Modular averaging with controlled tensor moments
An invariant target state and an accurate approximation at time zero do not, by themselves, guarantee an accurate approximation after time averaging. We first approximate an entire bounded interval of the modular orbit. The tool is a normal-state approximation which controls both marginal states in norm as well as finitely many tensor moments.
Let \(M\subset B(H)\) be in the standard form constructed from a faithful normal state \(\varphi\), with cyclic separating unit vector \(\xi\), modular operator \(\Delta\), and modular conjugation \(J\). Put
\[ \chi(b)=\langle b\xi,\xi\rangle\quad(b\in M'), \qquad K=H\otimes H,\qquad N=M\bar\otimes M'\subset B(K). \tag{1} \]The state \(\chi\) is faithful: \(\xi\) is separating for \(M'\) because it is cyclic for \(M\). Write
\[ C=M\otimes_{\min}M'\subset N \tag{2} \]for the norm closure of the algebraic tensor product in this spatial representation.
Our approximation hypothesis is the multiplication bound
\[ \left\|\sum_j a_jb_j\right\| \le \left\|\sum_j a_j\otimes b_j\right\|_{\min} \qquad(a_j\in M,\ b_j\in M'). \tag{3} \]This bound holds for semidiscrete algebras by Proposition 5.7 of Uniqueness of the injective II₁ factor; that proposition is stated and proved for arbitrary von Neumann algebras. In particular it holds for algebras generated by increasing directed families of finite-dimensional unital subalgebras, by Corollary 5.2 of Averaging and crossed products.
The local inputs are the predual duality and normal vector functionals, real convex separation, the faithful-state Tomita construction, and the positive-cone representative theorem with its norm estimate. All Hilbert spaces and preduals may be nonseparable.
1. The diagonal state and its invariant marginals
The commuting actions of \(M\) and \(M'\) give a unital *-homomorphism on the algebraic tensor product,
\[ \mu\left(\sum_j a_j\otimes b_j\right)=\sum_j a_jb_j . \]By (3) it extends continuously to \(C\). It is positive: if \(c\ge0\) in \(C\), then \(c=d^*d\) by continuous functional calculus and \(\mu(c)=\mu(d)^*\mu(d)\ge0\). Hence
\[ \lambda(c)=\langle\mu(c)\xi,\xi\rangle \tag{4} \]is a state of \(C\), with
\[ \lambda(a\otimes b)=\langle ab\xi,\xi\rangle,\qquad \lambda|_{M\otimes1}=\varphi,\qquad \lambda|_{1\otimes M'}=\chi . \tag{5} \]No normality of \(\lambda\) on \(N\) is asserted.
The strongly continuous unitary group
\[ V_t=\Delta^{it}\otimes\Delta^{it} \tag{6} \]normalizes \(N\) and \(C\). Indeed, \(\Delta^{it}\) normalizes \(M\) by the modular theorem, and therefore normalizes its commutant. Write
\[ \alpha_t=\operatorname{Ad}(V_t),\quad \sigma_t(a)=\Delta^{it}a\Delta^{-it},\quad \gamma_t(b)=\Delta^{it}b\Delta^{-it}. \tag{7} \]Then \(\alpha_t(a\otimes b)=\sigma_t(a)\otimes\gamma_t(b)\). Since \(\Delta^{it}\xi=\xi\),
\[ \lambda\circ\alpha_t=\lambda,\qquad \varphi\circ\sigma_t=\varphi,\qquad \chi\circ\gamma_t=\chi . \tag{8} \]For the first equality, compute on algebraic tensors:
\[ \lambda(\alpha_t(a\otimes b)) =\langle\Delta^{it}ab\Delta^{-it}\xi,\xi\rangle =\langle ab\xi,\xi\rangle . \]Linearity and norm density extend it to \(C\).
The group \(V_t\) is strongly continuous on \(K\): this holds on elementary tensors by strong continuity of each factor, and extends by their unitary norm bounds. We do not assume that \(t\mapsto\alpha_t(c)\) is norm continuous for every \(c\in C\).
2. Normal approximation with two marginals controlled in norm
For a normal state \(\rho\) of \(N\), denote its restrictions to the two factors by
\[ \rho_1(a)=\rho(a\otimes1),\qquad \rho_2(b)=\rho(1\otimes b). \]Lemma 2.1. For \(c_1,\ldots,c_m\in C\) and \(\eta>0\), there is a finite convex combination \(\rho\) of unit vector states on \(K\), restricted to \(N\), such that
\[ \|\rho_1-\varphi\|<\eta,\qquad \|\rho_2-\chi\|<\eta,\qquad |\rho(c_j)-\lambda(c_j)|<\eta\quad(1\le j\le m). \tag{9} \]Proof. Consider the Banach space
\[ X=M_*\oplus M'_*\oplus\mathbb C^m \]with the maximum norm. Let \(\mathcal S\subset X\) consist of the tuples
\[ (\rho_1,\rho_2,\rho(c_1),\ldots,\rho(c_m)) \tag{10} \]arising from finite convex combinations of unit vector states on \(K\). These restrictions are normal, and \(\mathcal S\) is convex.
We show that \(x_0=(\varphi,\chi,\lambda(c_1),\ldots,\lambda(c_m))\) belongs to its norm closure. Otherwise real Hahn–Banach separation gives a continuous real linear functional \(F\) on \(X\) with
\[ F(x_0)>\sup_{x\in\mathcal S}F(x). \tag{11} \]By the predual duality, \(F\) is the real part of a sum of evaluations at \(a\in M\), \(b\in M'\), and scalar multiples of the \(m\) coordinates. Consequently there is a self-adjoint \(z\in C\), obtained as the real part of
\[ a\otimes1+1\otimes b+\sum_j t_jc_j, \]such that \(F(x_0)=\lambda(z)\) and \(F((10))=\rho(z)\).
Let \(\beta=\sup_{\|\zeta\|=1}\langle z\zeta,\zeta\rangle\). Then \(\beta1-z\ge0\) in the concrete algebra \(C\), so positivity of \(\lambda\) gives \(\lambda(z)\le\beta\). All unit vector states of \(K\) occur among the allowed \(\rho\), whence
\[ \sup_{x\in\mathcal S}F(x) =\sup_{\|\zeta\|=1}\langle z\zeta,\zeta\rangle =\beta. \]This contradicts (11). Norm approximation in \(X\) now gives (9). \(\square\)
The proof stays inside \(C\) when evaluating \(\lambda\). It needs neither a normal extension of \(\lambda\) to \(N\) nor the interchange of a possibly singular state with an operator integral.
3. From a finite time grid to the entire orbit interval
Fix \(a\in M\), \(b\in M'\). For \(t,s\in\mathbb R\) put
\[ d_a=\sigma_t(a)-\sigma_s(a),\qquad d_b=\gamma_t(b)-\gamma_s(b). \]Suppose \(\rho\) is a normal state with \(\|\rho_1-\varphi\|<\eta\) and \(\|\rho_2-\chi\|<\eta\). The identity
\[ \alpha_t(a\otimes b)-\alpha_s(a\otimes b) =d_a\otimes\gamma_t(b)+\sigma_s(a)\otimes d_b \]and Cauchy–Schwarz for \(\rho\) give
\[ \begin{aligned} |\rho(\alpha_t(a\otimes b)-\alpha_s(a\otimes b))| &\le \|b\|\,\rho_1(d_ad_a^*)^{1/2} +\|a\|\,\rho_2(d_bd_b^*)^{1/2}\\ &\le \|b\|\bigl(\varphi(d_ad_a^*)+4\eta\|a\|^2\bigr)^{1/2}\\ &\quad+\|a\|\bigl(\chi(d_bd_b^*)+4\eta\|b\|^2\bigr)^{1/2}. \end{aligned} \tag{12} \]For example, the first summand is bounded by applying Cauchy–Schwarz to \(d_a\otimes1\) and \(1\otimes\gamma_t(b)\). The marginal bound then applies to the positive operator \(d_ad_a^*\), whose norm is at most \(4\|a\|^2\). The second summand uses the commuting tensor factors in the same way.
By invariance of \(\varphi\), the first state seminorm in (12) equals
\[ \varphi\bigl((\sigma_{t-s}(a)-a)(\sigma_{t-s}(a)-a)^*\bigr)^{1/2}. \tag{13} \]It tends to zero as \(t-s\to0\), because the modular group is strong* continuous. The corresponding assertion holds for \(\chi\) and \(\gamma\). Thus (12) supplies continuity estimates uniform in the base time \(s\) and in all \(\rho\) whose two marginals meet the given norm bounds.
Proposition 3.1. For \(L>0\), \(\varepsilon>0\), and finitely many pairs \(a_j\in M\), \(b_j\in M'\), there is a finite convex combination of unit vector states \(\rho\) on \(N\) such that
\[ \sup_{|t|\le L} |\rho(\alpha_t(a_j\otimes b_j)) -\lambda(a_j\otimes b_j)|<\varepsilon \qquad(1\le j\le m). \tag{14} \]The same state can satisfy \(\|\rho_1-\varphi\|<\varepsilon\) and \(\|\rho_2-\chi\|<\varepsilon\).
Proof. If \(m=0\), take the vector state of \(\xi\otimes\xi\), whose marginals are exactly \(\varphi\) and \(\chi\). Otherwise put \(B=\max(1,\max_j\|a_j\|\|b_j\|)\). Choose \(0<\eta<\varepsilon\) with \(4B\sqrt\eta<\varepsilon/4\). By (13), choose \(h>0\) such that, for every \(j\) and \(|t-s|<h\),
\[ \|b_j\|\varphi(d_{a_j}d_{a_j}^*)^{1/2} +\|a_j\|\chi(d_{b_j}d_{b_j}^*)^{1/2}<\varepsilon/4. \tag{15} \]Using \(\sqrt{x+y}\le\sqrt x+\sqrt y\), (12) is then less than \(\varepsilon/2\).
Choose a finite grid \(s_1,\ldots,s_q\) covering \([-L,L]\) to within distance \(h\). Apply Lemma 2.1 to all the elements \(\alpha_{s_\ell}(a_j\otimes b_j)\in C\), with an error parameter smaller than both \(\eta\) and \(\varepsilon/2\). Their \(\lambda\)-values are \(\lambda(a_j\otimes b_j)\) by (8). For each \(t\in[-L,L]\), select a grid point with \(|t-s_\ell|<h\). The grid error and (12) sum to less than \(\varepsilon\), proving (14). \(\square\)
The order of choices matters: the averaging interval is fixed first, and the normal state is then chosen to control its finite time grid and both marginals.
4. Averaging normal states without losing the moments
For a normal state \(\rho\) given by Proposition 3.1, define
\[ \Phi_L=\frac1{2L}\int_{-L}^{L}\rho\circ\alpha_t\,dt. \tag{16} \]This is a norm integral in \(N_*\). To justify it directly, write \(\rho=\sum_{\ell=1}^r c_\ell\omega_{\zeta_\ell}\), with \(c_\ell\ge0\), \(\sum c_\ell=1\), and \(\|\zeta_\ell\|=1\). Then
\[ \rho\circ\alpha_t =\sum_\ell c_\ell\omega_{V_{-t}\zeta_\ell}|_N. \]The estimate
\[ \|\omega_\zeta-\omega_\eta\| \le(\|\zeta\|+\|\eta\|)\|\zeta-\eta\| \tag{17} \]makes this map norm continuous. On a compact interval it is uniformly continuous, so its Riemann sums are Cauchy in the Banach space \(N_*\). Their limit is (16). Positive Riemann sums are normal states; positivity and evaluation at \(1\) pass to the norm limit. Thus \(\Phi_L\) is a normal state.
Lemma 4.1. For every \(s\in\mathbb R\),
\[ \|\Phi_L\circ\alpha_s-\Phi_L\| \le\min\{2,|s|/L\}. \tag{18} \]For the pairs selected in Proposition 3.1,
\[ |\Phi_L(a_j\otimes b_j)-\lambda(a_j\otimes b_j)| <\varepsilon. \tag{19} \]Its two marginals also remain within \(\varepsilon\) of \(\varphi\) and \(\chi\) in norm.
Proof. Translation replaces the interval \([-L,L]\) in (16) by \([-L+s,L+s]\). The overlap cancels, and the integrand has norm one. Their symmetric difference has length \(2|s|\) if \(|s|\le2L\), and \(4L\) otherwise. Dividing by \(2L\) gives (18). This uses only the triangle inequality for norm integrals.
Each continuous evaluation at \(a_j\otimes b_j\) commutes with the norm integral. The uniform bound (14) throughout its integration interval proves (19). For the first marginal, \(\|\rho_1\circ\sigma_t-\varphi\|=\|\rho_1-\varphi\|<\varepsilon\) by invariance; integrate this norm bound. The second marginal uses \(\chi\circ\gamma_t=\chi\) in the same way. \(\square\)
Only normal functionals are integrated. Invariance of the target state is used in the finite-grid argument, where all tested orbit elements belong to \(C\).
5. The product-state standard form and the correct implementer
Put \(\Omega=\xi\otimes\xi\). It is cyclic for \(N\), because the vectors \(a\xi\otimes b\xi\), \(a\in M\), \(b\in M'\), span a dense subspace of \(K\). It is separating as well: the algebraic tensor product \(M'\odot M\) commutes with \(N\) and also has dense orbit on \(\Omega\). If \(X\in N\) kills \(\Omega\), it therefore kills a dense subspace and is zero.
The vector state of \(\Omega\) is a faithful normal state of \(N\). Apply the faithful-state Tomita theorem and positive-cone construction on the concrete Hilbert space \(K\). Write
\[ S_N=\overline{\{X\Omega\mapsto X^*\Omega\}} =J_N D_N^{1/2}, \]and denote its natural cone by \(P_N\). Every normal positive functional of \(N\) has a unique representative in \(P_N\).
Lemma 5.1. The unitary \(V_t\) in (6) preserves \(P_N\), implements \(\alpha_t\), and fixes \(\Omega\). If \(\Phi\) is represented by \(\zeta\in P_N\), then \(\Phi\circ\alpha_{-t}\) is represented by \(V_t\zeta\).
Proof. The implementation and fixed-vector assertions follow from (6)–(7) and \(\Delta^{it}\xi=\xi\). On the Tomita core \(N\Omega\),
\[ S_NV_t(X\Omega) =S_N(\alpha_t(X)\Omega) =\alpha_t(X)^*\Omega =V_tS_N(X\Omega). \tag{20} \]Both \(V_t\) and its inverse preserve this core. Passing to graph closures gives \(V_tS_NV_t^*=S_N\), including the domain. Uniqueness of the polar decomposition gives
\[ V_tJ_NV_t^*=J_N,\qquad V_tD_N^{1/2}V_t^*=D_N^{1/2}. \tag{21} \]The quarter-power description of the natural cone,
\[ P_N=\overline{\{D_N^{1/4}X\Omega:X\in N_+\}}, \tag{22} \]therefore implies \(V_tP_N=P_N\). This is the SF04 quarter-power description: for the state Hilbert algebra, left squares are exactly \(X\Omega\), \(X\in N_+\), and the quarter-power graph estimate in that proof allows their closure to be taken after applying the quarter power. The vectors in (22) are in the quarter-power domain: \(X\Omega\) is in the Tomita domain \(D(D_N^{1/2})\), which is contained in \(D(D_N^{1/4})\) by the spectral integral test.
Finally, for \(X\in N\),
\[ \langle XV_t\zeta,V_t\zeta\rangle =\langle\alpha_{-t}(X)\zeta,\zeta\rangle. \tag{23} \]Together with cone preservation and uniqueness, this proves the assertion about the representing vector. \(\square\)
We have used the product state to construct a standard form on \(H\otimes H\). The group \(V_t=\Delta^{it}\otimes\Delta^{it}\) is the implementer of the simultaneous action (7); the argument does not identify it with the modular group of that product state.
6. Almost invariant vectors with the required tensor moments
Theorem 6.1. Assume the multiplication bound (3). Given \(a_1,\ldots,a_m\in M\), \(b_1,\ldots,b_m\in M'\), \(T\ge0\) and \(\varepsilon>0\), there is a unit vector \(\zeta\in P_N\subset H\otimes H\) such that
\[ \begin{aligned} \|(\Delta^{it}\otimes\Delta^{it})\zeta-\zeta\| &<\varepsilon &&(|t|\le T),\\ |\langle(a_j\otimes b_j)\zeta,\zeta\rangle -\langle a_jb_j\xi,\xi\rangle| &<\varepsilon &&(1\le j\le m). \end{aligned} \tag{24} \]The two marginal states of \(\omega_\zeta\) can also be chosen within \(\varepsilon\) of \(\varphi\) and \(\chi\) in predual norm.
Proof. Choose \(L>0\) with \(L>T/\varepsilon^2\). Use Proposition 3.1 with this interval and moment error \(\varepsilon\), and form \(\Phi_L\) as in (16). Let \(\zeta\in P_N\) be its representative. Since \(\Phi_L(1)=1\), we have \(\|\zeta\|=1\), and (19) gives the moment inequalities. The marginal conclusion is the last assertion of Lemma 4.1.
By Lemma 5.1, the vectors \(V_t\zeta\) and \(\zeta\) represent \(\Phi_L\circ\alpha_{-t}\) and \(\Phi_L\). The natural-cone norm estimate gives
\[ \|V_t\zeta-\zeta\|^2 \le\|\Phi_L\circ\alpha_{-t}-\Phi_L\| \le\min\{2,|t|/L\}. \tag{25} \]For \(|t|\le T\), the last bound is less than \(\varepsilon^2\) when \(T>0\). If \(T=0\), the vector difference is exactly zero. This proves (24). \(\square\)
Corollary 6.2. The conclusion holds for every semidiscrete von Neumann algebra with a faithful normal state, and in particular for every such algebra generated by an increasing directed family of finite-dimensional unital subalgebras.
Proof. The two earlier programme results cited after (3) give that bound under these hypotheses. Theorem 6.1 then applies. Their statements allow arbitrary directed systems, and nothing in the proof above uses separability. \(\square\)
For the time window in Takesaki's Lemma 4.13(i), set \(T=1/\varepsilon\). Any \(L>1/\varepsilon^3\) suffices in (25), after choosing the initial normal state as in Proposition 3.1 for that entire interval. This recovers the stated AFD type III₁ case, and also proves the broader conclusion in Corollary 6.2.
7. Why one-time moment agreement is insufficient
Here is a finite-dimensional check on the need for Proposition 3.1. On \(\mathbb C^3\), set
\[ U_t=\operatorname{diag}(1,e^{it},e^{2it}),\qquad \eta=3^{-1/2}(1,1,1), \]and let \(\rho=\omega_\eta\), while \(\lambda=\operatorname{Tr}/3\) is invariant under \(\operatorname{Ad}(U_t)\). For
\[ B=E_{12}+E_{21}-E_{13}-E_{31} \]we have
\[ \rho(B)=0=\lambda(B). \]Nevertheless,
\[ \rho(U_tBU_t^*) =\frac23(\cos t-\cos2t), \]so for \(L>0\),
\[ \frac1{2L}\int_{-L}^{L}\rho(U_tBU_t^*)\,dt =\frac23\left(\frac{\sin L}{L} -\frac{\sin2L}{2L}\right). \tag{26} \]At \(L=\pi/2\), this equals \(4/(3\pi)\), rather than the target value \(0\).
Thus even exact agreement on the tested observable at time zero can be lost under time averaging. The invariant target supplies the common value at every orbit point; the finite grid and marginal estimates ensure the normal approximation tracks those values throughout the interval before it is averaged.
Reference. M. Takesaki, Theory of Operator Algebras III, Chapter XVIII, Section 4, Lemma 4.13(i), printed pp.381–382. The argument above supplies explicit simultaneous control of normal marginals and orbit moments before the averaging step, constructs the relevant standard form directly on \(H\otimes H\), and proves the quantitative vector estimate (25). The normal product-state approximation in part (ii) is proved in QA06 under its bicentralizer-triviality input. The subsequent modular-automorphism theorem is proved in MI01–07, using the injective bicentralizer theorem and the local criterion.