Strongly invariant states and approximate eigenstates

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

Let \(X\) be a self-adjoint operator on a Hilbert space \(H\) and \(\sigma_t=\operatorname{Ad}e^{itX}\) the induced action of \(\mathbb R\) on \(B(H)\). A normal state of \(B(H)\) that is invariant under \(\sigma\) is a mixture of eigenvectors of \(X\). Non-normal states can be invariant without any such decomposition, because their values on the averages \(\int\sigma_t(T)\,d\mu(t)\) need not be the averages of their values. Marrakchi singled out the states for which this exchange holds, the strongly invariant states, and proved that they are exactly the barycentres of approximate eigenstates of \(X\) [Marrakchi, Section 3]. He also showed how to produce strongly invariant states extending a given state on the \(C^*\)-algebra generated by a von Neumann algebra and its commutant. Both results are proved in this lesson; the last lesson applies them to the modular group of a state.

We use Stone's theorem and the facts on generators in Sections 3–4 of Holomorphy in Banach spaces, Stone's theorem and resolvent convergence; the spectral calculus of self-adjoint operators with its domains, Spectral calculus with its domains retained; the identification of the predual of \(B(H)\) with the trace class, Compact and trace-class operators, the predual of B(H), and the operator topologies; and the Hahn–Banach separation theorem and Mazur's theorem from Weak topologies, Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian. States are positive unital linear functionals; \(S(B(H))\) denotes the state space with the weak\(^*\) topology, which is compact.

1. Averages over probability measures

Let \((U_t)\) be a strongly continuous unitary group on \(H\) and \(\sigma_t(T)=U_tTU_t^*\). For a finite signed Borel measure \(\nu\) on \(\mathbb R\) and \(T\in B(H)\), the bounded sesquilinear form \((h,k)\mapsto\int\langle\sigma_t(T)h,k\rangle\,d\nu(t)\) defines \(\sigma_\nu(T)\in B(H)\), with \(\|\sigma_\nu(T)\|\le\|\nu\|\,\|T\|\). We write \(\mathrm{Prob}(\mathbb R)\) for the Borel probability measures and \(\sigma_f=\sigma_\nu\) when \(d\nu=f\,dt\) with \(f\in L^1(\mathbb R)\).

Lemma 1.1. Let \(\mu,\nu\) be finite signed measures and \(T\in B(H)\).

  1. \(\sigma_\mu\circ\sigma_\nu=\sigma_{\mu*\nu}=\sigma_\nu\circ\sigma_\mu\); in particular \(\sigma_\mu\circ\sigma_s=\sigma_{\mu*\delta_s}\).
  2. For every normal functional \(\omega\) on \(B(H)\), \(\omega(\sigma_\mu(T))=\int\omega(\sigma_t(T))\,d\mu(t)\).
  3. If \(f\in L^1(\mathbb R)\), then \(\|\sigma_s(\sigma_f(T))-\sigma_f(T)\|\) is at most \(\|f(\cdot-s)-f\|_1\|T\|\), so \(s\mapsto\sigma_s(\sigma_f(T))\) is norm continuous.
  4. If \(s\mapsto\sigma_s(T)\) is norm continuous, then \(\Psi(\sigma_\mu(T))=\int\Psi(\sigma_s(T))\,d\mu(s)\) for every bounded linear functional \(\Psi\) on \(B(H)\).

Proof. (1) Test on vectors and use Fubini for the bounded continuous functions \((t,s)\mapsto\langle\sigma_{t+s}(T)h,k\rangle\). (2) A normal functional is \(\sum_n\langle\,\cdot\,h_n,k_n\rangle\) with \(\sum_n\|h_n\|\|k_n\|<\infty\); apply the definition to each term and sum by dominated convergence. (3) By (1), \(\sigma_s(\sigma_f(T))=\sigma_{f(\cdot-s)}(T)\). (4) Fix \(n\). On \([-n,n)\) the function \(s\mapsto\sigma_s(T)\) is uniformly norm continuous. For a partition of \([-n,n)\) into intervals \(I_j\) of length less than \(\delta\), with points \(s_j\in I_j\), the operator \(\int_{[-n,n)}\sigma_s(T)\,d\mu-\sum_j\sigma_{s_j}(T)\mu(I_j)\) has norm at most the oscillation of \(s\mapsto\sigma_s(T)\) over distances \(\delta\), and the part of \(\sigma_\mu(T)\) outside \([-n,n)\) has norm at most \(\|T\|\mu(\{|s|\ge n\})\). Apply \(\Psi\), use the same estimates for the scalar function \(s\mapsto\Psi(\sigma_s(T))\), and let \(\delta\to0\), then \(n\to\infty\). \(\square\)

Definition 1.2. A state \(\Psi\) of \(B(H)\) is strongly \(\sigma\)-invariant if \(\Psi\circ\sigma_\mu=\Psi\) for every \(\mu\in\mathrm{Prob}(\mathbb R)\).

Taking \(\mu=\delta_t\) shows that a strongly invariant state is invariant. For a normal state the converse holds by Lemma 1.1(2), but not in general.

2. Strongly invariant nets in a Banach space

Lemma 2.1. Let \(\alpha\) be a strongly continuous group of isometries of a Banach space \(E\), with generator \(G\), and \((x_i)\) a bounded net in \(E\). For \(\mu\in\mathrm{Prob}(\mathbb R)\) let \(\alpha_\mu(x)=\int\alpha_t(x)\,d\mu(t)\). The following are equivalent:

  1. for every \(T>0\), \(\sup_{|t|\le T}\|\alpha_t(x_i)-x_i\|\to0\);
  2. for every \(\mu\in\mathrm{Prob}(\mathbb R)\), \(\|\alpha_\mu(x_i)-x_i\|\to0\);
  3. there are \(y_i\) in the domain of \(G\) with \(\|y_i-x_i\|\to0\) and \(\|Gy_i\|\to0\).

Proof. Let \(C=\sup_i\|x_i\|\). (1)\(\Rightarrow\)(2): \(\|\alpha_\mu(x_i)-x_i\|\) is at most the sum of \(\sup_{|t|\le T}\|\alpha_t(x_i)-x_i\|\) and \(2C\mu(\{|t|>T\})\); choose \(T\) large.

(2)\(\Rightarrow\)(3): Let \(g(t)=(2\pi)^{-1/2}e^{-t^2/2}\) and \(g_\varepsilon(t)=\varepsilon g(\varepsilon t)\). For \(f\in L^1\cap C^1\) with \(f'\in L^1\), \(\alpha_f(x)=\int f(t)\alpha_t(x)\,dt\) lies in the domain of \(G\) and \(G\alpha_f(x)=-\alpha_{f'}(x)\): indeed \(h^{-1}(\alpha_h\alpha_f(x)-\alpha_f(x))\) equals \(\int h^{-1}(f(t-h)-f(t))\alpha_t(x)\,dt\), and \(h^{-1}(f(\cdot-h)-f)\to-f'\) in \(L^1\) by continuity of translation in \(L^1\). Put \(y_i=\alpha_g(x_i)\), so \(\|y_i\|\le C\) and \(Gy_i=-\alpha_{g'}(x_i)\). Then \(\|y_i-x_i\|\to0\) by (2). For \(\varepsilon\in(0,1]\), since the averages commute, \[ \begin{aligned} &Gy_i-G\alpha_{g_\varepsilon}(y_i)=-\alpha_{g'}(x_i)\\ &\quad +\alpha_{g'}\big(\alpha_{g_\varepsilon}(x_i)\big)\\ &\quad =-\alpha_{g'}\big(x_i-\alpha_{g_\varepsilon}(x_i)\big). \end{aligned} \] So \(\|Gy_i-G\alpha_{g_\varepsilon}(y_i)\|\) is at most \(\|g'\|_1\|x_i-\alpha_{g_\varepsilon}(x_i)\|\), which tends to \(0\) by (2), while \(\|G\alpha_{g_\varepsilon}(y_i)\|=\|\alpha_{g_\varepsilon'}(y_i)\|\), which is at most \(\|g_\varepsilon'\|_1C=\varepsilon\|g'\|_1C\). Hence \(\limsup_i\|Gy_i\|\le\varepsilon\|g'\|_1C\) for every \(\varepsilon\).

(3)\(\Rightarrow\)(1): for \(y\) in the domain of \(G\), \(\alpha_t(y)-y=\int_0^t\alpha_s(Gy)\,ds\), so \(\|\alpha_t(y)-y\|\le|t|\|Gy\|\). Hence \(\sup_{|t|\le T}\|\alpha_t(x_i)-x_i\|\) is at most \(2\|x_i-y_i\|+T\|Gy_i\|\). \(\square\)

A net satisfying these conditions is called strongly \(\alpha\)-invariant.

3. Approximate eigenstates

From now on \(X\) is self-adjoint on \(H\ne0\), \(U_t=e^{itX}\) and \(\sigma_t=\operatorname{Ad}U_t\). For a unit vector \(h\) let \(\omega_h=\langle\,\cdot\,h,h\rangle\).

Definition 3.1. A state \(\Phi\) of \(B(H)\) is an approximate eigenstate of \(X\) if there are a net of unit vectors \(h_i\) in the domain of \(X\) and reals \(\ell_i\) with \(\|(X-\ell_i)h_i\|\to0\) and \(\omega_{h_i}\to\Phi\) weak\(^*\). The set of approximate eigenstates is denoted \(\mathcal E(X)\), and \(\overline{\operatorname{conv}}\,\mathcal E(X)\) its weak\(^*\)-closed convex hull.

\(\mathcal E(X)\) is nonempty: for \(\ell\) in the spectrum of \(X\), unit vectors \(h_n\) in the range of the spectral projection of \((\ell-\frac1n,\ell+\frac1n)\) satisfy \(\|(X-\ell)h_n\|\le\frac1n\), and every weak\(^*\) cluster point of \(\omega_{h_n}\) lies in \(\mathcal E(X)\).

Lemma 3.2. Let \(T=T^*\in B(H)\) with \(\Phi(T)\le0\) for every \(\Phi\in\mathcal E(X)\), and let \(\varepsilon>0\). There is \(\kappa>0\) with the following property. Let \(K\) be a Hilbert space, \(Y\) a self-adjoint operator on \(K\), \(W_s=U_s\otimes e^{-isY}\) on \(H\otimes K\), and \(A\) the self-adjoint generator of \(W\), so that \(W_s=e^{isA}\). Then \[ \begin{gathered} \langle(T\otimes1)\zeta,\zeta\rangle\le\kappa\|A\zeta\|^2+\varepsilon\|\zeta\|^2\\ (\zeta\in\operatorname{dom}A). \end{gathered} \]

Proof. Step 1. There is \(\kappa_0\) with \(\langle Th,h\rangle\le\kappa_0\|(X-\ell)h\|^2+\frac\varepsilon2\|h\|^2\) for all \(\ell\in\mathbb R\) and \(h\in\operatorname{dom}X\). Otherwise there are unit vectors \(h_n\in\operatorname{dom}X\) and \(\ell_n\) with \(\langle Th_n,h_n\rangle>n\|(X-\ell_n)h_n\|^2+\frac\varepsilon2\); then \(\|(X-\ell_n)h_n\|^2<\|T\|/n\), and a weak\(^*\) cluster point \(\Phi\) of \(\omega_{h_n}\) lies in \(\mathcal E(X)\) and has \(\Phi(T)\ge\varepsilon/2\), a contradiction.

Step 2. Let \((k_\beta)\) be an orthonormal basis of \(K\). Then \(H\otimes K=\bigoplus_\beta H\otimes k_\beta\) and \(U_s\otimes1\) acts as \(U_s\) on each summand, so its generator \(X\otimes1\) is the direct sum of copies of \(X\). For \(\zeta=\sum_\beta h_\beta\otimes k_\beta\) in its domain and every \(\ell\), Step 1 gives \(\langle(T\otimes1)\zeta,\zeta\rangle=\sum_\beta\langle Th_\beta,h_\beta\rangle\), which is at most \(\kappa_0\|((X-\ell)\otimes1)\zeta\|^2+\frac\varepsilon2\|\zeta\|^2\).

Step 3. Choose \(\eta>0\) with \(2\kappa_0\eta^2\le\frac\varepsilon2\), let \(I_m=[m\eta,(m+1)\eta)\) for \(m\in\mathbb Z\), \(E_Y\) the spectral measure of \(Y\), and \(P_m=1\otimes E_Y(I_m)\). The \(P_m\) are pairwise orthogonal with sum \(1\), they commute with \(T\otimes1\) and with every \(W_s\), hence with \(A\). On \(E_Y(I_m)K\) the operator \(Y\) restricts to a bounded \(Y_m\) with \(\|Y_m-m\eta\|\le\eta\). Let \(\zeta\in\operatorname{dom}A\) and \(\zeta_m=P_m\zeta\in\operatorname{dom}A\). Since \((U_s\otimes1)\zeta_m=W_s(1\otimes e^{isY_m})\zeta_m\), \[ \begin{aligned} &\frac{(U_s\otimes1)\zeta_m-\zeta_m}{is}\\ &\quad =W_s\frac{(1\otimes e^{isY_m})\zeta_m-\zeta_m}{is}\\ &\quad +\frac{W_s\zeta_m-\zeta_m}{is}\longrightarrow(1\otimes Y_m)\zeta_m\\ &\quad +A\zeta_m\quad(s\to0). \end{aligned} \] By Stone's theorem \(\zeta_m\in\operatorname{dom}(X\otimes1)\) and \((X\otimes1)\zeta_m=A\zeta_m+(1\otimes Y_m)\zeta_m\), so \(\|((X-m\eta)\otimes1)\zeta_m\|\le\|A\zeta_m\|+\eta\|\zeta_m\|\). Step 2 with \(\ell=m\eta\) gives \[ \begin{aligned} &\langle(T\otimes1)\zeta_m,\zeta_m\rangle\le2\kappa_0\|A\zeta_m\|^2\\ &\quad +(2\kappa_0\eta^2+\tfrac\varepsilon2)\|\zeta_m\|^2\\ &\quad \le2\kappa_0\|A\zeta_m\|^2+\varepsilon\|\zeta_m\|^2 . \end{aligned} \] Summing over \(m\), and using \(\langle(T\otimes1)\zeta,\zeta\rangle=\sum_m\langle(T\otimes1)\zeta_m,\zeta_m\rangle\) and \(\|A\zeta\|^2=\sum_m\|A\zeta_m\|^2\), gives the claim with \(\kappa=2\kappa_0\). \(\square\)

Lemma 3.3. For positive trace-class operators \(\rho,\rho'\) on \(H\), \(\|\rho^{1/2}-\rho'^{1/2}\|_2^2\le\|\rho-\rho'\|_1\).

Proof. Put \(a=\rho^{1/2}\), \(b=\rho'^{1/2}\), let \(P\) be the spectral projection of \(a-b\) for \([0,\infty)\) and \(Q=1-P\), so \(|a-b|=(P-Q)(a-b)\), which is \((a-b)(P-Q)\). Since \(a^2-b^2\) is half of \((a-b)(a+b)+(a+b)(a-b)\) and the trace is cyclic, \[ \begin{aligned} &\|a^2-b^2\|_1\ge\operatorname{Tr}\big((P-Q)(a^2-b^2)\big)\\ &\quad =\operatorname{Tr}\big(|a-b|(a+b)\big)\\ &\quad =\operatorname{Tr}\big(|a-b|P(a+b)P\big)\\ &\quad +\operatorname{Tr}\big(|a-b|Q(a+b)Q\big). \end{aligned} \] On the range of \(P\), \(P(a+b)P\ge P(a-b)P=P|a-b|P\); on the range of \(Q\), \(Q(a+b)Q\ge Q(b-a)Q=Q|a-b|Q\). Since \(\operatorname{Tr}(c^{1/2}dc^{1/2})\ge0\) for \(c,d\ge0\), the right side is at least \(\operatorname{Tr}(|a-b|^2)=\|a-b\|_2^2\). \(\square\)

Lemma 3.4. Let \(\Psi\) be a strongly \(\sigma\)-invariant state. There is a net of normal states \(\Psi_i\) with \(\Psi_i\to\Psi\) weak\(^*\) and \(\|\Psi_i\circ\sigma_\mu-\Psi_i\|\to0\) for every \(\mu\in\mathrm{Prob}(\mathbb R)\).

Proof. The normal states are weak\(^*\) dense in \(S(B(H))\): if a state \(\Psi_0\) lay outside the weak\(^*\) closure of the convex set of normal states, the separation theorem would give \(T=T^*\) with \(\Psi_0(T)>\sup_{\|h\|=1}\langle Th,h\rangle\), whereas \(T\le(\sup_{\|h\|=1}\langle Th,h\rangle)1\) gives the reverse inequality. Index the net by triples \(i=(\mathcal K,\mathcal F,\varepsilon)\) with \(\mathcal K\subset B(H)\) and \(\mathcal F\subset\mathrm{Prob}(\mathbb R)\) finite and \(\varepsilon>0\). Let \(W_i\subset(B(H)_*)^{\mathcal F}\) be the set of the families \((\psi-\psi\circ\sigma_\mu)_{\mu\in\mathcal F}\) with \(\psi\) a normal state satisfying \(|\psi(T)-\Psi(T)|\le\varepsilon\) for \(T\in\mathcal K\), a convex set. Take normal states \(\psi_\alpha\to\Psi\) weak\(^*\). For each \(T\), \(\psi_\alpha(\sigma_\mu(T))\to\Psi(\sigma_\mu(T))=\Psi(T)\), so \((\psi_\alpha-\psi_\alpha\circ\sigma_\mu)_\mu\to0\) in the weak topology of the Banach space \((B(H)_*)^{\mathcal F}\), whose dual is \(B(H)^{\mathcal F}\); and these points lie in \(W_i\) eventually. By Mazur's theorem \(0\) is in the norm closure of \(W_i\): there is a normal state \(\Psi_i\) with \(\|\Psi_i-\Psi_i\circ\sigma_\mu\|\le\varepsilon\) for \(\mu\in\mathcal F\) and \(|\Psi_i(T)-\Psi(T)|\le\varepsilon\) for \(T\in\mathcal K\). \(\square\)

Theorem 3.5 ([Marrakchi, Theorem 3.2]). Every strongly \(\sigma\)-invariant state of \(B(H)\) lies in \(\overline{\operatorname{conv}}\,\mathcal E(X)\). Conversely, every state in \(\overline{\operatorname{conv}}\,\mathcal E(X)\) is strongly \(\sigma\)-invariant.

Proof. Let \(\Psi\) be strongly invariant and suppose \(\Psi\notin\overline{\operatorname{conv}}\,\mathcal E(X)\). The weak\(^*\) topology is locally convex with dual \(B(H)\), so the separation theorem gives \(T\) and a real \(c\) with \(\operatorname{Re}\Psi(T)>c\ge\operatorname{Re}\Phi(T)\) for all \(\Phi\in\mathcal E(X)\). States satisfy \(\Phi(T^*)=\overline{\Phi(T)}\), so replacing \(T\) by \(\frac12(T+T^*)-c1\) we may assume \(T=T^*\), \(\Psi(T)>0\) and \(\Phi(T)\le0\) on \(\mathcal E(X)\). Fix \(0<\varepsilon<\Psi(T)\).

Let \(\Psi_i\) be the net of Lemma 3.4, with densities \(\rho_i\), and \(Z_i=\rho_i^{1/2}\), a unit vector of the Hilbert–Schmidt space \(\mathrm{HS}(H)\). The group \(\psi\mapsto\psi\circ\sigma_t\) on \(B(H)_*\) is a strongly continuous group of isometries (approximate \(\rho\) by finite-rank operators), and \(\psi\circ\sigma_\mu\) is its average over \(\mu\) by Lemma 1.1(2). By Lemma 2.1, \(\sup_{|t|\le T_0}\|\Psi_i\circ\sigma_t-\Psi_i\|\to0\) for every \(T_0\). The density of \(\Psi_i\circ\sigma_t\) is \(U_t^*\rho_iU_t\), with square root \(V_tZ_i\), where \(V_t(Z)=U_t^*ZU_t\) is a strongly continuous unitary group on \(\mathrm{HS}(H)\). By Lemma 3.3, \(\|V_tZ_i-Z_i\|_2^2\le\|\Psi_i\circ\sigma_t-\Psi_i\|\), so \((Z_i)\) is strongly \(V\)-invariant, and Lemma 2.1 gives \(Z_i'\) in the domain of the generator of \(V\) with \(\|Z_i'-Z_i\|_2\to0\) and the generator applied to \(Z_i'\) tending to \(0\).

Let \(C\colon H\to\overline H\) be the canonical antiunitary and \(\Theta\colon H\otimes\overline H\to\mathrm{HS}(H)\) the unitary with \(\Theta(h\otimes Ck)=\langle\,\cdot\,,k\rangle h\). Then \(\Theta^{-1}(TZ)=(T\otimes1)\Theta^{-1}Z\), and \(\Theta^{-1}V_t\Theta=U_{-t}\otimes CU_{-t}C^{-1}\). By Stone's theorem \(CU_sC^{-1}=e^{-isY}\) for a self-adjoint \(Y\) on \(\overline H\), so \(\Theta^{-1}V_t\Theta=W_{-t}\) with \(W\) as in Lemma 3.2, and the generator of \(V\) corresponds to \(-A\). With \(\zeta_i=\Theta^{-1}Z_i'\) we get \(\|A\zeta_i\|\to0\), and by Lemma 3.2, \[ \begin{aligned} &\Psi(T)=\lim_i\operatorname{Tr}(\rho_iT)=\lim_i\langle TZ_i,Z_i\rangle_{\mathrm{HS}}\\ &\quad =\lim_i\langle(T\otimes1)\zeta_i,\zeta_i\rangle\\ &\quad \le\limsup_i\big(\kappa\|A\zeta_i\|^2+\varepsilon\|\zeta_i\|^2\big)=\varepsilon, \end{aligned} \] a contradiction.

Conversely, the strongly invariant states form a weak\(^*\)-closed convex set, so it suffices to treat \(\Phi\in\mathcal E(X)\), with \(h_i\), \(\ell_i\) as in Definition 3.1. Since \(|e^{itx}-e^{it\ell}|\le|t|\,|x-\ell|\), spectral calculus gives \(\|(U_t-e^{it\ell_i})h_i\|\le|t|\,\|(X-\ell_i)h_i\|\), and so \(\|\omega_{h_i}\circ\sigma_t-\omega_{h_i}\|\le2|t|\,\|(X-\ell_i)h_i\|\), because \(\omega_{h_i}\circ\sigma_t\) is the vector state of \(U_t^*h_i\) and phases do not change vector states. Lemma 2.1 makes \((\omega_{h_i})\) strongly invariant in the sense of (2), and then \(\Phi(\sigma_\mu(T))=\lim_i\omega_{h_i}(\sigma_\mu(T))\), which is \(\lim_i\omega_{h_i}(T)=\Phi(T)\). \(\square\)

4. Strongly invariant extensions

Proposition 4.1 ([Marrakchi, Proposition 3.10]). Let \(\mathcal K\subset S(B(H))\) be nonempty, weak\(^*\)-closed, and such that \(\Psi\circ\sigma_\mu\in\mathcal K\) for all \(\Psi\in\mathcal K\) and \(\mu\in\mathrm{Prob}(\mathbb R)\). Then \(\mathcal K\) contains a strongly \(\sigma\)-invariant state.

Proof. Take \(\Psi_0\in\mathcal K\), let \(\mu_n\) be the uniform probability measure on \([0,n]\), and \(\Psi_n=\Psi_0\circ\sigma_{\mu_n}\in\mathcal K\). For every \(s\), \(\|\mu_n*\delta_s-\mu_n\|\le2|s|/n\), so by Lemma 1.1(1), \(|\Psi_n(\sigma_s(T))-\Psi_n(T)|\le\frac{2|s|}n\|T\|\). A weak\(^*\) cluster point \(\Psi\in\mathcal K\) of \((\Psi_n)\) is therefore \(\sigma\)-invariant. Let \(f=1_{[0,1]}\) and \(\Psi'=\Psi\circ\sigma_f\in\mathcal K\). For \(T\in B(H)\), \(T'=\sigma_f(T)\) has a norm continuous orbit (Lemma 1.1(3)), so Lemma 1.1(4) and the invariance of \(\Psi\) give \(\Psi(\sigma_\mu(T'))=\int\Psi(\sigma_s(T'))\,d\mu(s)\), which is \(\Psi(T')\). By Lemma 1.1(1), \[ \begin{aligned} &\Psi'(\sigma_\mu(T))=\Psi(\sigma_\mu(\sigma_f(T)))\\ &\quad =\Psi(\sigma_\mu(T'))=\Psi(T')=\Psi'(T). \end{aligned} \] \(\square\)

Lemma 4.2 ([Marrakchi, Lemma 4.1]). Let \(M\subset B(H)\) be a von Neumann algebra with \(U_tMU_t^*=M\) for all \(t\), and \(C\) the \(C^*\)-algebra generated by \(M\) and \(M'\). Let \(\Phi\) be a state of \(C\) with \(\Phi\circ\sigma_t=\Phi\) on \(C\) for all \(t\), whose restrictions to \(M\) and to \(M'\) are normal. Then some strongly \(\sigma\)-invariant state of \(B(H)\) extends \(\Phi\).

Proof. \(U_tM'U_t^*=M'\) as well, so \(\sigma_t(C)=C\). Let \(\mathcal K\) be the set of states of \(B(H)\) extending \(\Phi\); it is nonempty by the Hahn–Banach theorem and weak\(^*\)-closed. By Proposition 4.1 it suffices to show \(\Psi'=\Psi\circ\sigma_\mu\in\mathcal K\) for \(\Psi\in\mathcal K\) and \(\mu\in\mathrm{Prob}(\mathbb R)\).

For \(x\in M\), \(\sigma_\mu(x)\in M\), and as \(\Phi|_M\) is normal, Lemma 1.1(2) gives \(\Psi'(x)=\Phi(\sigma_\mu(x))\), which is \(\int\Phi(\sigma_t(x))\,d\mu=\Phi(x)\); likewise on \(M'\). Let \(\mathcal A\subset M\) and \(\mathcal B\subset M'\) be the sets of elements with norm continuous orbits. For \(a\in\mathcal A\), \(b\in\mathcal B\), the orbit of \(ab\) is norm continuous and lies in \(C\), so Lemma 1.1(4) gives \(\Psi'(ab)=\int\Psi(\sigma_t(ab))\,d\mu\), which is \(\int\Phi(\sigma_t(ab))\,d\mu=\Phi(ab)\).

For \(x\in M\), \(x_n=\sigma_{f_n}(x)\) with \(f_n=n1_{[0,1/n]}\) lies in \(\mathcal A\) (Lemma 1.1(3)), \(\|x_n\|\le\|x\|\), and \(x_n\to x\) \(*\)-strongly, because \(t\mapsto U_txU_t^*h\) is continuous; similarly for \(M'\). Let \(a\in M\), \(b\in M'\) with such approximants \(a_n\in\mathcal A\), \(b_n\in\mathcal B\). For a state \(\chi\) of \(C\) whose restrictions to \(M\) and \(M'\) are normal, put \(\|z\|_\chi=\chi(z^*z)^{1/2}\). Then \(\|a(b-b_n)\|_\chi\le\|a\|\|b-b_n\|_\chi\to0\), and, since \((a-a_n)^*(a-a_n)\in M\) commutes with \(b_n^*b_n\in M'\), \[ \begin{aligned} &\|(a-a_n)b_n\|_\chi^2\\ &\quad =\chi\big((a-a_n)^*(a-a_n)\,b_n^*b_n\big)\\ &\quad \le\|b_n\|^2\chi\big((a-a_n)^*(a-a_n)\big)\to0 . \end{aligned} \] So \(\chi(a_nb_n)\to\chi(ab)\) by the Cauchy–Schwarz inequality. Applied to \(\chi=\Phi\) and \(\chi=\Psi'|_C\), this gives \(\Psi'(ab)=\Phi(ab)\). The products \(ab\) span a dense \(*\)-subalgebra of \(C\), so \(\Psi'|_C=\Phi\). \(\square\)

5. Exercises

Exercise 5.1. Show that the vector state of an eigenvector of \(X\) is strongly invariant, directly from Definition 1.2.

Exercise 5.2. Let \(X=0\). Show that every state of \(B(H)\) is strongly invariant, and that \(\overline{\operatorname{conv}}\,\mathcal E(0)=S(B(H))\).

Exercise 5.3. Show that the set of strongly \(\sigma\)-invariant states is weak\(^*\)-closed and convex.

Exercise 5.4. Let \(H=\mathbb C^n\) and \(X=\operatorname{diag}(x_1,\ldots,x_n)\) with distinct real \(x_k\), and let \(e_1,\ldots,e_n\) be the standard basis. Show that \(\mathcal E(X)=\{\omega_{e_1},\ldots,\omega_{e_n}\}\), and that the strongly invariant states are the states \(\operatorname{Tr}(\rho\,\cdot\,)\) with \(\rho\) diagonal. Check Theorem 3.5 in this case.

6. Solutions

5.1. If \(Xh=\ell h\), then \(U_t^*h=e^{-it\ell}h\), so \(\omega_h(\sigma_t(T))=\langle TU_t^*h,U_t^*h\rangle=\omega_h(T)\) for all \(t\), and \(\omega_h(\sigma_\mu(T))=\int\omega_h(\sigma_t(T))\,d\mu=\omega_h(T)\) by Lemma 1.1(2).

5.2. \(\sigma_t=\mathrm{id}\), so \(\sigma_\mu=\mathrm{id}\). Every unit vector \(h\) satisfies \(\|(X-0)h\|=0\), so all vector states lie in \(\mathcal E(0)\); their closed convex hull contains the normal states, which are weak\(^*\) dense (proof of Lemma 3.4).

5.3. For each \(\mu\) and \(T\), \(\Psi\mapsto\Psi(\sigma_\mu(T))-\Psi(T)\) is affine and weak\(^*\) continuous; the strongly invariant states are the common zeros.

5.4. Let \(d>0\) be the smallest distance between two of the \(x_k\). For a unit vector \(h=\sum_kc_ke_k\) and a real \(\ell\), \(\|(X-\ell)h\|^2=\sum_k|x_k-\ell|^2|c_k|^2\). If this tends to \(0\) along a net \((h_i,\ell_i)\), then \(\ell_i\) is eventually within \(d/2\) of a single \(x_k\) along a subnet, and the coefficients of \(h_i\) off \(e_k\) tend to \(0\), because \(|x_j-\ell_i|\ge d/2\) for \(j\ne k\); so \(\omega_{h_i}\to\omega_{e_k}\). Conversely \(\omega_{e_k}\in\mathcal E(X)\). Every state of \(B(\mathbb C^n)\) is normal, so strong invariance is invariance under \(\operatorname{Ad}e^{itX}\), that is \(e^{-itX}\rho e^{itX}=\rho\) for all \(t\); since the \(x_k\) are distinct, this says that \(\rho\) is diagonal. The diagonal density matrices are exactly the convex combinations of the \(\omega_{e_k}\), as Theorem 3.5 predicts.

References