Core commutants

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

The identity \(R=LD\) of the previous lesson determines how \(N\) and the relative bicentralizer \(\mathrm B\) sit together. On \(L^2(\mathrm B\vee N)\cong L^2(N)\otimes L^2(\mathrm B)\), the algebra \(\mathrm B\) acts on the second factor, and \(N\) acts on the first factor after conjugation by \(D=\exp(iX\otimes Y)\) (Proposition 1.1). In the continuous core \(c(M)=M\rtimes_{\sigma}\mathbb R\) the same coordinates separate \(N\) from the crossed product \(\mathrm B\rtimes_\sigma\mathbb R\), which carries a flow \(\gamma^c\) extending the relative bicentralizer flow. The fixed points of \(\gamma^c\) commute with \(N\) (Theorem 4.1) [OAI, Proposition 7.2].

Together with a relative commutant theorem for crossed products (Lemma 2.2), this gives a criterion (Theorem 5.1). Let \(P\subset N\) be globally invariant under the modular group, assume the centralizer of \(\varphi|_P\) has relative commutant in \(M\) inside \(\mathrm B\), and assume \(P\) contains complete rows of modular eigenoperators for a dense set of frequencies. Then \(P\) and \(N\) have the same relative commutant in the core: \(P'\cap c(M)=N'\cap c(M)\).

We use: Proposition 2.2 and the setting of The two relative products; Corollary 1.2 and Exercise 3.1 of Repeated tensor tests; Theorem 3.1 of Transition isomorphisms and the relative flow; Corollary 1.3 of Spectral shift maps; Theorem 2.3 of The relative bicentralizer; the commutant theorem (CCM2) of The normal crossed-product commutant; the normal independence of the regular representation from the initial representation, NR-4 of A normal regular construction on arbitrary Hilbert spaces; commutants of amplifications, Section 3 of The double commutant theorem; the modular expectation theorem (ME.1)–(ME.3) of Conditional expectations from modular invariance; and the spectral theorem for commuting self-adjoint operators.

1. The algebra generated by N and the relative bicentralizer

We keep the setting of the previous lessons: \(N\) a factor of type III₁ with separable predual, \(N\subset M\) with a faithful normal conditional expectation \(E\), \(M\) with separable predual, \(\varphi\) a faithful normal state on \(N\), \(\bar\varphi=\varphi\circ E\), \(\mathcal H=L^2(M,\bar\varphi)\), \(\Omega\), \(\Delta\), \(\sigma=\sigma^{\bar\varphi}\), \(\mathrm B=\mathrm B(N\subset M,\varphi)\), the flow \(\gamma\), \(H\), \(K\), \(X\), \(W_s=e^{isY}\), \(\Delta_K^{it}\), \(R\), \(L\) and \(D\). Let \(A=N\vee\mathrm B\) be the von Neumann algebra generated by \(N\) and \(\mathrm B\). It is globally invariant under \(\sigma\), since \(N\) and \(\mathrm B\) are. Let \(L^2(A)\) be the closure of \(A\Omega\); it is invariant under \(A\) and under \(\Delta^{it}\), and \(A\) acts on it faithfully because \(\Omega\) is separating.

Proposition 1.1. \(L^2(A)=R(H\otimes K)=L(H\otimes K)\). For \(y\in N\), \(b\in\mathrm B\) and \(t\in\mathbb R\), acting on \(L^2(A)\), \[ \begin{gathered} L^*yL=D(y\otimes1)D^*,\\ L^*bL=1\otimes b,\\ L^*\Delta^{it}L=e^{itX}\otimes\Delta_K^{it}. \end{gathered} \]

Proof. By Corollary 1.2 of the previous lesson, \(R\) and \(L\) have the same range \(\mathcal R\). It is the closure of the span of \(N\mathrm B\Omega\), hence invariant under left multiplication by \(N\), and the closure of the span of \(\mathrm BN\Omega\), hence invariant under \(\mathrm B\). The orthogonal projection onto \(\mathcal R\) therefore commutes with the self-adjoint sets \(N\) and \(\mathrm B\), hence with \(A\), so \(\mathcal R\) is \(A\)-invariant. It contains \(\Omega\), so \(L^2(A)\subset\mathcal R\), and \(\mathcal R\subset L^2(A)\) because \(\mathrm BN\Omega\subset A\Omega\). The first two formulas are Exercise 3.1 of the previous lesson. For the third, \(\Delta^{it}ax\Omega=\sigma_t(a)\sigma_t(x)\Omega\), which is \(L(e^{itX}x\Omega\otimes\Delta_K^{it}a\Omega)\). \(\square\)

So, in the coordinates given by \(L\), the left action of \(\mathrm B\) is \(1\otimes\mathrm B\), and the left action of \(N\) is \(N\otimes1\) conjugated by \(D\). On an eigenvector of \(X\) for the value \(h\), \(D\) acts as \(1\otimes W_h\), which is the relation \(xa=\gamma_h(a)x\) for eigenoperators.

2. Cores and relative commutants

Let \(c(M)\) be the continuous core of \(M\) in its regular realization on \(L^2(\mathbb R)\otimes\mathcal H=L^2(\mathbb R,\mathcal H)\): \[ \begin{gathered} (\pi(x)f)(s)=\sigma_{-s}(x)f(s),\\ (\lambda(t)f)(s)=f(s-t),\\ c(M)=\{\pi(M),\lambda(\mathbb R)\}''. \end{gathered} \] For a von Neumann subalgebra \(Q\subset M\) with \(\sigma_t(Q)=Q\) for all \(t\), put \(Q\rtimes\mathbb R=\{\pi(Q),\lambda(\mathbb R)\}''\subset c(M)\). This is the regular crossed product of \(Q\) built on its representation on \(\mathcal H\). By NR-4, for any faithful normal representation of \(Q\), the assignment \(\pi(x)\mapsto\pi'(x)\), \(\lambda(t)\mapsto\lambda'(t)\) extends to a normal \(*\)-isomorphism onto the regular crossed product built on that representation. If \(Q\) leaves a closed subspace \(\mathcal K\subset\mathcal H\) invariant, with \(\Delta^{it}\mathcal K=\mathcal K\) and \(Q\) acting faithfully on \(\mathcal K\), then restriction to \(L^2(\mathbb R,\mathcal K)\) is such an isomorphism: it is a normal \(*\)-homomorphism that agrees with the isomorphism of NR-4 on the generators. Changing the faithful normal state on \(N\) changes \(c(M)\) by an isomorphism that is the identity on \(\pi(M)\), so relative commutants of subalgebras of \(M\) in \(c(M)\) do not depend on the state; we use the state \(\bar\varphi\) throughout.

Lemma 2.1. Let \(Q_1\subset Q_2\subset M\) be \(\sigma\)-invariant von Neumann subalgebras. Then \[ \begin{aligned} &Q_1\rtimes\mathbb R\\ &\quad =(Q_2\rtimes\mathbb R)\cap\big(B(L^2(\mathbb R))\mathbin{\bar\otimes}Q_1\big). \end{aligned} \]

Proof. Apply the commutant theorem (CCM2) to \(Q\subset B(\mathcal H)\), \(G=\mathbb R\), the action \(\sigma|_Q\) and its implementation \(U_s=\Delta^{is}\). It gives \((Q\rtimes\mathbb R)'=\{1\otimes Q',\rho_s\}''\) with \((\rho_sf)(t)=\Delta^{is}f(t+s)\). Since \((1\otimes Q')'=B(L^2(\mathbb R))\mathbin{\bar\otimes}Q\) (commutant of an amplification), the bicommutant theorem gives \[ \begin{aligned} &Q\rtimes\mathbb R\\ &\quad =\big(B(L^2(\mathbb R))\mathbin{\bar\otimes}Q\big)\cap\{\rho_s:s\in\mathbb R\}' , \end{aligned} \] with the same unitaries \(\rho_s\) for \(Q_1\) and \(Q_2\). Intersect. \(\square\)

Lemma 2.2 (relative commutants in the core). Let \(Q_0\subset M\) be a von Neumann subalgebra with \(\sigma_t(q)=q\) for all \(q\in Q_0\) and \(t\in\mathbb R\). Then \[ Q_0'\cap c(M)=(Q_0'\cap M)\rtimes\mathbb R . \]

Proof. For \(q\in Q_0\), \(\pi(q)=1\otimes q\). The algebra \(Q_1=Q_0'\cap M\) is \(\sigma\)-invariant, because \(\sigma_t(Q_1)=\sigma_t(Q_0)'\cap M=Q_1\). Its crossed product commutes with \(\pi(Q_0)\): the operators \(\pi(x)\), \(x\in Q_1\), take values \(\sigma_{-s}(x)\in Q_1\), and \(\lambda(t)\) commutes with constant operators. Conversely, since \(c(M)\subset B(L^2(\mathbb R))\mathbin{\bar\otimes}M=(1\otimes M')'\), \[ \begin{aligned} &Q_0'\cap c(M)\subset(1\otimes M')'\cap(1\otimes Q_0)'\\ &\quad =\big(1\otimes(M'\vee Q_0)\big)'\\ &\quad =B(L^2(\mathbb R))\mathbin{\bar\otimes}(M\cap Q_0'), \end{aligned} \] and Lemma 2.1 with \(Q_2=M\) gives \(Q_0'\cap c(M)\subset Q_1\rtimes\mathbb R\). \(\square\)

3. The core flow

The subspace \(L^2(\mathbb R,K)\) is invariant under \(\mathrm B\rtimes\mathbb R\), and restriction to it is faithful. Conjugating by the unitary \((Vf)(u)=\Delta^{iu}f(u)\) of \(L^2(\mathbb R,K)\), as in Section 4 of the lesson on binormal states, gives the faithful normal twisted regular representation \(\pi_0\) of \(\mathrm B\rtimes\mathbb R\) on \(L^2(\mathbb R)\otimes K\): \[ \begin{gathered} \pi_0(\pi(b))=1\otimes b,\\ \pi_0(\lambda(t))=l(t)\otimes\Delta_K^{it}\\ (b\in\mathrm B,\ t\in\mathbb R), \end{gathered} \] where \(l(t)\) is translation by \(t\) on \(L^2(\mathbb R)\). Indeed \((V\pi(b)V^*f)(u)=\Delta^{iu}\sigma_{-u}(b)\Delta^{-iu}f(u)\), which is \(bf(u)\) and \((V\lambda(t)V^*f)(u)=\Delta^{iu}\Delta^{-i(u-t)}f(u-t)\), which is \(\Delta^{it}f(u-t)\). Let \(\mathcal U\) be multiplication by the variable on \(L^2(\mathbb R)\), \(M_s=e^{-is\mathcal U}\), and let \(\mathcal G\) be the self-adjoint operator \(\mathcal U\otimes1-1\otimes Y\) on \(L^2(\mathbb R)\otimes K\), defined by the joint spectral measure of the commuting operators \(\mathcal U\otimes1\) and \(1\otimes Y\). Then \(e^{-is\mathcal G}=M_s\otimes W_s\).

Lemma 3.1. For every \(s\), \(\operatorname{Ad}(M_s\otimes W_s)\) maps \(\pi_0(\mathrm B\rtimes\mathbb R)\) onto itself, and \(\gamma^c_s=\pi_0^{-1}\circ\operatorname{Ad}(M_s\otimes W_s)\circ\pi_0\) is a continuous action of \(\mathbb R\) on \(\mathrm B\rtimes\mathbb R\) by normal automorphisms with \[ \begin{gathered} \gamma^c_s(\pi(b))=\pi(\gamma_s(b)),\\ \gamma^c_s(\lambda(t))=e^{-ist}\lambda(t). \end{gathered} \]

Proof. Since \(W_sbW_s^*=\gamma_s(b)\) on \(K\) (both sides map \(a\Omega\) to \(\gamma_s(b)a\Omega\)), \((M_s\otimes W_s)(1\otimes b)(M_s\otimes W_s)^*\) equals \(1\otimes\gamma_s(b)\). Since \(W_s\) commutes with \(\Delta_K^{it}\) (Lemma 1.1 of the lesson on the two relative products) and \((M_sl(t)M_s^*f)(u)=e^{-isu}e^{is(u-t)}f(u-t)\), \[ \begin{aligned} &(M_s\otimes W_s)(l(t)\otimes\Delta_K^{it})(M_s\otimes W_s)^*\\ &\quad =e^{-ist}\,l(t)\otimes\Delta_K^{it}. \end{aligned} \] So the normal automorphism \(\operatorname{Ad}(M_s\otimes W_s)\) of \(B(L^2(\mathbb R)\otimes K)\) maps the generators of \(\pi_0(\mathrm B\rtimes\mathbb R)\) to generators, and its inverse does the same. It is a group in \(s\) and pointwise \(\sigma\)-weakly continuous, because \(s\mapsto M_s\otimes W_s\) is a strongly continuous unitary group. \(\square\)

The action \(\gamma^c\) extends the relative bicentralizer flow from \(\mathrm B\) to \(\mathrm B\rtimes\mathbb R\), and on the group unitaries it acts as the dual action.

4. The core fixed point theorem

Theorem 4.1 (OpenAI). If \(z\in\mathrm B\rtimes\mathbb R\) and \(\gamma^c_s(z)=z\) for all \(s\in\mathbb R\), then \(z\) commutes with \(\pi(N)\). Thus \((\mathrm B\rtimes\mathbb R)^{\gamma^c}\subset N'\cap c(M)\).

Proof. A faithful representation of \(A\rtimes\mathbb R\). The algebra \(A\rtimes\mathbb R\subset c(M)\) contains \(\pi(N)\) and \(\mathrm B\rtimes\mathbb R\). Restrict it to \(L^2(\mathbb R,L^2(A))\), conjugate by \((Vf)(u)=\Delta^{iu}f(u)\) as above, and transport \(L^2(A)\) to \(H\otimes K\) by \(L\). By Section 2 and Proposition 1.1 this gives a faithful normal representation \(\Pi_1\) of \(A\rtimes\mathbb R\) on \(L^2(\mathbb R)\otimes H\otimes K\) with \[ \begin{gathered} \Pi_1(\pi(y))=1\otimes D(y\otimes1)D^*,\\ \Pi_1(\pi(b))=1\otimes1\otimes b,\\ \Pi_1(\lambda(t))=l(t)\otimes e^{itX}\otimes\Delta_K^{it}. \end{gathered} \] Let \(\Pi=\operatorname{Ad}(1\otimes D^*)\circ\Pi_1\). Since \(D\) commutes with \(e^{itX}\otimes\Delta_K^{it}\), \(\Pi(\pi(y))=1\otimes y\otimes1\), \(\Pi(\lambda(t))=\Pi_1(\lambda(t))\), and \(\Pi(\pi(b))=(1\otimes D^*)(1\otimes1\otimes b)(1\otimes D)\).

Comparison with \(\pi_0\). Write \(\tilde\pi_0(z)\) for \(\pi_0(z)\) acting on the factors \(L^2(\mathbb R)\) and \(K\) of \(L^2(\mathbb R)\otimes H\otimes K\), and the identity on \(H\). Let \(\mathcal V=\exp(i\,\mathcal U\otimes X\otimes1)\). On the spectral subspace of \(X\) for a value \(h\), \(\mathcal V\) acts as \(e^{ih\mathcal U}\) on \(L^2(\mathbb R)\), and \(e^{ih\mathcal U}l(t)e^{-ih\mathcal U}=e^{iht}l(t)\). By the joint spectral calculus, \[ \begin{aligned} &\mathcal V\big(l(t)\otimes1\otimes\Delta_K^{it}\big)\mathcal V^*\\ &\quad =l(t)\otimes e^{itX}\otimes\Delta_K^{it},\\ &\mathcal V(1\otimes1\otimes b)\mathcal V^*=1\otimes1\otimes b . \end{aligned} \] So the normal \(*\)-homomorphisms \(\Pi_1\) and \(\operatorname{Ad}\mathcal V\circ\tilde\pi_0\) of \(\mathrm B\rtimes\mathbb R\) agree on its generators, hence everywhere. Therefore, for \(z\in\mathrm B\rtimes\mathbb R\), \[ \begin{gathered} \Pi(z)=\mathcal W\,\tilde\pi_0(z)\,\mathcal W^*,\\ \mathcal W=(1\otimes D^*)\mathcal V . \end{gathered} \] All of \(\mathcal U\otimes1\otimes1\), \(1\otimes X\otimes1\) and \(1\otimes1\otimes Y\) commute, and \(\mathcal W\) is the function \(e^{ix(u-y)}\) of them; that is, \(\mathcal W=\exp(iX\otimes\mathcal G)\), a function of \(X\) acting on \(H\) and of \(\mathcal G\) acting on \(L^2(\mathbb R)\otimes K\).

Fixed points. Let \(z\) be \(\gamma^c\)-fixed. Then \(\pi_0(z)\) commutes with \(e^{-is\mathcal G}=M_s\otimes W_s\) for every \(s\), hence with every bounded Borel function of \(\mathcal G\). So \(\tilde\pi_0(z)\) commutes with every spectral projection of the commuting pair formed by \(X\) on \(H\) and \(\mathcal G\) on \(L^2(\mathbb R)\otimes K\), and therefore with \(\mathcal W\). Hence \(\Pi(z)=\tilde\pi_0(z)\) acts trivially on \(H\), and it commutes with \(\Pi(\pi(y))=1\otimes y\otimes1\) for every \(y\in N\). Since \(\Pi\) is faithful, \(z\) commutes with \(\pi(N)\). \(\square\)

In [OAI, Proposition 7.2] the theorem is stated as the equality \((\mathrm B\rtimes\mathbb R)^{\gamma^c}=N'\cap c(M)\), together with the identification of \(\mathrm B\rtimes\mathbb R\) with the relative bicentralizer of \(N\subset c(M)\) [M1, Proposition 7.2]. The reverse inclusion \(N'\cap c(M)\subset(\mathrm B\rtimes\mathbb R)^{\gamma^c}\) is due to Marrakchi [M1, Propositions 8.1 and 8.4]; it is not used below.

5. A criterion for subalgebras

For a \(\sigma^\varphi\)-invariant von Neumann subalgebra \(P\subset N\), write \(P_\varphi=\{x\in P:\sigma^\varphi_t(x)=x\ \forall t\}\) for the centralizer of \(\varphi|_P\). By the modular expectation theorem there is a \(\varphi\)-preserving faithful normal conditional expectation of \(N\) onto \(P\).

Theorem 5.1. Let \(P\subset N\) be a von Neumann subalgebra with \(\sigma^\varphi_t(P)=P\) for all \(t\). Suppose that

  1. \(P_\varphi'\cap M\subset\mathrm B\);
  2. there is a dense subgroup \(S\subset\mathbb R\) such that for every \(s\in S\) there are \(v_1,v_2,\ldots\in P\) with \(\sigma^\varphi_t(v_k)=e^{its}v_k\) for all \(t\) and \(\sum_kv_kv_k^*=1\), the sum converging strongly.

Then \(P'\cap c(M)=N'\cap c(M)\).

Proof. The inclusion \(\supset\) holds because \(P\subset N\). Let \(z\in P'\cap c(M)\).

Step 1. The algebra \(P_\varphi\) is fixed by \(\sigma\), because \(\sigma\) restricts to \(\sigma^\varphi\) on \(N\). By Lemma 2.2 and hypothesis (1), \(z\in(P_\varphi'\cap M)\rtimes\mathbb R\subset\mathrm B\rtimes\mathbb R\), using Lemma 2.1.

Step 2. Let \(s\in S\) and \(v_k\) as in (2). By Corollary 1.3 of the lesson on spectral shift maps, \(v_k\varphi=e^{-s}\varphi v_k\), so by Theorem 3.1(2) of the lesson on the relative flow, \(v_kb=\gamma_s(b)v_k\) for \(b\in\mathrm B\). Also \(\pi(v_k)\lambda(t)=\lambda(t)\pi(\sigma_{-t}(v_k))\), which is \(e^{-its}\lambda(t)\pi(v_k)\). Define the normal map \(\Psi_s(w)=\sum_k\pi(v_k)w\pi(v_k)^*\) on \(c(M)\), the sum converging strongly. For each generator \(g\) of \(\mathrm B\rtimes\mathbb R\), that is \(g=\pi(b)\) or \(g=\lambda(t)\), the two relations give \(\pi(v_k)g=\gamma^c_s(g)\pi(v_k)\), hence \(\Psi_s(gw)=\gamma^c_s(g)\Psi_s(w)\) for all \(w\in c(M)\). By induction \(\Psi_s(g_1\cdots g_r)=\gamma^c_s(g_1\cdots g_r)\Psi_s(1)\), which is \(\gamma^c_s(g_1\cdots g_r)\) for products of generators. The normal maps \(\Psi_s\) and \(\gamma^c_s\) therefore agree on the \(\sigma\)-weakly dense \(*\)-algebra spanned by such products, hence on \(\mathrm B\rtimes\mathbb R\).

Step 3. Since \(z\) commutes with every \(\pi(v_k)\), \(\Psi_s(z)=z\sum_k\pi(v_kv_k^*)=z\). So \(\gamma^c_s(z)=z\) for \(s\in S\). The map \(s\mapsto\gamma^c_s(z)\) is \(\sigma\)-weakly continuous and \(S\) is dense, so \(z\) is \(\gamma^c\)-fixed. By Theorem 4.1, \(z\in N'\cap c(M)\). \(\square\)

Hypothesis (1) forces \(P_\varphi'\cap N\subset\mathrm B\cap N=\mathrm B(N,\varphi)=\mathbb C\) (Theorem 2.3 of the first lesson and the triviality of the bicentralizer of \(N\)): the centralizer of \(\varphi|_P\) must already be irreducible in \(N\). Conversely, irreducibility in \(N\) reduces hypothesis (1) to a condition that does not involve \(\varphi\) beyond \(P_\varphi\), because the algebra \(A=N\vee\mathrm B\) does not depend on the state (Theorem 4.1 of the lesson on the relative flow).

Proposition 5.2. Let \(P\subset N\) be a von Neumann subalgebra with \(\sigma^\varphi_t(P)=P\) for all \(t\) and \(P_\varphi'\cap N=\mathbb C\). Then \(P_\varphi'\cap A\subset\mathrm B\). Consequently hypothesis (1) of Theorem 5.1 holds as soon as \(P_\varphi'\cap M\subset A\).

Proof. In the coordinates of Proposition 1.1, \(L^*yL=D(y\otimes1)D^*\) for \(y\in N\). On the spectral subspace of \(Y\) for a value \(v\), \(D\) acts as \(e^{ivX}\otimes1\), so \(D(y\otimes1)D^*\) is decomposable over the spectral measure of \(1\otimes Y\) with values \(\sigma_v(y)\in N\); hence it commutes with \(y'\otimes1\) for every \(y'\) in the commutant \(N'\) of \(N\) on \(H\). The operators \(L^*bL=1\otimes b\) commute with \(N'\otimes1\) as well, so \[ L^*AL\subset(N'\otimes1)'=N\mathbin{\bar\otimes}B(K). \] For \(y\in P_\varphi\), \(\sigma_v(y)=y\), so \(L^*yL=y\otimes1\). Let \(w\in A\) commute with \(P_\varphi\). Then \(L^*wL\) lies in \[ \begin{aligned} &(N'\otimes1)'\cap(P_\varphi\otimes1)'\\ &\quad=\big((N'\vee P_\varphi)\otimes1\big)'\\ &\quad=(N\cap P_\varphi')\mathbin{\bar\otimes}B(K)=1\otimes B(K), \end{aligned} \] by the commutant of an amplification. Write \(L^*wL=1\otimes T\). Then \(w\Omega=wL(\Omega\otimes\Omega)\), which is \(L(\Omega\otimes T\Omega)=T\Omega\in K\). If \(e_{\mathrm B}\) denotes the projection onto \(K\), then \(E_{\mathrm B}(w)\Omega=e_{\mathrm B}w\Omega=w\Omega\), and since \(\Omega\) is separating, \(w=E_{\mathrm B}(w)\in\mathrm B\). \(\square\)

The main theorem of [OAI] asserts that every expected inclusion with separable preduals admits an amenable \(P\subset N\) with a faithful normal conditional expectation and \(P'\cap c(M)=N'\cap c(M)\). For factors \(N\) of type III₁, [OAI] obtains \(P\) from Theorem 4.1, the reverse inclusion quoted above, and Marrakchi's characterization of this property through the weak Dixmier property of \(N\subset c(M)\) [M1, Theorem D]. Theorem 5.1 and Proposition 5.2 give a direct route: it suffices to construct, for some faithful normal state \(\varphi\) of \(N\), an amenable \(\sigma^\varphi\)-invariant \(P\subset N\) with \(P_\varphi'\cap N=\mathbb C\), \(P_\varphi'\cap M\subset N\vee\mathrm B\), and the rows of hypothesis (2).

6. Exercises

Exercise 6.1. Show that, in the coordinates of Proposition 1.1, the modular operator of \(\bar\varphi|_A\) on \(L^2(A)\) is \(e^X\otimes\Delta_K\), and that \(D\) commutes with it.

Exercise 6.2. Suppose that \(N\) itself contains rows of exact eigenoperators as in hypothesis (2) of Theorem 5.1, for a dense subgroup of frequencies. Show that then \(N'\cap(\mathrm B\rtimes\mathbb R)\) consists of \(\gamma^c\)-fixed elements, so that \(N'\cap(\mathrm B\rtimes\mathbb R)=(\mathrm B\rtimes\mathbb R)^{\gamma^c}\).

Exercise 6.3. Let \(M=N\). Show that hypothesis (1) of Theorem 5.1 says exactly that \(P_\varphi'\cap N=\mathbb C\).

Exercise 6.4. Let \(Q_0\subset M\) be fixed by \(\sigma\). Show that \(Q_0'\cap c(M)\) is invariant under the dual action \(\theta_q=\operatorname{Ad}(e^{-iq\mathcal U}\otimes1)\) of \(c(M)\), and compute \(\theta_q\) on \((Q_0'\cap M)\rtimes\mathbb R\).

7. Solutions

6.1. By Proposition 1.1, \(L^*\Delta^{it}L=e^{itX}\otimes\Delta_K^{it}\), and \(\Delta^{it}\) restricted to \(L^2(A)\) is the modular group of \(\bar\varphi|_A\) because \(A\) is \(\sigma\)-invariant (the modular expectation theorem). By Stone's theorem the generator is \(X\otimes1+1\otimes\log\Delta_K\), so the modular operator is \(e^X\otimes\Delta_K\). The unitary \(D=\exp(iX\otimes Y)\) commutes with \(X\otimes1\) and with \(1\otimes\Delta_K^{it}\), since \(Y\) commutes with \(\Delta_K^{it}\).

6.2. Let \(z\in N'\cap(\mathrm B\rtimes\mathbb R)\) and \(s\) a frequency with a row \(v_k\in N\). Step 2 of the proof of Theorem 5.1 used only that the \(v_k\) are exact eigenoperators of \(N\) with \(\sum_kv_kv_k^*=1\); it gives \(\Psi_s=\gamma^c_s\) on \(\mathrm B\rtimes\mathbb R\). Since \(z\) commutes with the \(\pi(v_k)\), \(\gamma^c_s(z)=\Psi_s(z)=z\). By density and continuity \(z\) is \(\gamma^c\)-fixed. The reverse inclusion is Theorem 4.1.

6.3. For \(M=N\), the relative bicentralizer is the bicentralizer \(\mathrm B(N,\varphi)\), which is \(\mathbb C\) because \(N\) is a factor of type III₁ with separable predual. So (1) reads \(P_\varphi'\cap N\subset\mathbb C\), and the reverse inclusion always holds.

6.4. \(\theta_q\) fixes \(\pi(M)\), which consists of decomposable operators, and multiplies \(\lambda(t)\) by \(e^{-iqt}\), because \(e^{-iq\mathcal U}l(t)e^{iq\mathcal U}=e^{-iqt}l(t)\). So \(\theta_q\) is an automorphism of \(c(M)\) fixing \(\pi(Q_0)\), and it preserves \(Q_0'\cap c(M)\). On \((Q_0'\cap M)\rtimes\mathbb R\) it fixes \(\pi(x)\) and multiplies \(\lambda(t)\) by \(e^{-iqt}\).

References