Modular invariants of operator-valued weights
An operator-valued weight has modular data that lives on the relative commutant and does not depend on the scalar weight used to expose it. The mechanism is exact: scalar modular flows lift through the operator-valued weight, and a cocycle comparing two such weights commutes with the target algebra. Throughout, \(N\subseteq M\) is a unital inclusion of arbitrary von Neumann algebras and
\[ N^c=N'\cap M. \tag{OM.1} \]Lifting scalar modular flows and cocycles
Let \(T\in\mathcal W_0(M,N)\) be a faithful normal semifinite operator-valued weight. For \(\varphi,\psi\in\mathcal W_0(N)\), set
\[ \Phi=\widehat\varphi\circ T, \qquad \Psi=\widehat\psi\circ T. \tag{OM.2} \]The composition theorem makes \(\Phi\) and \(\Psi\) faithful normal semifinite weights on \(M\). The forward modular graph transfer, relative to its three stated component contracts, gives
\[ \sigma_t^\Phi(n)=\sigma_t^\varphi(n). \tag{OM.3} \]Here \(n\in N\) and \(t\in\mathbb R\). It also gives the stronger cocycle identity
\[ [D\Psi:D\Phi]_t=[D\psi:D\varphi]_t. \tag{OM.4} \]The common value belongs to \(N\).
The cocycle order matters: the numerator \(\Psi\) corresponds to \(\psi\), and the denominator \(\Phi\) to \(\varphi\). Equation (OM.4) follows by evaluating the equality of the two mixed isometric groups at \(1\); it is not inferred merely from (OM.3). This proves clauses (i) and (ii) of Takesaki II IX.4.22 at the same conditional public-proof boundary as OR-03.
Comparing two operator-valued weights
Let \(S,T\in\mathcal W_0(M,N)\), fix \(\varphi\in\mathcal W_0(N)\), and abbreviate
\[ \begin{aligned} \Phi_T&=\widehat\varphi\circ T,\\ \Phi_S&=\widehat\varphi\circ S,\\ u_t&=[D\Phi_T:D\Phi_S]_t. \end{aligned} \tag{OM.5} \]Thus \(u_t\in M\). The ordinary modular implementation formula says
\[ \sigma_t^{\Phi_T} =\operatorname{Ad}(u_t)\circ \sigma_t^{\Phi_S}. \tag{OM.6} \]Apply this identity to \(n\in N\). By (OM.3), both sides restrict to \(\sigma_t^\varphi(n)\), so
\[ u_t\,\sigma_t^\varphi(n)\,u_t^* =\sigma_t^\varphi(n). \tag{OM.7} \]Because \(\sigma_t^\varphi(N)=N\), equation (OM.7) gives \(u_t\in N'\). The scalar cocycle already lies in \(M\), hence
\[ u_t\in N'\cap M=N^c. \tag{OM.8} \]This proves the membership assertion of IX.4.22(iii) without a matrix amplification. It uses the full faithfulness assumptions: they make every cocycle above a whole-algebra unitary.
The same argument also shows that \(N^c\) is globally invariant under every \(\sigma_t^{\widehat\varphi\circ T}\). Indeed, (OM.3) maps \(N\) onto itself, and an automorphism preserving \(N\) preserves \(N'\cap M\).
Independence of the scalar reference
Choose another \(\psi\in\mathcal W_0(N)\), and put \(\Psi_T=\widehat\psi\circ T\) and \(\Psi_S=\widehat\psi\circ S\). Use (OM.4) separately for \(T\) and \(S\), and abbreviate
\[ \begin{aligned} a_t&:=[D\Psi_T:D\Phi_T]_t,\\ a_t&=[D\psi:D\varphi]_t,\\ b_t&:=[D\Phi_S:D\Psi_S]_t,\\ b_t&=[D\varphi:D\psi]_t. \end{aligned} \tag{OM.9} \]Apply the ordered cocycle chain rule twice, inserting the two \(\varphi\)-composites between the \(\psi\)-composites:
\[ [D\Psi_T:D\Psi_S]_t=a_tu_tb_t. \tag{OM.10} \]The first and last factors lie in \(N\), while \(u_t\in N^c\). They therefore commute with \(u_t\). The reverse-cocycle identity cancels the outer factors in their displayed order, and (OM.9) becomes
\[ [D\Psi_T:D\Psi_S]_t=u_t. \tag{OM.11} \]Thus the cocycle comparing \(T\) and \(S\) is independent of the faithful scalar reference. No two arbitrary cocycles were commuted; only the relative-commutant element \(u_t\) was moved past elements of \(N\).
The modular restriction is reference-independent as well. From (OM.4) and modular implementation,
\[ \begin{aligned} \sigma_t^{\Psi_T} &=\operatorname{Ad}(a_t)\\ &\quad{}\circ\sigma_t^{\Phi_T}. \end{aligned} \tag{OM.12} \]For \(x\in N^c\), the first factor belongs to \(N\) and commutes with \(\sigma_t^{\widehat\varphi\circ T}(x)\in N^c\). Hence the two modular actions agree on \(N^c\).
For \(T_j\), write \(\Phi_{T_j}=\widehat\varphi\circ T_j\), and abbreviate \(\delta_{12}(t)=(DT_1:DT_2)_t\). We may therefore make the intrinsic definitions
\[ \sigma_t^T :=\left.\sigma_t^{\Phi_T}\right|_{N^c}. \tag{OM.13a} \] \[ \begin{aligned} \delta_{12}(t) &:=[D\Phi_{T_1}:\\ &\qquad D\Phi_{T_2}]_t. \end{aligned} \tag{OM.13b} \]The first is the modular automorphism group of \(T\) on \(N^c\); the second is the cocycle derivative of \(T_1\) relative to \(T_2\), an element of \(N^c\). The scalar reference in (OM.13) is any member of \(\mathcal W_0(N)\). In the printed first clause of IX.4.23 the base algebra appears as \(M\), but the displayed composite, the second clause, and IX.4.22 require a weight on \(N\); (OM.13) records the type-correct quantifier.
Intrinsic laws and the exact boundary
All ordinary faithful cocycle identities descend to the relative commutant. If \(T_1,T_2,T_3\in\mathcal W_0(M,N)\), write \(\delta_{ij}(t)=(DT_i:DT_j)_t\). Then
\[ \delta_{31}(t)=\delta_{32}(t)\delta_{21}(t). \tag{OM.14a} \] \[ \delta_{12}(t)^*=\delta_{21}(t). \tag{OM.14b} \] \[ \begin{aligned} \sigma_t^{T_1} &=\operatorname{Ad}(\delta_{12}(t))\\ &\quad{}\circ\sigma_t^{T_2}. \end{aligned} \tag{OM.14c} \]Writing \(v_t=(DT_1:DT_2)_t\), its cocycle law is
\[ v_{s+t}=v_s\,\sigma_s^{T_2}(v_t). \tag{OM.15} \]Here \(s,t\in\mathbb R\). These formulas are obtained by fixing one faithful \(\varphi\in\mathcal W_0(N)\), applying the usual scalar identities to \(\widehat\varphi\circ T_j\), and then using (OM.13). They introduce no additional source assumption.
The human antecedents are Takesaki II IX.4.22–23, printed pp. 230–232 / PDF pp. 250–252. This proof is conditional on OR-03's mixed-graph, finite-matrix and analytic-generator contracts and on the provider closure of the faithful ordered chain rule. The adjacent commutant-duality bijection IX.4.24 and the scalar-weight/\(B(H)\) correspondence IX.4.25 require separate construction and proof. The stated dependencies are not proved here.