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 (Lemmas 1.4 and 1.5); Averaging and the completely positive criterion (Lemma 1.1 and Proposition 4.1); Preparing infinite factors (Definition 5.1); Rescaled Tomita graphs (Corollary 3.1, with its constant $C_{\rm cut}=2.1\cdot10^6$); Infinitesimal normalizers of type III factors (Lemma 3.1 and the group $\mathcal G_M$); Frequency nests and analytic multipliers (the notation $\mathcal K,P_a,C_s,M_F$, Lemmas 3.1 and 4.1 and Proposition 4.2); 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 $B(H)$.

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 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 $0<r\le\gamma$ put $\begin{gathered} A_r(t)=e^{-iz}B(z),\\ C_r(t)=e^{-iz}S(z). \end{gathered}$ Since $|e^{-iz}|=e^r\le e^\gamma$, Lemma 4.1 gives, uniformly in $r,t,k$ and $x\in M_k(M)_1$, $\begin{aligned} &\|A_r(t)\|,\ \|C_r(t)\|\le1+C\gamma,\\ &\|A_r(t)C_r(t)-1\|\le C\gamma,\\ &\|C_r(t)A_r(t)-1\|\le C\gamma,\\ &\operatorname{dist}\bigl(A_r(t)^{(k)}xC_r(t)^{(k)},M_k(M)\bigr)\\ &\qquad\le C\gamma,\quad(5.1) \end{aligned}$ and the same with $A_r$ and $C_r$ exchanged in the last line.

Polar parts. For small $\gamma$, $A_rC_r$ and $C_rA_r$ are invertible, so $A_r(t)$ has a right and a left inverse, which agree, and $\|A_r(t)^{-1}\|\le1+C\gamma$. With $\|A_r(t)\|\le1+C\gamma$, the spectrum of $|A_r(t)|$ lies in $[1-C\gamma,1+C\gamma]$, and the unitary polar part $V_r(t)=A_r(t)\bigl(A_r(t)^*A_r(t)\bigr)^{-1/2}$ satisfies $\|V_r(t)-A_r(t)\|=\|1-|A_r(t)|\|\le C\gamma$ and, comparing $A_r(t)^{-1}=|A_r(t)|^{-1}V_r(t)^*$ with $C_r(t)=A_r(t)^{-1}A_r(t)C_r(t)$, $\|V_r(t)^*-C_r(t)\|\le C\gamma$. By (5.1), at every matrix level, $V_r(t)xV_r(t)^*$ and $V_r(t)^*xV_r(t)$ lie within $C\gamma$ of $M_k(M)$ for $x\in M_k(M)_1$.

Continuity. For fixed $r$, $t\mapsto A_r(t)$ is norm continuous, by holomorphy of $B$, and its inverses are bounded, so $V_r$ is norm continuous by the continuous functional calculus. Put $E_r(t)=V_r(t)V_r(0)^*,$ a norm-continuous unitary path with $E_r(0)=1$. For $x\in M_1$, $V_r(0)^*xV_r(0)$ is within $C\gamma$ of some $m\in M$ with $\|m\|\le1+C\gamma$, and $V_r(t)mV_r(t)^*$ is within $C\gamma\|m\|$ of $M$; so $\sup_{x\in M_1}\operatorname{dist}\bigl(E_r(t)xE_r(t)^*,M\bigr)\le C\gamma .$ $M$ is a separably acting type III factor. Lemma 3.1 of Infinitesimal normalizers of type III factors, applied to the path on the interval between $0$ and $t$, gives $a_r(t)\in\mathcal U(M)$ and $b_r'(t)\in\mathcal U(M')$ with $\begin{aligned} &\|E_r(t)-a_r(t)b_r'(t)\|\\ &\quad \le C\gamma ,\quad(5.2) \end{aligned}$ where a slack of $\gamma$ allows for an infimum that is not attained. No continuous choice of the factors is needed.

6. Back to the real axis

The field $F_0(s)=e^{is}W(s)$ is strongly* continuous, since modular groups are strongly continuous. Put $D_r(t)=e^{-i(t+ir)}\int_{\mathbb R}\frac{r\,F_0(s)}{\pi((t-s)^2+r^2)}\,ds .$ By (3.2) of the analytic multiplier lesson and (4.1), $\|A_r(t)-D_r(t)\|\le e^r\delta$, so $\begin{gathered} \|V_r(t)-D_r(t)\|\le C\gamma,\\ \|D_r(t)\|\le e^r .\quad(6.1) \end{gathered}$ At every fixed $t$, $D_r(t)\to W(t)$ strongly* as $r\downarrow0$: on a fixed vector, split the integral into $|s-t|<\eta$, where strong continuity controls $F_0(s)\xi-F_0(t)\xi$, and the complement, whose Poisson mass tends to $0$; the same holds for the adjoint field, and $e^{-i(t+ir)}\to e^{-it}$. Since $W(0)=1$ and all the operators are uniformly bounded, $\begin{gathered} Q_r(t)=D_r(t)D_r(0)^*\longrightarrow W(t)\\ \text{strongly}^*, \end{gathered}$ and by (6.1), $\|E_r(t)-Q_r(t)\|\le C\gamma$. This compares $E_r(t)$ with a convergent family; it does not assert convergence of $E_r(t)$.

7. A completely positive limit

Proof of Theorem 1.1. Fix $t$ and a sequence $r\downarrow0$. The maps $\Phi_r(x)=a_r(t)xa_r(t)^*$ are UCP from $M$ to $M$, and $\Phi_r$ is the restriction of $\operatorname{Ad}(a_r(t)b_r'(t))$ because $b_r'(t)$ commutes with $M$. For operators $T_1,T_2$, the maps $x\mapsto T_1xT_1^*$ and $x\mapsto T_2xT_2^*$ differ in completely bounded norm by at most $(\|T_1\|+\|T_2\|)\|T_1-T_2\|$. With (5.2) and Section 6, $\begin{gathered} \begin{aligned} &\|\Phi_r^{(k)}(x)\\ &\qquad -Q_r(t)^{(k)}xQ_r(t)^{(k)*}\|\\ &\quad \le C\gamma\|x\| \end{aligned}\\ (x\in M_k(M)). \end{gathered}$ By Lemma 1.1 of the averaging lesson, a subnet of $(\Phi_r)$ converges pointwise in the weak operator topology to a UCP map $\Phi:M\to M$; it need not be normal. For fixed $x$, $Q_r(t)xQ_r(t)^*\to W(t)xW(t)^*$ strongly, also at every matrix level, and norm balls are weakly closed, so $\begin{gathered} \|\Phi-\rho_t\|_{\rm cb}\le C\gamma,\\ \rho_t(x)=W(t)xW(t)^* . \end{gathered}$ The representation $\rho_t$ of $M$ is faithful and normal, and by Lemma 2.1 every element of $M_1$ is within $2\gamma$ of $\rho_t(M)_1$. For small $\gamma$, Proposition 4.1 of the averaging lesson, with $Q=M$, gives a unitary $q_t$ with $q_t\rho_t(M)q_t^*=M$ and $\|q_t-1\|\le C\gamma$. Then $n_t=q_tW(t)$ satisfies $n_tMn_t^*=M$ and $\|n_t-W(t)\|=\|q_t-1\|\le C\gamma$. All thresholds and constants were absolute. $\square$

8. Exercises

Exercise 8.1. Verify $\mathcal U_P^*T_s\mathcal U_P=T_s\otimes U_P(s)$.

Exercise 8.2. Show that, for bounded operators $T_1,T_2$, $\|T_1xT_1^*-T_2xT_2^*\|$$\le(\|T_1\|+\|T_2\|)\|T_1-T_2\|\|x\|$, at every matrix level.

Exercise 8.3. Let $V\subseteq B(K)$ be a weak* closed subspace. Show that $\operatorname{dist}(y,V)=\sup\{|\omega(y)|\}$ over normal functionals $\omega$ of norm at most $1$ vanishing on $V$.

Exercise 8.4. Show that an invertible $A$ with $\|A\|\le1+\epsilon$ and $\|A^{-1}\|\le1+\epsilon$ has polar part $V$ with $\|V-A\|\le\epsilon$.

Exercise 8.5. Explain why the proof does not use a continuous or measurable choice of $t\mapsto n_t$, and where Section 2 of the next lesson removes the need for one.

9. Solutions

Solution 8.1. $(\mathcal U_P^*T_s\mathcal U_Pf)(t)$$=U_P(t)^*U_P(t+s)f(t+s)$$=U_P(s)f(t+s)$.

Solution 8.2. $T_1xT_1^*-T_2xT_2^*$$=(T_1-T_2)xT_1^*+T_2x(T_1-T_2)^*$; amplification preserves norms.

Solution 8.3. $V$ is the annihilator of its preannihilator $V_\perp$ in the predual, by the Hahn–Banach theorem in the predual, and the quotient $B(K)/V$ is isometrically the dual of $V_\perp$. So the quotient norm of $y$ is the norm of the functional $\omega\mapsto\omega(y)$ on $V_\perp$.

Solution 8.4. The spectrum of $|A|$ lies in $[(1+\epsilon)^{-1},1+\epsilon]\subseteq[1-\epsilon,1+\epsilon]$, and $V-A=V(1-|A|)$.

Solution 8.5. Each $n_t$ is produced separately, through a cluster point of a sequence in $r$ at that $t$. The next lesson averages the family $t\mapsto n_t$ with an invariant mean on $\mathbb R_d$, which needs no measurability, and recovers continuity at the end from an intertwining relation with the modular group of $N$.

References

[OpenAI-289] OpenAI, Universal strong Kadison–Kastler stability, preprint, 23 September 2026, Section 7. https://github.com/openai/math/tree/main/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026