $V_k^*V_l\in N_{\varphi,\omega}$ commutes with $x$, so $\beta^\varphi_\lambda(x)\beta^\varphi_\lambda(y)=\sum_{k,l}V_kxV_k^*V_lyV_l^*$, which is $\sum_{k,l}V_kV_k^*V_lxyV_l^*=\beta^\varphi_\lambda(xy)$. The map is linear, $*$-preserving, and unital since $\sum_kV_kV_k^*=1$.

(3) The constant sequence $1$ is an admissible $1$-family, so $\beta^\varphi_1=\mathrm{id}$. If $(v_{k,n})$ is admissible for $\lambda$ and $(w_{l,n})$ for $\mu$, then $(v_{k,n}w_{l,n})$ is admissible for $\lambda\mu$, since $\|vw\varphi-\lambda\mu\varphi vw\|$ is bounded by the sum of $\|v\|\,\|w\varphi-\mu\varphi w\|$ and $\mu\|v\varphi-\lambda\varphi v\|\,\|w\|$, and $\sum_{k,l}V_kW_lW_l^*V_k^*=1$; hence $\beta^\varphi_{\lambda\mu}=\beta^\varphi_\lambda\beta^\varphi_\mu$, and $\beta^\varphi_\lambda$ is an automorphism with inverse $\beta^\varphi_{1/\lambda}$. For state preservation, $V_k^*V_k\in N_{\varphi,\omega}$ and Lemma 1.1 give $\begin{aligned} &\bar\varphi^\omega(V_kxV_k^*)=(\bar\varphi^\omega V_k)(xV_k^*)\\ &\quad =\lambda^{-1}\bar\varphi^\omega(xV_k^*V_k)=\lambda^{-1}\bar\varphi(x)\tau(V_k^*V_k)\\ &\quad =\bar\varphi(x)\bar\varphi^\omega(V_kV_k^*), \end{aligned}$ using $\bar\varphi^\omega(V_k^*V_k)=(V_k\bar\varphi^\omega)(V_k^*)=\lambda\bar\varphi^\omega(V_kV_k^*)$. Summing over $k$, $\bar\varphi(\beta^\varphi_\lambda(x))=\bar\varphi(x)$.

(2) Let $W=\pi(w_n)$. By Lemma 1.2(1) of the flow lesson in $N$, $V_k^*W\in N_{\varphi,\omega}$, so $Wx=\sum_kV_kV_k^*Wx$, which is $\sum_kV_kxV_k^*W=\beta^\varphi_\lambda(x)W$ in every $M^\omega$; hence $w_nx-\beta^\varphi_\lambda(x)w_n\to0$ $*$-strongly along every $\omega$, so $*$-strongly. The last assertion is the case of a constant sequence.

(4) Let $\lambda_j\to1$, $\lambda_j\le1$. For each $j$ take a single sequence $(v_{j,n})_n$ from Proposition 4.1 of the asymptotic centralizer lesson (with $m=1$) and choose $n(j)$ so that $w_j=v_{j,n(j)}$ satisfies $\|w_j\varphi-\lambda_j\varphi w_j\|<1/j$, $\|1-w_jw_j^*\|^\sharp_\varphi<1/j$ and $\|w_jxw_j^*-\beta^\varphi_{\lambda_j}(x)\|^\sharp_{\bar\varphi}<1/j$. Then $\|[w_j,\varphi]\|\le1/j+(1-\lambda_j)\|w_j\|\to0$, so along every $\omega$ the class $W$ lies in $N_{\varphi,\omega}$ with $WW^*=1$, and $WxW^*=xWW^*=x$. Hence $\beta^\varphi_{\lambda_j}(x)\to x$ $*$-strongly. For $\lambda_j\ge1$, $\|\beta^\varphi_{\lambda_j}(x)-x\|_{\bar\varphi}=\|x-\beta^\varphi_{1/\lambda_j}(x)\|_{\bar\varphi}\to0$, because $\bar\varphi$-preserving automorphisms preserve $\|\cdot\|_{\bar\varphi}$. The $\bar\varphi$-preserving automorphisms $\beta^\varphi_\lambda$ are implemented on $L^2(\mathrm B(\varphi),\bar\varphi)$ by unitaries $U_\lambda$ with $U_\lambda x\bar\xi=\beta^\varphi_\lambda(x)\bar\xi$; these converge strongly to $1$, which is continuity in the $u$-topology.

(5) If $(v_{k,n})$ is admissible for $\varphi$ and $(u_n)$ is a $(\psi,\varphi)$-sequence, then $(u_nv_{k,n}u_n^*)$ is admissible for $\psi$, because $\|uvu^*\psi-\lambda\psi uvu^*\|$ is at most the sum of $\|v\varphi-\lambda\varphi v\|$ and $(1+\lambda)\|v\|\,\|u\varphi u^*-\psi\|$. In $M^\omega$, $\begin{aligned} &\beta^\psi_\lambda(\beta_{\psi,\varphi}(x))\\ &\quad =\sum_kUV_kU^*\,UxU^*\,UV_k^*U^*\\ &\quad =U\beta^\varphi_\lambda(x)U^*=\beta_{\psi,\varphi}(\beta^\varphi_\lambda(x)). \end{aligned}$

(6) By (1.1) of the previous lesson, $\sigma^{\bar\varphi}_t$ restricts to $\sigma^\varphi_t$ on $N$, and $\varphi\circ\sigma_t^\varphi=\varphi$ gives $\|\sigma^\varphi_t(v)\varphi-\lambda\varphi\sigma^\varphi_t(v)\|=\|v\varphi-\lambda\varphi v\|$ and $\|\sigma^\varphi_t(y)\|^\sharp_\varphi=\|y\|^\sharp_\varphi$. So if $(v_{k,n})$ is admissible with ordinary convergence, so is $(\sigma^\varphi_t(v_{k,n}))$. The normal automorphism $\sigma^{\bar\varphi}_t$ preserves $*$-strong convergence of bounded sequences (it preserves $\|\cdot\|^\sharp_{\bar\varphi}$), so applying it to the convergence in (1), $\begin{aligned} &\sum_k\sigma^\varphi_t(v_{k,n})\,\sigma^{\bar\varphi}_t(x)\,\sigma^\varphi_t(v_{k,n})^*\\ &\quad \to\sigma^{\bar\varphi}_t(\beta^\varphi_\lambda(x))\quad*\text{-strongly}. \end{aligned}$ Since $\sigma^{\bar\varphi}_t(x)\in\mathrm B(\varphi)$ (Theorem 2.3 of the previous lesson), the left side converges to $\beta^\varphi_\lambda(\sigma^{\bar\varphi}_t(x))$ by (1).

(7) For $x\in N'\cap M$, $\sum_kv_{k,n}xv_{k,n}^*=x\sum_kv_{k,n}v_{k,n}^*\to x$. $\square$

We also use the logarithmic parametrization $\gamma^\varphi_s=\beta^\varphi_{e^{-s}}\qquad(s\in\mathbb R). \tag{3.2}$ An element $v\in N$ with $v\varphi=e^{-s}\varphi v$ is said to have modular frequency $s$; Theorem 3.1(2) says that $vx=\gamma^\varphi_s(x)v$ for $x\in\mathrm B(\varphi)$. Corollary 1.3 of Spectral shift maps shows that this condition on $v$ is equivalent to $\sigma^\varphi_t(v)=e^{its}v$ for all $t$. By Theorem 3.1, $s\mapsto\gamma^\varphi_s$ is a continuous action of $\mathbb R$ on $\mathrm B(\varphi)$ by $\bar\varphi$-preserving automorphisms commuting with $\sigma^{\bar\varphi}$ and fixing $N'\cap M$.

4. A von Neumann algebra independent of the state

Theorem 4.1. For faithful normal states $\varphi,\psi$ on $N$, $N\vee\mathrm B(\varphi)=N\vee\mathrm B(\psi).$ We denote this von Neumann algebra by $\mathrm B^\sharp=\mathrm B^\sharp(N\subset M)$.

Proof. Let $y\in\mathrm B(\psi)$ and $x=\beta_{\varphi,\psi}(y)\in\mathrm B(\varphi)$, so $y=\beta_{\psi,\varphi}(x)$. For a $(\psi,\varphi)$-sequence $(u_n)$, Theorem 2.1 gives $y=\lim u_nxu_n^*$ $*$-strongly, and $u_nxu_n^*\in N\vee\mathrm B(\varphi)$, which is strongly closed. So $\mathrm B(\psi)\subset N\vee\mathrm B(\varphi)$, and by symmetry the two algebras agree. $\square$

Both $N$ and $\mathrm B(\varphi)$ are globally invariant under $\sigma^{\bar\varphi}$ (by (1.1) and Theorem 2.3 of the previous lesson), so $\mathrm B^\sharp$ is, and the modular expectation theorem gives a $\bar\varphi$-preserving faithful normal conditional expectation $E^\sharp\colon M\to\mathrm B^\sharp$.

Proposition 4.2. $E\circ E^\sharp=E$. Consequently $\bar\chi\circ E^\sharp=\bar\chi$ for every faithful normal state $\chi$ on $N$, and $E^{\chi}_{\mathrm B}\circ E^\sharp=E^\chi_{\mathrm B}$, where $E^\chi_{\mathrm B}$ is the $\bar\chi$-preserving expectation onto $\mathrm B(\chi)$. In particular, if $z\in M$ and $E^\sharp(z)=0$, then $E^\chi_{\mathrm B}(z)=0$ for every $\chi$.

Proof. $E\circ E^\sharp$ is a contractive retraction of $M$ onto $N\subset\mathrm B^\sharp$, and $\varphi\circ E\circ E^\sharp=\bar\varphi\circ E^\sharp=\bar\varphi$. By uniqueness in the modular expectation theorem, $E\circ E^\sharp=E$. Then $\bar\chi\circ E^\sharp=\chi\circ E\circ E^\sharp=\bar\chi$. Finally $E^\chi_{\mathrm B}\circ E^\sharp$ is a contractive retraction onto $\mathrm B(\chi)\subset\mathrm B^\sharp$ preserving $\bar\chi$, so it equals $E^\chi_{\mathrm B}$. $\square$

5. Matrix amplifications

Let $R\subset N$ be a subfactor isomorphic to $M_k(\mathbb C)$, with matrix units $(e_{ij})$ and the same unit as $N$. Put $N_1=R'\cap N$ and $M_1=R'\cap M$. The maps $x\mapsto\sum_{i,j}e_{ij}\otimes e_{1i}xe_{j1}$ identify $M$ with $R\otimes M_1$ and $N$ with $R\otimes N_1$ (here $e_{1i}xe_{j1}$ is read as an element of $M_1$ through the isomorphism $e_{11}Me_{11}\cong M_1$, $y\mapsto\sum_ie_{i1}ye_{1i}$). Since $E$ is $R$-bimodular, it maps $M_1$ into $N_1$ and corresponds to $\mathrm{id}_R\otimes E_1$ with $E_1=E|_{M_1}$. The algebra $N_1\cong e_{11}Ne_{11}\cong N$ is a factor of type III₁ with separable predual, since $e_{11}\sim1$ in $N$. Let $\tau_R$ be the normalized trace of $R$.

Proposition 5.1. For every faithful normal state $\chi_1$ on $N_1$, $\begin{aligned} &\mathrm B(N\subset M,\tau_R\otimes\chi_1)\\ &\quad =1\otimes\mathrm B(N_1\subset M_1,\chi_1),\\ &\mathrm B^\sharp(N\subset M)=R\otimes\mathrm B^\sharp(N_1\subset M_1). \end{aligned}$

Proof. Write $\chi=\tau_R\otimes\chi_1$. For $x=\sum_{i,j}e_{ij}\otimes x_{ij}\in N$ and $z=\sum_{a,b}e_{ab}\otimes z_{ab}$, $\begin{aligned} [x,\chi](z)&=\frac1k\sum_{i,j}\big(\chi_1(z_{ji}x_{ij})-\chi_1(x_{ij}z_{ji})\big)\\ &=\frac1k\sum_{i,j}[x_{ij},\chi_1](z_{ji}), \end{aligned}$ so $\frac1k\max_{i,j}\|[x_{ij},\chi_1]\|\le\|[x,\chi]\|$, which is at most $\frac1k\sum_{i,j}\|[x_{ij},\chi_1]\|$. Hence a bounded sequence in $N$ lies in $\mathrm{AC}(N,\chi)$ exactly when all its matrix entries lie in $\mathrm{AC}(N_1,\chi_1)$. In particular the constant sequences $e_{ij}\otimes1$ lie in $\mathrm{AC}(N,\chi)$, so every $a\in\mathrm B(N\subset M,\chi)$ commutes with $R$ and has the form $1\otimes a_1$, $a_1\in M_1$. For such $a$ and $x_n=\sum e_{ij}\otimes x_{ij,n}$, $\|[1\otimes a_1,x_n]\|^2_{\bar\chi}=\frac1k\sum_{i,j}\|[a_1,x_{ij,n}]\|^2_{\bar\chi_1},$ which tends to $0$ for every $(x_n)\in\mathrm{AC}(N,\chi)$ exactly when $a_1\in\mathrm B(N_1\subset M_1,\chi_1)$. This proves the first identity. By Theorem 4.1, applied to both inclusions, $\begin{aligned} &\mathrm B^\sharp(N\subset M)\\ &\quad =N\vee(1\otimes\mathrm B(N_1\subset M_1,\chi_1))\\ &\quad =R\otimes\big(N_1\vee\mathrm B(N_1\subset M_1,\chi_1)\big)\\ &\quad =R\otimes\mathrm B^\sharp(N_1\subset M_1). \end{aligned}$ $\square$

6. Exercises

Exercise 6.1. Show that $\beta_{\psi,\varphi}$ maps $N'\cap M$ identically and that $E^{\psi}_{N'\cap M}\circ\beta_{\psi,\varphi}=E^\varphi_{N'\cap M}$, where $E^\varphi_{N'\cap M}$ is the $\bar\varphi$-preserving expectation onto $N'\cap M$.

Exercise 6.2. In the situation of Exercise 4.2 of the previous lesson, $N\otimes1\subset N\bar\otimes Q$, show that $\beta^\varphi_\lambda=\mathrm{id}$ on $1\otimes Q$ and that $\mathrm B^\sharp=N\bar\otimes Q$.

Exercise 6.3. Let $v\in N$ with $v\varphi=\lambda\varphi v$ and $x\in\mathrm B(\varphi)$. Show that $xv^*=v^*\beta^\varphi_\lambda(x)$ and $v^*vx=v^*\beta^\varphi_\lambda(x)v$.

Exercise 6.4. Show that $\beta^\varphi_\lambda(E^\varphi_{N'\cap M}(x))=E^\varphi_{N'\cap M}(x)$ and $E^\varphi_{N'\cap M}(\beta^\varphi_\lambda(x))=E^\varphi_{N'\cap M}(x)$ for $x\in\mathrm B(\varphi)$.

7. Solutions

6.1. The first statement is in Theorem 2.1. Both $E^\psi_{N'\cap M}\circ\beta_{\psi,\varphi}$ and $E^\varphi_{N'\cap M}$ are maps $\mathrm B(\varphi)\to N'\cap M$ fixing $N'\cap M$ and $N'\cap M$-bimodular. For $y\in N'\cap M$ and $x\in\mathrm B(\varphi)$, $\bar\psi(y^*\beta_{\psi,\varphi}(x))=\bar\psi(\beta_{\psi,\varphi}(y^*x))=\bar\varphi(y^*x)$, using $\beta_{\psi,\varphi}(y)=y$, multiplicativity and state transport. On the other hand $\bar\varphi(y^*z)=\bar\psi(y^*z)$ for $y,z\in N'\cap M$: indeed $E(y^*z)\in N'\cap N=\mathbb C1$, so $\bar\varphi(y^*z)=E(y^*z)=\bar\psi(y^*z)$. Hence $E^\psi_{N'\cap M}(\beta_{\psi,\varphi}(x))$ and $E^\varphi_{N'\cap M}(x)$ have the same inner products with every $y\in N'\cap M$ for the common state $\bar\varphi|_{N'\cap M}=\bar\psi|_{N'\cap M}$, and they are equal.

6.2. By Exercise 4.2 of the previous lesson, $\mathrm B(\varphi)=1\otimes Q\subset N'\cap(N\bar\otimes Q)$, and Theorem 3.1(7) applies. Then $\mathrm B^\sharp=N\vee(1\otimes Q)=N\bar\otimes Q$.

6.3. By Theorem 3.1(2), $vx=\beta^\varphi_\lambda(x)v$ for every $x\in\mathrm B(\varphi)$; applied to $x^*$ and taking adjoints, $xv^*=v^*\beta^\varphi_\lambda(x)$. Multiplying $vx=\beta^\varphi_\lambda(x)v$ on the left by $v^*$ gives the second identity.

6.4. $E^\varphi_{N'\cap M}(x)\in N'\cap M$ is fixed by Theorem 3.1(7). For the second identity recall that $w\bar\xi\mapsto E^\varphi_{N'\cap M}(w)\bar\xi$ is the orthogonal projection onto the closure of $(N'\cap M)\bar\xi$, so it suffices that $\beta^\varphi_\lambda(x)$ and $x$ have the same inner products with every $y\in N'\cap M$. Since $N'\cap M\subset\mathrm B(\varphi)$ and $\beta^\varphi_\lambda(y^*)=y^*$, Theorem 3.1(3) gives $\bar\varphi(y^*\beta^\varphi_\lambda(x))=\bar\varphi(\beta^\varphi_\lambda(y^*x))=\bar\varphi(y^*x)$.

References

[AHHM] H. Ando, U. Haagerup, C. Houdayer, A. Marrakchi, Structure of bicentralizer algebras and inclusions of type III factors, Mathematische Annalen 376 (2020), 1145–1194. https://arxiv.org/abs/1804.05706
[M1] A. Marrakchi, Kadison's problem for type III subfactors and the bicentralizer conjecture, Inventiones Mathematicae 239 (2025), 79–163. https://arxiv.org/abs/2308.15163