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An injective subalgebra with the same core commutant

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

This lesson proves the main theorem of [OAI] when the smaller algebra is a factor of type III₁. Let \(N\subset M\) be von Neumann algebras with separable preduals and a faithful normal conditional expectation, with \(N\) a factor of type III₁. There are a faithful normal state \(\psi\) on \(N\) and an approximately finite-dimensional, hence injective, von Neumann subalgebra \(P\subset N\), globally invariant under \(\sigma^\psi\), such that \[ P'\cap c(M)=N'\cap c(M) \] in the continuous core \(c(M)\) (Theorem 5.2). [OAI] obtains \(P\) from Marrakchi's characterization of this property by a weak Dixmier property [M1, Theorem D]. Here \(P\) is constructed directly so that it satisfies the hypotheses of the criterion of Core commutants, Theorem 5.1 and Proposition 5.2. With \(M=N\) the same construction gives Haagerup's theorem that the faithful normal states whose centralizer has trivial relative commutant are norm dense (Corollary 5.3).

The construction follows the inductive scheme of Ando, Haagerup, Houdayer and Marrakchi [AHHM, Theorem C and Lemma 5.4]. The algebra \(P\) is generated by commuting matrix subalgebras \(F_1,F_2,\ldots\) of \(N\). At step \(n\) the state is replaced by the product of a fixed state on \(F_n\) and its restriction to the relative commutant, which moves it by at most \(2^{-n}\), and one prescribed target element loses a fixed fraction of its norm under the conditional expectation onto the relative commutant of \(F_n\). Two features are specific to the core. First, the criterion concerns the centralizer of the final state on \(P\), not \(P\) itself. So each \(F_n\) is a product \(T_n\otimes G_n\): a matrix algebra \(T_n\) with its trace, which lies in the centralizer and reduces the target, and a part \(G_n\cong(M_2(\mathbb C),\omega_\mu)\otimes(M_2(\mathbb C),\omega_\nu)\) of fixed shape, which supplies modular eigenoperators. Second, the final state is only known in the limit. So the targets are measured against the algebra \(\mathrm B^\sharp=N\vee\mathrm B(N\subset M,\varphi)\), which does not depend on the state, and against scalars; they never involve the relative bicentralizer of an intermediate state.

We use: Definition 2.1, Lemma 1.2, the remark after Theorem 2.3 (expectations as orthogonal projections), Corollary 2.6, Theorem 3.4(5) and the norm formula after Lemma 1.4 of The relative bicentralizer; Proposition 4.2 and Section 5 of Transition isomorphisms and the relative flow; Theorem 5.1 and Proposition 5.2 of Core commutants; all of Finite-dimensional subfactors and lifting; Lemma 1.4 of Ultraproducts and the asymptotic centralizer; result (B1) (separability) of Full factors; Corollary 5.2(c) of Injective von Neumann algebras; the modular expectation theorem (ME.1)–(ME.3) of Conditional expectations from modular invariance; the Schwarz inequality for conditional expectations, CE-006; the double commutant theorem, Section 4 of its lesson; supports of normal positive functionals, Lemma 11.1 of the universal enveloping algebra lesson; and Vigier's theorem on bounded monotone nets, Theorem 6.1 of the spectral theorem lesson.

1. Setting, amplified inclusions and slices

Throughout, \(N\subset M\) are von Neumann algebras with separable preduals and the same unit, \(E\colon M\to N\) is a faithful normal conditional expectation, \(N\) is a factor of type III₁, and \(\omega\) is a free ultrafilter. For a faithful normal state \(\chi\) on \(N\) we write \(\bar\chi=\chi\circ E\). We fix numbers \(0<\mu,\nu<1\) with \(\log\mu/\log\nu\notin\mathbb Q\), for example \(\mu=\frac12\), \(\nu=\frac13\).

Matrix subalgebras, matrix units diagonalizing a state, the induced expectations \(E_F\) and the states \(\omega_\lambda\) are those of Section 1 of the previous lesson. A matrix subalgebra isomorphic to \(M_{2^p}(\mathbb C)\) (\(p\ge0\)) always carries the normalized trace \(\tau\), one isomorphic to \(M_2(\mathbb C)\) carries \(\omega_\mu\) or \(\omega_\nu\) as specified, and the algebra generated by commuting matrix subalgebras carries the product state (Lemma 1.1(5) of the previous lesson). For a matrix subalgebra \(R\subset N\) put \(N_R=R'\cap N\) and \(M_R=R'\cap M\). By Section 5 of the lesson on the relative flow, \(N_R\) is a factor of type III₁ with separable predual; \(E\) maps \(M_R\) into \(N_R\), because it is \(N\)-bimodular; and \(E_R\) commutes with \(E\) (Lemma 1.1(6) of the previous lesson).

Lemma 1.1 (amplified inclusions). Let \(\mathcal M\) be a von Neumann algebra, \(G\subset\mathcal M\) a matrix subalgebra with matrix units \((g_{ab})\) diagonalizing a state \(\psi_G\) with weights \(k_a\), \(M_1=G'\cap\mathcal M\), and \(N_1\subset M_1\) a von Neumann subalgebra with a faithful normal conditional expectation \(E_1\colon M_1\to N_1\). Then \(\mathcal E=E_1\circ E_G\) is a faithful normal conditional expectation of \(\mathcal M\) onto \(N_1\). Let \(\chi\) be a faithful normal state on \(N_1\), \(\theta=\chi\circ\mathcal E\), \(\bar\chi_1=\chi\circ E_1\), and \(\mathrm B_1=\mathrm B(N_1\subset M_1,\chi)\) with its \(\bar\chi_1\)-preserving expectation \(E_{\mathrm B_1}\). Then \[ \begin{aligned} &\mathrm B(N_1\subset\mathcal M,\chi)\\ &\quad =\Big\{\sum_{a,b}g_{ab}b_{ab}:\ b_{ab}\in\mathrm B_1\Big\}, \end{aligned} \] and its \(\theta\)-preserving expectation is \(\sum_{a,b}g_{ab}z_{ab}\mapsto\sum_{a,b}g_{ab}E_{\mathrm B_1}(z_{ab})\) (coefficients \(z_{ab}\in M_1\) as in Lemma 1.1(1) of the previous lesson).

Proof. \(E_G\) is a faithful normal conditional expectation onto \(M_1\supset N_1\), so \(\mathcal E\) is one onto \(N_1\). For \(y\in M_1\), \(E_G(g_{cd}y)=\psi_G(g_{cd})y\), so \(\theta(g_{cd}y)=\delta_{cd}k_c\bar\chi_1(y)\). For \(v=\sum g_{ab}v_{ab}\) and \(w=\sum g_{ab}w_{ab}\) with coefficients in \(M_1\), \(v^*w=\sum_{a,b,d}g_{bd}v_{ab}^*w_{ad}\), hence \[ \begin{aligned} &\theta(v^*w)\\ &\quad =\sum_{a,b}k_b\,\bar\chi_1(v_{ab}^*w_{ab}) . \end{aligned}\tag{1.1} \] For \(x\in N_1\) and \(a=\sum g_{ab}a_{ab}\), \([a,x]=\sum g_{ab}[a_{ab},x]\), so by (1.1) \(\|[a,x_n]\|_\theta\to0\) exactly when \(\|[a_{ab},x_n]\|_{\bar\chi_1}\to0\) for all \(a,b\). By Definition 2.1 of the first lesson this gives the description of \(\mathrm B(N_1\subset\mathcal M,\chi)\). By the remark after Theorem 2.3 of the first lesson, the \(\theta\)-preserving expectation of \(z\) is the unique \(z'\) in the relative bicentralizer with \(\theta(b^*(z-z'))=0\) for all \(b\) in it. For \(z'=\sum g_{ab}E_{\mathrm B_1}(z_{ab})\) and \(b=\sum g_{ab}b_{ab}\), (1.1) shows that \(\theta(b^*(z-z'))\) equals \(\sum_{a,b}k_b\bar\chi_1\big(b_{ab}^*(z_{ab}-E_{\mathrm B_1}(z_{ab}))\big)\), which is \(0\). \(\square\)

Lemma 1.2 (\(E^\sharp\) in matrix coordinates). Let \(R\subset N\) be a matrix subalgebra with matrix units \((r_{ij})_{i,j=1}^k\). Let \(E^\sharp\) be the expectation of \(M\) onto \(\mathrm B^\sharp(N\subset M)\) and \(E^\sharp_R\) that of \(M_R\) onto \(\mathrm B^\sharp(N_R\subset M_R)\) (Proposition 4.2 of the lesson on the relative flow, for the inclusions with \(E\) and with \(E|_{M_R}\)). Then \(E^\sharp\big(\sum_{i,j}r_{ij}x_{ij}\big)=\sum_{i,j}r_{ij}E^\sharp_R(x_{ij})\) for \(x_{ij}\in M_R\).

Proof. By Proposition 5.1 of the lesson on the relative flow, \(\mathrm B^\sharp(N\subset M)\) consists of the elements \(\sum r_{ij}b_{ij}\) with \(b_{ij}\in\mathrm B^\sharp(N_R\subset M_R)\). Let \(\chi_1\) be a faithful normal state on \(N_R\) and \(\chi\) the state \(\sum r_{ij}y_{ij}\mapsto\frac1k\sum_i\chi_1(y_{ii})\) on \(N\). By Proposition 4.2 of that lesson, \(E^\sharp\) preserves \(\bar\chi\) and \(E^\sharp_R\) preserves \(\chi_1\circ E|_{M_R}\), so both are orthogonal projections as in the previous proof. Since \(\bar\chi(r_{cd}y)=\frac1k\delta_{cd}\chi_1(E(y))\) for \(y\in M_R\), formula (1.1) holds for \(\bar\chi\) with weights \(\frac1k\), and the computation of the previous proof shows that \(\sum r_{ij}E^\sharp_R(x_{ij})\) is the \(\bar\chi\)-orthogonal projection of \(x\) onto \(\mathrm B^\sharp(N\subset M)\). \(\square\)

Lemma 1.3 (slices). Let \(G\subset N\) be a matrix subalgebra with matrix units \((g_{ab})_{a,b=1}^k\) diagonalizing \(\psi_G\), and \(L\subset G'\cap N\) a von Neumann subalgebra. For \(y=\sum g_{ab}y_{ab}\) with \(y_{ab}\in L\) and \(\rho\in L_*\) put \(S_\rho(y)=\sum_{a,b}\rho(y_{ab})g_{ab}\in G\).

  1. \(\|S_\rho(y)\|\le\|\rho\|\,\|y\|\).
  2. If \(\psi\) is a faithful normal state on \(N\) with \(\psi\circ E_G=\psi\), then \(G\) is globally invariant under \(\sigma^\psi\), and \(S_{\psi|L}(y)=E^\psi_G(y)\), where \(E^\psi_G\) is the \(\psi\)-preserving conditional expectation of \(N\) onto \(G\).

Proof. (1) For unit vectors \(c,d\in\mathbb C^k\) let \(r_c=\sum_bc_bg_{b1}\). Then \(r_c^*r_c=g_{11}\), so \(\|r_c\|=1\), and \(r_d^*yr_c=g_{11}\sum_{a,b}\bar d_ac_by_{ab}\). The map \(y'\mapsto g_{11}y'\) is a \(*\)-homomorphism on \(G'\cap N\), injective because \(g_{11}y'=0\) gives \(g_{ll}y'=g_{l1}g_{11}y'g_{1l}=0\) for all \(l\); so it is isometric, and \(\|\sum\bar d_ac_by_{ab}\|\le\|y\|\). Identifying \(G\) with \(M_k(\mathbb C)\), the matrix of \(S_\rho(y)\) is \((\rho(y_{ab}))\), and \(|\sum_{a,b}\bar d_a\rho(y_{ab})c_b|=|\rho(\sum\bar d_ac_by_{ab})|\), which is at most \(\|\rho\|\|y\|\). Take the supremum over \(c,d\).

(2) Invariance is Lemma 1.2 of the previous lesson, which with the modular expectation theorem gives \(E^\psi_G\); it is characterized by \(E^\psi_G(y)\in G\) and \(\psi(g(y-E^\psi_G(y)))=0\) for \(g\in G\). By Lemma 1.1(3) of the previous lesson, \(\psi(g_{cd}y)=\sum_b\psi(g_{cb}y_{db})\), which is \(\sum_b\psi_G(g_{cb})\psi(y_{db})=k_c\psi(y_{dc})\), and the same computation gives \(\psi(g_{cd}S_\psi(y))=k_c\psi(y_{dc})\). \(\square\)

2. One step

Proposition 2.1. Let \(\mathcal M,G,\psi_G,M_1,N_1,E_1\) be as in Lemma 1.1, and assume moreover that \(N_1\) is a factor of type III₁ with separable predual. Let \(\chi\) be a faithful normal state on \(N_1\), \(\theta=\chi\circ E_1\circ E_G\), \(\mathcal B=\mathrm B(N_1\subset\mathcal M,\chi)\), \(z\in\mathcal M\) with \(E_{\mathcal B}(z)=0\), and \(\varepsilon>0\). There are \(p\ge0\), a matrix subalgebra \(T\subset N_1\) isomorphic to \(M_{2^p}(\mathbb C)\), and matrix subalgebras \(F^\mu\subset T'\cap N_1\) and \(F^\nu\subset(T\vee F^\mu)'\cap N_1\) isomorphic to \(M_2(\mathbb C)\), carrying \(\omega_\mu\) and \(\omega_\nu\), such that with \(K=F^\mu\vee F^\nu\):

  1. \(\|\chi\circ E_T-\chi\|\le\varepsilon\) and \(\|\chi\circ E_T\circ E_K-\chi\circ E_T\|\le\varepsilon\), as functionals on \(N_1\);
  2. \(\|E_T(z)\|^2_{\theta\circ E_T}\le\frac{15}{16}\|z\|^2_\theta+\varepsilon\).

Proof. Step 1: a unitary that moves \(z\). If \(z=0\), let \(T=\mathbb C1\) and go to Step 4. Otherwise apply Theorem 3.4(5) of the first lesson to the inclusion \(N_1\subset\mathcal M\) with the expectation \(\mathcal E=E_1\circ E_G\) of Lemma 1.1 and the state \(\chi\), so that \(\chi\circ\mathcal E=\theta\). It gives a unitary \(U\) of \(Q=(N_1)_{\chi,\omega}\) with \[ \|UzU^*-z\|_{\theta^\omega}\ge\|z\|_\theta . \] By Lemma 1.2 of the first lesson, \(Q\subset\mathcal M^\omega\) is a II₁ factor whose trace is the restriction of \(\theta^\omega\), and \(\theta^\omega(Xy)=\theta^\omega(yX)\) for \(X\in Q\), \(y\in\mathcal M^\omega\).

Step 2: a dyadic matrix algebra. By Lemma 5.1 of the previous lesson there are, for \(j\ge1\), integers \(p_j\), unital \(*\)-homomorphisms \(\Theta_j\colon M_{2^{p_j}}(\mathbb C)\to Q\) and unitaries \(c_j\) with \(\|U-U_j\|_\tau\le1/j\), where \(U_j=\Theta_j(c_j)\). Let \(\Omega\) be the cyclic and separating vector of \(\theta^\omega\) in \(L^2(\mathcal M^\omega,\theta^\omega)\), and \(D_j=U_j-U\). Then \(\|D_j\|\le2\) and \(\|D_j\Omega\|=\|D_j\|_\tau\to0\). For \(y'\) in the commutant of \(\mathcal M^\omega\), \(\|D_jy'\Omega\|=\|y'D_j\Omega\|\to0\); these vectors are dense and the \(D_j\) are bounded, so \(D_j\eta\to0\) for every vector \(\eta\), in particular \(\|D_jz\|_{\theta^\omega}\to0\). Unitaries \(W\in Q\) satisfy \(\|aW^*\|_{\theta^\omega}=\|a\|_{\theta^\omega}\), and \[ \begin{aligned} &\|zD_j^*\|^2_{\theta^\omega}=\theta^\omega(D_jz^*zD_j^*)\\ &\quad =\theta^\omega(z^*zD_j^*D_j)=\langle D_j^*D_j\Omega,z^*z\Omega\rangle\\ &\quad \le2\|z\|^2\|D_j\Omega\|\to0 . \end{aligned} \] Since \(U_jzU_j^*-UzU^*=D_jzU_j^*+UzD_j^*\), we get \(\|U_jzU_j^*-UzU^*\|_{\theta^\omega}\to0\). Fix \(j\) with \(\|U_jzU_j^*-z\|_{\theta^\omega}\ge\frac34\|z\|_\theta\), and write \(p,\Theta,c\) for \(p_j,\Theta_j,c_j\).

Step 3: lifting. Proposition 4.4 of the previous lesson, for \(N_1\) and \(\chi\), gives unital \(*\)-homomorphisms \(\theta_m\colon M_{2^p}(\mathbb C)\to N_1\) such that \((\theta_m(a))_m\in A_{\chi,\omega}(N_1)\) represents \(\Theta(a)\). Let \(T_m=\theta_m(M_{2^p}(\mathbb C))\), with matrix units \(\theta_m(\varepsilon_{ab})\), and \(u_m=\theta_m(c)\). By Lemma 1.2 of the first lesson the class of \((u_m)\) in \(\mathcal M^\omega\) is \(U_j\), so by the norm formula after Lemma 1.4 there, \[ \begin{aligned} &\lim_\omega\|u_mzu_m^*-z\|_\theta\\ &\quad =\|U_jzU_j^*-z\|_{\theta^\omega}\ge\tfrac34\|z\|_\theta . \end{aligned} \] By Lemma 1.1(4) of the previous lesson, with equal weights, \(\|\chi\circ E_{T_m}-\chi\|\) is at most \(2^p\max_{a,b}\|[\theta_m(\varepsilon_{ab}),\chi]\|\), which tends to \(0\) along \(\omega\). Put \(\varepsilon'=\min\big(\varepsilon,\varepsilon/(2\|z\|^2)\big)\) and choose \(m\) with \(\|u_mzu_m^*-z\|_\theta\ge\frac12\|z\|_\theta\) and \(\|\chi\circ E_{T_m}-\chi\|\le\varepsilon'\). Let \(T=T_m\) and \(u=u_m\). This gives the first half of (1).

The expectation \(E_T\) commutes with \(E_1\) and \(E_G\) (Lemma 1.1(5),(6) of the previous lesson), so \(\theta\circ E_T=(\chi\circ E_T)\circ E_1\circ E_G\) and \(\|\theta\circ E_T-\theta\|\le\varepsilon'\). Put \(\theta_T=\theta\circ E_T\). Then \(\theta_T\circ E_T=\theta_T\), and Lemma 1.3 of the previous lesson gives \[ \|E_T(z)\|^2_{\theta_T}\le\|z\|^2_{\theta_T}-\tfrac14\|uzu^*-z\|^2_{\theta_T}. \] For \(y\in\mathcal M\), \(|\,\|y\|^2_{\theta_T}-\|y\|^2_\theta|\le\|y\|^2\varepsilon'\); with \(\|uzu^*-z\|\le2\|z\|\) this gives \[ \begin{aligned} &\|E_T(z)\|^2_{\theta_T}\le\|z\|_\theta^2+\|z\|^2\varepsilon'\\ &\quad -\tfrac14\big(\tfrac14\|z\|^2_\theta-4\|z\|^2\varepsilon'\big)\\ &\quad =\tfrac{15}{16}\|z\|^2_\theta+2\|z\|^2\varepsilon', \end{aligned} \] which is (2).

Step 4: the eigen part. Let \(N_2=T'\cap N_1\) and \(\chi'\) the restriction of \(\chi\circ E_T\) to \(N_2\). Then \(\chi'\) is a faithful normal state, since \(E_T\) is faithful, and \(N_2\) is a factor of type III₁ with separable predual. Proposition 3.1 of the previous lesson in \(N_2\), with \(\lambda=\mu\) and \(\delta=\mu\varepsilon/4\), gives \(F^\mu\subset N_2\) carrying \(\omega_\mu\) with \(\|\chi'\circ E_{F^\mu}-\chi'\|\le\varepsilon/2\) on \(N_2\). With \(N_3=F^{\mu\prime}\cap N_2\) and \(\chi''\) the restriction of \(\chi'\circ E_{F^\mu}\) to \(N_3\), the same proposition with \(\lambda=\nu\) gives \(F^\nu\subset N_3\) carrying \(\omega_\nu\) with \(\|\chi''\circ E_{F^\nu}-\chi''\|\le\varepsilon/2\). For \(y\in N_1\), \(E_T(y)\in N_2\), and \(E_T\), \(E_{F^\mu}\) commute, so \[ \begin{aligned} &(\chi\circ E_T\circ E_{F^\mu}-\chi\circ E_T)(y)\\ &\quad =(\chi'\circ E_{F^\mu}-\chi')(E_T(y)), \end{aligned} \] whose modulus is at most \(\frac\varepsilon2\|y\|\). In the same way, since \(E_T\circ E_{F^\mu}\) maps \(N_1\) into \(N_3\), the functionals \(\chi\circ E_T\circ E_{F^\mu}\circ E_{F^\nu}\) and \(\chi\circ E_T\circ E_{F^\mu}\) differ in norm by at most \(\varepsilon/2\). As \(E_K=E_{F^\mu}\circ E_{F^\nu}\), this gives the second half of (1). \(\square\)

3. The construction

Fix a faithful normal state \(\varphi\) on \(N\) and a number \(0<\eta\le1\). By (B1) of the lesson on full factors, the unit balls of \(M\) and of \(N\) contain sequences \((a_j)_{j\ge1}\) and \((b_j)_{j\ge1}\) that are dense for the \(*\)-strong topology. Fix a sequence \(k(0),k(1),\ldots\) of positive integers in which every positive integer occurs infinitely often, and put \[ \begin{gathered} x_{2m}=a_{k(m)}\in M,\\ x_{2m+1}=b_{k(m)}\in N\qquad (m\ge0). \end{gathered} \]

Theorem 3.1. There are mutually commuting matrix subalgebras \(T_1,F^\mu_1,F^\nu_1,T_2,F^\mu_2,F^\nu_2,\ldots\) of \(N\), with \(T_n\cong M_{2^{p_n}}(\mathbb C)\) carrying \(\tau\), \(F^\mu_n\) carrying \(\omega_\mu\) and \(F^\nu_n\) carrying \(\omega_\nu\), with the following properties. Put \[ \begin{aligned} G_n&=F^\mu_n\vee F^\nu_n,\quad F_n=T_n\vee G_n,\quad R_n\\ &=F_1\vee\cdots\vee F_n,\quad T^{(n)}\\ &=T_1\vee\cdots\vee T_n,\quad G^{(n)}\\ &=G_1\vee\cdots\vee G_n, \end{aligned} \] with \(R_0=T^{(0)}=G^{(0)}=\mathbb C1\), \(N_n=R_n'\cap N\), \(M_n=R_n'\cap M\), and \(\varphi_n=\bar\varphi\circ E_{R_n}\). Define \(y_n=E_{T^{(n)}}(x_n)\) and \[ \begin{gathered} z_{2m}=y_{2m}-E^\sharp(y_{2m}),\\ z_{2m+1}=y_{2m+1}-S_{\varphi|N_{2m+1}}(y_{2m+1}), \end{gathered} \] where \(S\) is the slice of Lemma 1.3 for \(G=G^{(n)}\) and \(L=N_n\). Then \(\|z_n\|\le2\) and, for every \(n\ge0\),

  1. \(\|\varphi_n\circ E_{T_{n+1}}-\varphi_n\|\le\eta2^{-n-2}\) and \(\|\varphi_{n+1}-\varphi_n\circ E_{T_{n+1}}\|\le\eta2^{-n-2}\); in particular \(\|\varphi_{n+1}-\varphi_n\|\le\eta2^{-n-1}\);
  2. \(\|E_{T_{n+1}}(z_n)\|^2_{\varphi_n\circ E_{T_{n+1}}}\) is at most \(\frac{15}{16}\|z_n\|^2_{\varphi_n}+\eta2^{-n-2}\).

Proof. We construct \(T_{n+1},F^\mu_{n+1},F^\nu_{n+1}\) by induction, assuming the earlier ones constructed, so that \(R_n\) is a matrix subalgebra of \(N\) with the product state (Lemma 1.1(5) of the previous lesson), and \(E_{R_n}=E_{T^{(n)}}\circ E_{G^{(n)}}\).

The stage inclusion. Let \(\mathcal M=T^{(n)\prime}\cap M\), \(G=G^{(n)}\subset\mathcal M\), \(M_1=G'\cap\mathcal M=M_n\), \(N_1=N_n\), \(E_1=E|_{M_n}\) and \(\chi=\varphi|_{N_n}\). For \(w\in\mathcal M\), \(E_{R_n}(w)=E_G(w)\in M_n\), so \[ \begin{aligned} &\varphi_n(w)=\varphi(E(E_G(w)))\\ &\quad =\chi(E_1(E_G(w))): \end{aligned} \] on \(\mathcal M\), \(\varphi_n\) is the state \(\theta\) of Lemma 1.1 and Proposition 2.1. Also \(\varphi_n\) and \(\varphi\) agree on \(N_n\).

The target. \(E_{T^{(n)}}\) maps \(M\) into \(\mathcal M\) and \(N\) into \(T^{(n)\prime}\cap N\), so \(y_n=\sum_{a,b}g_{ab}y_{ab}\) with \(y_{ab}\in M_n\), and \(y_{ab}\in N_n\) when \(n\) is odd. In the matrix units of \(R_n\), the coefficients of \(y_n\) are \(y_{ab}\) on the diagonal of \(T^{(n)}\), so Lemma 1.2 gives \(E^\sharp(y_n)=\sum g_{ab}E^\sharp_n(y_{ab})\), with \(E^\sharp_n\) the expectation onto \(\mathrm B^\sharp(N_n\subset M_n)\). Hence \[ \begin{gathered} z_n=\sum_{a,b}g_{ab}\big(y_{ab}-E^\sharp_n(y_{ab})\big)\ (n\text{ even}),\\ z_n=\sum_{a,b}g_{ab}\big(y_{ab}-\chi(y_{ab})1\big)\ (n\text{ odd}). \end{gathered} \] The norm bound \(\|z_n\|\le2\) follows from \(\|y_n\|\le1\), \(\|E^\sharp(y_n)\|\le1\) and Lemma 1.3(1). Let \(\mathrm B_n=\mathrm B(N_n\subset M_n,\chi)\). For even \(n\), \(E_{\mathrm B_n}\circ E^\sharp_n=E_{\mathrm B_n}\) (Proposition 4.2 of the lesson on the relative flow), so \(E_{\mathrm B_n}(y_{ab}-E^\sharp_n(y_{ab}))=0\). For odd \(n\), Corollary 2.6 of the first lesson for the factor \(N_n\) gives \(E_{\mathrm B_n}(y_{ab})=\chi(y_{ab})1\). In both cases Lemma 1.1 gives \(E_{\mathcal B}(z_n)=0\) for \(\mathcal B=\mathrm B(N_n\subset\mathcal M,\chi)\).

The step. Proposition 2.1 with \(\varepsilon=\eta2^{-n-2}\) gives \(T_{n+1}\), \(F^\mu_{n+1}\), \(F^\nu_{n+1}\) inside \(N_n\), so they commute with \(R_n\). Statement (2) is Proposition 2.1(2), because \(E_{T_{n+1}}\) maps \(\mathcal M\) into itself and \(\varphi_n=\theta\) there. For (1), let \(x\in M\) and \(w=E_{R_n}(E(x))\in N_n\), so \(\|w\|\le\|x\|\). Since \(E\), \(E_{R_n}\), \(E_{T_{n+1}}\) and \(E_{G_{n+1}}\) commute and \(\varphi_n=\varphi\circ E\circ E_{R_n}\), \[ \begin{aligned} &\varphi_n(E_{T_{n+1}}(x))-\varphi_n(x)\\ &\quad =(\chi\circ E_{T_{n+1}}-\chi)(w), \end{aligned} \] and likewise \[ \begin{aligned} &\varphi_n(E_{T_{n+1}}E_{G_{n+1}}(x))-\varphi_n(E_{T_{n+1}}(x))\\ &\quad=(\chi\circ E_{T_{n+1}}\circ E_{G_{n+1}}-\chi\circ E_{T_{n+1}})(w). \end{aligned} \] As \(\varphi_{n+1}=\bar\varphi\circ E_{R_n}\circ E_{F_{n+1}}\), which is \(\varphi_n\circ E_{T_{n+1}}\circ E_{G_{n+1}}\), Proposition 2.1(1) gives (1). \(\square\)

Lemma 3.2 (the limit state). In the situation of Theorem 3.1, \(\varphi_n\) converges in norm to a faithful normal state \(\bar\psi\) on \(M\), with \(\|\bar\psi-\varphi_n\|\le\eta2^{-n}\); in particular \(\|\bar\psi-\bar\varphi\|\le\eta\). Moreover \(\bar\psi=\psi\circ E\) with \(\psi=\bar\psi|_N\), and \(\bar\psi\circ E_S=\bar\psi\) for \(S=R_n,F_n,T_n,F^\mu_n,F^\nu_n,G_n,T^{(n)},G^{(n)}\).

Proof. The sequence is Cauchy by Theorem 3.1(1), and the predual is norm closed; the limit is a state. For \(k\ge n\), \(E_{R_k}=E_{R_n}\circ E_{F_{n+1}\vee\cdots\vee F_k}\), and the factors commute, so \(E_{R_k}\circ E_{R_n}=E_{R_k}\) and \(\varphi_k\circ E_{R_n}=\varphi_k\); letting \(k\to\infty\) gives \(\bar\psi\circ E_{R_n}=\bar\psi\). Each \(S\) in the list is one of the commuting matrix subalgebras whose product is \(R_n\) for suitable \(n\), so \(E_{R_n}\circ E_S=E_{R_n}\) and \(\bar\psi\circ E_S=\bar\psi\circ E_{R_n}\circ E_S=\bar\psi\). Each \(\varphi_n=\varphi\circ E_{R_n}\circ E\) is the composition of its restriction to \(N\) with \(E\), hence so is \(\bar\psi\).

Faithfulness. Let \(s\) be the support of \(\bar\psi\); then \(\bar\psi(y^*y)=0\) if and only if \(ys=0\), since \(\bar\psi=\bar\psi s=s\bar\psi\) and \(\bar\psi\) is faithful on \(sMs\). Fix \(n\) and let \((r_{ij})\) be the matrix units of \(R_n\), with weights \(h_i\). Then \[ \begin{aligned} 0&=\bar\psi(1-s)=\bar\psi(E_{R_n}(1-s))\\ &=\sum_{i,j}h_i\,\bar\psi\big(((1-s)r_{ij})^*(1-s)r_{ij}\big), \end{aligned} \] so \((1-s)r_{ij}s=0\), that is \(r_{ij}s=sr_{ij}s\), for all \(i,j\); taking adjoints with \(i,j\) exchanged, \(sr_{ij}=sr_{ij}s\). Hence \(s\in R_n'\cap M\), so \(E_{R_n}(s)=s\) and \(\varphi_n(s)=\bar\varphi(s)\) for every \(n\). In the limit \(\bar\varphi(s)=\bar\psi(s)=1\), and \(s=1\) because \(\bar\varphi\) is faithful. \(\square\)

4. Relative commutants

Keep the situation of Theorem 3.1 and Lemma 3.2, and let \(T\), \(G\) and \(P\) be the von Neumann algebras generated by all \(T_n\), all \(G_n\), and all \(F_n\), respectively. For a \(\sigma^\psi\)-invariant von Neumann subalgebra \(Q\subset N\), the modular group of \(\psi|_Q\) is the restriction of \(\sigma^\psi\) (modular expectation theorem), so the centralizer of \(\psi|_Q\) is \(Q_\psi=Q\cap N_\psi\).

Lemma 4.1.

  1. \(\psi(ab)=\psi(a)\psi(b)\) for \(a\in R_n\) and \(b\in R_n'\cap N\).
  2. \(T\subset N_\psi\); every \(G_n\) is \(\sigma^\psi\)-invariant, and for its matrix units \(g_{\alpha\beta}\), which diagonalize \(\omega_\mu\otimes\omega_\nu\) with weights \(h_\alpha\), \(\sigma^\psi_t(g_{\alpha\beta})=(h_\alpha/h_\beta)^{it}g_{\alpha\beta}\). The algebras \(T\), \(G\), \(P\) are \(\sigma^\psi\)-invariant, and \(T\cup G_\psi\subset P_\psi\).
  3. \(E_{T^{(n)}}\), \(E_{T_n}\) and \(E^\sharp\) preserve \(\bar\psi\); consequently \(\|E_{T^{(n)}}(x)\|_{\bar\psi}\le\|x\|_{\bar\psi}\), and likewise for \(E_{T_n}\) and \(E^\sharp\).

Proof. (1) is Lemma 1.1(3) of the previous lesson, since \(\psi\circ E_{R_n}=\psi\) on \(N\). (2) Lemma 1.2 of the previous lesson, applied to \(\psi\) and \(T_n\) (equal weights) and to \(\psi\) and \(G_n\), using Lemma 3.2. The normal automorphism \(\sigma^\psi_t\) maps the generators of \(T\), \(G\), \(P\) into these algebras. Finally \(T\subset P\cap N_\psi=P_\psi\) and \(G_\psi=G\cap N_\psi\subset P_\psi\). (3) Lemma 3.2 and Proposition 4.2 of the lesson on the relative flow give the invariance of \(\bar\psi=\psi\circ E\). If \(F\) is a conditional expectation with \(\bar\psi\circ F=\bar\psi\), the Schwarz inequality \(F(x)^*F(x)\le F(x^*x)\) gives \(\|F(x)\|^2_{\bar\psi}\le\bar\psi(F(x^*x))=\|x\|^2_{\bar\psi}\). \(\square\)

The norm bounds of Theorem 3.1 transfer to \(\bar\psi\). For \(y\in M\) and normal states \(\rho,\rho'\), \(|\,\|y\|^2_\rho-\|y\|^2_{\rho'}|\le\|y\|^2\|\rho-\rho'\|\). By Lemma 3.2, Theorem 3.1(1) and \(\eta\le1\), \(\|\bar\psi-\varphi_n\|\le2^{-n}\) and \(\|\bar\psi-\varphi_n\circ E_{T_{n+1}}\|\le2^{1-n}\). Since \(\|z_n\|\le2\), Theorem 3.1(2) gives \[ \begin{aligned} &\|E_{T_{n+1}}(z_n)\|^2_{\bar\psi}\\ &\quad \le\tfrac{15}{16}\|z_n\|^2_{\bar\psi}+2^{4-n}. \end{aligned}\tag{4.1} \]

Lemma 4.2 (targets converge). Let \(w\in T'\cap M\) and \(n_1<n_2<\cdots\) with \(\|z_{n_k}-w\|_{\bar\psi}\to0\). Then \(w=0\).

Proof. Since \(w\in T_{n+1}'\cap M\), \(E_{T_{n+1}}(w)=w\), and by Lemma 4.1(3) \(\|E_{T_{n_k+1}}(z_{n_k})-w\|_{\bar\psi}\le\|z_{n_k}-w\|_{\bar\psi}\to0\). Letting \(k\to\infty\) in (4.1) gives \(\|w\|^2_{\bar\psi}\le\frac{15}{16}\|w\|^2_{\bar\psi}\), so \(w=0\) because \(\bar\psi\) is faithful. \(\square\)

For every \(w\) in the unit ball of \(M\) there are even indices \(n_1<n_2<\cdots\) with \(x_{n_k}\to w\) \(*\)-strongly: choose \(j_k\) with \(\|a_{j_k}-w\|^\sharp_{\bar\psi}<1/k\), which is possible because the \(\|\cdot\|^\sharp_{\bar\psi}\)-metric defines the \(*\)-strong topology on the unit ball (Lemma 1.4 of the ultraproduct lesson), and then \(n_k=2m_k\) with \(k(m_k)=j_k\) and \(n_k>n_{k-1}\). Likewise with odd indices for \(w\) in the unit ball of \(N\).

Proposition 4.3. \(T'\cap M\subset\mathrm B^\sharp(N\subset M)\).

Proof. Let \(x\in T'\cap M\), \(x\ne0\). For a unitary \(u\) of some \(T_n\), \(E^\sharp(x)=E^\sharp(uxu^*)=uE^\sharp(x)u^*\), because \(T\subset N\subset\mathrm B^\sharp\); so \(E^\sharp(x)\in T'\cap M\). Hence \(w=(x-E^\sharp(x))/(2\|x\|)\) lies in the unit ball of \(T'\cap M\) and \(E^\sharp(w)=0\). Choose even \(n_k\) with \(x_{n_k}\to w\) \(*\)-strongly. Since \(w\in T^{(n)\prime}\cap M\), Lemma 4.1(3) gives \(\|y_{n_k}-w\|_{\bar\psi}=\|E_{T^{(n_k)}}(x_{n_k}-w)\|_{\bar\psi}\to0\), and then \[ \begin{aligned} &\|z_{n_k}-w\|_{\bar\psi}\le\|y_{n_k}-w\|_{\bar\psi}\\ &\quad +\|E^\sharp(y_{n_k}-w)\|_{\bar\psi}\to0 . \end{aligned} \] By Lemma 4.2, \(w=0\), so \(x=E^\sharp(x)\in\mathrm B^\sharp\). \(\square\)

Proposition 4.4. \(T'\cap N=G\).

Proof. Each \(G_n\) commutes with \(T_n\) and with every other \(F_j\), so \(G\subset T'\cap N\). Conversely, let \(x\in T'\cap N\), \(x\ne0\). By Lemma 4.1(2) and the modular expectation theorem there is a \(\psi\)-preserving conditional expectation \(E^\psi_G\) of \(N\) onto \(G\). Then \(w=(x-E^\psi_G(x))/(2\|x\|)\) lies in the unit ball of \(T'\cap N\), and \(E^\psi_G(w)=0\). Choose odd \(n_k\) with \(x_{n_k}\to w\) \(*\)-strongly; as before \(\|y_{n_k}-w\|_\psi\to0\). For odd \(n\), Lemma 1.3, applied to \(G^{(n)}\), \(L=N_n\) and \(\psi\) (\(\psi\circ E_{G^{(n)}}=\psi\) by Lemma 3.2), gives \[ \begin{aligned} &z_n=y_n-E^\psi_{G^{(n)}}(y_n)\\ &\quad +S_{(\psi-\varphi)|N_n}(y_n),\qquad\|S_{(\psi-\varphi)|N_n}(y_n)\|\\ &\quad \le\|(\psi-\varphi_n)|_{N_n}\|\le2^{-n}, \end{aligned} \] because \(\varphi\) and \(\varphi_n\) agree on \(N_n\). The expectation \(E^\psi_{G^{(n)}}\circ E^\psi_G\) is \(\psi\)-preserving onto \(G^{(n)}\), so it equals \(E^\psi_{G^{(n)}}\); hence \(E^\psi_{G^{(n)}}(w)=0\) and \(\|E^\psi_{G^{(n)}}(y_n)\|_\psi=\|E^\psi_{G^{(n)}}(y_n-w)\|_\psi\), which is at most \(\|y_n-w\|_\psi\). Therefore \(\|z_{n_k}-w\|_\psi\to0\), and Lemma 4.2 gives \(w=0\), that is \(x=E^\psi_G(x)\in G\). \(\square\)

Lemma 4.5 (exchangeability). \(G_\psi'\cap G=\mathbb C1\).

Proof. All \(G_j\) carry the same state \(\omega_\mu\otimes\omega_\nu\); choose their matrix units \((g^{(j)}_{\alpha\beta})_{\alpha,\beta=1}^4\) as products of those of \(F^\mu_j\) and \(F^\nu_j\), with the same weights \(h_\alpha\) for every \(j\). For \(j\ne j'\) put \[ s_{jj'}=\sum_{\alpha,\beta}g^{(j)}_{\alpha\beta}\,g^{(j')}_{\beta\alpha}. \] Since the two systems commute, \(s_{jj'}^*=s_{jj'}\), \(s_{jj'}^2=\sum_{\alpha,\beta}g^{(j)}_{\alpha\alpha}g^{(j')}_{\beta\beta}=1\), and \(s_{jj'}g^{(j)}_{\gamma\delta}s_{jj'}=g^{(j')}_{\gamma\delta}\); and \(s_{jj'}\) commutes with every \(G_i\), \(i\notin\{j,j'\}\). By Lemma 4.1(2) each term \(g^{(j)}_{\alpha\beta}g^{(j')}_{\beta\alpha}\) is fixed by \(\sigma^\psi\), so \(s_{jj'}\in G_\psi\). For \(k\ge1\) the unitary \(\pi_k=s_{1,k+1}s_{2,k+2}\cdots s_{k,2k}\in G_\psi\) satisfies \(\pi_kg^{(i)}_{\gamma\delta}\pi_k^*=g^{(k+i)}_{\gamma\delta}\) for \(i\le k\), so \(\operatorname{Ad}\pi_k\) maps \(G^{(k)}\) into \(G_{k+1}\vee\cdots\vee G_{2k}\subset R_k'\cap N\).

Let \(x\in G_\psi'\cap G\) with \(\psi(x)=0\), and \(x_k=E^\psi_{G^{(k)}}(x)\), defined by Lemma 1.3(2). In \(L^2(N,\psi)\), with vector \(\xi_\psi\), \(x_k\xi_\psi\) is the orthogonal projection of \(x\xi_\psi\) onto the closure \(H_k\) of \(G^{(k)}\xi_\psi\). The subspaces \(H_k\) increase, and their union is dense in the closure of \(G\xi_\psi\), because \(\bigcup_kG^{(k)}\) is a unital \(*\)-algebra whose strong closure is \(G\) (double commutant theorem). Hence \(\|x-x_k\|_\psi\to0\). Also \(\psi(x_k)=\psi(x)=0\). Put \(y_k=\pi_kx_k\pi_k^*\). Since \(\pi_k\) commutes with \(x\) and lies in \(N_\psi\), \[ \begin{aligned} &\|x-y_k\|_\psi=\|\pi_k(x-x_k)\pi_k^*\|_\psi\\ &\quad =\|x-x_k\|_\psi\to0,\\ &\psi(y_k)=\psi(x_k)=0 . \end{aligned} \] By Lemma 4.1(1), \(\psi(x_k^*y_k)=\psi(x_k^*)\psi(y_k)=0\). Hence \[ \begin{aligned} &\|x\|_\psi^2=|\psi(x^*x)-\psi(x_k^*y_k)|\\ &\quad \le\|x-x_k\|_\psi\|x\|_\psi\\ &\quad +\|x_k\|_\psi\|x-y_k\|_\psi\to0, \end{aligned} \] and \(x=0\). For general \(x\in G_\psi'\cap G\) apply this to \(x-\psi(x)1\). \(\square\)

This is the operator-algebraic form of the Hewitt–Savage zero–one law: an element of an infinite tensor power that is invariant under all finite permutations of the factors is a scalar.

Corollary 4.6. \(P_\psi'\cap N=\mathbb C1\) and \(P_\psi'\cap M\subset\mathrm B^\sharp(N\subset M)\).

Proof. By Lemma 4.1(2), \(T\subset P_\psi\) and \(G_\psi\subset P_\psi\). If \(x\in P_\psi'\cap N\), then \(x\in T'\cap N=G\) by Proposition 4.4, and \(x\) commutes with \(G_\psi\), so \(x\in\mathbb C1\) by Lemma 4.5. If \(x\in P_\psi'\cap M\), then \(x\in T'\cap M\subset\mathrm B^\sharp\) by Proposition 4.3. \(\square\)

5. The theorem

Lemma 5.1 (rows of eigenoperators). Let \(S=\mathbb Z\log\mu+\mathbb Z\log\nu\subset\mathbb R\). For every \(s\in S\) there is a sequence \((v_k)\) in \(P\) with \(\sigma^\psi_t(v_k)=e^{its}v_k\) for all \(t\in\mathbb R\) and \(\sum_kv_kv_k^*=1\), the sum converging strongly. The subgroup \(S\) is dense in \(\mathbb R\).

Proof. Call such a sequence a row for \(s\). Let \((e^{(n)}_{ab})_{a,b=1,2}\) be the matrix units of \(F^\mu_n\), with the weights (1.1) of the previous lesson for \(\lambda=\mu\), so \(h_1/h_2=\mu\) and \(\psi(e^{(n)}_{22})=h_2=1/(1+\mu)\). By Lemma 1.2 of the previous lesson and Lemma 3.2, \(\sigma^\psi_t(e^{(n)}_{12})=\mu^{it}e^{(n)}_{12}\) and \(\sigma^\psi_t(e^{(n)}_{aa})=e^{(n)}_{aa}\). Put \[ v_k=e^{(1)}_{22}\cdots e^{(k-1)}_{22}\,e^{(k)}_{12}\qquad(k\ge1), \] a product of commuting elements. Then \(\sigma^\psi_t(v_k)=e^{it\log\mu}v_k\), and \(v_kv_k^*=e^{(1)}_{22}\cdots e^{(k-1)}_{22}e^{(k)}_{11}\). By induction on \(K\), \(\sum_{k\le K}v_kv_k^*=1-q_K\) with \(q_K=e^{(1)}_{22}\cdots e^{(K)}_{22}\), and by Lemma 4.1(1), \(\psi(q_K)=(1+\mu)^{-K}\). The projections \(q_K\) decrease to a projection \(q\) with \(\psi(q)=\lim_K\psi(q_K)=0\), so \(q=0\) and \((v_k)\) is a row for \(\log\mu\). With \(e^{(n)}_{21}\) and \(e^{(n)}_{11}\) in place of \(e^{(n)}_{12}\) and \(e^{(n)}_{22}\) one gets a row for \(-\log\mu\), and with \(F^\nu_n\) rows for \(\pm\log\nu\). The single element \(1\) is a row for \(0\). If \((v_k)\) is a row for \(s\) and \((w_l)\) a row for \(s'\), the countable family \((v_kw_l)_{k,l}\) is a row for \(s+s'\): \(\sigma^\psi_t(v_kw_l)=e^{it(s+s')}v_kw_l\), and the finite partial sums of the positive operators \(v_kw_lw_l^*v_k^*\) form an increasing net bounded by \(1\), which converges strongly to its supremum by Vigier's theorem. The partial sums over \(k\le K\), \(l\le L\) equal \(\sum_{k\le K}v_k\big(\sum_{l\le L}w_lw_l^*\big)v_k^*\), which tends strongly to \(\sum_{k\le K}v_kv_k^*\) as \(L\to\infty\); so the supremum is \(\sum_kv_kv_k^*=1\). So every element of \(S\) has a row.

A subgroup \(S\ne0\) of \(\mathbb R\) is dense or cyclic: let \(c=\inf(S\cap(0,\infty))\). If \(c=0\), \(S\) contains elements in \((0,\varepsilon)\) for every \(\varepsilon\), whose multiples are \(\varepsilon\)-dense. If \(c>0\), then \(c\in S\): otherwise there would be \(s_1>s_2>c\) in \(S\) with \(s_1<2c\), and \(s_1-s_2\in S\cap(0,c)\) would contradict the choice of \(c\). Then division with remainder gives \(S=c\mathbb Z\). In the cyclic case \(\log\mu=mc\) and \(\log\nu=m'c\) with integers \(m,m'\ne0\), and \(\log\mu/\log\nu=m/m'\) would be rational. \(\square\)

Theorem 5.2 (OpenAI, for factors of type III₁). Let \(N\subset M\) be von Neumann algebras with separable preduals and a faithful normal conditional expectation \(E\colon M\to N\), and suppose that \(N\) is a factor of type III₁. For every faithful normal state \(\varphi\) on \(N\) and every \(\eta\in(0,1]\) there are a faithful normal state \(\psi\) on \(N\) with \(\|\psi-\varphi\|\le\eta\) and a von Neumann subalgebra \(P\subset N\) such that:

  1. \(P\) is approximately finite-dimensional, hence injective;
  2. \(\sigma^\psi_t(P)=P\) for all \(t\), so there is a \(\psi\)-preserving faithful normal conditional expectation of \(N\) onto \(P\);
  3. \(P_\psi'\cap N=\mathbb C1\) and \(P_\psi'\cap M\subset\mathrm B(N\subset M,\psi)\);
  4. \(P'\cap c(M)=N'\cap c(M)\).

Proof. Construct the algebras of Theorem 3.1 for \(\varphi\) and \(\eta\), and let \(\psi\) be as in Lemma 3.2, so that \(\|\psi-\varphi\|\le\|\bar\psi-\bar\varphi\|\le\eta\), and \(P\) as in Section 4. (1) \(P\) is generated by the increasing sequence of finite-dimensional \(*\)-subalgebras \(R_n\), all containing \(1\); it is injective by Corollary 5.2(c) of the lesson on injective von Neumann algebras. (2) Lemma 4.1(2) and the modular expectation theorem. (3) By Corollary 4.6, \(P_\psi'\cap N=\mathbb C1\) and \(P_\psi'\cap M\subset\mathrm B^\sharp=N\vee\mathrm B(N\subset M,\psi)\); Proposition 5.2 of the lesson on core commutants, with \(\psi\) in place of \(\varphi\), gives \(P_\psi'\cap M\subset\mathrm B(N\subset M,\psi)\). (4) Theorem 5.1 of the lesson on core commutants, with \(\psi\) in place of \(\varphi\): hypothesis (1) is (3), and hypothesis (2) is Lemma 5.1. \(\square\)

Intersecting (4) with \(M\) gives \(P'\cap M=N'\cap M\), where \(M\) is identified with its image in the core. An approximately finite-dimensional \(P\subset N\) with expectation and \(P'\cap M=N'\cap M\) exists for every expected inclusion with separable preduals by a theorem of Marrakchi [M2, Theorem B]. When \(N\) is a factor of type III₁, the existence of such a \(P\) that is a factor is equivalent, by [AHHM, Theorem C], to the statement that the fixed points of the relative bicentralizer flow on \(\mathrm B(N\subset M,\varphi)\) form \(N'\cap M\), which Marrakchi proved [M2, Theorem C]. The proof of Theorem 5.2 does not use these results: in the core criterion, the eigenoperators of hypothesis (2) make every element of \(P'\cap c(M)\) invariant under the flow \(\gamma^c\) of the core.

[OAI, Theorem 1.1] asserts the conclusion of Theorem 5.2 for every expected inclusion with separable preduals, without assumptions on the type or the centre of \(N\). The reduction of that general statement to factors of type III₁ is not treated in this course.

For \(M=N\) the construction gives a classical theorem of Haagerup [Haagerup, Theorem 3.1], which he derived from the triviality of the bicentralizer, at that time a hypothesis.

Corollary 5.3 (irreducible centralizers). Let \(N\) be a factor of type III₁ with separable predual. The faithful normal states \(\psi\) on \(N\) whose centralizer satisfies \(N_\psi'\cap N=\mathbb C1\) are norm dense in the set of normal states of \(N\).

Proof. Let \(\rho\) be a normal state and \(\varepsilon\in(0,1]\). Choose a faithful normal state \(\varphi_0\) and put \(\varphi=(1-\frac\varepsilon4)\rho+\frac\varepsilon4\varphi_0\), a faithful normal state with \(\|\varphi-\rho\|\le\frac\varepsilon2\). Theorem 5.2 for the inclusion \(N\subset N\) with \(E=\mathrm{id}\), and \(\eta=\frac\varepsilon2\), gives \(\psi\) with \(\|\psi-\varphi\|\le\frac\varepsilon2\) and a \(\sigma^\psi\)-invariant \(P\subset N\) with \(P_\psi'\cap N=\mathbb C1\). Since \(P_\psi\subset N_\psi\), \(N_\psi'\cap N\subset P_\psi'\cap N=\mathbb C1\), and \(\|\psi-\rho\|\le\varepsilon\). \(\square\)

6. Exercises

Exercise 6.1. Show that the algebra \(P\) of Theorem 5.2 is a factor.

Exercise 6.2. Let \(\psi\) be a state as in Corollary 5.3. Show that the centralizer \(N_\psi\) is a factor of type II₁.

Exercise 6.3. Let \(M=N\) and \(E=\mathrm{id}\). Show that the targets \(z_{2m}\) vanish, and that the construction gives an approximately finite-dimensional \(\sigma^\psi\)-invariant \(P\subset N\) whose centralizer \(P_\psi\) has trivial relative commutant in \(N\).

Exercise 6.4. Verify the identity \(s_{jj'}g^{(j)}_{\gamma\delta}s_{jj'}=g^{(j')}_{\gamma\delta}\) used in the proof of Lemma 4.5.

Exercise 6.5. Suppose that the eigen parts are omitted, that is, every \(G_n\) is replaced by \(\mathbb C1\) in Theorem 3.1. Show that the resulting algebra \(P=T\) lies in \(N_\psi\), satisfies \(P_\psi'\cap N=\mathbb C1\) and hypothesis (1) of the core criterion, but that hypothesis (2) of that criterion fails.

7. Solutions

6.1. An element \(c\) of the centre of \(P\) commutes with \(P\supset P_\psi\) and lies in \(N\), so \(c\in P_\psi'\cap N=\mathbb C1\) by Theorem 5.2(3).

6.2. An element of the centre of \(N_\psi\) commutes with \(N_\psi\) and lies in \(N\), so it lies in \(N_\psi'\cap N=\mathbb C1\); thus \(N_\psi\) is a factor. The restriction of \(\psi\) is a faithful normal tracial state on it, since \(\psi(xy)=\psi(yx)\) for \(x\in N_\psi\), so \(N_\psi\) is finite. By (B8) of the ultraproduct lesson it is of type I\(_k\) for some \(k\), or of type II₁. In the first case \(N_\psi\cong M_k(\mathbb C)\) is a matrix subalgebra of \(N\), and Lemma 1.1(1) of the previous lesson writes every element of \(N\) as \(\sum f_{ab}x_{ab}\) with \(x_{ab}\in N_\psi'\cap N=\mathbb C1\); then \(N=N_\psi\) would be finite-dimensional, which a factor of type III is not.

6.3. For \(M=N\), \(\mathrm B^\sharp(N\subset N)=N\vee\mathrm B(N,\varphi)=N\), so \(E^\sharp=\mathrm{id}\) and \(z_{2m}=0\). Theorem 5.2(1)–(3) still apply, and (3) says \(P_\psi'\cap N=\mathbb C1\). (The even steps then only change the state and add eigen parts.)

6.4. Using that the two systems commute, \[ \begin{aligned} &s\,g^{(j)}_{\gamma\delta}=\sum_{\alpha,\beta}g^{(j)}_{\alpha\beta}g^{(j)}_{\gamma\delta}g^{(j')}_{\beta\alpha}\\ &\quad =\sum_\alpha g^{(j)}_{\alpha\delta}g^{(j')}_{\gamma\alpha},\\ &s\,g^{(j)}_{\gamma\delta}\,s=\sum_{\alpha,\epsilon,\zeta}g^{(j)}_{\alpha\delta}g^{(j)}_{\epsilon\zeta}\,g^{(j')}_{\gamma\alpha}g^{(j')}_{\zeta\epsilon}\\ &\quad =\sum_\alpha g^{(j)}_{\alpha\alpha}g^{(j')}_{\gamma\delta}=g^{(j')}_{\gamma\delta}, \end{aligned} \] since \(g^{(j)}_{\alpha\delta}g^{(j)}_{\epsilon\zeta}=\delta_{\delta\epsilon}g^{(j)}_{\alpha\zeta}\) and \(g^{(j')}_{\gamma\alpha}g^{(j')}_{\zeta\epsilon}=\delta_{\alpha\zeta}g^{(j')}_{\gamma\epsilon}\).

6.5. Each \(T_n\) carries the trace, so \(T\subset N_\psi\) by Lemma 4.1(2), and \(P_\psi=P=T\). Propositions 4.3 and 4.4 hold with \(G=\mathbb C1\) (Lemma 1.3 with \(G=\mathbb C1\) is the scalar slice \(y\mapsto\rho(y)1\)), so \(T'\cap N=\mathbb C1\) and \(T'\cap M\subset\mathrm B^\sharp\); Proposition 5.2 of the lesson on core commutants then gives hypothesis (1). But \(\sigma^\psi\) is trivial on \(P\), so an element \(v\in P\) with \(\sigma^\psi_t(v)=e^{its}v\) for all \(t\) and some \(s\ne0\) satisfies \(v=e^{its}v\) for all \(t\), hence \(v=0\); no row exists for \(s\ne0\).

References