Spectral shift maps

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

The previous lesson proved the binormal identity for states of $B(L^2(M))$ concentrated at modular energy zero. This lesson moves the energy. For each $s\in\mathbb R$ we construct a unital completely positive map $\theta_s$ of $B(L^2(M))$, in general not normal, that fixes the Jones projection and the right action of $M$, acts on the left action of $M$ by the relative bicentralizer flow composed with the expectation onto the relative bicentralizer, and translates the modular energy by $s$ [M2, Lemma 2.2]. Composing a state concentrated at energy $s$ with $\theta_s$ gives a state concentrated at energy zero, and the binormal identity becomes $\Phi(\gamma_s(a)\rho(b))=\langle a\xi b,\xi\rangle$ (Corollary 5.1).

The maps are limits of averaged conjugations: by unitaries of $N$ that almost commute with $\varphi$ for $s=0$ (Proposition 3.1), and by rows of elements of $N$ that almost satisfy $v\varphi=e^{-s}\varphi v$ in general (Theorem 4.1). The modular energy is controlled by an explicit estimate: an element $v$ with $v\varphi$ close to $e^{-s}\varphi v$ in norm is close to a modular eigenvector, $\sigma_t(v)\approx e^{its}v$ (Proposition 1.2). It follows from the estimate for elements almost commuting with a state, proved in the ultraproduct lesson after Connes, by a two-by-two matrix trick. It also gives the equivalence of the two descriptions of modular frequency announced in the lesson on the relative flow (Corollary 1.3).

We use: Theorem 3.4(3) and formula (1.1) of The relative bicentralizer; Theorem 3.1, formula (3.2) and the admissible families of Section 3 of Transition isomorphisms and the relative flow; Section 1 and Theorem 5.1 of Binormal states and the relative bicentralizer; Corollary 6.3 and Theorem 6.5 of Ultraproducts and the asymptotic centralizer; Proposition 4.1 of Asymptotic centralizers of type III₁ factors; Lemma 1.1 of Modular averaging and bounded recovery; the modular theorem, The modular group and its analytic algebra, §MF-06; the multiplicative domain theorem, Theorem 4.1(3) of Completely positive maps; and Tychonoff's and the Banach–Alaoglu theorems, Weak topologies, Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian.

1. Approximate eigenoperators

Throughout this section $N$ is a von Neumann algebra with a faithful normal state $\varphi$, GNS space $H_\varphi$, cyclic and separating vector $\xi_\varphi$, modular operator $\Delta_\varphi$ and modular group $\sigma_t=\sigma^\varphi_t$. We write $\|y\|_\varphi=\|y\xi_\varphi\|$, $\|y\|^\sharp_\varphi=(\|y\|_\varphi^2+\|y^*\|_\varphi^2)^{1/2}$, and use the bimodule notation $(y\varphi)(x)=\varphi(xy)$, $(\varphi y)(x)=\varphi(yx)$ for the predual.

Let $\lambda>0$, $P=N\otimes M_2(\mathbb C)$, whose elements are matrices $Y=\sum_{i,j}y_{ij}\otimes e_{ij}$ with $y_{ij}\in N$, and $\begin{gathered} \theta(Y)=h_1\varphi(y_{11})+h_2\varphi(y_{22}),\\ h_1=\frac1{1+\lambda},\quad h_2=\frac\lambda{1+\lambda}. \end{gathered}$

Lemma 1.1. $\theta$ is a faithful normal state of $P$, and for $y\in N$, $t\in\mathbb R$ and $i,j\in\{1,2\}$, $\begin{gathered} \sigma^\theta_t(y\otimes e_{ij})=(h_i/h_j)^{it}\,\sigma_t(y)\otimes e_{ij};\\ \text{in particular}\\ \sigma^\theta_t(y\otimes e_{21})=\lambda^{it}\sigma_t(y)\otimes e_{21}. \end{gathered}$

Proof. Let $K$ be the direct sum of four copies $H_{ij}$ of $H_\varphi$, indexed by $i,j\in\{1,2\}$, and let $P$ act on $K$ by matrix multiplication: $(Y\eta)_{ij}=\sum_ky_{ik}\eta_{kj}$. This is a faithful normal representation, and its image is a von Neumann algebra. Let $\Xi\in K$ have components $\Xi_{ij}=\delta_{ij}h_j^{1/2}\xi_\varphi$. Then $(Y\Xi)_{ij}=h_j^{1/2}y_{ij}\xi_\varphi$, so $\langle Y\Xi,\Xi\rangle=\sum_ih_i\varphi(y_{ii})=\theta(Y)$. The vectors $Y\Xi$ fill the dense subspace $\bigoplus_{i,j}N\xi_\varphi$, and $Y\Xi=0$ forces every $y_{ij}\xi_\varphi=0$, hence $Y=0$. So $\Xi$ is cyclic and separating, $\theta$ is a faithful normal state, and $(K,\Xi)$ is its GNS representation.

The Tomita map $S\colon Y\Xi\mapsto Y^*\Xi$ sends the component $h_i^{1/2}y_{ji}\xi_\varphi$ in $H_{ji}$ to the component $h_j^{1/2}y_{ji}^*\xi_\varphi$ in $H_{ij}$. So $S$ is the direct sum, over the ordered pairs $(j,i)$, of the maps $(h_j/h_i)^{1/2}S_\varphi\colon H_{ji}\to H_{ij}$, where $S_\varphi y\xi_\varphi=y^*\xi_\varphi$. A finite direct sum of closable operators has as closure the direct sum of the closures and as adjoint the direct sum of the adjoints. Hence $\bar S^*\bar S$ acts on $H_{ji}$ as $(h_j/h_i)\bar S_\varphi^*\bar S_\varphi=(h_j/h_i)\Delta_\varphi$, and the modular operator of $\theta$ is $\Delta_\theta=\bigoplus_{i,j}(h_i/h_j)\Delta_\varphi$, with $(h_i/h_j)\Delta_\varphi$ acting on $H_{ij}$. By the modular theorem, $\sigma^\theta_t(Y)\Xi=\Delta_\theta^{it}Y\Xi$. For $Y=y\otimes e_{ij}$ the vector $Y\Xi$ is $h_j^{1/2}y\xi_\varphi$ in $H_{ij}$, so $\begin{aligned} &\Delta_\theta^{it}Y\Xi=(h_i/h_j)^{it}h_j^{1/2}\Delta_\varphi^{it}y\xi_\varphi\\ &\quad =(h_i/h_j)^{it}h_j^{1/2}\sigma_t(y)\xi_\varphi\ \text{ in }H_{ij}, \end{aligned}$ which is the vector of $(h_i/h_j)^{it}\sigma_t(y)\otimes e_{ij}$. Since $\Xi$ is separating, this proves the formula. Finally $h_2/h_1=\lambda$. $\square$

Proposition 1.2 (approximate eigenoperators). Let $v\in N$, $\lambda>0$ and $\varepsilon=\|v\varphi-\lambda\varphi v\|$. For every $t\in\mathbb R$, $w=\lambda^{it}\sigma_t(v)-v$ satisfies $\|w\|_\varphi^2+\lambda\|w^*\|_\varphi^2\le512\,(1+|t|)^2\,\|v\|\,\varepsilon .$ In the logarithmic parametrization $\lambda=e^{-s}$, $\begin{aligned} &\|\sigma_t(v)-e^{its}v\|_\varphi^2\\ &\quad +e^{-s}\big\|(\sigma_t(v)-e^{its}v)^*\big\|_\varphi^2\\ &\quad \le512\,(1+|t|)^2\,\|v\|\,\|v\varphi-e^{-s}\varphi v\| . \end{aligned}$

Proof. We may assume $v\ne0$. Let $V=v\otimes e_{21}\in P$ and $Y=\sum_{i,j}y_{ij}\otimes e_{ij}\in P$. Then $VY=\sum_jvy_{1j}\otimes e_{2j}$ and $YV=\sum_iy_{i2}v\otimes e_{i1}$, so $\begin{aligned} &(V\theta-\theta V)(Y)=\theta(YV)-\theta(VY)\\ &\quad =h_1\big(\varphi(y_{12}v)-\lambda\varphi(vy_{12})\big)\\ &\quad =h_1(v\varphi-\lambda\varphi v)(y_{12}). \end{aligned}$ Since $\|y_{12}\|\le\|Y\|$, $\|[V,\theta]\|\le h_1\varepsilon$. Theorem 6.5 of the ultraproduct lesson, applied to $(P,\theta)$ and $V$, shows that $\|\sigma^\theta_t(V)-V\|^\sharp_\theta$ is at most $16\sqrt2\,(1+|t|)\,\|v\|^{1/2}(h_1\varepsilon)^{1/2}$. By Lemma 1.1, $\sigma^\theta_t(V)-V=w\otimes e_{21}$. Its squared $\sharp$-norm is $\begin{aligned} &\theta(w^*w\otimes e_{11})+\theta(ww^*\otimes e_{22})\\ &\quad =h_1\|w\|_\varphi^2+h_2\|w^*\|_\varphi^2\\ &\quad =h_1\big(\|w\|_\varphi^2+\lambda\|w^*\|_\varphi^2\big). \end{aligned}$ Divide by $h_1$. The second form follows because $\sigma_t(v)-e^{its}v=e^{its}w$ when $\lambda=e^{-s}$. $\square$

Corollary 1.3 (modular frequency). For $v\in N$ and $s\in\mathbb R$, $v\varphi=e^{-s}\varphi v$ if and only if $\sigma_t(v)=e^{its}v$ for all $t\in\mathbb R$.

Proof. If $v\varphi=e^{-s}\varphi v$, Proposition 1.2 gives $\sigma_t(v)\xi_\varphi=e^{its}v\xi_\varphi$, and $\xi_\varphi$ is separating. Conversely, if $\sigma_t(v)=e^{its}v$ for all $t$, then $\Delta_\varphi^{it}v\xi_\varphi=e^{its}v\xi_\varphi$, so $v\xi_\varphi$ has spectral support $\{s\}$ for $\log\Delta_\varphi$. Lemma 1.1 of the modular averaging lesson shows that $\|\varphi v-e^sv\varphi\|$ is at most $(1+e^{s/2})\|(e^{\log\Delta_\varphi/2}-e^{s/2})v\xi_\varphi\|$, which is $0$, that is $v\varphi=e^{-s}\varphi v$. $\square$

Corollary 1.4. Let $(v_n)$ be a bounded sequence in $N$ with $\|v_n\varphi-e^{-s}\varphi v_n\|\to0$. Then for every $T>0$, $\sup_{|t|\le T}\|\sigma_t(v_n)-e^{its}v_n\|^\sharp_\varphi\to0 .$

Proof. Proposition 1.2, with $\|v_n\|$ bounded and $(1+|t|)\le1+T$. $\square$

2. Completely positive maps on B(H)

For a Hilbert space $H$ let $\mathrm{UCP}(B(H))$ be the set of unital completely positive linear maps $B(H)\to B(H)$, not required to be normal. Such a map $\Theta$ is contractive: for $T\in B(H)$ the operator matrix $\begin{pmatrix}\|T\|&T\\T^*&\|T\|\end{pmatrix}$ is positive, hence so is $\begin{pmatrix}\|T\|&\Theta(T)\\\Theta(T)^*&\|T\|\end{pmatrix}$, and testing it on vectors $(\eta,-\Theta(T)^*\eta/\|T\|)$ gives $\|\Theta(T)^*\eta\|\le\|T\|\,\|\eta\|$. A completely positive map $\Theta$, not necessarily unital, satisfies in the same way $\|\Theta(T)\|\le\|\Theta(1)\|\,\|T\|$. We give $\mathrm{UCP}(B(H))$ the point-weak$^*$ topology: $\Theta_i\to\Theta$ when $\Theta_i(T)\to\Theta(T)$ $\sigma$-weakly for every $T$. On bounded sets of $B(H)$ the $\sigma$-weak topology is the weak operator topology.

Lemma 2.1. $\mathrm{UCP}(B(H))$ is compact in the point-weak$^*$ topology. If $\Theta_i\to\Theta$ point-weak$^*$ and $A,B\in B(H)$, then $A\Theta_i(T)B\to A\Theta(T)B$ $\sigma$-weakly for every $T$. If $\Theta_n$ are completely positive maps with $\sup_n\|\Theta_n(1)\|<\infty$ and $\Theta_n(1)\to1$ strongly, and $\omega$ is a free ultrafilter on $\mathbb N$, then $\Theta(T)=\sigma\text{-weak}\lim_{n\to\omega}\Theta_n(T)$ defines an element of $\mathrm{UCP}(B(H))$.

Proof. By contractivity, $\mathrm{UCP}(B(H))$ is a subset of the product $\prod_T\{S:\|S\|\le\|T\|\}$ of $\sigma$-weakly compact balls (Banach–Alaoglu), which is compact by Tychonoff's theorem. It is closed: linearity, $\Theta(1)=1$, and positivity of the matrices $(\Theta(T_{kl}))_{k,l}$ for positive $(T_{kl})\in M_n(B(H))$ are preserved by pointwise $\sigma$-weak limits, positivity being tested on vectors. The second statement holds because $S\mapsto ASB$ is $\sigma$-weakly continuous. For the third, the values $\Theta_n(T)$ lie in a bounded set, so the limit along $\omega$ exists; the limit map is linear and completely positive by the same tests, and $\Theta(1)=1$ because $\Theta_n(1)\to1$ strongly. $\square$

3. The zero-shift map

In this section and the next we use the setting of Section 5 of the previous lesson: $N\subset M$ with a faithful normal conditional expectation $E$, $M$ countably decomposable, $\varphi$ a faithful normal state on $N$, $\bar\varphi=\varphi\circ E$, $H=L^2(M,\bar\varphi)$ with vector $\xi$, $X=\log\Delta$, $\rho(b)=Jb^*J$, $e_N$ the Jones projection, $\mathrm B=\mathrm B(N\subset M,\varphi)$ and $E_{\mathrm B}$ its $\bar\varphi$-preserving expectation. Elements of $M$ act on $H$ on the left, and $N'$ denotes the commutant of $N$ in $B(H)$; it contains $\rho(M)$ and $e_N$, because the closure of $N\xi$ is invariant under $N$. By (1.1) of the first lesson, $\sigma^{\bar\varphi}_t$ restricts to $\sigma^\varphi_t$ on $N$, and $\|y\|_{\bar\varphi}=\|y\|_\varphi$ for $y\in N$. We write $\sigma_t=\sigma^{\bar\varphi}_t$.

Proposition 3.1 (Marrakchi). There is $\zeta\in\mathrm{UCP}(B(H))$ such that

$\zeta(e_N)=e_N$;
$\zeta(a)=E_{\mathrm B}(a)$ for $a\in M$;
$\zeta(T_1TT_2)=T_1\zeta(T)T_2$ for $T\in B(H)$ and $T_1,T_2$ in the commutant $N'$ of the left action of $N$ on $H$; in particular $\zeta\circ\rho=\rho$;
$\zeta(f(X))=f(X)$ for $f\in C_0(\mathbb R)$;
$\zeta(\Delta^{it})=\Delta^{it}$ for $t\in\mathbb R$.

Proof. An averaging net. Let $I$ be the set of triples $i=(\delta,S,\varepsilon)$ with $\delta,\varepsilon>0$ and $S\subset M$ finite, directed by $(\delta,S,\varepsilon)\le(\delta',S',\varepsilon')$ when $\delta'\le\delta$, $S\subset S'$ and $\varepsilon'\le\varepsilon$. By Theorem 3.4(3) of the first lesson, for each $i$ there is a finitely supported probability measure $\mu_i$ on $\mathcal U_\delta=\{u\in\mathcal U(N):\|[u,\varphi]\|<\delta\}$ with $\|\int uxu^*\,d\mu_i(u)-E_{\mathrm B}(x)\|^\sharp_{\bar\varphi}<\varepsilon$ for $x\in S$. For each $x\in M$, the bounded net $\int uxu^*\,d\mu_i(u)$ converges to $E_{\mathrm B}(x)$ in $\|\cdot\|^\sharp_{\bar\varphi}$, hence $*$-strongly: for $y'\in M'$, $\|(z_i-z)y'\xi\|\le\|y'\|\,\|(z_i-z)\xi\|$, and $M'\xi$ is dense.

Put $\Theta_i(T)=\int uTu^*\,d\mu_i(u)$, a finite convex combination of unitary conjugations, so $\Theta_i\in\mathrm{UCP}(B(H))$. By Lemma 2.1, a subnet of $(\Theta_i)$ converges point-weak$^*$ to some $\zeta\in\mathrm{UCP}(B(H))$.

(1) Unitaries of $N$ commute with $e_N$, so $\Theta_i(e_N)=e_N$. (2) $\Theta_i(a)=\int uau^*\,d\mu_i\to E_{\mathrm B}(a)$ strongly. (3) Unitaries of $N$ commute with $T_1,T_2\in N'$, so $\Theta_i(T_1TT_2)=T_1\Theta_i(T)T_2$; pass to the limit with Lemma 2.1. The right action $\rho(M)$ lies in $N'$.

(4) First let $f(x)=\int h(t)e^{itx}dt$ with $h\in L^1(\mathbb R)$, so that $f(X)=\int h(t)\Delta^{it}dt$ strongly. Since $\Delta^{it}u^*=\sigma_t(u^*)\Delta^{it}$, $\begin{gathered} \Theta_i(f(X))=\int h(t)A_i(t)\Delta^{it}\,dt,\\ A_i(t)=\int u\,\sigma_t(u)^*\,d\mu_i(u). \end{gathered}$ For $u\in\mathcal U_\delta$ and $y'\in M'$, $(u\sigma_t(u)^*-1)y'\xi=y'u(\sigma_t(u)-u)^*\xi$, so by Corollary 6.3 of the ultraproduct lesson, applied to $(N,\varphi)$, $\begin{aligned} &\|(A_i(t)-1)y'\xi\|\\ &\quad \le\|y'\|\sup_{u\in\mathcal U_\delta}\|\sigma_t(u)-u\|^\sharp_\varphi\\ &\quad \le2\,C_t^{1/2}\delta^{1/2}\,\|y'\| , \end{aligned}$ where $C_t=8t^2+5|t|+8$. Since $\|A_i(t)-1\|\le2$ and $M'\xi$ is dense, $A_i(t)\to1$ strongly, uniformly for $|t|\le T$ on each fixed vector, for every $T$. For $\eta\in H$, the set $\{\Delta^{it}\eta:|t|\le T\}$ is norm compact; covering it by finitely many balls of radius $r$ gives $\limsup_i\sup_{|t|\le T}\|(A_i(t)-1)\Delta^{it}\eta\|\le2r$ for every $r>0$. Hence $\begin{aligned} &\limsup_i\|(\Theta_i(f(X))-f(X))\eta\|\\ &\quad\le\limsup_i\int_{|t|\le T}|h(t)|\\ &\qquad\qquad\cdot\|(A_i(t)-1)\Delta^{it}\eta\|\,dt\\ &\quad+2\|\eta\|\int_{|t|>T}|h|=2\|\eta\|\int_{|t|>T}|h|, \end{aligned}$ which tends to $0$ as $T\to\infty$. So $\Theta_i(f(X))\to f(X)$ strongly and $\zeta(f(X))=f(X)$. Such $f$ include every $f\in C_c^\infty(\mathbb R)$, whose inverse Fourier transform is integrable, and $C_c^\infty(\mathbb R)$ is uniformly dense in $C_0(\mathbb R)$. Both sides of (4) are contractive in $f$ for the supremum norm, so (4) holds for all $f\in C_0(\mathbb R)$.

(5) $\Theta_i(\Delta^{it})=A_i(t)\Delta^{it}\to\Delta^{it}$ strongly, as shown in the proof of (4). $\square$

4. The shift maps

From now on $N$ is moreover a factor of type III₁ with separable predual, so that the relative bicentralizer flow $\gamma_s=\gamma^\varphi_s=\beta^\varphi_{e^{-s}}$ of the lesson on the relative flow is defined on $\mathrm B$. Recall that an admissible $\lambda$-family with ordinary convergence is a finite family of bounded sequences $(v_{k,n})_n$ in $N$, $1\le k\le m$, with $\|v_{k,n}\varphi-\lambda\varphi v_{k,n}\|\to0$ and $\|1-\sum_kv_{k,n}v_{k,n}^*\|^\sharp_\varphi\to0$. Such families exist for every $\lambda>0$ by Proposition 4.1 of the asymptotic centralizer lesson, and by Theorem 3.1(1) of the lesson on the relative flow, $\sum_kv_{k,n}xv_{k,n}^*\to\beta^\varphi_\lambda(x)$ $*$-strongly for $x\in\mathrm B$.

Theorem 4.1 (Marrakchi). For every $s\in\mathbb R$ there is $\theta_s\in\mathrm{UCP}(B(H))$ such that

$\theta_s(e_N)=e_N$;
$\theta_s(a)=\gamma_s(E_{\mathrm B}(a))$ for $a\in M$;
$\theta_s(T_1TT_2)=T_1\theta_s(T)T_2$ for $T\in B(H)$ and $T_1,T_2\in N'$; in particular $\theta_s\circ\rho=\rho$;
$\theta_s(f(X))=f(X-s)$ for $f\in C_0(\mathbb R)$;
$\theta_s(\Delta^{it})=e^{-its}\Delta^{it}$ for $t\in\mathbb R$.

Proof. Let $\zeta$ be as in Proposition 3.1, let $(v_{k,n})$ be an admissible $e^{-s}$-family with ordinary convergence, $\|v_{k,n}\|\le c$, and $\omega$ a free ultrafilter on $\mathbb N$. Put $\begin{gathered} \Theta_n(T)=\sum_{k=1}^mv_{k,n}\,\zeta(T)\,v_{k,n}^*,\\ \theta_s(T)=\sigma\text{-weak}\lim_{n\to\omega}\Theta_n(T). \end{gathered}$ Each $\Theta_n$ is completely positive with $\Theta_n(1)=\sum_kv_{k,n}v_{k,n}^*$, of norm at most $mc^2$. This sequence tends to $1$ strongly: for $y'\in M'$, $\|(1-\sum_kv_{k,n}v_{k,n}^*)y'\xi\|$ is at most $\|y'\|\,\|1-\sum_kv_{k,n}v_{k,n}^*\|_\varphi$, which tends to $0$, and the sequence is bounded. By Lemma 2.1, $\theta_s\in\mathrm{UCP}(B(H))$.

(1) $\Theta_n(e_N)=\sum_kv_{k,n}e_Nv_{k,n}^*$, which is $e_N\Theta_n(1)\to e_N$ strongly. (2) $\Theta_n(a)=\sum_kv_{k,n}E_{\mathrm B}(a)v_{k,n}^*$ tends strongly to $\beta^\varphi_{e^{-s}}(E_{\mathrm B}(a))=\gamma_s(E_{\mathrm B}(a))$. (3) The $v_{k,n}$ commute with $T_1,T_2\in N'$, and $\zeta$ has property (3).

(4) Let $f(x)=\int h(t)e^{itx}dt$ with $h\in L^1(\mathbb R)$. By Proposition 3.1(4) and $\Delta^{it}v^*=\sigma_t(v)^*\Delta^{it}$, $\begin{aligned} &\Theta_n(f(X))=\sum_kv_{k,n}f(X)v_{k,n}^*\\ &\quad =\int h(t)A_n(t)\Delta^{it}\,dt,\\ &A_n(t)=\sum_kv_{k,n}\sigma_t(v_{k,n})^* . \end{aligned}$ Now $A_n(t)-e^{-its}\Theta_n(1)$ equals $\sum_kv_{k,n}\big(\sigma_t(v_{k,n})-e^{its}v_{k,n}\big)^*$. For $y'\in M'$, Corollary 1.4 shows that the supremum over $|t|\le T$ of $\big\|\big(A_n(t)-e^{-its}\Theta_n(1)\big)y'\xi\big\|$ is at most $\|y'\|\,c\sum_k\sup_{|t|\le T}\big\|\sigma_t(v_{k,n})-e^{its}v_{k,n}\big\|^\sharp_\varphi,$ which tends to $0$. Also $\|A_n(t)\|$ is at most the product of $\|\sum_kv_{k,n}v_{k,n}^*\|^{1/2}$ and $\|\sum_k\sigma_t(v_{k,n}v_{k,n}^*)\|^{1/2}$, hence at most $mc^2$, and $\Theta_n(1)\to1$ strongly. As in the proof of Proposition 3.1, $\sup_{|t|\le T}\|(A_n(t)-e^{-its})\Delta^{it}\eta\|\to0$ for every $\eta\in H$ and $T>0$, and therefore $\begin{aligned} &\Theta_n(f(X))\to\int h(t)e^{-its}\Delta^{it}\,dt\\ &\quad =f(X-s)\quad\text{strongly}. \end{aligned}$ Hence $\theta_s(f(X))=f(X-s)$ for these $f$, and by density for all $f\in C_0(\mathbb R)$.

(5) By Proposition 3.1(5), $\Theta_n(\Delta^{it})=\sum_kv_{k,n}\Delta^{it}v_{k,n}^*=A_n(t)\Delta^{it}$, which tends strongly to $e^{-its}\Delta^{it}$. $\square$

By the multiplicative domain theorem, Completely positive maps, Theorem 4.1(3), properties (4) and (5) give more: $\theta_s(f(X)^*f(X))=\theta_s(f(X))^*\theta_s(f(X))$, and likewise for $f(X)f(X)^*$ and for the unitaries $\Delta^{it}$, so $\begin{aligned} &\theta_s(f(X)T)\\ &\quad =f(X-s)\theta_s(T),\\ &\theta_s(\Delta^{it}T\Delta^{it'})\\ &\quad =e^{-i(t+t')s}\Delta^{it}\theta_s(T)\Delta^{it'} \end{aligned}\tag{4.1}$ for $f\in C_0(\mathbb R)$, $T\in B(H)$ and $t,t'\in\mathbb R$, and similarly with $f(X)$ on the right.

The maps $\theta_s$ depend on the choices made, and the theorem asserts only their existence. For $s=0$ one may take $\theta_0=\zeta$, because $\gamma_0=\mathrm{id}$.

5. The binormal identity at a nonzero frequency

Corollary 5.1. Let $N$ be a factor of type III₁ with separable predual, $N\subset M$ with a faithful normal conditional expectation, $M$ with separable predual, and $s\in\mathbb R$. Let $\Phi$ be a state of $B(H)$ such that $\Phi(y)=\Phi(\rho(y))=\bar\varphi(y)$ for $y\in M$, $\Phi(e_N)=1$, and $\Phi(g(X))=g(s)$ for $g\in C_0(\mathbb R)$. Then $\begin{gathered} \Phi\big(\gamma_s(a)\rho(b)\big)=\langle a\xi b,\xi\rangle\\ (a\in\mathrm B,\ b\in M). \end{gathered}$

Proof. Let $\theta_s$ be as in Theorem 4.1 and $\Phi'=\Phi\circ\theta_s$. For $y\in M$, $\Phi'(y)=\bar\varphi(\gamma_s(E_{\mathrm B}(y)))=\bar\varphi(y)$, since $\gamma_s$ preserves $\bar\varphi$ on $\mathrm B$ (Theorem 3.1(3) of the lesson on the relative flow) and $E_{\mathrm B}$ preserves $\bar\varphi$. Also $\Phi'(\rho(y))=\Phi(\rho(y))=\bar\varphi(y)$, $\Phi'(e_N)=\Phi(e_N)=1$, and for $g\in C_0(\mathbb R)$, $\Phi'(g(X))=\Phi(g(X-s))=g(0)$, because $x\mapsto g(x-s)$ lies in $C_0(\mathbb R)$. By Theorem 5.1 of the previous lesson, $\Phi'(a\rho(b))=\langle a\xi b,\xi\rangle$ for $a\in\mathrm B$ and $b\in M$. By Theorem 4.1(2),(3), $\Phi'(a\rho(b))=\Phi(\theta_s(a)\rho(b))$, which is $\Phi(\gamma_s(a)\rho(b))$, because $E_{\mathrm B}(a)=a$. $\square$

6. Exercises

Exercise 6.1. Let $N=M_2(\mathbb C)$ and $\varphi=\operatorname{Tr}(k\,\cdot)$ with $k=\operatorname{diag}(k_1,k_2)$, $k_1,k_2>0$, $k_1+k_2=1$. Compute $e_{12}\varphi$ and $\varphi e_{12}$, and check Corollary 1.3 for $v=e_{12}$, using $\sigma_t(x)=k^{it}xk^{-it}$.

Exercise 6.2. Show that $\theta_s\circ\theta_{s'}$ satisfies (1)–(4) of Theorem 4.1 with $s+s'$ in place of $s$.

Exercise 6.3. Let $N$ be a factor of type III₁ with separable predual. Show that $\zeta(x)=\varphi(x)1$ for $x\in N$, and that $\Phi(x)=\varphi(x)$ for $x\in N$ whenever $\Phi$ is a state with $\Phi\circ\zeta=\Phi$.

Exercise 6.4. In the setting of Section 1, let $v\in N$, $s\in\mathbb R$ and $d>0$, and suppose that $v\xi_\varphi$ has spectral support in $[s-d,s+d]$ for $\log\Delta_\varphi$. Show that $\|v\varphi-e^{-s}\varphi v\|$ is at most $(1+e^{-s/2})(e^{d/2}-1)\|v\|_\varphi$. Compare with Proposition 1.2.

7. Solutions

6.1. With $\operatorname{Tr}(e_{12}A)=A_{21}$: $(e_{12}\varphi)(y)=\varphi(ye_{12})$, which is $\operatorname{Tr}(e_{12}ky)=k_2y_{21}$ and $(\varphi e_{12})(y)=\operatorname{Tr}(ke_{12}y)=k_1y_{21}$. So $e_{12}\varphi=(k_2/k_1)\varphi e_{12}$, that is $s=\log(k_1/k_2)$. On the other hand $\sigma_t(e_{12})=k_1^{it}k_2^{-it}e_{12}=e^{its}e_{12}$.

6.2. Composition of unital completely positive maps is unital completely positive. (1) and (3) are clear. For $a\in M$, $\theta_s(\theta_{s'}(a))=\gamma_s(E_{\mathrm B}(\gamma_{s'}(E_{\mathrm B}(a))))$, which is $\gamma_{s+s'}(E_{\mathrm B}(a))$, because $\gamma_{s'}(E_{\mathrm B}(a))\in\mathrm B$. For $f\in C_0(\mathbb R)$, $\theta_s(\theta_{s'}(f(X)))=\theta_s(g(X))$ with $g(x)=f(x-s')$, which is $g(X-s)=f(X-s-s')$.

6.3. By Corollary 2.6 of the first lesson, $E_{\mathrm B}(x)=\varphi(x)1$ for $x\in N$, and $\zeta(x)=E_{\mathrm B}(x)$. Then $\Phi(x)=\Phi(\zeta(x))=\varphi(x)$.

6.4. By Lemma 1.1 of the modular averaging lesson, $\|\varphi v-e^sv\varphi\|$ is at most $(1+e^{s/2})\|(e^{\log\Delta_\varphi/2}-e^{s/2})v\xi_\varphi\|$. For $|x-s|\le d$, $|e^{x/2}-e^{s/2}|=e^{s/2}|e^{(x-s)/2}-1|$, which is at most $e^{s/2}(e^{d/2}-1)$, so the right side is at most $(1+e^{s/2})e^{s/2}(e^{d/2}-1)\|v\|_\varphi$. Multiplying by $e^{-s}$ shows that $\|v\varphi-e^{-s}\varphi v\|$ is at most $(1+e^{-s/2})(e^{d/2}-1)\|v\|_\varphi$. Spectral localization therefore controls the twisted commutator linearly in $e^{d/2}-1$, while Proposition 1.2 recovers localization from a small twisted commutator only with a square root.

References

[M2] A. Marrakchi, Kadison's problem and ergodicity of the bicentralizer flow (2026). https://arxiv.org/abs/2606.23636