# Near normalizers at every time *Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).* Let \(M,N\) be a prepared pair with common vector \(\Omega\) and modular groups \(U_M(t)=\Delta_M^{it}\), \(U_N(t)=\Delta_N^{it}\). The unitaries \(W(t)=U_N(t)U_M(t)^*\) nearly normalize \(M\): conjugation by \(W(t)\) moves \(M\) by at most twice the completely bounded distance. This lesson proves that each \(W(t)\) is close to an exact normalizer of \(M\), with a bound uniform in \(t\) (Theorem 1.1). The proof makes the comparison of modular cuts into a statement about the frequency nest, replaces the field \(W\) by analytic multipliers, and smooths them into a norm-continuous path of approximate normalizers. The path is then factored by the infinitesimal estimates for type III factors, and a completely positive limit recovers an exact normalizer at each time [OpenAI-289, Section 7]. We use: [Close operator algebras](close-operator-algebras.md) (Lemmas 1.4 and 1.5); [Averaging and the completely positive criterion](averaging-and-the-completely-positive-criterion.md) (Lemma 1.1 and Proposition 4.1); [Preparing infinite factors](preparing-infinite-factors.md) (Definition 5.1); [Rescaled Tomita graphs](rescaled-tomita-graphs.md) (Corollary 3.1, with its constant \(C_{\rm cut}=2.1\cdot10^6\)); [Infinitesimal normalizers of type III factors](infinitesimal-normalizers-of-type-iii-factors.md) (Lemma 3.1 and the group \(\mathcal G_M\)); [Frequency nests and analytic multipliers](frequency-nests-and-analytic-multipliers.md) (the notation \(\mathcal K,P_a,C_s,M_F\), Lemmas 3.1 and 4.1 and Proposition 4.2); [Stone's theorem](course:analytic-elements-strips-and-kms/holomorphy-in-banach-spaces-stone-s-theorem-and-resolvent-convergence#4-stone-s-theorem) (a strongly continuous unitary group \(s\mapsto e^{isA}\) determines the spectral measure of its generator); and the [weak compactness of the unit ball of \(B(H)\)](course:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#OA-FND-LT-10). ## 1. The statement In this lesson \(C\) denotes absolute constants, whose values may change from one occurrence to the next, and \(\gamma_0>0\) an absolute threshold; all smallness conditions below are met by decreasing \(\gamma_0\). **Theorem 1.1** (near normalizers). There are absolute constants \(\gamma_0>0\) and \(C\) such that, for every prepared pair \(M,N\subseteq B(H)\) with \(d_{\rm cb}(M,N)\le\gamma<\gamma_0\), and every \(t\in\mathbb R\), there is a unitary \(n_t\) with \[ n_tMn_t^*=M,\qquad\|n_t-W(t)\|\le C\gamma . \] No measurability or continuity of \(t\mapsto n_t\) is claimed. Throughout the proof write \(L_P=\log\Delta_P\), so \(U_P(t)=e^{itL_P}\). ## 2. Initial approximate normalizers **Lemma 2.1.** For every \(t\), \(k\ge1\) and \(x\in M_k(M)_1\), \[ \begin{gathered} \begin{aligned} &\operatorname{dist}\bigl(W(t)^{(k)}xW(t)^{(k)*},M_k(M)_1\bigr)\\ &\quad \le2\gamma, \end{aligned}\\ \begin{aligned} &\operatorname{dist}\bigl(W(t)^{(k)*}xW(t)^{(k)},M_k(M)_1\bigr)\\ &\quad \le2\gamma . \end{aligned} \end{gathered} \] Consequently \(d_{\rm cb}(W(t)MW(t)^*,M)\le2\gamma\). **Proof.** \(U_M(t)\) normalizes \(M\) and \(U_N(t)\) normalizes \(N\). Given \(x\), the element \(U_M(t)^*xU_M(t)\) lies in \(M_k(M)_1\); approximate it by \(y\in M_k(N)_1\) within \(\gamma\) plus a small slack; then \(U_N(t)yU_N(t)^*\in M_k(N)_1\) is within \(\gamma\) plus slack of some \(z\in M_k(M)_1\). So \(W(t)xW(t)^*\) is within \(2\gamma\) plus slack of \(z\). The second estimate approximates \(x\) by \(y\in M_k(N)_1\), then \(U_N(t)^*yU_N(t)\) by \(z\in M_k(M)_1\), and uses \(U_M(t)zU_M(t)^*\in M_k(M)_1\). The two estimates give the two directed distances between the unit balls of \(M_k(W(t)MW(t)^*)\) and \(M_k(M)\). \(\square\) ## 3. The frequency nest sees the modular comparison On \(\mathcal K=L^2(\mathbb R,H)\) let \(\mathcal U_P\) be multiplication by the field \(t\mapsto U_P(t)\), for \(P=M,N\), and \(\mathcal W=\mathcal U_N\mathcal U_M^*\), multiplication by \(W\). Let \(T_s\) be the translation \((T_sf)(t)=f(t+s)\), so that \(P_a\) is the spectral projection of the generator of \(s\mapsto T_s\) for \((a,\infty)\). **Lemma 3.1.** For every \(a\in\mathbb R\), \[ \begin{gathered} \|(1-P_a)C_1\mathcal WP_a\|\le C_{\rm cut}\gamma,\\ \|(1-P_a)C_1\mathcal W^*P_a\|\le C_{\rm cut}\gamma . \end{gathered} \] **Proof.** A direct computation gives \(\mathcal U_P^*T_s\mathcal U_P=T_s\otimes U_P(s)\), because \(U_P(t)^*U_P(t+s)=U_P(s)\). So conjugation by \(\mathcal U_P\) carries the spectral projections of the generator of \(s\mapsto T_s\) to those of the generator of \(V_P(s)=T_s\otimes e^{isL_P}\). Under the Fourier transform in \(t\), \(T_s\) becomes multiplication by \(e^{is\xi}\), so \(V_P(s)\) becomes the decomposable unitary field \(\xi\mapsto e^{is(\xi+L_P)}\). The decomposable field of spectral projections \(\xi\mapsto1_B(\xi+L_P)\), \(B\) a Borel set, is a projection-valued measure whose unitary group is \(V_P\). By Stone's theorem it is the spectral measure of the generator of \(V_P\). Hence \[ \begin{gathered} \mathcal U_P^*P_a\mathcal U_P\ \text{has fibers}\\ 1_{(a,\infty)}(\xi+L_P)=1_{(a-\xi,\infty)}(L_P). \end{gathered} \] By (3.1) of the analytic multiplier lesson, \(C_1^*(1-P_a)C_1=1-P_{a-1}\), and so \[ \begin{aligned} &(1-P_a)C_1\mathcal WP_a\\ &\quad=C_1\mathcal U_N\bigl[\mathcal U_N^*(1-P_{a-1})\mathcal U_N\bigr]\\ &\qquad\cdot\bigl[\mathcal U_M^*P_a\mathcal U_M\bigr]\mathcal U_M^* . \end{aligned} \] The bracketed product is decomposable with fibers \(1_{(-\infty,a-1-\xi]}(L_N)\,1_{(a-\xi,\infty)}(L_M)\). With \(b=a-\frac12-\xi\), this is a product of the cuts of Corollary 3.1 of [Rescaled Tomita graphs](rescaled-tomita-graphs.md) at \(b\mp\frac12\) (the open half-line gives a smaller projection), so each fiber has norm at most \(C_{\rm cut}\gamma\). By Lemma 4.1 of the analytic multiplier lesson the norm of a decomposable operator is the essential supremum of the norms of its fibers. The second estimate is the same with \(M\) and \(N\) exchanged, since \(\mathcal W^*=\mathcal U_M\mathcal U_N^*\). \(\square\) ## 4. Analytic multipliers and their extensions Put \(\delta=C_{\rm cut}\gamma\), \(F(s)=e^{is}W(s)\) and \(\tilde F(s)=e^{is}W(s)^*\); then \(C_1\mathcal W=M_F\) and \(C_1\mathcal W^*=M_{\tilde F}\). By Lemma 3.1 and Proposition 4.2 of the analytic multiplier lesson there are analytic bounded fields \(B\) and \(S\) with \[ \begin{gathered} \|B-F\|_\infty\le\delta,\\ \|S-\tilde F\|_\infty\le\delta .\quad(4.1) \end{gathered} \] Let \(B(z)\) and \(S(z)\), \(\operatorname{Im}z>0\), be their extensions (Lemma 3.1 there). **Lemma 4.1.** For \(\operatorname{Im}z>0\), every \(k\) and every \(x\in M_k(M)_1\), \[ \begin{aligned} &\|B(z)S(z)-e^{2iz}\|\le2\delta+\delta^2,\\ &\|S(z)B(z)-e^{2iz}\|\le2\delta+\delta^2,\\ &\operatorname{dist}\bigl(B(z)^{(k)}xS(z)^{(k)},M_k(M)\bigr)\le\beta,\\ &\operatorname{dist}\bigl(S(z)^{(k)}xB(z)^{(k)},M_k(M)\bigr)\le\beta, \end{aligned} \] where \(\beta=2\gamma+2\delta+\delta^2\). **Proof.** On the boundary, \(F\tilde F=e^{2is}\), \(\|S\|_\infty\le1+\delta\), and \(BS-e^{2is}=(B-F)S+F(S-\tilde F)\) has norm at most \(2\delta+\delta^2\) almost everywhere. The scalar \(e^{2is}\) is an analytic field with extension \(e^{2iz}\), so by the product rule of Lemma 3.1 of the analytic multiplier lesson the extension of the analytic field \(BS-e^{2is}\) is \(B(z)S(z)-e^{2iz}\), and its norm is bounded by the boundary bound. The same holds for \(SB\). For the second estimate, \(F^{(k)}x\tilde F^{(k)}=e^{2is}W(s)^{(k)}xW(s)^{(k)*}\) is within \(2\gamma\) of \(M_k(M)\) by Lemma 2.1, and \(\|B^{(k)}xS^{(k)}-F^{(k)}x\tilde F^{(k)}\|\le2\delta+\delta^2\); so on the boundary the distance is at most \(\beta\) almost everywhere. Let \(\omega\) be a normal functional on \(B(H^k)\) of norm at most \(1\) vanishing on \(M_k(M)\). The fields \(B^{(k)},S^{(k)}\) are analytic on \(L^2(\mathbb R,H^k)\), so the extension of \(B^{(k)}xS^{(k)}\) is \(B(z)^{(k)}xS(z)^{(k)}\), the Poisson integral of its boundary values; testing \(\omega\), a norm-convergent sum of vector functionals, gives \[ \begin{aligned} &|\omega(B(z)^{(k)}xS(z)^{(k)})|\\ &\quad \le\int\frac{r\,|\omega(B(s)^{(k)}xS(s)^{(k)})|}{\pi((t-s)^2+r^2)}\,ds\le\beta \end{aligned} \] for \(z=t+ir\), since \(|\omega(y)|\le\operatorname{dist}(y,M_k(M))\). As \(M_k(M)\) is weak\* closed, the distance to it is the supremum of \(|\omega(y)|\) over such \(\omega\), by the Hahn–Banach theorem applied in the predual. The case \(S\,x\,B\) is the same. \(\square\) ## 5. A norm-continuous path For \(z=t+ir\) with \(0