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The bicentralizer

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

The centralizer \(M_\varphi\) of a faithful normal state consists of the elements that commute exactly with \(\varphi\). Connes introduced a weaker requirement: an element \(a\) lies in the bicentralizer of \(\varphi\) when it asymptotically commutes with every bounded sequence that asymptotically commutes with \(\varphi\). The bicentralizer always contains the centre of \(M\), and Connes asked whether, for a type III₁ factor with separable predual, it is always trivial. This lesson develops its basic structure: three equivalent descriptions, one of them inside the Ocneanu ultrapower; the fact that it is a von Neumann subalgebra with a state-preserving conditional expectation; the theorem of Houdayer and Isono that it is its own bicentralizer [HI]; and, for type III₁ factors, the triviality of its centre and of the centralizer of the restricted state.

We use Asymptotic centralizers of type III₁ factors (Lemma 1.1 and Theorem 3.1); the conventions of Section 1, Lemmas 1.1 and 1.3 and the results (B2), (B6) listed in Ultraproducts and the asymptotic centralizer; the normal copy of \(M\) in the Ocneanu ultrapower from Section 4 of Multiplier ultraproducts and normal embeddings; and the existence and uniqueness of state-preserving conditional expectations onto modular-invariant subalgebras, §§ME-01 and ME-08 of Conditional expectations from modular invariance. The notation \(M^\omega\), \(\varphi^\omega\), \(A_{\varphi,\omega}\), \(M_{\varphi,\omega}\) is that of the previous lesson.

1. Three descriptions

Let \(M\) be a countably decomposable von Neumann algebra and \(\varphi\) a faithful normal state on \(M\).

Definition 1.1. The asymptotic centralizer of \(\varphi\) is \[ \begin{aligned} &\mathrm{AC}(M,\varphi)\\ &\quad =\{(x_n)\in\ell^\infty(M):\ \|[x_n,\varphi]\|\to0\}, \end{aligned} \] and the bicentralizer \(\mathrm B(M,\varphi)\) is the set of \(a\in M\) with \(\|[a,x_n]\|_\varphi\to0\) for every \((x_n)\in\mathrm{AC}(M,\varphi)\). For a free ultrafilter \(\omega\) let \(\mathrm B_\omega(M,\varphi)\) be the set of \(a\in M\) with \(\lim_\omega\|[a,x_n]\|_\varphi=0\) for every \((x_n)\in A_{\varphi,\omega}\).

Here \([a,x]=ax-xa\), and \([x,\varphi]=x\varphi-\varphi x\) in the bimodule notation of the previous lesson.

Lemma 1.2. \(\mathrm{AC}(M,\varphi)\) is a unital \(C^*\)-subalgebra of \(\ell^\infty(M)\). Its unitaries are the sequences of unitaries \((u_n)\) with \(\|[u_n,\varphi]\|\to0\), and every element is a linear combination of four of them.

Proof. By Lemma 1.1(a) and (d) of the ultraproduct lesson, \(\|[x^*,\varphi]\|=\|[x,\varphi]\|\) and \(\|[xy,\varphi]\|\le\|x\|\|[y,\varphi]\|+\|y\|\|[x,\varphi]\|\); and \(\|[x,\varphi]-[y,\varphi]\|\le2\|x-y\|\) gives norm closure. A unitary of \(\ell^\infty(M)\) is a sequence of unitaries. Every element of a unital \(C^*\)-algebra is a linear combination of four unitaries of that algebra. \(\square\)

Proposition 1.3. For every free ultrafilter \(\omega\) and every \(a\in M\), the following are equivalent:

  1. \(a\in\mathrm B(M,\varphi)\);
  2. \(a\in\mathrm B_\omega(M,\varphi)\);
  3. \(aX=Xa\) in \(M^\omega\) for every \(X\in M_{\varphi,\omega}\);
  4. for every \(\varepsilon>0\) there is \(\delta>0\) such that every unitary \(u\in M\) with \(\|[u,\varphi]\|<\delta\) satisfies \(\|u^*au-a\|_\varphi<\varepsilon\).

Proof. (2)\(\Leftrightarrow\)(3). Let \(X=[x_n]\in M_{\varphi,\omega}\), with \((x_n)\in A_{\varphi,\omega}\subset N_\omega\). The constant sequence \(a\) lies in \(N_\omega\), so \(aX-Xa\) is the class of \((ax_n-x_na)\), and \(\|aX-Xa\|_{\varphi^\omega}=\lim_\omega\|[a,x_n]\|_\varphi\). Since \(\varphi^\omega\) is faithful, \(aX=Xa\) exactly when this limit is \(0\).

(2)\(\Rightarrow\)(1). Let \((x_n)\in\mathrm{AC}(M,\varphi)\) and suppose \(\|[a,x_{n_k}]\|_\varphi\ge c>0\) along a subsequence. The sequence \(y_k=x_{n_k}\) still belongs to \(\mathrm{AC}(M,\varphi)\subset A_{\varphi,\omega}\), and \(\lim_\omega\|[a,y_k]\|_\varphi\ge c\), contradicting (2).

(1)\(\Rightarrow\)(2). Let \((x_n)\in A_{\varphi,\omega}\) with \(c=\lim_\omega\|[a,x_n]\|_\varphi>0\). For each \(j\) the set \(\{n:\|[x_n,\varphi]\|<1/j,\ \|[a,x_n]\|_\varphi>c/2\}\) belongs to \(\omega\) and is therefore infinite, so we can choose \(n_1<n_2<\cdots\) with \(n_j\) in the \(j\)-th set. Then \((x_{n_j})_j\in\mathrm{AC}(M,\varphi)\) and \(\|[a,x_{n_j}]\|_\varphi>c/2\) for all \(j\), contradicting (1).

(1)\(\Leftrightarrow\)(4). For a unitary \(u\), \([a,u]=u(u^*au-a)\), and \(\|uy\|_\varphi^2=\varphi(y^*u^*uy)=\|y\|_\varphi^2\); so \(\|[a,u]\|_\varphi=\|u^*au-a\|_\varphi\). If (4) fails, there are \(\varepsilon>0\) and unitaries \(u_j\) with \(\|[u_j,\varphi]\|<1/j\) and \(\|[a,u_j]\|_\varphi\ge\varepsilon\); then \((u_j)\in\mathrm{AC}(M,\varphi)\), and (1) fails. Conversely, (4) gives \(\|[a,u_n]\|_\varphi\to0\) for every unitary \((u_n)\in\mathrm{AC}(M,\varphi)\), and Lemma 1.2 extends this to every element of \(\mathrm{AC}(M,\varphi)\). \(\square\)

So \(\mathrm B_\omega(M,\varphi)\) does not depend on \(\omega\), and \[ \mathrm B(M,\varphi)=(M_{\varphi,\omega})'\cap M , \] the relative commutant being taken inside \(M^\omega\). In (4), the condition \(\|[u,\varphi]\|<\delta\) is equivalent to \(\|u\varphi u^*-\varphi\|<\delta\).

2. Structure

Theorem 2.1. \(\mathrm B(M,\varphi)\) is a von Neumann subalgebra of \(M\) containing the centre of \(M\), globally invariant under the modular group \(\sigma^\varphi\). There is a unique \(\varphi\)-preserving normal conditional expectation \(E\) of \(M\) onto \(\mathrm B(M,\varphi)\), and it is faithful. Moreover \(\mathrm B(M,\varphi)\subset(M_\varphi)'\cap M\).

Proof. By Proposition 1.3, \(\mathrm B(M,\varphi)\) is the set of constants commuting with the self-adjoint set \(M_{\varphi,\omega}\subset M^\omega\), so it is a unital \(*\)-subalgebra. If \(a_i\in\mathrm B(M,\varphi)\) is a bounded net converging \(\sigma\)-weakly to \(a\in M\), then \(a_i\to a\) \(\sigma\)-weakly in \(M^\omega\), because the constant embedding is normal; the commutant of \(M_{\varphi,\omega}\) in \(M^\omega\) is \(\sigma\)-weakly closed, so \(a\in\mathrm B(M,\varphi)\). By Kaplansky's density theorem (B2), the unit ball of the \(\sigma\)-weak closure of a \(*\)-algebra is the strong closure of its unit ball, so \(\mathrm B(M,\varphi)\) is \(\sigma\)-weakly closed. Central elements commute with everything, so they lie in \(\mathrm B(M,\varphi)\). A constant \(x\in M_\varphi\) gives \((x)\in\mathrm{AC}(M,\varphi)\), so every \(a\in\mathrm B(M,\varphi)\) satisfies \(\|[a,x]\|_\varphi=0\), that is \([a,x]=0\); hence \(\mathrm B(M,\varphi)\subset(M_\varphi)'\cap M\).

Since \(\varphi\circ\sigma_t^\varphi=\varphi\), for \(x,y\in M\) we have \((\sigma_t^\varphi(x)\varphi)(y)=\varphi(y\sigma_t^\varphi(x))\), which is \((x\varphi)(\sigma^\varphi_{-t}(y))\), and similarly on the other side; so \(\|[\sigma_t^\varphi(x),\varphi]\|=\|[x,\varphi]\|\), and \(\mathrm{AC}(M,\varphi)\) is invariant under \(\sigma_t^\varphi\) applied termwise. If \(a\in\mathrm B(M,\varphi)\) and \((x_n)\in\mathrm{AC}(M,\varphi)\), then \[ \begin{aligned} &\|[\sigma_t^\varphi(a),x_n]\|_\varphi=\|\sigma_t^\varphi([a,\sigma_{-t}^\varphi(x_n)])\|_\varphi\\ &\quad =\|[a,\sigma^\varphi_{-t}(x_n)]\|_\varphi\to0 . \end{aligned} \] So \(\mathrm B(M,\varphi)\) is invariant under \(\sigma^\varphi\), and §ME-01 of the modular expectation lesson gives a \(\varphi\)-preserving faithful normal conditional expectation onto it, unique by §ME-08 there. \(\square\)

Write \(\mathrm B=\mathrm B(M,\varphi)\), \(\varphi_{\mathrm B}=\varphi|_{\mathrm B}\), and \(E\) for the expectation of Theorem 2.1. Since \(E\) is a \(\mathrm B\)-bimodule map with \(\varphi\circ E=\varphi\), \[ \begin{gathered} \varphi(yb)=\varphi_{\mathrm B}(E(y)b),\\ \varphi(by)=\varphi_{\mathrm B}(bE(y))\\ (y\in M,\ b\in\mathrm B). \end{gathered}\tag{2.1} \]

Proposition 2.2 (Houdayer–Isono). \(\mathrm B(\mathrm B,\varphi_{\mathrm B})=\mathrm B\).

Proof. Always \(\mathrm B(\mathrm B,\varphi_{\mathrm B})\subset\mathrm B\). Let \((x_n)\in\mathrm{AC}(\mathrm B,\varphi_{\mathrm B})\). By (2.1), for \(y\in M\), \[ \begin{aligned} [x_n,\varphi](y)&=\varphi(yx_n)-\varphi(x_ny)\\ &=\varphi_{\mathrm B}(E(y)x_n-x_nE(y))\\ &=[x_n,\varphi_{\mathrm B}](E(y)), \end{aligned} \] so \(\|[x_n,\varphi]\|\le\|[x_n,\varphi_{\mathrm B}]\|\to0\), and \((x_n)\in\mathrm{AC}(M,\varphi)\). Every \(a\in\mathrm B\) therefore satisfies \(\|[a,x_n]\|_{\varphi_{\mathrm B}}=\|[a,x_n]\|_\varphi\to0\). So \(\mathrm B\subset\mathrm B(\mathrm B,\varphi_{\mathrm B})\). \(\square\)

Lemma 2.3. The predual of \(\mathrm B\) embeds isometrically in \(M_*\) by \(\rho\mapsto\rho\circ E\). In particular \(\mathrm B\) has separable predual when \(M\) does.

Proof. \(\|\rho\circ E\|\le\|\rho\|\) since \(E\) is a contraction, and \((\rho\circ E)|_{\mathrm B}=\rho\). \(\square\)

3. Type III1 factors

Theorem 3.1. Let \(M\) be a type III₁ factor with separable predual and \(\varphi\) a faithful normal state. Then \[ \mathrm B\cap M_\varphi=\mathbb C1 . \] Consequently the centralizer of \(\varphi_{\mathrm B}\) in \(\mathrm B\) is \(\mathbb C1\), and \(\mathrm B\) is a factor. If \(\mathrm B\ne\mathbb C1\), then \(\mathrm B\) is a factor with separable predual, \(\varphi_{\mathrm B}\) is a faithful normal state with trivial centralizer, and \(\mathrm B(\mathrm B,\varphi_{\mathrm B})=\mathrm B\).

Proof. Let \(a\in\mathrm B\cap M_\varphi\) and fix a free ultrafilter \(\omega\). The constant sequence \(a\) belongs to \(A_{\varphi,\omega}\), so its class lies in \(M_{\varphi,\omega}\); by Proposition 1.3 it commutes with \(M_{\varphi,\omega}\), so it lies in the centre of \(M_{\varphi,\omega}\), which is \(\mathbb C1\) by Theorem 3.1 of the previous lesson. Thus \(a-c\in I_\omega\) for a scalar \(c\); for a constant sequence this means \(\|a-c\|_\varphi=0\), so \(a=c\).

If \(b\in\mathrm B\) satisfies \([b,\varphi_{\mathrm B}]=0\), then by (2.1), \(\varphi(yb)=\varphi_{\mathrm B}(E(y)b)\), which is \(\varphi_{\mathrm B}(bE(y))=\varphi(by)\) for all \(y\in M\), so \(b\in M_\varphi\) by (B6), and \(b\) is a scalar. A central element \(z\) of \(\mathrm B\) satisfies \([z,\varphi_{\mathrm B}]=0\), so it is a scalar. The remaining statements are Lemma 2.3 and Proposition 2.2. \(\square\)

Connes' bicentralizer problem asks whether \(\mathrm B(M,\varphi)=\mathbb C1\) for every type III₁ factor with separable predual. Theorem 3.1 reduces it to the question whether a self-bicentralizing factor, one with \(\mathrm B(M,\varphi)=M\), must be \(\mathbb C1\). For such a factor the centralizer of \(\varphi\) is trivial, so \(\varphi\) is ergodic for its modular group.

4. Exercises

Exercise 4.1. Let \(M\) be a finite factor with faithful normal tracial state \(\tau\). Show that \(\mathrm{AC}(M,\tau)=\ell^\infty(M)\) and \(\mathrm B(M,\tau)=\mathbb C1\).

Exercise 4.2. Show that \(\mathrm B(M,\varphi)\) is contained in the centre of \(M\) whenever \(M_\varphi'\cap M\) is contained in the centre. (In a factor this gives \(\mathrm B(M,\varphi)=\mathbb C1\) when \(\varphi\) has an irreducible centralizer, \(M_\varphi'\cap M=\mathbb C1\).)

Exercise 4.3. Let \(\alpha\) be an automorphism of \(M\). Show that \(\alpha(\mathrm B(M,\varphi))=\mathrm B(M,\varphi\circ\alpha^{-1})\).

Exercise 4.4. Show that, in Proposition 1.3(4), the conclusion \(\|u^*au-a\|_\varphi<\varepsilon\) may be replaced by \(\|uau^*-a\|_\varphi<\varepsilon\).

5. Solutions

4.1. \([x,\tau]=0\) for every \(x\), since \(\tau(yx)=\tau(xy)\); so every bounded sequence is in \(\mathrm{AC}(M,\tau)\). Taking constant sequences, every \(a\in\mathrm B(M,\tau)\) commutes with all of \(M\), so it is central, hence scalar.

4.2. By Theorem 2.1, \(\mathrm B(M,\varphi)\subset M_\varphi'\cap M\).

4.3. Put \(\psi=\varphi\circ\alpha^{-1}\). For \(x\in M\), \([\alpha(x),\psi]=[x,\varphi]\circ\alpha^{-1}\), so \((x_n)\in\mathrm{AC}(M,\varphi)\) exactly when \((\alpha(x_n))\in\mathrm{AC}(M,\psi)\). Also \(\|\alpha(y)\|_\psi=\|y\|_\varphi\). Hence \(\|[\alpha(a),\alpha(x_n)]\|_\psi=\|[a,x_n]\|_\varphi\), and \(a\in\mathrm B(M,\varphi)\) exactly when \(\alpha(a)\in\mathrm B(M,\psi)\).

4.4. Apply Proposition 1.3(4) to \(u^*\): \(\|[u^*,\varphi]\|=\|[u,\varphi]\|\) by Lemma 1.1(a) of the ultraproduct lesson, and \((u^*)^*au^*=uau^*\).

References