Sources and hypotheses. The restricted-state identity in [bc:reduction] is (Houdayer and Isono 2017, Proposition 3.4); the type dichotomy and expectation are (Houdayer and Isono 2017, Theorem 3.5). The bicentralizer is globally invariant under the modular group by (Houdayer and Isono 2017, Proposition 3.3), so its canonical expectation preserves \(\phi\). These results impose no amenability or fullness hypothesis.

For [bc:flow], apply (Ando et al. 2020, Theorem A(ii)) to the identity inclusion \(M\subset M\). Continuity is part of the flow construction. The expectation-preservation identity in (Ando et al. 2020, Theorem A(iv)) specializes to preservation of \(\phi\), because the relative commutant is \(\mathbb C1\).

Finally, (Marrakchi 2020, Theorem D) states, for any \(\sigma\)-finite type \(\mathrm{III}_1\) factor, that every \(\beta^\phi_\lambda\) with \(\lambda\ne1\) is ergodic. This is stronger than [bc:ergodicity]. Fullness is not assumed in this Theorem. ◻

Proof of 1, assuming 2. Suppose that \(\mathop{\mathrm{BC}}(M,\phi)\) is nonscalar. Replace \((M,\phi)\) by \((B,\phi|_B)\) using 3[bc:reduction]. The restricted state is faithful and normal. The predual is still separable: if \(E:M\to B\) is the normal expectation, \(B_*\ni\rho\mapsto\rho\circ E\in M_*\) is an isometric embedding. We may therefore assume \[\mathop{\mathrm{BC}}(M,\phi)=M,\qquad M\ne\mathbb C1.\]

Every unitary \(u\in M_\phi\) centralizes \(\phi\) exactly. The unitary definition of the bicentralizer gives \(u^*au=a\) for every \(a\in M\). Since unitaries span \(M_\phi\), and \(M\) is a factor, it follows that \(M_\phi=\mathbb C1\).

Set \(b_s=\beta^\phi_{e^{-s}}\). The premise of (1) is equivalent to \[\norm{w_n\phi-e^{-s}\phi w_n}_{M_*}\longrightarrow0.\] 3[bc:flow] therefore gives exactly (1), and also continuity and state preservation. By 2, \(b\) is trivial. By 3[bc:ergodicity], it is ergodic. Hence \(M=\mathbb C1\), contradicting the reduction. The contradiction applies to every original pair \((M,\phi)\). ◻

Proof. Put \(c=e^{s/2}\) and \(\eta=\xi w-cw\xi\). By (2), \(\eta=(e^{D/2}-c)w\xi\), and \(J\eta=w^*\xi-c\xi w^*\). For \(\norm{y}\le1\), commutation of the left and right actions gives \[\langle y\xi,\xi w^*\rangle =\langle y(\xi w),\xi\rangle.\] Starting from \(\phi(wy)=\langle y\xi,w^*\xi\rangle\), replace \(w^*\xi\) by \(c\xi w^*\), and then \(\xi w\) by \(cw\xi\). The total error is at most \((1+c)\norm{\eta}\), while the resulting expression is \(c^2\phi(yw)\). Taking the supremum over \(y\) proves the estimate. The last assertion follows from \[\norm{(e^{D/2}-e^{s/2})w_n\xi} \le \sup_{\abs{u-s}\le d_n}\abs{e^{u/2}-e^{s/2}}\, \norm{w_n\xi}. \qedhere\] ◻

Proof. The inverse Fourier transform \(g\) is integrable. The bounded weak integral is defined by integration against normal functionals, and its operator norm has the stated bound. Applying it to \(\xi\) and using the spectral theorem gives \(y_f\xi=f(D)y\xi\). This uses no operator norm continuity of the orbit.

If \(K\) is a compact support for \(a\xi\), choose \(f\in C_c^\infty(\mathbb R)\) equal to \(e^{-u/2}\) on a neighborhood of \(K\), and put \(a^\flat=a_f\). Then (2) and functional calculus give \[\xi a^\flat=e^{D/2}a^\flat\xi =e^{D/2}f(D)a\xi=a\xi.\] All vectors in this computation belong to the domain of \(e^{D/2}\); in particular the filtered vector has compact spectral support. The Fourier argument for another state-preserving action is identical. ◻

Proof. The unitary mean ergodic theorem gives a norm limit for the vectors in (5). The corresponding operator averages have norm at most \(\norm{y}\). For each fixed \(u\), translating the interval changes their averages in operator norm by at most \(\abs{u}\norm{y}/k\) for all sufficiently large \(k\). Every weak operator cluster point is consequently fixed by \(\sigma\), hence scalar. Evaluation under \(\phi\) identifies it as \(\phi(y)1\). This identifies the Hilbert norm limit and proves (5).

The action on \(M_*\) is norm continuous on each functional. For positive normal functionals this follows from their realization as vector functionals in standard form and the strong continuity of \(U\); linear decomposition gives the general case. The predual averages are therefore Bochner integrals and are contractions. Let \[\mathcal N=\overline{\mathop{\mathrm{span}}}^{\,\norm{\cdot}} \{\rho\circ\sigma_u-\rho:\rho\in M_*,\ u\in\mathbb R\}.\] Its annihilator in \((M_*)^*=M\) is \(M_\phi=\mathbb C1\). The Hahn–Banach theorem implies \(\mathcal N=\{\psi\in M_*:\psi(1)=0\}\). The averages tend to zero in norm on each generator of \(\mathcal N\), by the interval boundary estimate, and hence on all of \(\mathcal N\) by contractivity. They fix \(\phi\). Apply this to \(\psi-\psi(1)\phi\) to obtain (6). ◻

Proof. The bounded sesquilinear form defines \(P(S)\), with \(\norm{P(S)}\le\norm{S}\). Positivity and unitality follow directly from the mean. Translation invariance gives \(U_u^*P(S)U_u=P(S)\) for every \(u\), hence commutation with the spectral projections of \(D\). For \(S=L(a)\) the averages and their weak limits belong to the weakly closed algebra \(L(M)\), since \(U_t^*L(a)U_t=L(\sigma_{-t}(a))\). The fixed-point algebra is scalar, and evaluation at \(\xi\) gives (7). Equation (8) is the definition applied to \(S^*S\). No normality of \(P\) is asserted or needed. ◻

Proof. We use the same family of Fourier filters to localize the initial bounded approximations and to restore localization after polar truncation. Choose \(f_0\in C_c^\infty(\mathbb R)\) such that \[0\le f_0\le1,\qquad f_0=1\text{ on }[-1,1],\qquad \mathop{\mathrm{supp}}f_0\subset[-2,2],\] and write \(f_0(u)=\int_\mathbb Rg_0(t)e^{itu}\,dt\), with \(g_0\) a Schwartz function. For \(\rho>0\), set \[f_\rho(u)=f_0\bigl((u-s)/\rho\bigr),\qquad g_\rho(t)=\rho e^{-ist}g_0(\rho t).\] Then \(f_\rho(u)=\int_\mathbb Rg_\rho(t)e^{itu}\,dt\) and \[ \norm{g_\rho}_1=\norm{g_0}_1=:C_1 \quad\text{for every }\rho>0. \tag{10}\] By 5, the weak integral \[\mathcal F_\rho(x) =\int_\mathbb Rg_\rho(t)\sigma_t(x)\,dt\in M\] satisfies \[ \mathcal F_\rho(x)\xi=f_\rho(D)x\xi, \qquad \norm{\mathcal F_\rho(x)}\le C_1\norm{x}. \tag{11}\] In particular, its vector has spectral support in \([s-2\rho,s+2\rho]\).

Approximation by bounded representatives. By [eq:recovery-positive], after passing to a subsequence and reindexing, there is \(\gamma>0\) such that \[m_t\norm{TU_th_n}^{2}\ge4\gamma\qquad(n\ge1).\] Since \(M\xi\) is dense in \(H\), approximate \(h_n\) by \(x_n\xi\) and apply \(f_{\delta_n}(D)\). This multiplier fixes \(h_n\) and is a contraction. Thus the vectors \(\mathcal F_{\delta_n}(x_n)\xi\) approximate \(h_n\) as closely as desired. Normalize them, obtaining \(y_n\in M\) with \(\norm{y_n\xi}=1\) and spectral support in \([s-d_n,s+d_n]\), where \(d_n=2\delta_n\). Choose the approximations so that \(\norm{y_n\xi-h_n}\to0\) and \[ m_t\norm{TU_ty_n\xi}^{2}\ge2\gamma\qquad(n\ge1). \tag{12}\] This is possible because the averaged seminorm in 7 gives \[\abs{\norm{t\mapsto TU_ty_n\xi}_{\mathrm{av}} -\norm{t\mapsto TU_th_n}_{\mathrm{av}}} \le\norm{T}\norm{y_n\xi-h_n}.\] Discard finitely many terms so that \(d_n\le1\). The modular identity in [eq:modular-identities] gives \[ \phi(y_n^*y_n)=1,\qquad \phi(y_ny_n^*)=\norm{e^{D/2}y_n\xi}^{2}\le e^{s+d_n}. \tag{13}\] There is no asserted uniform bound on \(\norm{y_n}\).

Simultaneous defect and moment bounds. Fix \(n\) for the moment, and write \(y=y_n\). With \(\mu_k\) the uniform probability measure on \([-k,k]\), define \[\begin{aligned} A_k&=\int\sigma_t(y^*y)\,d\mu_k(t),& B_k&=\int\sigma_t(yy^*)\,d\mu_k(t),\\ \psi(x)&=\phi(y^*xy),& \psi_k&=\int\psi\circ\sigma_{-t}\,d\mu_k(t). \end{aligned}\] We seek simultaneous control of \(\phi(A_k^2)\) and \(\psi_k(B_k)\), which will bound the fourth moment after discretization. Here \(\psi\) is a normal state. State invariance gives the exact identity \[ \psi_k(x)= \int\phi\bigl(\sigma_t(y)^*x\sigma_t(y)\bigr)\,d\mu_k(t). \tag{14}\] By the two ordinary norm limits in 6, \[A_k\xi\longrightarrow\xi,\qquad \norm{\psi_k-\phi}_{M_*}\longrightarrow0.\] Consequently \(\phi(A_k^2)=\norm{A_k\xi}^{2}\to1\). Although \(B_k\) depends on \(k\), it satisfies \[\norm{B_k}\le\norm{y}^{2},\qquad \phi(B_k)=\phi(yy^*)\le e^{s+1},\] and hence \[ \abs{\psi_k(B_k)-\phi(B_k)} \le\norm{\psi_k-\phi}_{M_*}\norm{y}^{2}\longrightarrow0. \tag{15}\] All these limits are taken with \(n\) fixed. For every sufficiently large \(k\) we may impose \(\phi(A_k^2)\le2\) and \(\norm{\psi_k-\phi}_{M_*}\le\norm{y}^{-2}\) simultaneously. It follows that \[ \phi(A_k^2)+\psi_k(B_k)\le3+e^{s+1}. \tag{16}\] On the other hand, [eq:recovery-defect] and the definition of \(m\) imply that the set of integers \(k\) for which \[\int\norm{T\sigma_t(y)\xi}^{2}\,d\mu_k(t)>\frac32\gamma\] belongs to the fixed free ultrafilter. It therefore meets the cofinite set on which [eq:recovery-continuous-moments] holds. Choose one such \(k\), still for this fixed \(n\).

We next replace this continuous average by a finite probability average. Put \(z_t=\sigma_t(y)\). The three scalar functions \[t\longmapsto\norm{Tz_t\xi}^{2},\qquad (t,u)\longmapsto\phi(z_t^*z_tz_u^*z_u),\qquad (t,u)\longmapsto\phi(z_t^*z_uz_u^*z_t)\] are continuous on the relevant compact interval or square. Indeed, \(t\mapsto z_t\) is bounded and strong-star continuous, and products on bounded sets preserve this continuity. A common sufficiently fine partition of \([-k,k]\), using its normalized interval lengths as weights, therefore approximates all three integrals. The product weights approximate the double integrals defining \(\phi(A_k^2)\) and \(\psi_k(B_k)\). Thus there are finitely many times \(t_i\) and weights \(p_i\ge0\), with \(\sum_i p_i=1\), such that, writing \[z_i=\sigma_{t_i}(y),\qquad A=\sum_i p_i z_i^*z_i,\qquad B=\sum_i p_i z_iz_i^*,\] we have \[ \sum_i p_i\norm{Tz_i\xi}^{2}\ge\gamma, \qquad \phi(A^2)+\sum_i p_i\phi(z_i^*Bz_i)\le C, \qquad C:=4+e^{s+1}. \tag{17}\] The finite averages may depend on \(n\); \(\gamma\) and \(C\) do not. In particular, no estimate on the rate in [eq:recovery-varying-test] uniform in \(n\) has been used.

Randomization and truncation. For this finite average, let \(\epsilon_i\) be independent random variables uniformly distributed on the complex unit circle, and set \[z=\sum_i\sqrt{p_i}\,\epsilon_i z_i.\] The vector \(z\xi\) is supported in \([s-d_n,s+d_n]\). Independence and vanishing first moments give \[\mathbb E\norm{Tz\xi}^{2} =\sum_i p_i\norm{Tz_i\xi}^{2}\ge\gamma.\] The following expansion is the Steinhaus variant of the fourth-moment identity in (Haagerup and Musat 2007, equation (2.10) and Remark 2.8), with coefficients \(\sqrt{p_i}z_i\). We include the expansion to keep track of the order of the noncommuting factors. The two surviving pairings give \[ \mathbb E[(z^*z)^2] =A^2+\sum_i p_i z_i^*Bz_i -\sum_i p_i^2(z_i^*z_i)^2. \tag{18}\] The last sum subtracts the overlap of the pairings, where all four indices agree. Its summands are positive, so [eq:recovery-finite] implies \[ \mathbb E\phi((z^*z)^2)\le C. \tag{19}\]

We next use the functional-calculus truncation underlying (Haagerup and Musat 2007, Lemma 2.6, equations (2.15)–(2.20)). For \(K>0\), write \(z=u\abs{z}\) in its polar decomposition and put \(z^{[K]}=u\min(\abs{z},K)\). Then \(\norm{z^{[K]}}\le K\) and \[ \begin{aligned} \mathbb E\norm{(z-z^{[K]})\xi}^{2} &=\mathbb E\phi\bigl((\abs{z}-K)_+^2\bigr)\\ &\le K^{-2}\mathbb E\phi(\abs{z}^{4}) \le CK^{-2}. \end{aligned} \tag{20}\] This controls the vector error on the side required for the bounded operator \(T\). No estimate for the adjoint truncation error is needed.

Restoring the spectral band and selecting realizations. Define the random element \(v=\mathcal F_{d_n}(z^{[K]})\in M\). By [eq:recovery-filter], every realization satisfies \[ \norm{v}\le C_1K, \qquad v\xi=f_{d_n}(D)z^{[K]}\xi, \tag{21}\] and \(v\xi\) is supported in \([s-2d_n,s+2d_n]\). Since \(f_{d_n}(D)\) fixes \(z\xi\) and has norm at most one, [eq:recovery-truncation] gives \[\mathbb E\norm{v\xi-z\xi}^{2}\le CK^{-2}.\] The triangle inequality in the Hilbert space of square-integrable random \(H\)-vectors now yields \[ \bigl(\mathbb E\norm{Tv\xi}^{2}\bigr)^{1/2} \ge\sqrt\gamma-\frac{\norm{T}\sqrt C}{K}. \tag{22}\] Fix once and for all \[K\ge2\norm{T}\sqrt{C/\gamma}.\] The hypothesis implies \(T\ne0\), so \(K\) can be chosen positive. This choice is independent of \(n\) and ensures that the left-hand side of [eq:recovery-fixed-cutoff] is at least \(\sqrt\gamma/2\).

To select a realization, note that \(z\) depends norm-continuously on the finite phase torus and \(z^{[K]}=zq_K(z^*z)\), where \[q_K(r)=\min\{1,K/\sqrt r\}\quad(r>0),\qquad q_K(0)=1.\] Since \(q_K\) is continuous on \([0,\infty)\), continuous functional calculus makes the cutoff norm-continuous. The filter \(\mathcal F_{d_n}\) is bounded and linear, so \(\norm{Tv\xi}^2\) is continuous on the compact phase torus. It therefore attains a maximum at least its expectation. Choose such a realization \(v_n\) for each \(n\). Then \[\norm{v_n}\le C_1K, \qquad \norm{Tv_n\xi}\ge\frac{\sqrt\gamma}{2},\] and its spectral support lies in \([s-2d_n,s+2d_n]=[s-4\delta_n,s+4\delta_n]\). Restoring the subsequence notation proves 8 with \(C_*=C_1K\) and \(\eta=\sqrt\gamma/2\). ◻