The relative bicentralizer

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

Let $N\subset M$ be von Neumann algebras with a faithful normal conditional expectation $E\colon M\to N$, and let $\varphi$ be a faithful normal state on $N$. The relative bicentralizer $\mathrm B(N\subset M,\varphi)$ consists of the elements of $M$ that asymptotically commute with every bounded sequence of $N$ that asymptotically commutes with $\varphi$. Masuda introduced it for subfactors of finite index, and Ando, Haagerup, Houdayer and Marrakchi studied it for arbitrary inclusions with expectation [AHHM]. For $N=M$ it is Connes' bicentralizer $\mathrm B(M,\varphi)$ of the course Bicentralizers of type III₁ factors.

This lesson proves the basic structure: four equivalent descriptions, one inside the Ocneanu ultrapower (Proposition 2.2); that $\mathrm B(N\subset M,\varphi)$ is a von Neumann subalgebra with a state-preserving conditional expectation $E_{\mathrm B}$ (Theorem 2.3); that $E_{\mathrm B}$ is a limit of averages of conjugations by unitaries of $N$ that almost commute with $\varphi$ (Theorem 3.4); and that every element of $M$ orthogonal to $\mathrm B(N\subset M,\varphi)$ is moved by some unitary of the asymptotic centralizer of $N$ (Theorem 3.4(5)). When $N$ is a factor of type III₁ with separable predual, the triviality of its own bicentralizer gives $E(b)=\varphi(E(b))1$ for $b\in\mathrm B(N\subset M,\varphi)$ and $E_{\mathrm B}(x)=\varphi(x)1$ for $x\in N$ (Corollary 2.6). These two identities make the products $xb$ and $bx$ behave like elementary tensors, which is the starting point of the later lessons.

We use: the conventions of Section 1 and Lemmas 1.1 and 1.3 of Ultraproducts and the asymptotic centralizer, together with its Theorem 5.1 (the finite von Neumann algebra $M_{\varphi,\omega}$); the Ocneanu ultrapower $M^\omega$, its faithful normal state (Lemma 2.1 and Theorem 3.2) and the normal copy of $M$ (Section 4) of Multiplier ultraproducts and normal embeddings; Lemma 1.1 (asymptotic centralizers inside $M^\omega$) and Theorem 3.1 of Asymptotic centralizers of type III₁ factors, and equation (1.1) there; Lemma 1.2 (four unitaries) of The bicentralizer; Theorem 6.1 of The bicentralizer of a type III₁ factor is trivial; the modular expectation theorem (ME.1)–(ME.3) of Conditional expectations from modular invariance, including the uniqueness of a state-preserving contractive retraction; the Schwarz inequality for conditional expectations, CE-006; the Borel functional calculus in a von Neumann algebra, Theorem 3.1 of the spectral theorem lesson; and Mazur's theorem (the weak and norm closures of a convex set in a Hilbert space agree), Weak topologies, Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian.

1. Setting

Throughout, $M$ is a countably decomposable von Neumann algebra, $N\subset M$ a von Neumann subalgebra with the same unit, $E\colon M\to N$ a faithful normal conditional expectation, and $\omega$ a free ultrafilter on $\mathbb N$. For a faithful normal state $\varphi$ on $N$ put $\bar\varphi=\varphi\circ E ,$ a faithful normal state on $M$. We work in the GNS space $H=L^2(M,\bar\varphi)$ with cyclic and separating vector $\bar\xi$, write $\|x\|_{\bar\varphi}=\|x\bar\xi\|$ and $\|x\|^\sharp_{\bar\varphi}=(\|x\|_{\bar\varphi}^2+\|x^*\|_{\bar\varphi}^2)^{1/2}$, and use the bimodule notation $(x\psi)(y)=\psi(yx)$, $(\psi x)(y)=\psi(xy)$, $[x,\psi]=x\psi-\psi x$ of the ultraproduct lesson. Since $E$ is a $\bar\varphi$-preserving contractive retraction onto $N$, the modular expectation theorem gives $\begin{gathered} \sigma^{\bar\varphi}_t(N)=N,\\ \sigma^{\bar\varphi}_t|_N=\sigma^\varphi_t,\\ E\circ\sigma_t^{\bar\varphi}=\sigma^\varphi_t\circ E\\ (t\in\mathbb R). \end{gathered}\tag{1.1}$ Write $\mathcal U(N)$ for the unitary group of $N$, $\mathrm{AC}(N,\varphi)$ for the set of sequences $(x_n)\in\ell^\infty(N)$ with $\|[x_n,\varphi]\|\to0$, $A_{\varphi,\omega}(N)$ for the set of those with $\lim_{n\to\omega}\|[x_n,\varphi]\|=0$, and $\mathcal U_\delta=\{u\in\mathcal U(N):\ \|[u,\varphi]\|<\delta\}$ for $\delta>0$. Here $\|[x,\varphi]\|$ is the norm in $N_*$. By (1.1) of the asymptotic centralizer lesson, $\|[u,\varphi]\|=\|u\varphi u^*-\varphi\|$ for a unitary $u$.

Lemma 1.1. For $x\in N$, $\begin{gathered} \|[x,\bar\varphi]\|_{M_*}=\|[x,\varphi]\|_{N_*},\\ \|x\|_{\bar\varphi}=\|x\|_\varphi,\qquad \|x\|^\sharp_{\bar\varphi}=\|x\|^\sharp_\varphi . \end{gathered}$ Consequently $\mathrm{AC}(N,\varphi)\subset\mathrm{AC}(M,\bar\varphi)$, $A_{\varphi,\omega}(N)\subset A_{\bar\varphi,\omega}(M)$, and a bounded sequence in $N$ tends to $0$ $*$-strongly along $\omega$ as a sequence in $N$ exactly when it does so as a sequence in $M$.

Proof. Since $E$ is $N$-bimodular, for $y\in M$ $\begin{aligned} &[x,\bar\varphi](y)=\varphi(E(yx)-E(xy))\\ &\quad =\varphi(E(y)x-xE(y))=[x,\varphi](E(y)). \end{aligned}$ $E$ maps the unit ball of $M$ onto the unit ball of $N$ (it is contractive and fixes $N$), so the two norms agree. Also $\|x\|^2_{\bar\varphi}=\varphi(E(x^*x))=\varphi(x^*x)$. The last assertion is Lemma 1.3 of the ultraproduct lesson, applied in $N$ with $\varphi$ and in $M$ with $\bar\varphi$. $\square$

Write $I_\omega$, $N_\omega$ and $M^\omega=N_\omega/I_\omega$ for the Ocneanu ultrapower of $M$, with its faithful normal state $\bar\varphi^\omega$, and $\pi\colon N_\omega\to M^\omega$ for the quotient map; we identify $M$ with the constant sequences. By Theorem 5.1 of the ultraproduct lesson applied to $(N,\varphi)$, $N_{\varphi,\omega}=A_{\varphi,\omega}(N)/I_\omega(N)$ is a finite von Neumann algebra with faithful normal tracial state $\tau_{\varphi,\omega}([x_n])=\lim_\omega\varphi(x_n)$.

Lemma 1.2. The inclusion $A_{\varphi,\omega}(N)\subset A_{\bar\varphi,\omega}(M)\subset N_\omega$ induces an injective normal unital $*$-homomorphism $N_{\varphi,\omega}\to M^\omega$ with $\bar\varphi^\omega\circ\iota=\tau_{\varphi,\omega}$. Every element of its image commutes with $\bar\varphi^\omega$. If $N$ is a factor of type III₁ with separable predual, $N_{\varphi,\omega}$ is a factor of type II₁.

Proof. $A_{\bar\varphi,\omega}(M)\subset N_\omega$ by Lemma 1.1(1) of the asymptotic centralizer lesson, and the first inclusion is Lemma 1.1. The kernel of $A_{\varphi,\omega}(N)\to M^\omega$ is $A_{\varphi,\omega}(N)\cap I_\omega=I_\omega(N)$ by Lemma 1.1, so the induced map $\iota$ is injective, and $\bar\varphi^\omega(\iota[x_n])=\lim_\omega\bar\varphi(x_n)=\lim_\omega\varphi(x_n)$. If $a_i\uparrow a$ in $N_{\varphi,\omega}$, then $Z=\sup_i\iota(a_i)\le\iota(a)$ and $\bar\varphi^\omega(\iota(a)-Z)=\tau_{\varphi,\omega}(a)-\lim_i\tau_{\varphi,\omega}(a_i)$, which is $0$, so $Z=\iota(a)$ by faithfulness: $\iota$ is normal. Commutation with $\bar\varphi^\omega$ is Lemma 1.1(2) of the asymptotic centralizer lesson, applied in $M$ with $\bar\varphi$. The last statement is Theorem 3.1 of that lesson applied to $(N,\varphi)$. $\square$

We identify $N_{\varphi,\omega}$ with its image in $M^\omega$.

Let $\mathcal G_\varphi$ be the set of classes $\pi(u_n)$ of sequences $(u_n)\in A_{\varphi,\omega}(N)$ consisting of unitaries of $N$. It is a subgroup of the unitary group of $N_{\varphi,\omega}$, since products and adjoints of such sequences are again such sequences.

Lemma 1.3. Every element of $N_{\varphi,\omega}$ is a linear combination of four elements of $\mathcal G_\varphi$. Consequently an element of $M^\omega$ commutes with $N_{\varphi,\omega}$ as soon as it commutes with $\mathcal G_\varphi$.

Proof. $A_{\varphi,\omega}(N)$ is a unital $C^*$-algebra (Theorem 5.1 of the ultraproduct lesson), and every element of a unital $C^*$-algebra is a linear combination of four of its unitaries (Proposition 7.3 of the $C^*$-algebra lesson). A unitary of $A_{\varphi,\omega}(N)$ is a sequence of unitaries of $N$. Apply $\pi$. $\square$

Lemma 1.4 (the canonical expectation). For $X=\pi(x_n)\in M^\omega$ the $\sigma$-weak limit $E_\omega(X)=\lim_{n\to\omega}x_n$ exists in $M$ and depends only on $X$. The map $E_\omega\colon M^\omega\to M$ is a faithful normal conditional expectation onto $M$, with $\bar\varphi\circ E_\omega=\bar\varphi^\omega$ and $E_\omega(aXb)=aE_\omega(X)b$ for $a,b\in M$.

Proof. Bounded sets of $M$ are $\sigma$-weakly compact, so the limit along $\omega$ exists. A sequence in $I_\omega$ tends to $0$ strongly along $\omega$, hence $\sigma$-weakly, so $E_\omega$ is well defined; it is linear, unital and $M$-bimodular, and it is the identity on constants. If $X\ge0$, write $X=Y^*Y$ with $Y=\pi(y_n)$; then $X=\pi(y_n^*y_n)$ and $E_\omega(X)=\lim_\omega y_n^*y_n\ge0$. By $\sigma$-weak continuity of $\bar\varphi$, $\bar\varphi(E_\omega(X))=\lim_\omega\bar\varphi(x_n)=\bar\varphi^\omega(X)$. If $X\ge0$ and $E_\omega(X)=0$, then $\bar\varphi^\omega(X)=0$ and $X=0$. If $X_i\uparrow X$, then $E_\omega(X_i)\uparrow Z\le E_\omega(X)$ and $\bar\varphi(E_\omega(X)-Z)=\bar\varphi^\omega(X)-\lim_i\bar\varphi^\omega(X_i)$, which is $0$ by normality of $\bar\varphi^\omega$, so $Z=E_\omega(X)$. $\square$

For $X=\pi(x_n)$ we have $\|X\|_{\bar\varphi^\omega}=\lim_\omega\|x_n\|_{\bar\varphi}$, by the definition of $\bar\varphi^\omega$ and multiplicativity of $\pi$; in particular $\langle X,Y\rangle_{\bar\varphi^\omega}=\bar\varphi^\omega(Y^*X)=\lim_\omega\bar\varphi(y_n^*x_n)$.

2. The relative bicentralizer

Definition 2.1. The relative bicentralizer $\mathrm B(N\subset M,\varphi)$ of $\varphi$ is the set of $a\in M$ with $\|[a,x_n]\|_{\bar\varphi}\to0$ for every $(x_n)\in\mathrm{AC}(N,\varphi)$.

By Lemma 1.3 of the ultraproduct lesson, the condition says that $[a,x_n]\to0$ strongly; it does not depend on the faithful normal state used to measure it, so $\mathrm B(N\subset M,\varphi)$ does not depend on $E$. For $N=M$ and $E=\mathrm{id}$ it is the bicentralizer $\mathrm B(M,\varphi)$.

Proposition 2.2. For every $a\in M$ the following are equivalent:

$a\in\mathrm B(N\subset M,\varphi)$;
$\lim_\omega\|[a,x_n]\|_{\bar\varphi}=0$ for every $(x_n)\in A_{\varphi,\omega}(N)$;
$aX=Xa$ in $M^\omega$ for every $X\in N_{\varphi,\omega}$;
for every $\varepsilon>0$ there is $\delta>0$ with $\|u^*au-a\|_{\bar\varphi}<\varepsilon$ for every $u\in\mathcal U_\delta$.

Proof. (2)$\Leftrightarrow$(3). For $X=\pi(x_n)$ with $(x_n)\in A_{\varphi,\omega}(N)$, $aX-Xa=\pi(ax_n-x_na)$ and $\|aX-Xa\|_{\bar\varphi^\omega}=\lim_\omega\|[a,x_n]\|_{\bar\varphi}$; $\bar\varphi^\omega$ is faithful.

(2)$\Rightarrow$(1). If $(x_n)\in\mathrm{AC}(N,\varphi)$ and $\|[a,x_{n_k}]\|_{\bar\varphi}\ge c>0$ along a subsequence, then $(x_{n_k})_k\in\mathrm{AC}(N,\varphi)\subset A_{\varphi,\omega}(N)$ contradicts (2).

(1)$\Rightarrow$(2). Let $(x_n)\in A_{\varphi,\omega}(N)$ with $c=\lim_\omega\|[a,x_n]\|_{\bar\varphi}>0$. The sets $\{n:\|[x_n,\varphi]\|<1/j,\ \|[a,x_n]\|_{\bar\varphi}>c/2\}$ belong to $\omega$ and are infinite, so there are $n_1<n_2<\cdots$ with $n_j$ in the $j$-th set; then $(x_{n_j})_j\in\mathrm{AC}(N,\varphi)$ contradicts (1).

(1)$\Leftrightarrow$(4). For a unitary $u$, $[a,u]=u(u^*au-a)$ and $\|uy\|_{\bar\varphi}=\|y\|_{\bar\varphi}$, so $\|[a,u]\|_{\bar\varphi}=\|u^*au-a\|_{\bar\varphi}$. If (4) fails, there are $\varepsilon>0$ and $u_j\in\mathcal U_{1/j}$ with $\|[a,u_j]\|_{\bar\varphi}\ge\varepsilon$, against (1). Conversely (4) gives $\|[a,u_n]\|_{\bar\varphi}\to0$ for every unitary $(u_n)\in\mathrm{AC}(N,\varphi)$; by Lemma 1.2 of the bicentralizer lesson, applied to $(N,\varphi)$, every element of the $C^*$-algebra $\mathrm{AC}(N,\varphi)$ is a linear combination of four of its unitaries, which gives (1). $\square$

So $\mathrm B(N\subset M,\varphi)=(N_{\varphi,\omega})'\cap M$, the relative commutant being taken in $M^\omega$, for every free ultrafilter $\omega$.

Theorem 2.3. $\mathrm B=\mathrm B(N\subset M,\varphi)$ is a von Neumann subalgebra of $M$, and $\begin{gathered} N'\cap M\subset\mathrm B\subset(N_\varphi)'\cap M,\\ \sigma_t^{\bar\varphi}(\mathrm B)=\mathrm B\ (t\in\mathbb R),\\ \mathrm B\cap N=\mathrm B(N,\varphi). \end{gathered}$ There is a unique $\bar\varphi$-preserving normal conditional expectation $E_{\mathrm B}$ of $M$ onto $\mathrm B$; it is faithful.

Proof. By Proposition 2.2, $\mathrm B$ is the set of constants commuting with the self-adjoint set $N_{\varphi,\omega}\subset M^\omega$: a unital $*$-subalgebra. If $a_i\in\mathrm B$ is a bounded net converging $\sigma$-weakly to $a\in M$, it converges $\sigma$-weakly in $M^\omega$ (the constant embedding is normal), and commutants are $\sigma$-weakly closed, so $a\in\mathrm B$. By Kaplansky's density theorem, exactly as in the proof of Theorem 2.1 of the bicentralizer lesson, a unital $*$-algebra with this property is $\sigma$-weakly closed. Elements of $N'\cap M$ commute with every $x_n\in N$. A constant $x\in N_\varphi$ gives $(x)\in\mathrm{AC}(N,\varphi)$, so $[a,x]=0$ for $a\in\mathrm B$.

By (1.1) and $\varphi\circ\sigma^\varphi_t=\varphi$, $\|[\sigma_t^{\varphi}(x),\varphi]\|=\|[x,\varphi]\|$ for $x\in N$, so $\mathrm{AC}(N,\varphi)$ is invariant under $\sigma^\varphi_t$ termwise. For $a\in\mathrm B$ and $(x_n)\in\mathrm{AC}(N,\varphi)$, $\begin{aligned} &\|[\sigma_t^{\bar\varphi}(a),x_n]\|_{\bar\varphi}=\|\sigma_t^{\bar\varphi}([a,\sigma^{\varphi}_{-t}(x_n)])\|_{\bar\varphi}\\ &\quad =\|[a,\sigma^\varphi_{-t}(x_n)]\|_{\bar\varphi}\to0, \end{aligned}$ since $\bar\varphi\circ\sigma^{\bar\varphi}_t=\bar\varphi$. So $\mathrm B$ is $\sigma^{\bar\varphi}$-invariant, and the modular expectation theorem gives a unique $\bar\varphi$-preserving contractive retraction onto $\mathrm B$, which is a faithful normal conditional expectation. Finally, for $a\in N$, $\|[a,x_n]\|_{\bar\varphi}=\|[a,x_n]\|_\varphi$ by Lemma 1.1, which gives $\mathrm B\cap N=\mathrm B(N,\varphi)$. $\square$

Since $E_{\mathrm B}$ is $\mathrm B$-bimodular and $\bar\varphi$-preserving, $\langle z,b\rangle_{\bar\varphi}=\bar\varphi(b^*z)=\bar\varphi(b^*E_{\mathrm B}(z))$ for $z\in M$, $b\in\mathrm B$: the map $z\bar\xi\mapsto E_{\mathrm B}(z)\bar\xi$ is the orthogonal projection of $H$ onto the closure of $\mathrm B\bar\xi$. In particular $E_{\mathrm B}(z)=0$ exactly when $z\bar\xi\perp\mathrm B\bar\xi$.

Lemma 2.4 (uniform modulus). Let $a\in\mathrm B(N\subset M,\varphi)$ and $\varepsilon>0$. There is $\delta>0$ such that $\|[a,y]\|^\sharp_{\bar\varphi}\le\varepsilon$ for every $y\in N$ with $\|y\|\le1$ and $\|[y,\varphi]\|\le\delta$.

Proof. Otherwise there are contractions $y_k\in N$ with $\|[y_k,\varphi]\|\le1/k$ and $\|[a,y_k]\|^\sharp_{\bar\varphi}>\varepsilon$. Then $(y_k)$ and $(y_k^*)$ belong to $\mathrm{AC}(N,\varphi)$ (Lemma 1.1(a) of the ultraproduct lesson), and $a,a^*\in\mathrm B(N\subset M,\varphi)$ by Theorem 2.3, so $\|[a,y_k]\|_{\bar\varphi}\to0$ and $\|[a,y_k]^*\|_{\bar\varphi}=\|[a^*,y_k^*]\|_{\bar\varphi}\to0$, a contradiction. $\square$

Proposition 2.5. $E(\mathrm B(N\subset M,\varphi))\subset\mathrm B(N,\varphi)$.

Proof. Let $a\in\mathrm B(N\subset M,\varphi)$ and $(x_n)\in\mathrm{AC}(N,\varphi)$. Since $E$ is $N$-bimodular, $[E(a),x_n]=E([a,x_n])$, and the Schwarz inequality $E(z)^*E(z)\le E(z^*z)$ gives $\begin{aligned} &\|[E(a),x_n]\|_\varphi^2\\ &\quad =\varphi(E([a,x_n])^*E([a,x_n]))\\ &\quad \le\bar\varphi([a,x_n]^*[a,x_n])=\|[a,x_n]\|^2_{\bar\varphi}\to0 . \end{aligned}$ $\square$

Corollary 2.6. Suppose $N$ is a factor of type III₁ with separable predual, and write $\mathrm B=\mathrm B(N\subset M,\varphi)$. Then $\begin{gathered} E(b)=\bar\varphi(b)1\quad(b\in\mathrm B),\\ E_{\mathrm B}(x)=\varphi(x)1\quad(x\in N), \end{gathered}$ and consequently $\bar\varphi(xb)=\bar\varphi(bx)=\varphi(x)\bar\varphi(b)$ for $x\in N$, $b\in\mathrm B$.

Proof. By Proposition 2.5 and Theorem 6.1 of the trivial bicentralizer lesson, $E(b)\in\mathrm B(N,\varphi)=\mathbb C1$, and the scalar is $\varphi(E(b))=\bar\varphi(b)$. For $x\in N$ and $b\in\mathrm B$, bimodularity of $E$ gives $\begin{gathered} \bar\varphi(xb)=\varphi(xE(b))=\varphi(x)\bar\varphi(b),\\ \bar\varphi(bx)=\varphi(E(b)x)=\bar\varphi(b)\varphi(x). \end{gathered}$ The second identity, with $b^*$ in place of $b$, says $\langle x,b\rangle_{\bar\varphi}=\langle\varphi(x)1,b\rangle_{\bar\varphi}$ for all $b\in\mathrm B$; since $\varphi(x)1\in\mathrm B$, this is $E_{\mathrm B}(x)=\varphi(x)1$. $\square$

3. Averaging by asymptotically centralizing unitaries

Lemma 3.1 (averaging over state-preserving automorphisms). Let $P$ be a von Neumann algebra with a faithful normal state $\psi$, $G$ a group of $\psi$-preserving $*$-automorphisms of $P$, and $P^G$ its fixed-point algebra. For $X\in P$ let $K_G(X)$ be the $\sigma$-weakly closed convex hull of $\{\alpha(X):\alpha\in G\}$.

$K_G(X)\cap P^G$ consists of exactly one element $E_G(X)$, which is also the unique element of $K_G(X)$ of minimal $\|\cdot\|_\psi$-norm. The map $E_G$ is the unique $\psi$-preserving conditional expectation of $P$ onto $P^G$; it is normal and faithful, and $\langle X-E_G(X),Z\rangle_\psi=0$ for $Z\in P^G$.
If $E_G(X)=0$, there is $\alpha\in G$ with $\|\alpha(X)-X\|_\psi\ge\|X\|_\psi$.

Proof. (1) $K_G(X)$ is a $\sigma$-weakly compact convex set, and $Y\mapsto Y\xi_\psi$ is injective and continuous from the $\sigma$-weak to the weak topology, so $K_G(X)\xi_\psi$ is weakly compact and convex and has a unique vector $Y_0\xi_\psi$ of minimal norm, $Y_0\in K_G(X)$. Automorphisms of von Neumann algebras are $\sigma$-weakly continuous, so each $\alpha\in G$ maps $K_G(X)$ onto itself, and $\|\alpha(Y)\|_\psi=\|Y\|_\psi$; hence $\alpha(Y_0)=Y_0$, and $Y_0\in P^G$. For $Z\in P^G$ and $\alpha\in G$, $\langle\alpha(X),Z\rangle_\psi=\psi(\alpha(Z^*X))=\langle X,Z\rangle_\psi$, so $Y\mapsto\langle Y,Z\rangle_\psi$ is constant on $K_G(X)$. If $Y,Y'\in K_G(X)\cap P^G$, then $\langle Y-Y',Z\rangle_\psi=0$ for all $Z\in P^G$, in particular for $Z=Y-Y'$, so $Y=Y'$. Thus $K_G(X)\cap P^G=\{Y_0\}$; put $E_G(X)=Y_0$. Then $E_G(X)\xi_\psi$ is the orthogonal projection of $X\xi_\psi$ onto the closure of $P^G\xi_\psi$, so $E_G$ is linear. It is positive ($K_G(X)\subset P_+$ when $X\ge0$), unital, $\psi$-preserving (take $Z=1$), and $P^G$-bimodular, because $Z_1K_G(X)Z_2=K_G(Z_1XZ_2)$ for $Z_1,Z_2\in P^G$ and $Z_1E_G(X)Z_2\in K_G(Z_1XZ_2)\cap P^G$. So $E_G$ is a $\psi$-preserving conditional expectation onto $P^G$; by the modular expectation theorem it is unique, normal and faithful.

(2) Suppose $X\ne0$ and $\|\alpha(X)-X\|_\psi<\|X\|_\psi$ for all $\alpha$. Since $\|\alpha(X)\|_\psi=\|X\|_\psi$, this says $\operatorname{Re}\langle\alpha(X),X\rangle_\psi>\frac12\|X\|_\psi^2$. The set of $Y\in P$ with $\operatorname{Re}\langle Y,X\rangle_\psi\ge\frac12\|X\|^2_\psi$ is convex and $\sigma$-weakly closed, so it contains $K_G(X)\ni E_G(X)=0$, which is absurd. $\square$

Apply Lemma 3.1 to $P=M^\omega$, $\psi=\bar\varphi^\omega$ and $G=\{\operatorname{Ad}U:U\in\mathcal G_\varphi\}$. These automorphisms preserve $\bar\varphi^\omega$ by Lemma 1.2, and by Lemma 1.3 the fixed-point algebra is $Q_\varphi=(N_{\varphi,\omega})'\cap M^\omega .$ Write $F_\varphi$ for the $\bar\varphi^\omega$-preserving conditional expectation of $M^\omega$ onto $Q_\varphi$. By Proposition 2.2, $Q_\varphi\cap M=\mathrm B(N\subset M,\varphi)$.

Lemma 3.2 (uniformity). Let $X=\pi(x_n)\in M^\omega$. Then $X\in Q_\varphi$ if and only if for every $\varepsilon>0$ there are $\delta>0$ and $A\in\omega$ such that $\begin{gathered} \|ux_nu^*-x_n\|_{\bar\varphi}\le\varepsilon\\ (n\in A,\ u\in\mathcal U_\delta). \end{gathered}$

Proof. Suppose the condition holds and let $U=\pi(u_n)\in\mathcal G_\varphi$, with unitaries $u_n$ and $\lim_\omega\|[u_n,\varphi]\|=0$. Given $\varepsilon$, with $\delta,A$ as in the condition, the set of $n\in A$ with $u_n\in\mathcal U_\delta$ belongs to $\omega$, so $\|UXU^*-X\|_{\bar\varphi^\omega}$, which equals $\lim_\omega\|u_nx_nu_n^*-x_n\|_{\bar\varphi}$, is at most $\varepsilon$. Hence $UXU^*=X$, and $X\in Q_\varphi$ by Lemma 1.3.

Conversely suppose the condition fails for some $\varepsilon$. For each $k$ the set $B_k$ of the $n$ for which some $u\in\mathcal U_{1/k}$ has $\|ux_nu^*-x_n\|_{\bar\varphi}>\varepsilon$ belongs to $\omega$ (otherwise its complement would serve as $A$ with $\delta=1/k$), and $B_1\supset B_2\supset\cdots$. For $n\in B_1$ let $k(n)$ be the largest $k\le n$ with $n\in B_k$, and choose $u_n\in\mathcal U_{1/k(n)}$ with $\|u_nx_nu_n^*-x_n\|_{\bar\varphi}>\varepsilon$; for $n\notin B_1$ put $u_n=1$. For every $K$, $\{n:k(n)\ge K\}\supset B_K\cap\{n\ge K\}\in\omega$, so $\lim_\omega\|[u_n,\varphi]\|=0$, and $U=\pi(u_n)\in\mathcal G_\varphi$. But $\|UXU^*-X\|_{\bar\varphi^\omega}\ge\varepsilon$ because $B_1\in\omega$, so $X\notin Q_\varphi$. $\square$

For $x\in M$ and $\delta>0$ let $K_\delta(x)$ be the $\sigma$-weakly closed convex hull in $M$ of $\{uxu^*:u\in\mathcal U_\delta\}$. As in the proof of Lemma 3.1, $K_\delta(x)$ has a unique element $y_\delta(x)$ of minimal $\|\cdot\|_{\bar\varphi}$-norm. We use the elementary estimate, valid for a unitary $v\in N$ and $w\in M$: $\begin{aligned} &\|wv^*\|^2_{\bar\varphi}=\bar\varphi(vw^*wv^*)\\ &\quad =(v^*\bar\varphi v)(w^*w)\\ &\quad \le\|w\|^2_{\bar\varphi}+\|w\|^2\,\|[v,\varphi]\| , \end{aligned}\tag{3.1}$ because $\|v^*\bar\varphi v-\bar\varphi\|=\|\bar\varphi-v\bar\varphi v^*\|$, which is $\|[v,\bar\varphi]\|=\|[v,\varphi]\|$ by Lemma 1.1; the same bound holds for $\|wv\|^2_{\bar\varphi}$. Also $\|vw\|_{\bar\varphi}=\|w\|_{\bar\varphi}$.

Proposition 3.3. For every $x\in M$ and $\delta>0$, $y_\delta(x)\in\mathrm B(N\subset M,\varphi)$.

Proof. Put $y=y_\delta(x)$, $m=\|y\|_{\bar\varphi}$, and let $C_\eta$ be the convex hull of $\{uxu^*:u\in\mathcal U_\eta\}$. Since $\mathcal U_\delta=\bigcup_{\eta<\delta}\mathcal U_\eta$, $C_\delta$ is the increasing union of the $C_\eta$, $\eta<\delta$. The vector $y\bar\xi$ is a weak limit of vectors in $C_\delta\bar\xi$, so by Mazur's theorem it is a norm limit of such vectors: for every $k$ there are $\eta_k<\delta$ and $z^{(k)}\in C_{\eta_k}$ with $\|z^{(k)}-y\|_{\bar\varphi}<1/k$, and we may take $\eta_1\le\eta_2\le\cdots$.

For $w\in K_\delta(x)$, $(w+y)/2\in K_\delta(x)$, so the parallelogram law gives $\begin{aligned} &\|w-y\|^2_{\bar\varphi}=2\|w\|^2_{\bar\varphi}\\ &\quad +2m^2-4\|(w+y)/2\|^2_{\bar\varphi}\\ &\quad \le2\|w\|^2_{\bar\varphi}-2m^2 . \end{aligned}\tag{3.2}$

Let $(v_n)$ be unitaries of $N$ with $\theta_n=\|[v_n,\varphi]\|\to0$. For $n$ with $\theta_n<\delta-\eta_1$ let $k(n)$ be the largest $k\le n$ with $\theta_n<\delta-\eta_k$; then $k(n)\to\infty$. Put $z_n=z^{(k(n))}$. If $u\in\mathcal U_{\eta_{k(n)}}$, then $\|[v_nu,\varphi]\|\le\theta_n+\|[u,\varphi]\|<\delta$ by Lemma 1.1(d) of the ultraproduct lesson, so $v_nz_nv_n^*\in C_\delta\subset K_\delta(x)$. By (3.1), $\|v_nz_nv_n^*\|^2_{\bar\varphi}\le\|z_n\|^2_{\bar\varphi}+\|x\|^2\theta_n\to m^2$, so (3.2) gives $\|v_nz_nv_n^*-y\|_{\bar\varphi}\to0$. Also by (3.1), $\|v_n(y-z_n)v_n^*\|^2_{\bar\varphi}$ is at most $\|y-z_n\|^2_{\bar\varphi}+4\|x\|^2\theta_n$, which tends to $0$. Hence $\|v_nyv_n^*-y\|_{\bar\varphi}\to0$, and again by (3.1) $\begin{aligned} &\|[v_n,y]\|^2_{\bar\varphi}=\|(v_nyv_n^*-y)v_n\|^2_{\bar\varphi}\\ &\quad \le\|v_nyv_n^*-y\|^2_{\bar\varphi}+4\|x\|^2\theta_n\to0 . \end{aligned}$ This holds for every unitary sequence in $\mathrm{AC}(N,\varphi)$, hence, by the four-unitary decomposition used in Proposition 2.2, for every sequence in $\mathrm{AC}(N,\varphi)$. So $y\in\mathrm B(N\subset M,\varphi)$. $\square$

Theorem 3.4. Write $\mathrm B=\mathrm B(N\subset M,\varphi)$.

$E_\omega(Q_\varphi)\subset\mathrm B$.
$E_{\mathrm B}=E_\omega\circ F_\varphi|_M$.
For every $\delta>0$, every finite set $S\subset M$ and every $\varepsilon>0$ there is a probability measure $\mu$ on $\mathcal U_\delta$ with finite support such that $\begin{gathered} \Big\|\int uxu^*\,d\mu(u)-E_{\mathrm B}(x)\Big\|^\sharp_{\bar\varphi}<\varepsilon\\ (x\in S). \end{gathered}$
In particular $E_{\mathrm B}(x)\in K_\delta(x)$ for every $x\in M$ and $\delta>0$.
If $z\in M$ and $E_{\mathrm B}(z)=0$, then $F_\varphi(z)=0$, and there is $U\in\mathcal G_\varphi$ with $\|UzU^*-z\|_{\bar\varphi^\omega}\ge\|z\|_{\bar\varphi}$.

Proof. (1) Let $X=\pi(x_n)\in Q_\varphi$, $a=E_\omega(X)$, and $\varepsilon>0$; take $\delta,A$ from Lemma 3.2. For $u\in\mathcal U_\delta$, $uau^*-a$ is the $\sigma$-weak limit along $\omega$ of $ux_nu^*-x_n$, and the map $y\mapsto y\bar\xi$ is $\sigma$-weak to weak continuous, so $\|uau^*-a\|_{\bar\varphi}\le\lim_\omega\|ux_nu^*-x_n\|_{\bar\varphi}$, which is at most $\varepsilon$. Since $u^*\in\mathcal U_\delta$ whenever $u\in\mathcal U_\delta$, Proposition 2.2(4) gives $a\in\mathrm B$.

(2) The map $E_\omega\circ F_\varphi|_M$ takes values in $\mathrm B$ by (1), and is the identity on $\mathrm B\subset Q_\varphi$. It is a positive unital $\mathrm B$-bimodular map, hence a contractive retraction onto $\mathrm B$, and $\bar\varphi\circ E_\omega\circ F_\varphi=\bar\varphi^\omega\circ F_\varphi=\bar\varphi^\omega$ restricts to $\bar\varphi$ on $M$. By the uniqueness in the modular expectation theorem it equals $E_{\mathrm B}$.

(3) Let $S=\{x_1,\ldots,x_k\}$ and enlarge it to contain the adjoints. Apply Lemma 3.1 to $P=(M^\omega)^{\oplus k}$ with the state $\frac1k\sum_i\bar\varphi^\omega$ and the diagonal action of $\mathcal G_\varphi$: its fixed-point algebra is $Q_\varphi^{\oplus k}$, and by uniqueness its expectation is $F_\varphi^{\oplus k}$. So $(F_\varphi(x_i))_i$ is a $\sigma$-weak limit of convex combinations $\big(\sum_j\lambda_jU_jx_iU_j^*\big)_i$ with $U_j\in\mathcal G_\varphi$. Applying the normal map $E_\omega$ and (2), $(E_{\mathrm B}(x_i))_i$ is a $\sigma$-weak limit of the tuples $\big(E_\omega(\sum_j\lambda_jU_jx_iU_j^*)\big)_i$. Write $U_j=\pi(u_{j,n})$ with unitaries $u_{j,n}\in N$ and $\lim_\omega\|[u_{j,n},\varphi]\|=0$. Then $E_\omega(\sum_j\lambda_jU_jx_iU_j^*)$ is the $\sigma$-weak limit along $\omega$ of $\sum_j\lambda_ju_{j,n}x_iu_{j,n}^*$, and for $n$ in a set belonging to $\omega$ all $u_{j,n}$ lie in $\mathcal U_\delta$. Consequently $(E_{\mathrm B}(x_i))_i$ lies in the $\sigma$-weak closure, in $M^{\oplus k}$, of the convex set of tuples $\big(\int ux_iu^*d\mu(u)\big)_i$ with $\mu$ finitely supported on $\mathcal U_\delta$. Passing to the vectors $(y_i\bar\xi)_i\in H^{\oplus k}$, Mazur's theorem turns the weak approximation into a norm approximation; since the adjoints are in $S$, this is the $\|\cdot\|^\sharp_{\bar\varphi}$ estimate.

(4) is the case $S=\{x\}$ of the argument in (3), before Mazur's theorem.

(5) Let $X=\pi(x_n)\in Q_\varphi$ and $\varepsilon>0$, with $\delta,A$ from Lemma 3.2. For $n\in A$ every element of $C_\delta(x_n)$, hence of $K_\delta(x_n)$, lies within $\varepsilon$ of $x_n$ in $\|\cdot\|_{\bar\varphi}$ (weak lower semicontinuity of the norm), so $\|y_\delta(x_n)-x_n\|_{\bar\varphi}\le\varepsilon$. By Proposition 3.3, $y_\delta(x_n)\in\mathrm B$, so $\langle y_\delta(x_n),z\rangle_{\bar\varphi}=0$ since $E_{\mathrm B}(z)=0$. Hence $|\langle x_n,z\rangle_{\bar\varphi}|\le\varepsilon\|z\|_{\bar\varphi}$ for $n\in A$, and $|\langle X,z\rangle_{\bar\varphi^\omega}|=\lim_\omega|\bar\varphi(z^*x_n)|\le\varepsilon\|z\|_{\bar\varphi} .$ So $z\perp Q_\varphi$ in $L^2(M^\omega,\bar\varphi^\omega)$, and $\|F_\varphi(z)\|^2_{\bar\varphi^\omega}=\langle z,F_\varphi(z)\rangle_{\bar\varphi^\omega}=0$ by Lemma 3.1(1). Lemma 3.1(2) gives the unitary. $\square$

Part (3) is the averaging statement used by Marrakchi to construct the zero-shift maps of a later lesson [M2, Lemma 2.2(1)]; part (5) is the form of the argument of [AHHM, Proposition 3.3] used in the construction of amenable subalgebras.

4. Exercises

Exercise 4.1. Let $Q$ be a von Neumann algebra with faithful normal state $\chi$, and consider $N\otimes1\subset N\bar\otimes Q$ with the expectation $\mathrm{id}\otimes\chi$. Show that $\mathrm B(N,\varphi)\bar\otimes Q\subset\mathrm B(N\otimes1\subset N\bar\otimes Q,\varphi)$.

Exercise 4.2. In the setting of Exercise 4.1, assume that $N$ is a factor of type III₁ with separable predual. Show that $\mathrm B(N\otimes1\subset N\bar\otimes Q,\varphi)=1\otimes Q$.

Exercise 4.3. Show that $\mathrm B(N\subset M,\varphi)$ contains $\mathrm B(N,\varphi)\vee(N'\cap M)$, and that for $N$ a factor of type III₁ with separable predual every element $b$ of $\mathrm B(N\subset M,\varphi)$ satisfies $E(b^*b)=\bar\varphi(b^*b)1$.

Exercise 4.4. Let $P=M_2(\mathbb C)$, $\psi=\operatorname{Tr}/2$, $G=\{\mathrm{id},\operatorname{Ad}u\}$ with $u=\operatorname{diag}(1,-1)$, and $X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. Compute $E_G(X)$ and $\|uXu^*-X\|_\psi$, and compare with Lemma 3.1(2).

5. Solutions

4.1. For $b\in\mathrm B(N,\varphi)$, $q\in Q$ and $(x_n)\in\mathrm{AC}(N,\varphi)$, $[b\otimes q,x_n\otimes1]=[b,x_n]\otimes q$, and $\|[b,x_n]\otimes q\|_{\varphi\otimes\chi}\le\|q\|\,\|[b,x_n]\|_\varphi\to0$. So the algebraic tensor product lies in the relative bicentralizer, which is a von Neumann algebra by Theorem 2.3.

4.2. Let $a$ be in the relative bicentralizer, $\eta$ a vector in the GNS space $K$ of $\chi$ and $\omega_\eta=\langle\,\cdot\,\eta,\eta\rangle$. For bounded $z_n\to0$ strongly, the slice $(\mathrm{id}\otimes\omega_\eta)(z_n)$ satisfies $\|(\mathrm{id}\otimes\omega_\eta)(z_n)\zeta\|\le\|\eta\|\,\|z_n(\zeta\otimes\eta)\|\to0$ for every vector $\zeta$, so it tends to $0$ strongly. Applied to $z_n=[a,x_n\otimes1]$ this shows $[(\mathrm{id}\otimes\omega_\eta)(a),x_n]\to0$ strongly, so the slice lies in $\mathrm B(N,\varphi)=\mathbb C1$. Slices by vector functionals determine $a$, and an element whose slices are all scalar lies in $1\otimes Q$. The reverse inclusion is Exercise 4.1.

4.3. $\mathrm B(N,\varphi)\subset\mathrm B(N\subset M,\varphi)$ by Lemma 1.1, $N'\cap M\subset\mathrm B(N\subset M,\varphi)$ by Theorem 2.3, and the latter is an algebra. For the second statement apply Corollary 2.6 to $b^*b\in\mathrm B(N\subset M,\varphi)$.

4.4. The fixed-point algebra is the diagonal, and $E_G(X)=\frac12(X+uXu^*)=0$. Here $uXu^*=-X$, so $\|uXu^*-X\|_\psi=2\|X\|_\psi\ge\|X\|_\psi$, as Lemma 3.1(2) predicts.

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
[M2] A. Marrakchi, Kadison's problem and ergodicity of the bicentralizer flow (2026). https://arxiv.org/abs/2606.23636