Modular automorphisms are approximately inner
The modular group of an approximately finite-dimensional factor of type III₁ can be approximated by inner automorphisms. The approximation is in the pointwise norm topology of the predual. Its proof joins two constructions: the covariant C*-algebra of modular integrals, and the criterion that turns local vector intertwiners into implementing unitaries.
The spectral argument below retains the domain of the modular square root. It also treats finite families whose sum state is not faithful; adding an extra vector would change the relative error being estimated.
1. Statement and exact inputs
Theorem 1.1. Let \(M\) be an approximately finite-dimensional factor of type III₁ with separable predual. For every faithful normal state \(\psi\) and every \(\tau\in\mathbb R\), the automorphism \(\sigma_\tau^\psi\) is approximately inner. More generally, the conclusion holds for a semidiscrete factor of type III₁ with separable predual.
Here approximately finite-dimensional means that an increasing directed family of finite-dimensional unital subalgebras generates \(M\) as a von Neumann algebra. In Theorem 1.1 the predual is separable, so approximate innerness can be expressed by a sequence of unitaries \(u_n\in M\) such that
\[ \|\omega\circ\operatorname{Ad}(u_n)^{-1} -\omega\circ(\sigma_\tau^\psi)^{-1}\|\longrightarrow0 \qquad(\omega\in M_*). \tag{1} \]The sequence may depend on \(\psi\) and \(\tau\).
We will also prove a conditional version with no separable-predual assumption: if a type III factor has a faithful normal state, satisfies the minimal-tensor multiplication bound, and has scalar bicentralizer for every faithful normal state, then every modular automorphism of each such state is approximately inner. In this version the implementing unitaries form a net. The proof below applies to this general setting; injectivity and separable predual will supply the bicentralizer hypothesis for Theorem 1.1.
We use the following earlier proofs at their stated generalities.
- QV01 supplies the minimal-tensor multiplication bound for semidiscrete algebras, including the approximately finite-dimensional case.
- The injective-factor bicentralizer theorem, BSI06–07, proves \(\mathrm B(M,\varphi)=\mathbb C1\) for every faithful normal state on an injective type III₁ factor with separable predual. Its proof uses finite-dimensional averages in the dominant centralizer and the earlier finite injective-factor theorem. The class in Theorem 1.1 meets its injectivity hypothesis: Corollary 5.2 of the averaging lesson proves this for an increasing directed finite-dimensional family, and Theorem 4.2 of the finite-model lesson proves it for every semidiscrete algebra. The latter extends the incoming completely positive matrix maps to the ambient operator algebra and takes a point-ultraweak limit of the resulting completely positive maps. The limit fixes the algebra and is a retraction; normality of that retraction is not needed.
- CV05–CV08 construct the coefficient and integrated C*-algebras and their modular covariance under those two hypotheses.
- DQI07 proves the local criterion for approximate innerness on every type III factor with a faithful normal state. IU06 supplies the sequence specialization when the predual is separable.
We work in a standard form \((M,H,J,P)\). Write \(\zeta a=Ja^*J\zeta\) for the right action. For an automorphism \(\theta\), write \(\theta\zeta\) for its canonical action on the cone. The criterion in DQI07 asks: for every finite family \(\xi_1,\ldots,\xi_m\in P\) and every \(\varepsilon>0\), find a nonzero \(x\in M\) such that
\[ \left(\sum_j\|x\xi_j-(\theta\xi_j)x\|^2\right)^{1/2} \leq\varepsilon\left(\sum_j\|x\xi_j\|^2\right)^{1/2}. \tag{2} \]Sections 2–6 establish (2) for \(\theta=\sigma_\tau^\psi\).
2. The state supplied by the test family
Put \(\Phi=\sum_j\omega_{\xi_j}\), and let \(p\) be its support, as constructed in CP12.
If \(p<1\), both \(r=1-p\) and \(s=1-\theta(p)\) are nonzero. The factor-corner argument in GI06 proves \(sMr\ne0\). Take \(0\ne x\in sMr\). Since
\[ (1-p)\xi_j=0=\xi_j(1-p), \]and the transported vector has support \(\theta(p)\) on both sides, we have \(x\xi_j=0=(\theta\xi_j)x\). Thus (2) holds exactly. This includes an empty or entirely zero family.
Suppose henceforth that \(p=1\). Multiply all the \(\xi_j\) by the same positive constant so that \(\sum_j\|\xi_j\|^2=1\). Both sides of (2) are multiplied by that constant. Now \(\varphi=\Phi\) is a faithful normal state. Let \(\xi\in P\) be its representing unit vector; \(\xi\) is cyclic and separating. These standard-form facts are proved in SF10–SF12.
For \(a\in M\),
\[ \|a\xi_j\|^2\leq\sum_k\|a\xi_k\|^2=\|a\xi\|^2. \tag{3} \]Therefore \(a\xi\mapsto a\xi_j\) extends from the dense space \(M\xi\) to a contraction \(T_j\in M'\). Indeed it commutes with each left multiplier on that dense space, hence on \(H\). Set \(b_j=JT_j^*J\in M\). Since \(J\xi=\xi\) and \(J\xi_j=\xi_j\),
\[ \begin{gathered} \|b_j\|\leq1,\\ \xi_j=\xi b_j=b_j^*\xi. \end{gathered} \tag{4} \]In particular, for every \(y\in M\),
\[ \sum_j\|y\xi_j\|^2=\varphi(y^*y)=\|y\xi\|^2. \tag{5} \]The cocycle identity SI14, equation (SI.61) gives a unitary \(w=[D\psi:D\varphi]_\tau\in M\) with
\[ \sigma_\tau^\psi=\operatorname{Ad}(w)\circ\sigma_\tau^\varphi. \tag{6} \]Write \(\gamma=\sigma_\tau^\varphi\). If \(y\ne0\) satisfies (2) with \(\gamma\), then \(x=wy\) satisfies it with \(\theta\): LC06 proves the exact identities
\[ (wy)\xi_j-(\theta\xi_j)(wy) =w\bigl(y\xi_j-(\gamma\xi_j)y\bigr), \qquad \|wy\xi_j\|=\|y\xi_j\|. \tag{7} \]It remains to produce \(y\) for the state \(\varphi\) just constructed. This is why the bicentralizer input must hold for every faithful normal state.
3. A positive operator and a spectral cutoff
Let \(\Delta=\Delta_\varphi\), \(U_t=\Delta^{it}\) and \(\alpha_t=\operatorname{Ad}(U_t)\). By FS01,
\[ \begin{gathered} U_t\xi=\xi,\\ \Delta^{1/2}y\xi=\xi y\quad(y\in M),\\ M\xi\text{ is a core for }\Delta^{1/2}. \end{gathered} \tag{8} \]The core assertion follows because the Tomita operator is the closure of \(y\xi\mapsto y^*\xi\), and multiplication by the antiunitary \(J\) preserves graph norms.
Put \(R_b=Jb^*J\), and define
\[ X=\sum_{j=1}^m(R_{b_j}-b_j^*)^*(R_{b_j}-b_j^*). \tag{9} \]Then \(X\in C^*(M,M')\), \(X\xi=0\), and \(0\leq X\leq4mI\). The first equality on \(\xi\) is (4); the bound follows from \(\|R_{b_j}-b_j^*\|\leq2\).
Let \(\mathcal C\) be the coefficient algebra of CV02–CV05, which contains \(C^*(M,M')\). Its isometric automorphism \(\bar\gamma\), supplied by CV05, acts by \(\gamma\) on \(M\) and fixes \(M'\). Consequently
\[ \begin{gathered} \bar\gamma(X)=\sum_j A_j^*A_j,\\ A_j=R_{b_j}-\gamma(b_j^*),\\ \|A_j\|\leq2. \end{gathered} \tag{10} \]It also commutes with \(\alpha\) and its scalar averages.
Fix \(0<\kappa\leq1/4\). Choose \(f\in C_c^\infty((0,\infty))\) such that
\[ \begin{gathered} 0\leq f\leq1,\\ f(1)=1,\\ \sup_{\lambda>0}|(\sqrt\lambda-1)f(\lambda)|\leq\kappa. \end{gathered} \tag{11} \]For example, choose its support inside a neighborhood of 1 where \(|\sqrt\lambda-1|<\kappa\), using the smooth cutoffs proved in BI03.
Set \(r=f(\Delta)-I\) and \(C=4m+1\). Spectral calculus gives
\[ \begin{gathered} r=r^*,\\ \|r\|\leq1,\\ r\xi=0,\\ 0\leq X+r^2\leq CI. \end{gathered} \tag{12} \]The operator \(CI-X-r^2\) is positive and has norm exactly \(C\), since it sends \(\xi\) to \(C\xi\). The same upper bound and positivity hold with \(X\) replaced by \(\bar\gamma(X)\); equality of that latter norm is the point of the next section.
4. Transporting the norm through the integral algebra
Let \(\mathcal A\subset\mathcal C\) be the norm-continuous coefficient algebra of CV05. Write \(\mathcal B\) for the norm closure of
\[ \int_{\mathbb R}y(t)U_t\,dt,\qquad y\in C_c(\mathbb R,\mathcal A). \tag{13} \]By CV08 there is an isometric automorphism \(\beta\) of \(\mathcal B\) taking (13) to the same integral with coefficient \(\bar\gamma(y(t))\). In particular it fixes every scalar integral \(\int k(t)U_t\,dt\) with \(k\in C_c(\mathbb R)\), and by the bound \(\|\int k(t)U_tdt\|\leq\|k\|_1\), every such integral with \(k\in L^1\).
Here is the required membership for the spectral cutoff. The function \(F(s)=f(e^s)\) lies in \(C_c^\infty(\mathbb R)\). Fourier inversion in BI03 gives an integrable inverse kernel \(k\) with
\[ \begin{gathered} F(s)=\int k(t)e^{its}\,dt,\\ f(\Delta)=F(\log\Delta)=\int k(t)U_t\,dt. \end{gathered} \tag{14} \]The last equality follows by scalar spectral integration and Fubini, tested on matrix coefficients; SK04–SK08 supply that calculus. Approximation of \(k\) in \(L^1\) by compact continuous kernels puts (14) in \(\mathcal B\). Thus \(\beta\) fixes \(f(\Delta)\). We do not need to assert that \(I\) or \(r\) itself lies in \(\mathcal B\).
Choose \(h\in C_c^\infty(\mathbb R)\), \(h\geq0\), \(\int h=1\). Put
\[ \begin{gathered} G=\int h(t)U_t\,dt,\\ X_h=\int h(t)\alpha_t(X)\,dt. \end{gathered} \tag{15} \]Then \(\|G\|\leq1\), \(G\xi=\xi\), \(X_h\in\mathcal A\), \(X_h\xi=0\), and \(0\leq X_h\leq4mI\). CV05 proves the membership of the strong operator average \(X_h\) in \(\mathcal A\), and proves that \(\bar\gamma(X_h)=\alpha_h(\bar\gamma(X))\).
Consider
\[ Y=(CI-X_h-r^2)G. \tag{16} \]It belongs to \(\mathcal B\). Indeed, \(X_hG\) is (13) with coefficient \(h(t)X_h\), while
\[ (CI-r^2)G=(C-1)G+2f(\Delta)G-f(\Delta)^2G \]is in \(\mathcal B\) by (14) and its algebra property. Positivity of \(CI-X_h-r^2\) and its upper bound \(CI\) give \(\|Y\|\leq C\), whereas \(Y\xi=C\xi\). Hence \(\|Y\|=C\).
The covariance formula and the fixed scalar integrals give
\[ \beta(Y)=\bigl(CI-\alpha_h(\bar\gamma(X))-r^2\bigr)G. \tag{17} \]Since \(r\) commutes with \(U_t\), the first factor in (17) is the average \(\alpha_h(CI-\bar\gamma(X)-r^2)\). This average is contractive, so
\[ \begin{gathered} C=\|\beta(Y)\|,\\ \|\beta(Y)\|\leq\|CI-\bar\gamma(X)-r^2\|,\\ \|CI-\bar\gamma(X)-r^2\|\leq C. \end{gathered} \tag{18} \]Let \(Q=\bar\gamma(X)+r^2\). We have \(0\leq Q\leq CI\) and \(\|CI-Q\|=C\). It follows that
\[ \inf_{\|\eta\|=1}\langle Q\eta,\eta\rangle=0. \tag{19} \]For if all these quadratic values were at least \(d>0\), then \(dI\leq Q\leq CI\), whence \(0\leq CI-Q\leq(C-d)I\), contradicting (18).
5. Returning to the Tomita core
Choose a unit vector \(\eta\) using (19) with
\[ \sum_j\|A_j\eta\|^2+\|r\eta\|^2<\kappa^2. \tag{20} \]The squared tolerance is explicit. In particular, each \(\|A_j\eta\|<\kappa\) and \(\|r\eta\|<\kappa\).
Set \(\zeta=f(\Delta)\eta=\eta+r\eta\). It belongs to \(D(\Delta^{1/2})\), since \(f\) has compact support in \((0,\infty)\). Equations (10)–(11) and (20) give
\[ \begin{gathered} \|\zeta\|>1-\kappa,\\ \|A_j\zeta\|<3\kappa,\\ \|(\Delta^{1/2}-I)\zeta\|\leq\kappa. \end{gathered} \tag{21} \]The middle inequality uses \(A_j\zeta=A_j\eta+A_jr\eta\); it does not require \(A_j\) to commute with the cutoff.
The core statement (8) supplies \(y\in M\) such that
\[ \begin{gathered} \|y\xi-\zeta\|<\kappa/2,\\ \|\Delta^{1/2}y\xi-\Delta^{1/2}\zeta\|<\kappa/2. \end{gathered} \tag{22} \]Consequently
\[ \begin{gathered} \|y\xi\|>1-2\kappa,\\ \|y\xi-\xi y\|<2\kappa,\\ \|A_jy\xi\|<4\kappa. \end{gathered} \tag{23} \]The first inequality shows \(y\ne0\). The second is obtained by adding the two graph errors in (22) to the last bound in (21); norm approximation alone would not prove it.
6. The local error and implementing unitaries
The canonical implementation of \(\gamma=\sigma_\tau^\varphi\) is \(U_\tau\); it preserves the cone and fixes \(\xi\). Therefore (4) implies
\[ \gamma\xi_j=\gamma(b_j^*)\xi. \tag{24} \]Left and right actions commute. Using (10) and (24),
\[ \begin{aligned} y\xi_j-(\gamma\xi_j)y &=A_jy\xi+\gamma(b_j^*)(y\xi-\xi y),\\ \|y\xi_j-(\gamma\xi_j)y\|&<6\kappa. \end{aligned} \tag{25} \]Together with (5) and (23), this gives
\[ \begin{gathered} \left(\sum_j\|y\xi_j-(\gamma\xi_j)y\|^2\right)^{1/2} <6\sqrt m\,\kappa,\\ \left(\sum_j\|y\xi_j\|^2\right)^{1/2}>1-2\kappa. \end{gathered} \tag{26} \]Choose
\[ 0<\kappa\leq\min\{1/4,\varepsilon/(12\sqrt m)\}. \tag{27} \]The first bound in (26) is then at most \(\varepsilon/2\), strictly before the comparison, while the second is greater than \(1/2\). This proves (2) for \(\gamma\); (7) proves it for \(\theta\). Undoing the common normalization preserves that inequality. The proper-support case was already settled in Section 2.
Thus every finite test family satisfies the local criterion. DQI07 applies because \(M\) is a type III factor with a faithful normal state. It supplies an implementing net. If the predual is separable, choose a norm-dense sequence of functionals, and for each \(n\) choose an implementing unitary with error less than \(1/n\) on its first \(n\) terms. Both predual actions are isometric, so approximation of an arbitrary functional by a dense test bounds its additional error by twice that norm distance. The resulting sequence satisfies (1). This proves Theorem 1.1 and the conditional version stated in Section 1. No sign restriction was placed on \(\tau\). \(\square\)
The conditional conclusion uses neither a separable predual nor a type III₁ hypothesis. It requires the multiplication bound and scalar bicentralizer for every faithful normal state, as well as a faithful normal state to apply DQI07. The unconditional class in Theorem 1.1 is the separable-predual semidiscrete type III₁ class for which Section 1 verifies these inputs.
7. Faithful semifinite weights and finite families of times
Corollary 7.1. Under the hypotheses of Theorem 1.1, every modular automorphism of a faithful normal semifinite weight is approximately inner.
The same conclusion holds under the conditional hypotheses of Section 1, with a net in place of a sequence.
Proof. Choose a faithful normal state \(\varphi\). In Theorem 1.1 it exists because the predual is separable, as proved in IU01; in the conditional version its existence is a hypothesis. If \(\chi\) is the weight, SI14 gives
\[ \sigma_\tau^\chi =\operatorname{Ad}([D\chi:D\varphi]_\tau)\circ\sigma_\tau^\varphi. \tag{28} \]Let \((u_i)\) implement the approximation for \(\sigma_\tau^\varphi\), and put \(w=[D\chi:D\varphi]_\tau\). Then \((wu_i)\) implements the approximation for (28): for every \(\omega\in M_*\), the norm of the difference of the two predual values is unchanged by their final composition with the isometry \(\operatorname{Ad}(w)^{-1}\). The same calculation applies to a sequence or a net. \(\square\)
For a fixed \(\chi\), a finite set of times \(\tau_1,\ldots,\tau_q\), a finite set of normal functionals, and a positive tolerance, (1) supplies one unitary for each time satisfying all those functional tests. No compatibility between these finitely many chosen unitaries has been proved. In particular, pointwise approximate innerness does not by this argument give a single one-parameter unitary group or convergence uniform in time.
8. Checks on support, cutoffs, and normalization
Why retain the original family? If a new faithful cone vector were appended, the new right side in (2) would contain its extra positive squared norm. An estimate relative to that larger denominator need not imply the required estimate for the original family. The complementary corner in Section 2 resolves this case with no change of denominator.
Why use a squared tolerance? Equation (20) bounds squared errors by \(\kappa^2\). This gives the linear estimates \(3\kappa,2\kappa,6\kappa\) in (21), (23), and (25). A bound by \(\kappa\) in (20) would instead give errors of order \(\sqrt\kappa\), requiring a different choice in (27).
Why include the scalar integral factor? The positive operator \(CI-X-r^2\) is not asserted to lie in \(\mathcal B\). Smoothing \(X\), then multiplying by \(G\), produces the actual element (16). The identity \(G\xi=\xi\) retains its full norm, while \(\|G\|\leq1\) permits the comparison after applying \(\beta\).
Finite matrix check. In the Hilbert–Schmidt standard form of \(M_n(\mathbb C)\), let \(\varphi\) be the normalized trace and \(\xi=n^{-1/2}I\). Then \(\Delta=I\), \(\gamma=\mathrm{id}\), \(r=0\), and \(\eta=\xi\) makes (20) exact. Taking \(y=1\) makes every error zero. For another faithful state with density \(d>0\), its modular automorphism is already \(\operatorname{Ad}(d^{i\tau})\), in agreement with the cocycle transport. This checks the formulas, without applying the type III criterion to a finite algebra.
Reference. M. Takesaki, Theory of Operator Algebras III, Chapter XVIII, Section 4, Theorem 4.12, printed p.381 and proof on pp.386–389. The covariant-algebra preparation is proved in CV; BSI06–07 supplies the injective-factor bicentralizer input. The proof above includes the support reduction, the integral-algebra membership, the squared-error choice, and the Tomita graph approximation. DQI07 extends its conditional conclusion to faithful-state type III factors without a separable predual. The unconditional application retains the separable-predual hypothesis of the injective bicentralizer theorem.