Introduction
For a classical Markov–Feller semigroup ( P t ) t ≥ 0 (P_t)_{t\geq0} ( P t ) t ≥ 0 X X X e e e { P t f : t ≥ 0 } \{P_t f:t\geq0\} { P t f : t ≥ 0 } f f f e e e [15 , 16 , 18 , 17 , 30 , 29 , 35 , 37 , 45 , 46 ] . The classical e e e
The quantum framework has two layers. A quantum Markov semigroup acts normally and weak-∗ ^* ∗ M \mathcal{M} M C ∗ C^* C ∗ A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M
RVO: 67985840.
The second-named author acknowledges with thanks support from NCN SONATA BIS 13 (2023/50/E/ST1/00067). whereas the Feller algebra A \mathcal{A} A ∗ * ∗ e e e C ∗ C^* C ∗ [14 , 9 ] .
For a = a ∗ ∈ A a=a^* \in\mathcal{A} a = a ∗ ∈ A
S ( A ) ∋ φ ⟼ φ ( T t ( a ) ) ∈ R , t ≥ 0. S(\mathcal{A}) \ni\varphi\longmapsto\varphi(\mathcal{T}_t(a)) \in\mathbb{R}, \qquad t \ge0. S ( A ) ∋ φ ⟼ φ ( T t ( a )) ∈ R , t ≥ 0. Their weak-∗ * ∗ e e e
Theorem A. For every a = a ∗ ∈ A a=a^* \in\mathcal{A} a = a ∗ ∈ A ε T ( a ) \varepsilon_{\mathcal{T}}(a) ε T ( a ) χ T ( a ) \chi_{\mathcal{T}}(a) χ T ( a )
1 2 ε T ( a ) ≤ χ T ( a ) ≤ ε T ( a ) . \frac{1}{2}\varepsilon_{\mathcal{T}}(a) \le\chi_{\mathcal{T}}(a) \le\varepsilon_{\mathcal{T}}(a). 2 1 ε T ( a ) ≤ χ T ( a ) ≤ ε T ( a ) . Both quantities are unchanged by deleting a bounded initial time interval and are Lipschitz under perturbations of a a a e e e
The factor two is optimal for bounded families on compact spaces. We also study Cesàro averages: the ordinary e e e e e e
The second theme is global mixing. The classical Doeblin condition dominates one time step by a reference probability. Here the minorising map itself may already contract the state space.
Theorem B. Let T : A → A T:\mathcal{A}\to\mathcal{A} T : A → A
T ( b ) ≥ λ Ψ ( b ) ( b ∈ A + ) , T(b) \ge\lambda\Psi(b) \qquad(b \in\mathcal{A}_+), T ( b ) ≥ λ Ψ ( b ) ( b ∈ A + ) , where Ψ \Psi Ψ λ ∈ ( 0 , 1 ] \lambda\in(0,1] λ ∈ ( 0 , 1 ]
∥ φ ∘ Ψ − ψ ∘ Ψ ∥ ≤ β ∥ φ − ψ ∥ ( φ , ψ ∈ S ( A ) ) \lVert\varphi\circ\Psi-\psi\circ\Psi\rVert\le\beta\lVert\varphi-\psi\rVert\qquad(\varphi,\psi\in S(\mathcal{A})) ∥ φ ∘ Ψ − ψ ∘ Ψ ∥ ≤ β ∥ φ − ψ ∥ ( φ , ψ ∈ S ( A )) for some β < 1 \beta<1 β < 1 T T T 1 − λ ( 1 − β ) 1-\lambda(1-\beta) 1 − λ ( 1 − β ) e e e
Positive return profiles, composable minorizations, positivity-improving finite-dimensional channels and measure-prepare maps are used to produce rank-one lower bounds. This global theory is distinct from the local corner mechanism developed in the companion paper [32 ] .
When every observable has the e e e [28 , 2 , 44 ] ; we use it as the structural meaning of the e e e
Theorem C. If every orbit in A \mathcal{A} A P P P
A = P ( A ) ⊕ ker P , ∥ T t ( a ) ∥ ⟶ 0 ( a ∈ ker P ) , \mathcal{A}=P(\mathcal{A})\oplus\ker P,\qquad\lVert T_t(a)\rVert\longrightarrow0\quad(a\in\ker P), A = P ( A ) ⊕ ker P , ∥ T t ( a )∥ ⟶ 0 ( a ∈ ker P ) , and the point-norm closure of the restricted dynamics on P ( A ) P(\mathcal{A}) P ( A ) C ∗ C^* C ∗ C ∗ C^* C ∗ P ( A ) ∩ MD ( P ) P(\mathcal{A})\cap\operatorname{MD}(P) P ( A ) ∩ MD ( P )
We give norm spectral synthesis, describe invariant states and provide infinite-dimensional examples where the Choi–Effros and ambient products differ.
Theorem D. Let E : A → B E:\mathcal{A}\to\mathcal{B} E : A → B C ∗ C^* C ∗
η ( a ) = lim sup t → ∞ ∥ T t ( a − E ( a ) ) ∥ . \eta(a)=\limsup_{t\to\infty}\lVert T_t(a-E(a))\rVert. η ( a ) = t → ∞ lim sup ∥ T t ( a − E ( a ))∥ . Then the orbit non-compactness measures of a a a E ( a ) E(a) E ( a ) η ( a ) \eta(a) η ( a ) e e e 2 η ( a ) 2\eta(a) 2 η ( a ) η ( a ) = 0 \eta(a)=0 η ( a ) = 0
This yields lifting results for Cartan masas and finite-dimensional spectral gaps. It also treats decoherence-free subalgebras. Throughout the paper, the e e e
The von Neumann–Feller framework
Definition 2.1. Let M \mathcal{M} M quantum Markov semigroup on M \mathcal{M} M ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0 M \mathcal{M} M T 0 = id \mathcal{T}_0=\operatorname{id} T 0 = id T s + t = T s T t \mathcal{T}_{s+t}=\mathcal{T}_s\mathcal{T}_t T s + t = T s T t s , t ≥ 0 s,t\geq0 s , t ≥ 0 x ∈ M x\in\mathcal{M} x ∈ M ω ∈ M ∗ \omega\in\mathcal{M}_* ω ∈ M ∗ t ↦ ω ( T t ( x ) ) t\mapsto\omega(\mathcal{T}_t(x)) t ↦ ω ( T t ( x )) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
Definition 2.3. Let ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M C ∗ C^* C ∗ A \mathcal{A} A Feller algebra for ( T t ) (\mathcal{T}_t) ( T t ) T t ( A ) ⊆ A \mathcal{T}_t(\mathcal{A})\subseteq\mathcal{A} T t ( A ) ⊆ A t ≥ 0 t\geq0 t ≥ 0 a ∈ A a\in\mathcal{A} a ∈ A t ↦ T t ( a ) t\mapsto\mathcal{T}_t(a) t ↦ T t ( a ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) A \mathcal{A} A ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0 quantum Markov–Feller semigroup on A \mathcal{A} A
compactness underlying the e -property live on A \mathcal{A} A C ∗ C^* C ∗ A \mathcal{A} A
Remark 2.5. We keep A \mathcal{A} A C 0 ( X ) C_0(X) C 0 ( X ) M ( C 0 ( X ) ) = C b ( X ) M(C_0(X))=C_b(X) M ( C 0 ( X )) = C b ( X ) C 0 C_0 C 0 M ( A ) M(\mathcal{A}) M ( A )
Let S ( A ) \mathcal{S}(\mathcal{A}) S ( A ) A \mathcal{A} A ∗ * ∗ T t ∗ : A ∗ → A ∗ T_t^*:\mathcal{A}^*\to\mathcal{A}^* T t ∗ : A ∗ → A ∗ T t ∣ A T_t|_{\mathcal{A}} T t ∣ A T t ∗ ( φ ) = φ ∘ T t T_t^*(\varphi)=\varphi\circ T_t T t ∗ ( φ ) = φ ∘ T t T t , ∗ T_{t,*} T t , ∗ M ∗ \mathcal{M}_* M ∗ L ⊆ A s a \mathcal{L}\subseteq\mathcal{A}_{\mathrm{sa}} L ⊆ A sa L \mathcal{L} L
Definition 2.6. Let ( T t ) t ≥ 0 (T_t)_{t\geq0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M
(a) We say that ( T t ) (T_t) ( T t ) e -property at φ ∈ S ( A ) \varphi\in\mathcal{S}(\mathcal{A}) φ ∈ S ( A ) L \mathcal{L} L a ∈ L a\in\mathcal{L} a ∈ L ε > 0 \varepsilon>0 ε > 0 ∗ * ∗ U U U φ \varphi φ S ( A ) \mathcal{S}(\mathcal{A}) S ( A )
sup ψ ∈ U t ≥ 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ < ε . \sup_{\substack{\psi\in U\\ t\geq0}}\left|\psi(T_t(a))-\varphi(T_t(a))\right|<\varepsilon. ψ ∈ U t ≥ 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ < ε . (b) We say that ( T t ) (T_t) ( T t ) eventual e -property at φ ∈ S ( A ) \varphi\in\mathcal{S}(\mathcal{A}) φ ∈ S ( A ) L \mathcal{L} L a ∈ L a\in\mathcal{L} a ∈ L ε > 0 \varepsilon>0 ε > 0 ∗ * ∗ U U U φ \varphi φ S ( A ) \mathcal{S}(\mathcal{A}) S ( A ) t 0 ≥ 0 t_0\geq0 t 0 ≥ 0
sup ψ ∈ U t ≥ t 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ < ε . \sup_{\substack{\psi\in U\\ t\geq t_0}}\left|\psi(T_t(a))-\varphi(T_t(a))\right|<\varepsilon. ψ ∈ U t ≥ t 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ < ε . If the corresponding property holds at every φ ∈ S ( A ) \varphi\in\mathcal{S}(\mathcal{A}) φ ∈ S ( A ) ( T t ) (T_t) ( T t ) e -property, respectively the eventual e -property, on L \mathcal{L} L
Remark 2.7. The dual maps T t ∗ T_t^* T t ∗ A ∗ \mathcal{A}^* A ∗ t ≥ 0 t\geq0 t ≥ 0 φ , ψ ∈ S ( A ) \varphi,\psi\in\mathcal{S}(\mathcal{A}) φ , ψ ∈ S ( A )
∥ T t ∗ ( φ ) − T t ∗ ( ψ ) ∥ = ∥ φ ∘ T t − ψ ∘ T t ∥ ≤ ∥ φ − ψ ∥ . \left\|T_t^*(\varphi)-T_t^*(\psi)\right\|=\left\|\varphi\circ T_t-\psi\circ T_t\right\|\leq\left\|\varphi-\psi\right\|. ∥ T t ∗ ( φ ) − T t ∗ ( ψ ) ∥ = ∥ φ ∘ T t − ψ ∘ T t ∥ ≤ ∥ φ − ψ ∥ . Consequently the family of maps
S ( A ) ∋ φ ⟼ φ ∘ T t ∈ S ( A ) \mathcal{S}(\mathcal{A})\ni\varphi\longmapsto\varphi\circ T_t\in\mathcal{S}(\mathcal{A}) S ( A ) ∋ φ ⟼ φ ∘ T t ∈ S ( A ) is always uniformly equicontinuous with respect to the norm topology on the state space. This is not the e -property. In infinite dimensions the norm topology on S ( A ) \mathcal{S}(\mathcal{A}) S ( A ) e -property, and our non-commutative formulation, concern instead the much coarser weak-∗ * ∗
Remark 2.8. The preceding remark also explains why trace-norm or fidelity-based metrics do not by themselves capture the phenomenon studied here. On normal states of a von Neumann algebra, the predual maps of a quantum Markov semigroup are contractions in trace norm. In finite dimensions, and more generally in the usual normal-state settings where fidelity is available, the data-processing inequality gives
F ( φ ∘ T t , ψ ∘ T t ) ≥ F ( φ , ψ ) , F(\varphi\circ T_t,\psi\circ T_t) \ge F(\varphi,\psi), F ( φ ∘ T t , ψ ∘ T t ) ≥ F ( φ , ψ ) , and hence the corresponding Bures distance is non-increasing under the dual dynamics. Thus trace distance and Bures distance give automatic global equicontinuity for every quantum Markov semigroup. The non-trivial issue is instead equicontinuity for the weak-∗ \ast ∗ ∗ \ast ∗
Remark 2.9. The classical literature distinguishes several eventual or averaged equicontinuity notions; see, for example, [29 , 16 , 35 , 37 ] . Besides the ordinary e e e e e e
lim x → z sup t ≥ τ ∣ P t f ( x ) − P t f ( z ) ∣ = 0 ; \lim_{x\to z}\sup_{t\geq\tau}|P_t f(x)-P_t f(z)|=0; x → z lim t ≥ τ sup ∣ P t f ( x ) − P t f ( z ) ∣ = 0 ; (ii) the weaker ε \varepsilon ε dependent eventual e e e , where the threshold time may depend on the tolerance ε \varepsilon ε
(iii) eventual continuity , where one only asks that the large-time tails become equicontinuous in the limit, without requiring one fixed threshold time to work uniformly;
(iv) the Cesàro e e e , obtained by replacing the orbit by its time averages.
Our definition is the version in (ii): for each observable and each tolerance ε \varepsilon ε ∗ \ast ∗ t ≥ t 0 ( ε ) t\geq t_0(\varepsilon) t ≥ t 0 ( ε )
The relations most relevant for the present paper may be summarised schematically as
fixed-threshold eventual e ⟹ eventual e used here ⟹ eventual continuity , \text{fixed-threshold eventual }e \Longrightarrow\text{ eventual }e\text{ used here} \Longrightarrow\text{ eventual continuity}, fixed-threshold eventual e ⟹ eventual e used here ⟹ eventual continuity , eventual e used here → strong stochastic continuity e ⟹ Ces a ˋ ro e . \text{eventual }e\text{ used here} \xrightarrow{\text{strong stochastic continuity}} e \Longrightarrow\text{ Cesàro }e. eventual e used here strong stochastic continuity e ⟹ Ces a ˋ ro e . The second implication is classical for strongly stochastically continuous Markov–Feller semigroups [35 ] ; in the present non-commutative framework it is replaced by Theorem 3.2 , where norm continuity on the Feller algebra plays the corresponding rôle. For Markov operators, several equicontinuity notions become equivalent under additional Schur-like hypotheses on spaces of measures [29 ] , while the relation with eventual continuity is analysed further in [37 ] . In general the reverse arrows fail, as shown by examples in [30 , 16 , 35 ] . Historically, the terminology is also connected with the theory of e e e [38 ] , although the present paper follows the more recent e e e
Remark 2.10. This definition is deliberately formulated on the full state space of A \mathcal{A} A M \mathcal{M} M
Remark 2.11. Let X X X A = C b ( X , C ) \mathcal{A}=C_b(X,\mathbb{C}) A = C b ( X , C ) L = Lip b ( X , R ) \mathcal{L}=\operatorname{Lip}_b(X,\mathbb{R}) L = Lip b ( X , R ) ( P t ) t ≥ 0 (P_t)_{t\ge0} ( P t ) t ≥ 0 x ∈ X x\in X x ∈ X δ x ∈ S ( A ) \delta_x\in S(\mathcal{A}) δ x ∈ S ( A )
δ y ( P t f ) − δ x ( P t f ) = P t f ( y ) − P t f ( x ) . \delta_y(P_tf)-\delta_x(P_tf)=P_tf(y)-P_tf(x). δ y ( P t f ) − δ x ( P t f ) = P t f ( y ) − P t f ( x ) . Hence the classical e e e { δ x : x ∈ X } ⊆ S ( A ) \{\delta_x:x\in X\}\subseteq S(\mathcal{A}) { δ x : x ∈ X } ⊆ S ( A )
Two remarks are worth making. First, for a general Polish metric space X X X Lip b ( X , R ) \operatorname{Lip}_b(X,\mathbb{R}) Lip b ( X , R ) C b ( X , R ) C_b(X,\mathbb{R}) C b ( X , R ) X X X Lip ( X , R ) \operatorname{Lip}(X,\mathbb{R}) Lip ( X , R ) C ( X , R ) C(X,\mathbb{R}) C ( X , R ) C b ( X ) C_b(X) C b ( X ) X X X
From the eventual e e e e e e
The key compactness input is completely elementary.
Lemma 3.1. Let A \mathcal{A} A C ∗ C^* C ∗ K ⊆ A K\subseteq\mathcal{A} K ⊆ A ( φ i ) i (\varphi_i)_i ( φ i ) i S ( A ) S(\mathcal{A}) S ( A ) φ \varphi φ ∗ * ∗
sup a ∈ K ∣ φ i ( a ) − φ ( a ) ∣ ⟶ 0. \sup_{a\in K}\left|\varphi_i(a)-\varphi(a)\right|\longrightarrow0. a ∈ K sup ∣ φ i ( a ) − φ ( a ) ∣ ⟶ 0. Proof. Fix ε > 0 \varepsilon>0 ε > 0 a 1 , … , a n ∈ K a_1,\ldots,a_n\in K a 1 , … , a n ∈ K a ∈ K a\in K a ∈ K min k ∥ a − a k ∥ < ε / 3 \min_k\|a-a_k\|<\varepsilon/3 min k ∥ a − a k ∥ < ε /3 φ i → φ \varphi_i\to\varphi φ i → φ ∗ * ∗ i 0 i_0 i 0
∣ φ i ( a k ) − φ ( a k ) ∣ < ε / 3 , i ≥ i 0 , 1 ≤ k ≤ n . \left|\varphi_i(a_k)-\varphi(a_k)\right|<\varepsilon/3,\qquad i\ge i_0,\ 1\le k\le n. ∣ φ i ( a k ) − φ ( a k ) ∣ < ε /3 , i ≥ i 0 , 1 ≤ k ≤ n . Take i ≥ i 0 i\ge i_0 i ≥ i 0 a ∈ K a\in K a ∈ K k k k ∥ a − a k ∥ < ε / 3 \|a-a_k\|<\varepsilon/3 ∥ a − a k ∥ < ε /3
∣ φ i ( a ) − φ ( a ) ∣ ≤ ∣ φ i ( a − a k ) ∣ + ∣ φ i ( a k ) − φ ( a k ) ∣ + ∣ φ ( a k − a ) ∣ ≤ 2 ∥ a − a k ∥ + ε / 3 < ε . \left|\varphi_i(a)-\varphi(a)\right|
\le\left|\varphi_i(a-a_k)\right|+\left|\varphi_i(a_k)-\varphi(a_k)\right|+\left|\varphi(a_k-a)\right|
\le2\|a-a_k\|+\varepsilon/3<\varepsilon. ∣ φ i ( a ) − φ ( a ) ∣ ≤ ∣ φ i ( a − a k ) ∣ + ∣ φ i ( a k ) − φ ( a k ) ∣ + ∣ φ ( a k − a ) ∣ ≤ 2∥ a − a k ∥ + ε /3 < ε . This proves the claim. □
Theorem 3.2. Let ( T t ) t ≥ 0 (T_t)_{t\ge0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M ( T t ) (T_t) ( T t ) e e e L \mathcal{L} L e e e L \mathcal{L} L
Proof. Fix φ ∈ S ( A ) \varphi\in\mathcal{S}(\mathcal{A}) φ ∈ S ( A ) a ∈ L a \in\mathcal{L} a ∈ L ε > 0 \varepsilon> 0 ε > 0 e e e ∗ * ∗ U 1 U_1 U 1 φ \varphi φ t 0 ≥ 0 t_0 \ge0 t 0 ≥ 0
sup ψ ∈ U 1 sup t ≥ t 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ < ε / 2. \sup_{\psi\in U_1}\sup_{t \ge t_0} \left|\psi(\mathcal{T}_t(a))-\varphi(\mathcal{T}_t(a))\right| < \varepsilon/2. ψ ∈ U 1 sup t ≥ t 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ < ε /2. Since A \mathcal{A} A [ 0 , t 0 ] ∋ t ↦ T t ( a ) ∈ A [0,t_0] \ni t \mapsto\mathcal{T}_t(a) \in\mathcal{A} [ 0 , t 0 ] ∋ t ↦ T t ( a ) ∈ A
K : = { T t ( a ) : 0 ≤ t ≤ t 0 } K := \{\mathcal{T}_t(a): 0 \le t \le t_0\} K := { T t ( a ) : 0 ≤ t ≤ t 0 } is norm compact. We claim that there exists a weak-∗ * ∗ U 2 U_2 U 2 φ \varphi φ
sup ψ ∈ U 2 sup b ∈ K ∣ ψ ( b ) − φ ( b ) ∣ < ε / 2. \sup_{\psi\in U_2}\sup_{b \in K} \left|\psi(b)-\varphi(b)\right| < \varepsilon/2. ψ ∈ U 2 sup b ∈ K sup ∣ ψ ( b ) − φ ( b ) ∣ < ε /2. Indeed, otherwise one could find a net ( ψ i ) (\psi_i) ( ψ i ) S ( A ) \mathcal{S}(\mathcal{A}) S ( A ) ψ i → φ \psi_i \to\varphi ψ i → φ ∗ * ∗
sup b ∈ K ∣ ψ i ( b ) − φ ( b ) ∣ ≥ ε / 2 \sup_{b \in K}\left|\psi_i(b)-\varphi(b)\right| \ge\varepsilon/2 b ∈ K sup ∣ ψ i ( b ) − φ ( b ) ∣ ≥ ε /2 for all i i i U : = U 1 ∩ U 2 U := U_1 \cap U_2 U := U 1 ∩ U 2 ψ ∈ U \psi\in U ψ ∈ U
sup t ≥ 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ ≤ max { sup 0 ≤ t ≤ t 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ , sup t ≥ t 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ } < ε . \sup_{t \ge0}\left|\psi(\mathcal{T}_t(a))-\varphi(\mathcal{T}_t(a))\right|
\le\max\left\{\sup_{0 \le t \le t_0}\left|\psi(\mathcal{T}_t(a))-\varphi(\mathcal{T}_t(a))\right|,\sup_{t \ge t_0}\left|\psi(\mathcal{T}_t(a))-\varphi(\mathcal{T}_t(a))\right|\right\}<\varepsilon. t ≥ 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ ≤ max { 0 ≤ t ≤ t 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ , t ≥ t 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ } < ε . Hence ( T t ) (\mathcal{T}_t) ( T t ) e e e φ \varphi φ φ \varphi φ
Remark 3.3. The proof uses only norm continuity on A \mathcal{A} A L \mathcal{L} L ( T t ) (\mathcal{T}_t) ( T t ) a ∈ L a \in\mathcal{L} a ∈ L { T t ( a ) : 0 ≤ t ≤ t 0 } \{\mathcal{T}_t(a): 0 \le t \le t_0\} { T t ( a ) : 0 ≤ t ≤ t 0 } A \mathcal{A} A
Corollary 3.4. Let ( T t ) t ≥ 0 (\mathcal{T}_t)_{t \ge0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A} \subseteq\mathcal{M} A ⊆ M a ∈ A s a a \in\mathcal{A}_{\mathrm{sa}} a ∈ A sa
the family { T t ( a ) : t ≥ 0 } \{\mathcal{T}_t(a): t \ge0\} { T t ( a ) : t ≥ 0 } A \mathcal{A} A
the family of functions S ( A ) ∋ φ ↦ φ ( T t ( a ) ) ∈ R \mathcal{S}(\mathcal{A}) \ni\varphi\mapsto\varphi(\mathcal{T}_t(a)) \in\mathbb{R} S ( A ) ∋ φ ↦ φ ( T t ( a )) ∈ R t ≥ 0 t \ge0 t ≥ 0 ∗ * ∗ S ( A ) \mathcal{S}(\mathcal{A}) S ( A )
Consequently, ( T t ) (\mathcal{T}_t) ( T t ) e e e L \mathcal{L} L a ∈ L a \in\mathcal{L} a ∈ L A \mathcal{A} A
Proof. Let
Γ : A s a ⟶ C ( S ( A ) , R ) , Γ ( b ) ( φ ) : = b ^ ( φ ) : = φ ( b ) . \Gamma:\mathcal{A}_{\mathrm{sa}} \longrightarrow C(\mathcal{S}(\mathcal{A}),\mathbb{R}), \qquad\Gamma(b)(\varphi):=\widehat{b}(\varphi):=\varphi(b). Γ : A sa ⟶ C ( S ( A ) , R ) , Γ ( b ) ( φ ) := b ( φ ) := φ ( b ) . For self-adjoint b b b
∥ b ∥ = sup φ ∈ S ( A ) ∣ φ ( b ) ∣ , \lVert b\rVert=\sup_{\varphi\in\mathcal{S}(\mathcal{A})}|\varphi(b)|, ∥ b ∥ = φ ∈ S ( A ) sup ∣ φ ( b ) ∣ , so Γ \Gamma Γ { T t ( a ) : t ≥ 0 } \{\mathcal{T}_t(a):t \ge0\} { T t ( a ) : t ≥ 0 } ∗ * ∗ ⇒ \Rightarrow ⇒ Γ ( T t ( a ) ) \Gamma(T_t(a)) Γ ( T t ( a )) ∥ a ∥ \lVert a\rVert ∥ a ∥ T t T_t T t S ( A ) S(A) S ( A ) ∗ \ast ∗
{ Γ ( T t ( a ) ) : t ≥ 0 } \{\Gamma(T_t(a)) : t \ge0\} { Γ ( T t ( a )) : t ≥ 0 } is relatively compact in C ( S ( A ) , R ) C(S(A),\mathbb{R}) C ( S ( A ) , R ) Γ \Gamma Γ A s a \mathcal{A}_{\mathrm{sa}} A sa C ( S ( A ) , R ) C(S(A),\mathbb{R}) C ( S ( A ) , R ) A \mathcal{A} A □ \square □
Corollary 3.5. Let L ⊆ A s a \mathcal{L} \subseteq\mathcal{A}_{\mathrm{sa}} L ⊆ A sa ( T t ) (T_t) ( T t ) e e e L \mathcal{L} L e e e A s a \mathcal{A}_{\mathrm{sa}} A sa
Proof. Fix b ∈ A s a b \in\mathcal{A}_{\mathrm{sa}} b ∈ A sa φ ∈ S ( A ) \varphi\in S(\mathcal{A}) φ ∈ S ( A ) ε > 0 \varepsilon> 0 ε > 0 a ∈ L a \in\mathcal{L} a ∈ L ∥ b − a ∥ < ε / 6 \lVert b-a\rVert< \varepsilon/6 ∥ b − a ∥ < ε /6 e e e a a a ∗ \ast ∗ U U U φ \varphi φ
sup ψ ∈ U , t ≥ 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ < ε / 3. \sup_{\psi\in U,\ t\ge0}\lvert\psi(T_t(a))-\varphi(T_t(a))\rvert<\varepsilon/3. ψ ∈ U , t ≥ 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a ))∣ < ε /3. For ψ ∈ U \psi\in U ψ ∈ U t ≥ 0 t \ge0 t ≥ 0 T t T_t T t
∣ ψ ( T t ( b ) ) − φ ( T t ( b ) ) ∣ ≤ ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ + 2 ∥ T t ( b − a ) ∥ < ε / 3 + 2 ∥ b − a ∥ < ε . \lvert\psi(T_t(b))-\varphi(T_t(b))\rvert
\le\lvert\psi(T_t(a))-\varphi(T_t(a))\rvert+2\lVert T_t(b-a)\rVert
<\varepsilon/3+2\lVert b-a\rVert<\varepsilon. ∣ ψ ( T t ( b )) − φ ( T t ( b ))∣ ≤ ∣ ψ ( T t ( a )) − φ ( T t ( a ))∣ + 2 ∥ T t ( b − a )∥ < ε /3 + 2 ∥ b − a ∥ < ε . Thus the e e e b b b □ \square □
Remark 3.6. Corollary 3.4 is specific to the full state-space formulation. In the classical Polish-space definition one usually tests equicontinuity only on the embedded Dirac states, which need not form a compact space. Passing to the whole compact state space converts the e e e
A quantitative full-state-space e e e
Definition 4.1. Let X X X F ⊆ C ( X , R ) \mathcal{F} \subseteq C(X,\mathbb{R}) F ⊆ C ( X , R ) equicontinuity defect is
e X ( F ) : = inf U max U ∈ U sup f ∈ F osc U ( f ) , e_X(\mathcal{F}) := \inf_{\mathcal{U}}\max_{U\in\mathcal{U}}\sup_{f\in\mathcal{F}}\operatorname{osc}_U(f), e X ( F ) := U inf U ∈ U max f ∈ F sup osc U ( f ) , where U \mathcal{U} U X X X Hausdorff measure of non-compactness is
χ ( F ) : = inf { r > 0 : F has a finite r -net in C ( X , R ) } . \chi(\mathcal{F}) := \inf\{r>0:\mathcal{F}\text{ has a finite }r\text{-net in }C(X,\mathbb{R})\}. χ ( F ) := inf { r > 0 : F has a finite r -net in C ( X , R )} . The defect vanishes exactly for equicontinuous families. The measure of non-compactness vanishes exactly for relatively compact families.
Proposition 4.2. For every bounded family F ⊆ C ( X , R ) \mathcal{F} \subseteq C(X,\mathbb{R}) F ⊆ C ( X , R )
1 2 e X ( F ) ≤ χ ( F ) ≤ e X ( F ) . (1) \frac{1}{2}e_X(\mathcal{F})\le\chi(\mathcal{F})\le e_X(\mathcal{F}). \tag*{(1)} 2 1 e X ( F ) ≤ χ ( F ) ≤ e X ( F ) . (1) If F F F G G G
ε X ( F ∪ G ) = max { ε X ( F ) , ε X ( G ) } . (2) \varepsilon_X(F \cup G) = \max\{\varepsilon_X(F), \varepsilon_X(G)\}. \tag*{(2)} ε X ( F ∪ G ) = max { ε X ( F ) , ε X ( G )} . (2) Proof. Fix r > χ ( F ) r > \chi(F) r > χ ( F ) g 1 , … , g m ∈ C ( X , R ) g_1,\ldots,g_m \in C(X,\mathbb{R}) g 1 , … , g m ∈ C ( X , R ) f ∈ F f \in F f ∈ F ∥ f − g j ∥ < r \lVert f-g_j\rVert< r ∥ f − g j ∥ < r j j j ( g j ) (g_j) ( g j ) ε > 0 \varepsilon> 0 ε > 0 U \mathcal{U} U
osc U ( g j ) < ε ( U ∈ U , 1 ≤ j ≤ m ) . \operatorname{osc}_{U}(g_j) < \varepsilon\qquad(U \in\mathcal{U},\ 1 \leq j \leq m). osc U ( g j ) < ε ( U ∈ U , 1 ≤ j ≤ m ) . If ∥ f − g j ∥ < r \lVert f-g_j\rVert< r ∥ f − g j ∥ < r
osc U ( f ) ≤ osc U ( g j ) + 2 r < ε + 2 r . \operatorname{osc}_{U}(f) \leq\operatorname{osc}_{U}(g_j) + 2r < \varepsilon+ 2r. osc U ( f ) ≤ osc U ( g j ) + 2 r < ε + 2 r . Thus ε X ( F ) ≤ 2 r \varepsilon_X(F) \leq2r ε X ( F ) ≤ 2 r r ↓ χ ( F ) r \downarrow\chi(F) r ↓ χ ( F ) (1) .
For the reverse inequality, fix d > ε X ( F ) d > \varepsilon_X(F) d > ε X ( F ) U = ( U j ) j = 1 m \mathcal{U} = (U_j)_{j=1}^{m} U = ( U j ) j = 1 m
sup f ∈ F osc U j ( f ) < d ( 1 ≤ j ≤ m ) . \sup_{f \in F}\operatorname{osc}_{U_j}(f) < d \qquad(1 \leq j \leq m). f ∈ F sup osc U j ( f ) < d ( 1 ≤ j ≤ m ) . Choose a partition of unity ( h j ) j = 1 m (h_j)_{j=1}^{m} ( h j ) j = 1 m x j ∈ U j x_j \in U_j x j ∈ U j h j ≠ 0 h_j \neq0 h j = 0
Q f : = ∑ j = 1 m f ( x j ) h j . Qf := \sum_{j=1}^{m} f(x_j)h_j. Q f := j = 1 ∑ m f ( x j ) h j . If h j ( x ) > 0 h_j(x) > 0 h j ( x ) > 0 x , x j ∈ U j x,x_j \in U_j x , x j ∈ U j
∣ f ( x ) − Q f ( x ) ∣ ≤ ∑ j h j ( x ) ∣ f ( x ) − f ( x j ) ∣ < d . \lvert f(x)-Qf(x)\rvert\leq\sum_j h_j(x)\lvert f(x)-f(x_j)\rvert< d. ∣ f ( x ) − Q f ( x )∣ ≤ j ∑ h j ( x ) ∣ f ( x ) − f ( x j )∣ < d . Hence ∥ f − Q f ∥ ≤ d \lVert f-Qf\rVert\leq d ∥ f − Q f ∥ ≤ d f ∈ F f \in F f ∈ F Q ( F ) Q(F) Q ( F ) ε \varepsilon ε ε > 0 \varepsilon> 0 ε > 0 F F F ( d + ε ) (d+\varepsilon) ( d + ε ) ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 d ↓ ε X ( F ) d \downarrow\varepsilon_X(F) d ↓ ε X ( F )
The lower bound in (2) is immediate. For the converse, choose finite covers that realise the two defects up to ε \varepsilon ε ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0
Remark 4.3. Quantitative measures of non-equicontinuity go back to Nussbaum and have been developed in much greater generality; see [39 , 8 ] . Proposition 4.2 is the scalar compact-Hausdorff form needed here. Its factor two is sharp for general bounded families. Indeed, if X X X B B B C ( X , R ) C(X,\mathbb{R}) C ( X , R ) χ ( B ) = 1 \chi(B)=1 χ ( B ) = 1 X X X X X X B B B 1 1 1 − 1 -1 − 1 ε X ( B ) = 2 \varepsilon_X(B)=2 ε X ( B ) = 2
Put
O T ( a ) : = { T t ( a ) : t ≥ 0 } , \mathcal{O}_T(a) := \{T_t(a):t \geq0\}, O T ( a ) := { T t ( a ) : t ≥ 0 } , and define
χ T ( a ) : = χ ( O T ( a ) ) , ε T ( a ) : = ε S ( A ) ( { T ‾ t ( a ) : t ≥ 0 } ) , \chi_T(a) := \chi(\mathcal{O}_T(a)), \qquad\varepsilon_T(a) := \varepsilon_{S(A)}\left(\{\overline{T}_t(a):t \geq0\}\right), χ T ( a ) := χ ( O T ( a )) , ε T ( a ) := ε S ( A ) ( { T t ( a ) : t ≥ 0 } ) , where b ^ = Γ ( b ) \hat{b}=\Gamma(b) b ^ = Γ ( b ) b ∈ A s a b \in\mathcal{A}_{\mathrm{sa}} b ∈ A sa χ \chi χ A s a \mathcal{A}_{\mathrm{sa}} A sa
Theorem 4.4. For a , b ∈ A s a a,b \in\mathcal{A}_{\mathrm{sa}} a , b ∈ A sa
1 2 ε T ( a ) ≤ χ T ( a ) ≤ ε T ( a ) , (3) \frac{1}{2}\varepsilon_T(a) \leq\chi_T(a) \leq\varepsilon_T(a), \tag*{(3)} 2 1 ε T ( a ) ≤ χ T ( a ) ≤ ε T ( a ) , (3) ∣ χ T ( a ) − χ T ( b ) ∣ ≤ ∥ a − b ∥ , ∣ ε T ( a ) − ε T ( b ) ∣ ≤ 2 ∥ a − b ∥ . (4) \begin{aligned}
|\chi_T(a)-\chi_T(b)| &\leq\lVert a-b\rVert,\\
|\varepsilon_T(a)-\varepsilon_T(b)| &\leq2\lVert a-b\rVert. \tag*{(4)}
\end{aligned} ∣ χ T ( a ) − χ T ( b ) ∣ ∣ ε T ( a ) − ε T ( b ) ∣ ≤ ∥ a − b ∥ , ≤ 2 ∥ a − b ∥ . (4) For every T ≥ 0 T \geq0 T ≥ 0 [ 0 , T ] [0,T] [ 0 , T ]
Proof. Kadison’s map
Γ : A s a ⟶ C ( S ( A ) , R ) , Γ ( c ) ( φ ) = φ ( c ) , \Gamma:\mathcal{A}_{\mathrm{sa}} \longrightarrow C(S(\mathcal{A}),\mathbb{R}), \qquad\Gamma(c)(\varphi)=\varphi(c), Γ : A sa ⟶ C ( S ( A ) , R ) , Γ ( c ) ( φ ) = φ ( c ) , is an isometry. Hence the Hausdorff measure of non-compactness of O T ( a ) \mathcal{O}_T(a) O T ( a ) Γ \Gamma Γ (3) .
For every t ≥ 0 t \geq0 t ≥ 0
∥ T t ( a ) − T t ( b ) ∥ ≤ ∥ a − b ∥ . \lVert\mathcal{T}_t(a)-\mathcal{T}_t(b)\rVert\leq\lVert a-b\rVert. ∥ T t ( a ) − T t ( b )∥ ≤ ∥ a − b ∥ . Thus the two orbit sets are at Hausdorff distance at most ∥ a − b ∥ \lVert a-b\rVert ∥ a − b ∥ (4) . On every subset U ⊆ S ( A ) U \subseteq S(\mathcal{A}) U ⊆ S ( A )
osc U ( T t ( a ) ) ≤ osc U ( T t ( b ) ) + 2 ∥ a − b ∥ . \operatorname{osc}_{U}(\mathcal{T}_t(a)) \leq\operatorname{osc}_{U}(\mathcal{T}_t(b)) + 2\lVert a-b\rVert. osc U ( T t ( a )) ≤ osc U ( T t ( b )) + 2 ∥ a − b ∥ . Taking the relevant suprema and infima, and then interchanging a a a b b b
Strong continuity makes the initial orbit segment
K T : = { T t ( a ) : 0 ≤ t ≤ T } K_T:=\{\mathcal{T}_t(a):0\leq t\leq T\} K T := { T t ( a ) : 0 ≤ t ≤ T } norm compact. The Hausdorff measure of non-compactness has the maximum property under finite unions, so χ ( K T ) = 0 \chi(K_T)=0 χ ( K T ) = 0 K T K_T K T Γ ( K T ) \Gamma(K_T) Γ ( K T ) (2) shows that adjoining it to the tail does not change the equicontinuity defect. □
Corollary 4.5. For a = a ∗ ∈ A a=a^* \in\mathcal{A} a = a ∗ ∈ A
(i) the full-state-space e e e a a a
(ii) the eventual full-state-space e e e a a a
(iii) O T ( a ) \mathcal{O}_T(a) O T ( a )
(iv) ε T ( a ) = 0 \varepsilon_T(a)=0 ε T ( a ) = 0
(v) χ T ( a ) = 0 \chi_T(a)=0 χ T ( a ) = 0
Proof. The equivalence of (i), (iv) and (v) follows from Theorem 4.4. The full property implies the eventual one. Conversely, assume the eventual property. Fix ε > 0 \varepsilon>0 ε > 0 φ \varphi φ U φ U_\varphi U φ T φ T_\varphi T φ ε \varepsilon ε 2 ε 2\varepsilon 2 ε ε \varepsilon ε □ \square □
Remark 4.6. On a compact classical phase space X X X e e e X X X S ( C ( X ) ) S(C(X)) S ( C ( X )) C b ( X ) C_b(X) C b ( X )
Example 4.7. Let A = BUC ( R ) A = \operatorname{BUC}(\mathbb{R}) A = BUC ( R )
( T t f ) ( x ) = f ( x + t ) . (T_t f)(x) = f(x+t). ( T t f ) ( x ) = f ( x + t ) . This is a strongly continuous group of unital ∗ \ast ∗ h ( x ) = e − x 2 h(x) = e^{-x^2} h ( x ) = e − x 2 ( T n h ) n ≥ 1 (T_nh)_{n\geq1} ( T n h ) n ≥ 1 m − n m-n m − n − n -n − n
∥ T n h − T m h ∥ ≥ 1 − e − ( m − n ) 2 . \lVert T_nh-T_mh\rVert\geq1-e^{-(m-n)^2}. ∥ T n h − T m h ∥ ≥ 1 − e − ( m − n ) 2 . Thus the orbit is not relatively compact and the full-state-space e e e h h h
The hierarchy of eventual equicontinuity variants
To place Corollary 4.5 in the context of the classical comparison theory [16 , 29 , 35 , 37 ] , we record the natural weak-∗ \ast ∗ φ ∈ S ( A ) \varphi\in S(A) φ ∈ S ( A ) U ( φ ) \mathcal{U}(\varphi) U ( φ ) ∗ \ast ∗ φ \varphi φ S ( A ) S(A) S ( A )
Definition 4.8. Let a ∈ A s a a \in A_{\mathrm{sa}} a ∈ A sa φ ∈ S ( A ) \varphi\in S(A) φ ∈ S ( A )
( T t ) (T_t) ( T t ) fixed-threshold eventual e e e at φ \varphi φ a a a t 0 ≥ 0 t_0 \geq0 t 0 ≥ 0 ε > 0 \varepsilon> 0 ε > 0 U ∈ U ( φ ) U \in\mathcal{U}(\varphi) U ∈ U ( φ )
sup ψ ∈ U sup t ≥ t 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ < ε . \sup_{\psi\in U}\sup_{t\geq t_0}\lvert\psi(T_t(a))-\varphi(T_t(a))\rvert< \varepsilon. ψ ∈ U sup t ≥ t 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a ))∣ < ε . ( T t ) (T_t) ( T t ) locally uniformly eventually continuous at φ \varphi φ a a a
inf U ∈ U ( φ ) lim sup t → ∞ sup ψ ∈ U ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ = 0. \inf_{U\in\mathcal{U}(\varphi)}\limsup_{t\to\infty}\sup_{\psi\in U}\lvert\psi(T_t(a))-\varphi(T_t(a))\rvert=0. U ∈ U ( φ ) inf t → ∞ lim sup ψ ∈ U sup ∣ ψ ( T t ( a )) − φ ( T t ( a ))∣ = 0. ( T t ) (T_t) ( T t ) pointwise eventually continuous at φ \varphi φ a a a
inf U ∈ U ( φ ) sup ψ ∈ U lim sup t → ∞ ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ = 0. \inf_{U\in\mathcal{U}(\varphi)}\sup_{\psi\in U}\limsup_{t\to\infty}\lvert\psi(T_t(a))-\varphi(T_t(a))\rvert=0. U ∈ U ( φ ) inf ψ ∈ U sup t → ∞ lim sup ∣ ψ ( T t ( a )) − φ ( T t ( a ))∣ = 0. If the corresponding condition holds at every φ ∈ S ( A ) \varphi\in S(A) φ ∈ S ( A ) S ( A ) S(A) S ( A ) a a a
Remark 4.9. The distinction between (2) and (3) lies entirely in the order of lim sup t → ∞ \limsup_{t\to\infty} lim sup t → ∞ sup ψ ∈ U \sup_{\psi\in U} sup ψ ∈ U ∗ \ast ∗ U U U ε \varepsilon ε ε \varepsilon ε e e e § 2 . Condition (3) is the direct state-space analogue of the pointwise classical condition lim x → z lim sup t → ∞ ∣ P t f ( x ) − P t f ( z ) ∣ = 0 \lim_{x\to z}\limsup_{t\to\infty}\lvert P_tf(x)-P_tf(z)\rvert=0 lim x → z lim sup t → ∞ ∣ P t f ( x ) − P t f ( z )∣ = 0 [37 ] . As we show below, Condition (2) collapses completely to the ordinary e e e S ( A ) S(A) S ( A )
Theorem 4.10. Let ( T t ) t ≥ 0 (T_t)_{t\geq0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M A\subseteq\mathcal{M} A ⊆ M a ∈ A s a a\in A_{\mathrm{sa}} a ∈ A sa
(i) ( T t ) (T_t) ( T t ) e e e S ( A ) S(\mathcal{A}) S ( A ) a a a ( T t ) (T_t) ( T t ) e e e S ( A ) S(\mathcal{A}) S ( A ) a a a ( T t ) (T_t) ( T t ) e e e S ( A ) S(\mathcal{A}) S ( A ) a a a ( T t ) (T_t) ( T t ) S ( A ) S(\mathcal{A}) S ( A ) a a a O T ( a ) = { T t ( a ) : t ≥ 0 } \mathcal{O}_T(a)=\{T_t(a):t\ge0\} O T ( a ) = { T t ( a ) : t ≥ 0 } A A A ε T ( a ) = 0 \varepsilon_T(a)=0 ε T ( a ) = 0 χ T ( a ) = 0 \chi_T(a)=0 χ T ( a ) = 0
Each of these conditions implies that ( T t ) (T_t) ( T t ) S ( A ) S(\mathcal{A}) S ( A ) a a a
Proof. The equivalences (i) ⟺ \Longleftrightarrow ⟺ ⟺ \Longleftrightarrow ⟺ ⟺ \Longleftrightarrow ⟺ ⟺ \Longleftrightarrow ⟺ ⟹ \Longrightarrow ⟹ ⟹ \Longrightarrow ⟹
(iii) ⟹ \Longrightarrow ⟹ φ ∈ S ( A ) \varphi\in S(\mathcal{A}) φ ∈ S ( A ) ε > 0 \varepsilon>0 ε > 0 t 0 ≥ 0 t_0\ge0 t 0 ≥ 0 U ∈ U ( φ ) U\in\mathcal{U}(\varphi) U ∈ U ( φ )
sup ψ ∈ U sup t ≥ t 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ ≤ ε . \sup_{\psi\in U}\sup_{t\ge t_0}\left|\psi(T_t(a))-\varphi(T_t(a))\right|\le\varepsilon. ψ ∈ U sup t ≥ t 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ ≤ ε . Hence
lim sup t → ∞ sup ψ ∈ U ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ ≤ ε . \limsup_{t\to\infty}\sup_{\psi\in U}\left|\psi(T_t(a))-\varphi(T_t(a))\right|\le\varepsilon. t → ∞ lim sup ψ ∈ U sup ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ ≤ ε . Taking the infimum over all U ∈ U ( φ ) U\in\mathcal{U}(\varphi) U ∈ U ( φ ) ε \varepsilon ε ε > 0 \varepsilon>0 ε > 0
(iv) ⟹ \Longrightarrow ⟹ f t : = Γ ( T t ( a ) ) ∈ C ( S ( A ) , R ) f_t:=\Gamma(T_t(a))\in C(S(\mathcal{A}),\mathbb{R}) f t := Γ ( T t ( a )) ∈ C ( S ( A ) , R ) Γ \Gamma Γ ε > 0 \varepsilon>0 ε > 0 φ ∈ S ( A ) \varphi\in S(\mathcal{A}) φ ∈ S ( A ) ∗ ^* ∗ U φ U_\varphi U φ φ \varphi φ τ φ ≥ 0 \tau_\varphi\ge0 τ φ ≥ 0
sup ψ ∈ U φ ∣ f t ( ψ ) − f t ( φ ) ∣ ≤ ε 2 ( t ≥ τ φ ) . \sup_{\psi\in U_\varphi}|f_t(\psi)-f_t(\varphi)|\le\frac{\varepsilon}{2}\qquad(t\ge\tau_\varphi). ψ ∈ U φ sup ∣ f t ( ψ ) − f t ( φ ) ∣ ≤ 2 ε ( t ≥ τ φ ) . By the triangle inequality, osc U φ ( f t ) ≤ ε \operatorname{osc}_{U_\varphi}(f_t)\le\varepsilon osc U φ ( f t ) ≤ ε t ≥ τ φ t\ge\tau_\varphi t ≥ τ φ ∗ ^* ∗ S ( A ) S(\mathcal{A}) S ( A ) { U φ 1 , … , U φ m } \{U_{\varphi_1},\ldots,U_{\varphi_m}\} { U φ 1 , … , U φ m } τ max : = max 1 ≤ j ≤ m τ φ j \tau_{\max}:=\max_{1\le j\le m}\tau_{\varphi_j} τ m a x := max 1 ≤ j ≤ m τ φ j F τ max : = { f t : t ≥ τ max } \mathcal{F}_{\tau_{\max}}:=\{f_t:t\ge\tau_{\max}\} F τ m a x := { f t : t ≥ τ m a x }
ε S ( A ) ( F τ max ) ≤ max 1 ≤ j ≤ m sup t ≥ τ max osc U φ j ( f t ) ≤ ε . \varepsilon_{S(\mathcal{A})}(\mathcal{F}_{\tau_{\max}})\le\max_{1\le j\le m}\sup_{t\ge\tau_{\max}}\operatorname{osc}_{U_{\varphi_j}}(f_t)\le\varepsilon. ε S ( A ) ( F τ m a x ) ≤ 1 ≤ j ≤ m max t ≥ τ m a x sup osc U φ j ( f t ) ≤ ε . By Theorem 4.4 , deleting the initial interval [ 0 , τ max ] [0,\tau_{\max}] [ 0 , τ m a x ] ε T ( a ) = ε S ( A ) ( F τ max ) ≤ ε \varepsilon_T(a)=\varepsilon_{S(\mathcal{A})}(\mathcal{F}_{\tau_{\max}})\le\varepsilon ε T ( a ) = ε S ( A ) ( F τ m a x ) ≤ ε ε > 0 \varepsilon>0 ε > 0 ε T ( a ) = 0 \varepsilon_T(a)=0 ε T ( a ) = 0
Cesàro averages and the mean-ergodic mechanism
For a ∈ A a\in A a ∈ A T > 0 T>0 T > 0
M T ( a ) : = 1 T ∫ 0 T T t ( a ) d t , M_T(a):=\frac{1}{T}\int_0^T T_t(a)\,\mathrm{d}t, M T ( a ) := T 1 ∫ 0 T T t ( a ) d t , where the integral is the Bochner integral in A A A
Definition 5.1. Let ( T t ) t ≥ 0 (T_t)_{t\geq0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M ( T t ) (T_t) ( T t ) e e e φ ∈ S ( A ) \varphi\in S(\mathcal{A}) φ ∈ S ( A ) L \mathcal{L} L a ∈ L a\in\mathcal{L} a ∈ L ε > 0 \varepsilon>0 ε > 0 ∗ * ∗ U U U φ \varphi φ
sup ψ ∈ U sup T > 0 ∣ ψ ( M T ( a ) ) − φ ( M T ( a ) ) ∣ < ε . \sup_{\psi\in U}\sup_{T>0}\left|\psi(M_T(a))-\varphi(M_T(a))\right|<\varepsilon. ψ ∈ U sup T > 0 sup ∣ ψ ( M T ( a )) − φ ( M T ( a )) ∣ < ε . If this holds at every state, we say that ( T t ) (T_t) ( T t ) e e e L \mathcal{L} L
Lemma 5.2. For every a ∈ A a\in\mathcal{A} a ∈ A ( 0 , ∞ ) ∋ T ↦ M T ( a ) (0,\infty)\ni T\mapsto M_T(a) ( 0 , ∞ ) ∋ T ↦ M T ( a ) T = 0 T=0 T = 0 M 0 ( a ) : = a M_0(a):=a M 0 ( a ) := a
Proof. For T > 0 T>0 T > 0 t ↦ T t ( a ) t\mapsto T_t(a) t ↦ T t ( a )
∥ M T ( a ) − a ∥ ≤ 1 T ∫ 0 T ∥ T t ( a ) − a ∥ d t ⟶ 0 ( T ↓ 0 ) , \left\|M_T(a)-a\right\|\leq\frac{1}{T}\int_0^T\left\|T_t(a)-a\right\|\,dt\longrightarrow0\qquad(T\downarrow0), ∥ M T ( a ) − a ∥ ≤ T 1 ∫ 0 T ∥ T t ( a ) − a ∥ d t ⟶ 0 ( T ↓ 0 ) , again by norm continuity. □ \square □
Proposition 5.3. If ( T t ) (T_t) ( T t ) e e e L \mathcal{L} L e e e L \mathcal{L} L
Proof. Fix a ∈ L a\in\mathcal{L} a ∈ L φ ∈ S ( A ) \varphi\in S(\mathcal{A}) φ ∈ S ( A ) ε > 0 \varepsilon>0 ε > 0 e e e ∗ * ∗ U U U φ \varphi φ
sup ψ ∈ U sup t ≥ 0 ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ < ε . \sup_{\psi\in U}\sup_{t\geq0}\left|\psi(T_t(a))-\varphi(T_t(a))\right|<\varepsilon. ψ ∈ U sup t ≥ 0 sup ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ < ε . Therefore, for every ψ ∈ U \psi\in U ψ ∈ U T > 0 T>0 T > 0
∣ ψ ( M T ( a ) ) − φ ( M T ( a ) ) ∣ ≤ 1 T ∫ 0 T ∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ d t < ε . \begin{aligned}
\left|\psi(M_T(a))-\varphi(M_T(a))\right|
&\leq\frac{1}{T}\int_0^T\left|\psi(T_t(a))-\varphi(T_t(a))\right|\,dt\\
&<\varepsilon.
\end{aligned} ∣ ψ ( M T ( a )) − φ ( M T ( a )) ∣ ≤ T 1 ∫ 0 T ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ d t < ε . Hence the Cesàro e e e □ \square □
Theorem 5.4. Let ( T t ) t ≥ 0 (T_t)_{t\geq0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M a ∈ L a\in\mathcal{L} a ∈ L E ( a ) ∈ A E(a)\in\mathcal{A} E ( a ) ∈ A
∥ M T ( a ) − E ( a ) ∥ ⟶ 0 ( T → ∞ ) . \left\|M_T(a)-E(a)\right\|\longrightarrow0\qquad(T\to\infty). ∥ M T ( a ) − E ( a ) ∥ ⟶ 0 ( T → ∞ ) . Then for every s ≥ 0 s\geq0 s ≥ 0 a ∈ L a\in\mathcal{L} a ∈ L
T s ( E ( a ) ) = E ( a ) . T_s(E(a))=E(a). T s ( E ( a )) = E ( a ) . In particular, ( T t ) (T_t) ( T t ) e e e L \mathcal{L} L
Proof. Fix a ∈ L a\in\mathcal{L} a ∈ L s ≥ 0 s\geq0 s ≥ 0 T > 0 T>0 T > 0
T s ( M T ( a ) ) − M T ( a ) = 1 T ∫ T T + s T r ( a ) d r − 1 T ∫ 0 s T r ( a ) d r , T_s(M_T(a))-M_T(a)=\frac{1}{T}\int_T^{T+s}T_r(a)\,dr-\frac{1}{T}\int_0^sT_r(a)\,dr, T s ( M T ( a )) − M T ( a ) = T 1 ∫ T T + s T r ( a ) d r − T 1 ∫ 0 s T r ( a ) d r , whence, using the contractivity of the positive unital maps T r T_r T r
∥ T s ( M T ( a ) ) − M T ( a ) ∥ ≤ 2 s ∥ a ∥ T . \left\|T_s(M_T(a))-M_T(a)\right\|\leq\frac{2s\|a\|}{T}. ∥ T s ( M T ( a )) − M T ( a ) ∥ ≤ T 2 s ∥ a ∥ . Therefore
∥ T s ( E ( a ) ) − E ( a ) ∥ ≤ ∥ T s ( E ( a ) − M T ( a ) ) ∥ + ∥ T s ( M T ( a ) ) − M T ( a ) ∥ + ∥ M T ( a ) − E ( a ) ∥ \lVert\mathcal{T}_s(E(a))-E(a)\rVert\le\lVert\mathcal{T}_s(E(a)-M_T(a))\rVert+\lVert\mathcal{T}_s(M_T(a))-M_T(a)\rVert+\lVert M_T(a)-E(a)\rVert ∥ T s ( E ( a )) − E ( a )∥ ≤ ∥ T s ( E ( a ) − M T ( a ))∥ + ∥ T s ( M T ( a )) − M T ( a )∥ + ∥ M T ( a ) − E ( a )∥ ≤ 2 ∥ M T ( a ) − E ( a ) ∥ + 2 s ∥ a ∥ T . \le2\lVert M_T(a)-E(a)\rVert+\frac{2s\lVert a\rVert}{T}. ≤ 2 ∥ M T ( a ) − E ( a )∥ + T 2 s ∥ a ∥ . Letting T → ∞ T\to\infty T → ∞ T s ( E ( a ) ) = E ( a ) \mathcal{T}_s(E(a))=E(a) T s ( E ( a )) = E ( a ) φ ∈ S ( A ) \varphi\in\mathcal{S}(\mathcal{A}) φ ∈ S ( A ) ε > 0 \varepsilon>0 ε > 0 T 0 > 0 T_0>0 T 0 > 0
∥ M T ( a ) − E ( a ) ∥ < ε / 3 , T ≥ T 0 . \lVert M_T(a)-E(a)\rVert<\varepsilon/3,\qquad T\ge T_0. ∥ M T ( a ) − E ( a )∥ < ε /3 , T ≥ T 0 . Since evaluation at the fixed element E ( a ) E(a) E ( a ) ∗ * ∗ ∗ * ∗ U 1 U_1 U 1 φ \varphi φ
∣ ψ ( E ( a ) ) − φ ( E ( a ) ) ∣ < ε / 3 , ψ ∈ U 1 . |\psi(E(a))-\varphi(E(a))|<\varepsilon/3,\qquad\psi\in U_1. ∣ ψ ( E ( a )) − φ ( E ( a )) ∣ < ε /3 , ψ ∈ U 1 . By Lemma 5.2 , the set
K : = { M T ( a ) : 0 ≤ T ≤ T 0 } K:=\{M_T(a):0\le T\le T_0\} K := { M T ( a ) : 0 ≤ T ≤ T 0 } is norm compact. Lemma 3.1 therefore yields a weak-∗ * ∗ U 2 U_2 U 2 φ \varphi φ
sup ψ ∈ U 2 sup b ∈ K ∣ ψ ( b ) − φ ( b ) ∣ < ε . \sup_{\psi\in U_2}\sup_{b\in K}|\psi(b)-\varphi(b)|<\varepsilon. ψ ∈ U 2 sup b ∈ K sup ∣ ψ ( b ) − φ ( b ) ∣ < ε . Let U : = U 1 ∩ U 2 U:=U_1\cap U_2 U := U 1 ∩ U 2 ψ ∈ U \psi\in U ψ ∈ U 0 < T ≤ T 0 0<T\le T_0 0 < T ≤ T 0 M T ( a ) ∈ K M_T(a)\in K M T ( a ) ∈ K
∣ ψ ( M T ( a ) ) − φ ( M T ( a ) ) ∣ < ε . |\psi(M_T(a))-\varphi(M_T(a))|<\varepsilon. ∣ ψ ( M T ( a )) − φ ( M T ( a )) ∣ < ε . If instead T ≥ T 0 T\ge T_0 T ≥ T 0
∣ ψ ( M T ( a ) ) − φ ( M T ( a ) ) ∣ ≤ ∣ ψ ( M T ( a ) − E ( a ) ) ∣ + ∣ ψ ( E ( a ) ) − φ ( E ( a ) ) ∣ + ∣ φ ( E ( a ) − M T ( a ) ) ∣ ≤ 2 ∥ M T ( a ) − E ( a ) ∥ + ∣ ψ ( E ( a ) ) − φ ( E ( a ) ) ∣ < ε . \begin{aligned}
|\psi(M_T(a))-\varphi(M_T(a))|
&\le|\psi(M_T(a)-E(a))|\\
&\quad+|\psi(E(a))-\varphi(E(a))|\\
&\quad+|\varphi(E(a)-M_T(a))|\\
&\le2\lVert M_T(a)-E(a)\rVert+|\psi(E(a))-\varphi(E(a))|\\
&<\varepsilon.
\end{aligned} ∣ ψ ( M T ( a )) − φ ( M T ( a )) ∣ ≤ ∣ ψ ( M T ( a ) − E ( a )) ∣ + ∣ ψ ( E ( a )) − φ ( E ( a )) ∣ + ∣ φ ( E ( a ) − M T ( a )) ∣ ≤ 2 ∥ M T ( a ) − E ( a )∥ + ∣ ψ ( E ( a )) − φ ( E ( a )) ∣ < ε . Hence the Cesàro e e e φ \varphi φ S ( A ) \mathcal{S}(\mathcal{A}) S ( A )
Corollary 5.5. Under the assumptions of Theorem 5.4 , assume in addition that there exists a state φ ∗ ∈ S ( A ) \varphi_*\in\mathcal{S}(\mathcal{A}) φ ∗ ∈ S ( A )
∥ M T ( a ) − φ ∗ ( a ) 1 ∥ ⟶ 0 ( T → ∞ ) \lVert M_T(a)-\varphi_*(a)1\rVert\longrightarrow0\qquad(T\to\infty) ∥ M T ( a ) − φ ∗ ( a ) 1 ∥ ⟶ 0 ( T → ∞ ) for every a ∈ L a\in\mathcal{L} a ∈ L a ∈ L a\in\mathcal{L} a ∈ L
sup φ , ψ ∈ S ( A ) ∣ φ ( M T ( a ) ) − ψ ( M T ( a ) ) ∣ ≤ 2 ∥ M T ( a ) − φ ∗ ( a ) 1 ∥ ⟶ 0. \sup_{\varphi,\psi\in\mathcal{S}(\mathcal{A})}|\varphi(M_T(a))-\psi(M_T(a))|\le2\lVert M_T(a)-\varphi_*(a)1\rVert\longrightarrow0. φ , ψ ∈ S ( A ) sup ∣ φ ( M T ( a )) − ψ ( M T ( a )) ∣ ≤ 2 ∥ M T ( a ) − φ ∗ ( a ) 1 ∥ ⟶ 0. In particular, the Cesàro e e e
Proof. For any states φ , ψ \varphi,\psi φ , ψ
∣ φ ( M T ( a ) ) − ψ ( M T ( a ) ) ∣ ≤ ∣ φ ( M T ( a ) − φ ∗ ( a ) 1 ) ∣ + ∣ ψ ( M T ( a ) − φ ∗ ( a ) 1 ) ∣ ≤ 2 ∥ M T ( a ) − φ ∗ ( a ) 1 ∥ . \left|\varphi(M_T(a))-\psi(M_T(a))\right|
\leq\left|\varphi(M_T(a)-\varphi_*(a)1)\right|+\left|\psi(M_T(a)-\varphi_*(a)1)\right|
\leq2\left\|M_T(a)-\varphi_*(a)1\right\|. ∣ φ ( M T ( a )) − ψ ( M T ( a )) ∣ ≤ ∣ φ ( M T ( a ) − φ ∗ ( a ) 1 ) ∣ + ∣ ψ ( M T ( a ) − φ ∗ ( a ) 1 ) ∣ ≤ 2 ∥ M T ( a ) − φ ∗ ( a ) 1 ∥ . The conclusion follows. □
Remark 5.6. Theorem 5.4 is the abstract place where a C ∗ C^* C ∗ e e e e e e
Quantitative Cesàro defects and strict separation from the e e e
For a ∈ A s a a\in\mathcal{A}_{\mathrm{sa}} a ∈ A sa O T C e s ( a ) : = { M T ( a ) : T > 0 } \mathcal{O}^{\mathrm{Ces}}_T(a):=\{M_T(a):T>0\} O T Ces ( a ) := { M T ( a ) : T > 0 }
χ T C e s ( a ) : = χ ( O T C e s ( a ) ) , ϵ T C e s ( a ) : = ϵ S ( A ) ( { M T ( a ) ^ : T > 0 } ) . \chi^{\mathrm{Ces}}_T(a):=\chi\left(\mathcal{O}^{\mathrm{Ces}}_T(a)\right),\qquad
\epsilon^{\mathrm{Ces}}_T(a):=\epsilon_{S(\mathcal{A})}\left(\{\widehat{M_T(a)}:T>0\}\right). χ T Ces ( a ) := χ ( O T Ces ( a ) ) , ϵ T Ces ( a ) := ϵ S ( A ) ( { M T ( a ) : T > 0 } ) . Proposition 5.7. For every a ∈ A s a a\in\mathcal{A}_{\mathrm{sa}} a ∈ A sa
1 2 ϵ T C e s ( a ) ≤ χ T C e s ( a ) ≤ ϵ T C e s ( a ) , \frac{1}{2}\epsilon^{\mathrm{Ces}}_T(a)\leq\chi^{\mathrm{Ces}}_T(a)\leq\epsilon^{\mathrm{Ces}}_T(a), 2 1 ϵ T Ces ( a ) ≤ χ T Ces ( a ) ≤ ϵ T Ces ( a ) , and
χ T C e s ( a ) ≤ χ T ( a ) , ϵ T C e s ( a ) ≤ ϵ T ( a ) . \chi^{\mathrm{Ces}}_T(a)\leq\chi_T(a),\qquad
\epsilon^{\mathrm{Ces}}_T(a)\leq\epsilon_T(a). χ T Ces ( a ) ≤ χ T ( a ) , ϵ T Ces ( a ) ≤ ϵ T ( a ) . Proof. The first chain of inequalities follows from Proposition 4.2 applied to the family { M T ( a ) : T > 0 } \{M_T(a):T>0\} { M T ( a ) : T > 0 } U ⊆ S ( A ) U\subseteq S(\mathcal{A}) U ⊆ S ( A ) T > 0 T>0 T > 0
osc U ( M T ( a ) ^ ) = sup φ , ψ ∈ U ∣ 1 T ∫ 0 T ( φ ( T t ( a ) ) − ψ ( T t ( a ) ) ) d t ∣ ≤ sup t ≥ 0 osc U ( T t ( a ) ) , \operatorname{osc}_U\left(\widehat{M_T(a)}\right)
=\sup_{\varphi,\psi\in U}\left|\frac{1}{T}\int_0^T\left(\varphi(\mathcal{T}_t(a))-\psi(\mathcal{T}_t(a))\right)\,dt\right|
\leq\sup_{t\geq0}\operatorname{osc}_U\left(\mathcal{T}_t(a)\right), osc U ( M T ( a ) ) = φ , ψ ∈ U sup T 1 ∫ 0 T ( φ ( T t ( a )) − ψ ( T t ( a )) ) d t ≤ t ≥ 0 sup osc U ( T t ( a ) ) , which gives ϵ T C e s ( a ) ≤ ϵ T ( a ) \epsilon^{\mathrm{Ces}}_T(a)\leq\epsilon_T(a) ϵ T Ces ( a ) ≤ ϵ T ( a )
To compare the measures of non-compactness, fix r > χ T ( a ) r>\chi_T(a) r > χ T ( a ) r r r F = { b 1 , … , b m } ⊆ A s a F=\{b_1,\ldots,b_m\}\subseteq\mathcal{A}_{\mathrm{sa}} F = { b 1 , … , b m } ⊆ A sa O T ( a ) \mathcal{O}_T(a) O T ( a ) O T ( a ) \mathcal{O}_T(a) O T ( a ) conv ( F ) + B ‾ A s a ( 0 , r ) \operatorname{conv}(F)+\overline{B}_{\mathcal{A}_{\mathrm{sa}}}(0,r) conv ( F ) + B A sa ( 0 , r ) M T ( a ) M_T(a) M T ( a ) O T ( a ) \mathcal{O}_T(a) O T ( a )
O T C e s ( a ) ⊆ conv ( F ) + B ‾ A s a ( 0 , r ) . \mathcal{O}^{\mathrm{Ces}}_T(a)\subseteq\operatorname{conv}(F)+\overline{B}_{\mathcal{A}_{\mathrm{sa}}}(0,r). O T Ces ( a ) ⊆ conv ( F ) + B A sa ( 0 , r ) . Because conv ( F ) \operatorname{conv}(F) conv ( F ) ε \varepsilon ε ε > 0 \varepsilon>0 ε > 0 O T C e s ( a ) \mathcal{O}^{\mathrm{Ces}}_T(a) O T Ces ( a ) ( r + ε ) (r+\varepsilon) ( r + ε ) ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 r ↓ χ T ( a ) r\downarrow\chi_T(a) r ↓ χ T ( a ) χ T C e s ( a ) ≤ χ T ( a ) \chi^{\mathrm{Ces}}_T(a)\leq\chi_T(a) χ T Ces ( a ) ≤ χ T ( a )
We now show that both inequalities in Proposition 5.7 can be strict (χ T C e s ( a ) = ϵ T C e s ( a ) = 0 \chi^{\mathrm{Ces}}_T(a)=\epsilon^{\mathrm{Ces}}_T(a)=0 χ T Ces ( a ) = ϵ T Ces ( a ) = 0 ϵ T ( a ) > 0 \epsilon_T(a)>0 ϵ T ( a ) > 0
Proposition 5.8 (Matrix Jamison System). Let S = { 0 } ∪ { 1 / n : n ∈ N } ⊆ [ 0 , 1 ] S=\{0\}\cup\{1/n:n\in\mathbb{N}\}\subseteq[0,1] S = { 0 } ∪ { 1/ n : n ∈ N } ⊆ [ 0 , 1 ] A = C ( S , M m ( C ) ) \mathcal{A}=C(S,M_m(\mathbb{C})) A = C ( S , M m ( C )) m ≥ 1 m\geq1 m ≥ 1 b ∈ M m ( C ) b\in M_m(\mathbb{C}) b ∈ M m ( C ) x ↦ b x\mapsto b x ↦ b A \mathcal{A} A T : S → S T:S\to S T : S → S
T ( 0 ) = 0 , T ( 1 ) = 0 , T ( 1 / n ) = 1 n − 1 ( n ≥ 2 ) . T(0)=0,\qquad T(1)=0,\qquad T(1/n)=\frac{1}{n-1}\quad(n\geq2). T ( 0 ) = 0 , T ( 1 ) = 0 , T ( 1/ n ) = n − 1 1 ( n ≥ 2 ) . and let P : A → A P:\mathcal{A}\to\mathcal{A} P : A → A P ( f ) = f ∘ T P(f)=f\circ T P ( f ) = f ∘ T
(i) the discrete ergodic averages M N ( P ) ( f ) = 1 N ∑ k = 0 N − 1 P k ( f ) M_N(P)(f)=\frac{1}{N}\sum_{k=0}^{N-1}P^k(f) M N ( P ) ( f ) = N 1 ∑ k = 0 N − 1 P k ( f )
lim N → ∞ ∥ M N ( P ) ( f ) − f ( 0 ) ∥ A = 0 ( f ∈ A ) , \lim_{N\to\infty}\left\|M_N(P)(f)-f(0)\right\|_{\mathcal{A}}=0\qquad(f\in\mathcal{A}), N → ∞ lim ∥ M N ( P ) ( f ) − f ( 0 ) ∥ A = 0 ( f ∈ A ) , so P P P e e e
(ii) the discrete semigroup ( P k ) k ≥ 0 (P^k)_{k\geq0} ( P k ) k ≥ 0 e e e A s a \mathcal{A}_{\mathrm{sa}} A sa
Proof. (i) Let f ∈ A f\in\mathcal{A} f ∈ A f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 ε > 0 \varepsilon>0 ε > 0 n 0 ∈ N n_0\in\mathbb{N} n 0 ∈ N ∥ f ( 1 / n ) ∥ M m < ε / 2 \|f(1/n)\|_{M_m}<\varepsilon/2 ∥ f ( 1/ n ) ∥ M m < ε /2 n ≥ n 0 n\geq n_0 n ≥ n 0 n , N ∈ N n,N\in\mathbb{N} n , N ∈ N T k ( 1 / n ) = 1 / ( n − k ) T^k(1/n)=1/(n-k) T k ( 1/ n ) = 1/ ( n − k ) k < n k<n k < n T k ( 1 / n ) = 0 T^k(1/n)=0 T k ( 1/ n ) = 0 k ≥ n k\geq n k ≥ n f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 f ( T k ( 1 / n ) ) = 0 f(T^k(1/n))=0 f ( T k ( 1/ n )) = 0 k ≥ n k\geq n k ≥ n
∥ ∑ k = 0 N − 1 f ( T k ( 1 / n ) ) ∥ M m = ∥ ∑ j = max { 1 , n − N + 1 } n f ( 1 / j ) ∥ M m ≤ n 0 ∥ f ∥ ∞ + N ε 2 , \left\|\sum_{k=0}^{N-1}f\left(T^k(1/n)\right)\right\|_{M_m}
=\left\|\sum_{j=\max\{1,n-N+1\}}^n f(1/j)\right\|_{M_m}
\leq n_0\|f\|_\infty+N\frac{\varepsilon}{2}, k = 0 ∑ N − 1 f ( T k ( 1/ n ) ) M m = j = m a x { 1 , n − N + 1 } ∑ n f ( 1/ j ) M m ≤ n 0 ∥ f ∥ ∞ + N 2 ε , because at most n 0 n_0 n 0 j ≤ n 0 j\leq n_0 j ≤ n 0 N N N N ≥ 2 n 0 ∥ f ∥ ∞ / ε N\geq2n_0\|f\|_\infty/\varepsilon N ≥ 2 n 0 ∥ f ∥ ∞ / ε
sup x ∈ S ∥ 1 N ∑ k = 0 N − 1 f ( T k ( x ) ) ∥ M m ≤ n 0 ∥ f ∥ ∞ N + ε 2 ≤ ε . \sup_{x\in S}\left\|\frac{1}{N}\sum_{k=0}^{N-1}f\left(T^k(x)\right)\right\|_{M_m}
\leq\frac{n_0\|f\|_\infty}{N}+\frac{\varepsilon}{2}\leq\varepsilon. x ∈ S sup N 1 k = 0 ∑ N − 1 f ( T k ( x ) ) M m ≤ N n 0 ∥ f ∥ ∞ + 2 ε ≤ ε . Thus ∥ M N ( P ) ( f ) ∥ A → 0 \|M_N(P)(f)\|_{\mathcal{A}}\to0 ∥ M N ( P ) ( f ) ∥ A → 0 f ∈ A f\in\mathcal{A} f ∈ A f = ( f − f ( 0 ) ) + f ( 0 ) f=(f-f(0))+f(0) f = ( f − f ( 0 )) + f ( 0 ) M N ( P ) ( f ) M_N(P)(f) M N ( P ) ( f ) f ( 0 ) f(0) f ( 0 ) e e e
(ii) Choose g ∈ C ( S , R ) g\in C(S,\mathbb{R}) g ∈ C ( S , R ) g ( 0 ) = 0 g(0)=0 g ( 0 ) = 0 g ( 1 ) = 1 g(1)=1 g ( 1 ) = 1 0 ≤ g ≤ 1 0\leq g\leq1 0 ≤ g ≤ 1 e 11 ∈ M m ( C ) e_{11}\in M_m(\mathbb{C}) e 11 ∈ M m ( C ) a ( x ) = g ( x ) e 11 ∈ A s a a(x)=g(x)e_{11}\in\mathcal{A}_{\mathrm{sa}} a ( x ) = g ( x ) e 11 ∈ A sa φ n ( f ) = tr m ( f ( 1 / n ) ) \varphi_n(f)=\operatorname{tr}_m(f(1/n)) φ n ( f ) = tr m ( f ( 1/ n )) φ 0 ( f ) = tr m ( f ( 0 ) ) \varphi_0(f)=\operatorname{tr}_m(f(0)) φ 0 ( f ) = tr m ( f ( 0 )) tr m \operatorname{tr}_m tr m φ n → φ 0 \varphi_n\to\varphi_0 φ n → φ 0 S ( A ) S(\mathcal{A}) S ( A ) n → ∞ n\to\infty n → ∞ n ≥ 2 n\geq2 n ≥ 2 k = n − 1 k=n-1 k = n − 1 T n − 1 ( 1 / n ) = 1 T^{n-1}(1/n)=1 T n − 1 ( 1/ n ) = 1
φ n ( P n − 1 ( a ) ) = tr m ( a ( 1 ) ) = tr m ( e 11 ) = 1 m , \varphi_n(P^{n-1}(a))=\operatorname{tr}_m(a(1))=\operatorname{tr}_m(e_{11})=\frac{1}{m}, φ n ( P n − 1 ( a )) = tr m ( a ( 1 )) = tr m ( e 11 ) = m 1 , whereas φ 0 ( P k ( a ) ) = tr m ( a ( 0 ) ) = 0 \varphi_0(P^k(a))=\operatorname{tr}_m(a(0))=0 φ 0 ( P k ( a )) = tr m ( a ( 0 )) = 0 k ≥ 0 k\geq0 k ≥ 0
sup k ≥ 0 ∣ φ n ( P k ( a ) ) − φ 0 ( P k ( a ) ) ∣ ≥ 1 m > 0 ( n ≥ 2 ) , \sup_{k\geq0}\left|\varphi_n(P^k(a))-\varphi_0(P^k(a))\right|\geq\frac{1}{m}>0\qquad(n\geq2), k ≥ 0 sup φ n ( P k ( a )) − φ 0 ( P k ( a )) ≥ m 1 > 0 ( n ≥ 2 ) , and the ordinary e e e φ 0 \varphi_0 φ 0
Continuous-time homoclinic matrix flow
Example 5.9 (Continuous-time homoclinic matrix flow). Identify the unit circle T \mathbb{T} T R / 2 π Z \mathbb{R}/2\pi\mathbb{Z} R /2 π Z μ \mu μ T \mathbb{T} T B ( T ) \mathcal{B}(\mathbb{T}) B ( T ) σ \sigma σ m ≥ 1 m\geq1 m ≥ 1
M = L ∞ ( T , B ( T ) , μ ) ⊗ ‾ M m ( C ) , A = C ( T , M m ( C ) ) . (5) \mathcal{M}=L^\infty(\mathbb{T},\mathcal{B}(\mathbb{T}),\mu)\,\overline{\otimes}\,M_m(\mathbb{C}),\qquad\mathcal{A}=C(\mathbb{T},M_m(\mathbb{C})). \tag*{(5)} M = L ∞ ( T , B ( T ) , μ ) ⊗ M m ( C ) , A = C ( T , M m ( C )) . (5) Let ( ϕ t ) t ∈ R (\phi_t)_{t\in\mathbb{R}} ( ϕ t ) t ∈ R T \mathbb{T} T θ ˙ = 1 − cos θ = 2 sin 2 ( θ / 2 ) ≥ 0 \dot{\theta}=1-\cos\theta=2\sin^2(\theta/2)\ge0 θ ˙ = 1 − cos θ = 2 sin 2 ( θ /2 ) ≥ 0
( T t f ) ( z ) : = f ( ϕ t ( z ) ) ( f ∈ M , t ≥ 0 ) . (T_t f)(z):=f(\phi_t(z)) \qquad(f\in\mathcal{M},\ t\ge0). ( T t f ) ( z ) := f ( ϕ t ( z )) ( f ∈ M , t ≥ 0 ) . Because each ϕ t \phi_t ϕ t T \mathbb{T} T μ \mu μ ( T t ) t ≥ 0 (T_t)_{t\ge0} ( T t ) t ≥ 0 ∗ * ∗ M \mathcal{M} M A \mathcal{A} A
(i) The point z = 1 z=1 z = 1 ( θ = 0 ) (\theta=0) ( θ = 0 ) lim t → ∞ ϕ t ( z ) = 1 \lim_{t\to\infty}\phi_t(z)=1 lim t → ∞ ϕ t ( z ) = 1 z ∈ T z\in\mathbb{T} z ∈ T f ∈ A f\in\mathcal{A} f ∈ A f ( 1 ) = 0 f(1)=0 f ( 1 ) = 0 ε > 0 \varepsilon>0 ε > 0 U ⊆ T U\subseteq\mathbb{T} U ⊆ T 1 1 1 ∥ f ( z ) ∥ M m < ε / 2 \lVert f(z)\rVert_{M_m}<\varepsilon/2 ∥ f ( z ) ∥ M m < ε /2 T ∖ U \mathbb{T}\setminus U T ∖ U 2 sin 2 ( θ / 2 ) 2\sin^2(\theta/2) 2 sin 2 ( θ /2 ) c U > 0 c_U>0 c U > 0 τ U : = 2 π / c U < ∞ \tau_U:=2\pi/c_U<\infty τ U := 2 π / c U < ∞ T ∖ U \mathbb{T}\setminus U T ∖ U T > 0 T>0 T > 0
∥ M T ( f ) ∥ A ≤ sup z ∈ T ( 1 T ∫ I o u t ( z ) ∥ f ( ϕ t ( z ) ) ∥ M m d t + 1 T ∫ I i n ( z ) ∥ f ( ϕ t ( z ) ) ∥ M m d t ) ≤ τ U ∥ f ∥ ∞ T + ε 2 , \lVert M_T(f)\rVert_{\mathcal{A}}\le\sup_{z\in\mathbb{T}}\left(\frac{1}{T}\int_{I_{\mathrm{out}}(z)}\lVert f(\phi_t(z))\rVert_{M_m}\,dt+\frac{1}{T}\int_{I_{\mathrm{in}}(z)}\lVert f(\phi_t(z))\rVert_{M_m}\,dt\right)
\le\frac{\tau_U\lVert f\rVert_\infty}{T}+\frac{\varepsilon}{2}, ∥ M T ( f ) ∥ A ≤ z ∈ T sup ( T 1 ∫ I out ( z ) ∥ f ( ϕ t ( z )) ∥ M m d t + T 1 ∫ I in ( z ) ∥ f ( ϕ t ( z )) ∥ M m d t ) ≤ T τ U ∥ f ∥ ∞ + 2 ε , where I i n ( z ) : = { t ∈ [ 0 , T ] : ϕ t ( z ) ∈ U } I_{\mathrm{in}}(z):=\{t\in[0,T]:\phi_t(z)\in U\} I in ( z ) := { t ∈ [ 0 , T ] : ϕ t ( z ) ∈ U } I o u t ( z ) : = { t ∈ [ 0 , T ] : ϕ t ( z ) ∈ T ∖ U } I_{\mathrm{out}}(z):=\{t\in[0,T]:\phi_t(z)\in\mathbb{T}\setminus U\} I out ( z ) := { t ∈ [ 0 , T ] : ϕ t ( z ) ∈ T ∖ U } T → ∞ T\to\infty T → ∞ ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 ∥ M T ( f ) ∥ A → 0 \lVert M_T(f)\rVert_{\mathcal{A}}\to0 ∥ M T ( f ) ∥ A → 0 M T ( f ) → f ( 1 ) M_T(f)\to f(1) M T ( f ) → f ( 1 ) f ∈ A f\in\mathcal{A} f ∈ A Theorem 5.4 , ( T t ) (T_t) ( T t ) e e e A s a \mathcal{A}_{\mathrm{sa}} A sa χ A s a C e s ( a ) = ε A s a C e s ( a ) = 0 \chi_{\mathcal{A}_{\mathrm{sa}}}^{\mathrm{Ces}}(a)=\varepsilon_{\mathcal{A}_{\mathrm{sa}}}^{\mathrm{Ces}}(a)=0 χ A sa Ces ( a ) = ε A sa Ces ( a ) = 0 a ∈ A s a a\in\mathcal{A}_{\mathrm{sa}} a ∈ A sa
(ii) For a ( e i θ ) : = sin 2 ( θ / 2 ) e 11 ∈ A s a a(e^{i\theta}):=\sin^2(\theta/2)e_{11}\in\mathcal{A}_{\mathrm{sa}} a ( e i θ ) := sin 2 ( θ /2 ) e 11 ∈ A sa φ n ( f ) : = tr m ( f ( e i / n ) ) \varphi_n(f):=\operatorname{tr}_m(f(e^{i/n})) φ n ( f ) := tr m ( f ( e i / n )) ∗ * ∗ φ 0 ( f ) : = tr m ( f ( 1 ) ) \varphi_0(f):=\operatorname{tr}_m(f(1)) φ 0 ( f ) := tr m ( f ( 1 )) n ∈ N n\in\mathbb{N} n ∈ N t n > 0 t_n>0 t n > 0 ϕ t n ( e i / n ) = − 1 \phi_{t_n}(e^{i/n})=-1 ϕ t n ( e i / n ) = − 1
sup t ≥ 0 ∣ φ n ( T t ( a ) ) − φ 0 ( T t ( a ) ) ∣ ≥ φ n ( T t n ( a ) ) = 1 m > 0. \sup_{t\ge0}\lvert\varphi_n(T_t(a))-\varphi_0(T_t(a))\rvert\ge\varphi_n(T_{t_n}(a))=\frac{1}{m}>0. t ≥ 0 sup ∣ φ n ( T t ( a )) − φ 0 ( T t ( a ))∣ ≥ φ n ( T t n ( a )) = m 1 > 0. Hence ( T t ) (T_t) ( T t ) e e e A s a \mathcal{A}_{\mathrm{sa}} A sa
Reversible and stable dynamics
Write
AP ( T ) : = { a ∈ A : O T ( a ) is relatively compact in norm } . \operatorname{AP}(T):=\{a\in\mathcal{A}: \mathcal{O}_T(a)\text{ is relatively compact in norm}\}. AP ( T ) := { a ∈ A : O T ( a ) is relatively compact in norm } . It is a closed invariant operator system. If every T t T_t T t ∗ * ∗ C ∗ C^* C ∗
The point-norm topology on a family of maps is the topology of pointwise norm convergence; we abbreviate it by pn. We also write
Fix ( T ) : = { a ∈ A : T t ( a ) = a for every t ≥ 0 } , \operatorname{Fix}(T):=\{a\in\mathcal{A}:T_t(a)=a\text{ for every }t\ge0\}, Fix ( T ) := { a ∈ A : T t ( a ) = a for every t ≥ 0 } , and
Inv ( T ) : = { φ ∈ S ( A ) : φ ∘ T t = φ for every t ≥ 0 } . \operatorname{Inv}(T):=\{\varphi\in S(\mathcal{A}):\varphi\circ T_t=\varphi\text{ for every }t\ge0\}. Inv ( T ) := { φ ∈ S ( A ) : φ ∘ T t = φ for every t ≥ 0 } . Theorem 6.1. Let ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M AP ( T ) = A \operatorname{AP}(\mathcal{T})=\mathcal{A} AP ( T ) = A P : A → A P:\mathcal{A}\to\mathcal{A} P : A → A
(i) P T t = T t P P\mathcal{T}_t=\mathcal{T}_tP P T t = T t P t ≥ 0 t\geq0 t ≥ 0
(ii)
A = R ⊕ ker P , R = P ( A ) ; \mathcal{A}=\mathcal{R}\oplus\ker P,\qquad\mathcal{R}=P(\mathcal{A}); A = R ⊕ ker P , R = P ( A ) ; (iii) ∥ T t ( a ) ∥ → 0 \lVert\mathcal{T}_t(a)\rVert\to0 ∥ T t ( a )∥ → 0 a ∈ ker P a\in\ker P a ∈ ker P
(iv)
G : = { T t ∣ R : t ≥ 0 } ‾ p n G:=\overline{\{\mathcal{T}_t|_{\mathcal{R}}:t\geq0\}}^{\mathrm{pn}} G := { T t ∣ R : t ≥ 0 } pn is a compact abelian group of surjective complete order isomorphisms of R \mathcal{R} R
(v) with
x ∘ y : = P ( x y ) , x , y ∈ R , x\circ y:=P(xy),\qquad x,y\in\mathcal{R}, x ∘ y := P ( x y ) , x , y ∈ R , the operator system R \mathcal{R} R C ∗ \mathrm{C}^{*} C ∗ g ∈ G g\in G g ∈ G ∗ * ∗
Proof. By Corollary 4.5, all Banach-space orbits are norm relatively compact. The Jacobs–de Leeuw–Glicksberg theorem for strongly almost periodic semigroups gives a bounded projection P P P [19 , 21 ] . On the stable kernel, zero belongs to the norm closure of every orbit. Contractivity upgrades this to convergence of the whole orbit: given ε > 0 \varepsilon>0 ε > 0 s s s ∥ T s ( a ) ∥ < ε \lVert\mathcal{T}_s(a)\rVert<\varepsilon ∥ T s ( a )∥ < ε
∥ T t ( a ) ∥ = ∥ T t − s ( T s ( a ) ) ∥ ≤ ∥ T s ( a ) ∥ < ε ( t ≥ s ) . \lVert\mathcal{T}_t(a)\rVert=\lVert\mathcal{T}_{t-s}(\mathcal{T}_s(a))\rVert\leq\lVert\mathcal{T}_s(a)\rVert<\varepsilon\qquad(t\geq s). ∥ T t ( a )∥ = ∥ T t − s ( T s ( a ))∥ ≤ ∥ T s ( a )∥ < ε ( t ≥ s ) . This proves (iii) explicitly.
Point-norm limits of unital completely positive maps are unital and completely positive, so P P P G G G G G G C ∗ \mathrm{C}^{*} C ∗ x ∘ y = P ( x y ) x\circ y=P(xy) x ∘ y = P ( x y ) [13 ] , and complete order automorphisms are ∗ * ∗
Remark 6.2. Theorem 6.1 is a C ∗ \mathrm{C}^{*} C ∗ ∗ * ∗ [28 ] ; Bátkai, Groh, Kunszenti-Kovács and Schreiber obtained structural decompositions on W ∗ W^{*} W ∗ [2 ] ; Runde and Viselter developed almost-periodic operators for quantum semigroups [44 ] . Earlier long-time asymptotics were established by Frigerio and Frigerio–Verri [24 , 25 ] , while modern decoherence-free decompositions include [10 , 22 ] . The Choi–Effros and peripheral viewpoints are further developed in [23 , 5 , 4 , 1 ] . We do not claim this architecture as new; it is the structural setting for the quantitative and multiplicative results of the present paper.
Theorem 6.3. Let P P P R \mathcal{R} R G G G Theorem 6.1 . Restriction induces an affine weak-∗ * ∗
Inv ( T ) ⟶ S ( R ) G : = { ψ ∈ S ( R ) : ψ ∘ g = ψ for every g ∈ G } . \operatorname{Inv}(\mathcal{T})\longrightarrow\mathcal{S}(\mathcal{R})^{G}:=\{\psi\in\mathcal{S}(\mathcal{R}):\psi\circ g=\psi\text{ for every }g\in G\}. Inv ( T ) ⟶ S ( R ) G := { ψ ∈ S ( R ) : ψ ∘ g = ψ for every g ∈ G } . Its inverse is ψ ↦ ψ ∘ P \psi\mapsto\psi\circ P ψ ↦ ψ ∘ P
Proof. Let φ \varphi φ a ∈ ker P a \in\ker P a ∈ ker P
φ ( a ) = φ ( T t ( a ) ) ⟶ 0 , \varphi(a) = \varphi(T_t(a)) \longrightarrow0, φ ( a ) = φ ( T t ( a )) ⟶ 0 , so φ = φ ∘ P \varphi= \varphi\circ P φ = φ ∘ P R \mathcal{R} R T t ∣ R T_t|_{\mathcal{R}} T t ∣ R G G G
Conversely, if ψ ∈ S ( R ) G \psi\in\mathcal{S}(\mathcal{R})^G ψ ∈ S ( R ) G ψ ∘ P \psi\circ P ψ ∘ P A \mathcal{A} A P P P P P P T t T_t T t T t T_t T t G G G
( ψ ∘ P ) ∘ T t = ψ ∘ ( T t ∣ R ) ∘ P = ψ ∘ P . (\psi\circ P) \circ T_t = \psi\circ(T_t|_{\mathcal{R}}) \circ P = \psi\circ P. ( ψ ∘ P ) ∘ T t = ψ ∘ ( T t ∣ R ) ∘ P = ψ ∘ P . The two maps are inverse. Weak-∗ * ∗ P P P □ \square □
For λ ∈ R \lambda\in\mathbb{R} λ ∈ R
A λ : = { a ∈ A : T t ( a ) = e i λ t a for every t ≥ 0 } . \mathcal{A}_{\lambda} := \{a \in\mathcal{A} : T_t(a) = e^{i\lambda t}a \text{ for every } t \ge0\}. A λ := { a ∈ A : T t ( a ) = e iλ t a for every t ≥ 0 } . Theorem 6.4. Under the assumptions of Theorem 6.1 .
R = span { A λ : λ ∈ R } . \mathcal{R} = \operatorname{span}\{\mathcal{A}_{\lambda} : \lambda\in\mathbb{R}\}. R = span { A λ : λ ∈ R } . Proof. For a character γ ∈ G ^ \gamma\in\widehat{G} γ ∈ G
P γ a : = ∫ G γ ( g ) ‾ g ( a ) d m G ( g ) , a ∈ R , P_{\gamma}a := \int_G \overline{\gamma(g)}\,g(a)\,dm_G(g), \qquad a \in\mathcal{R}, P γ a := ∫ G γ ( g ) g ( a ) d m G ( g ) , a ∈ R , where m G m_G m G G G G g ↦ g ( a ) g \mapsto g(a) g ↦ g ( a ) [43 Chapter 2] . Consequently, the linear span of the Fourier subspaces P γ ( R ) P_{\gamma}(\mathcal{R}) P γ ( R ) R \mathcal{R} R
The one-parameter group t ↦ T t ∣ R t \mapsto T_t|_{\mathcal{R}} t ↦ T t ∣ R G G G γ ∈ G ^ \gamma\in\widehat{G} γ ∈ G t ↦ γ ( T t ∣ R ) t \mapsto\gamma(T_t|_{\mathcal{R}}) t ↦ γ ( T t ∣ R ) R \mathbb{R} R e i λ t e^{i\lambda t} e iλ t λ ∈ R \lambda\in\mathbb{R} λ ∈ R P γ ( R ) ⊆ A λ P_{\gamma}(\mathcal{R}) \subseteq\mathcal{A}_{\lambda} P γ ( R ) ⊆ A λ
Conversely, if a ∈ A λ a \in\mathcal{A}_{\lambda} a ∈ A λ { z a : ∣ z ∣ = 1 } \{za : |z| = 1\} { z a : ∣ z ∣ = 1 } a = P ( a ) + ( id − P ) ( a ) a = P(a) + (\operatorname{id} - P)(a) a = P ( a ) + ( id − P ) ( a ) λ \lambda λ ( id − P ) ( a ) = 0 (\operatorname{id} - P)(a) = 0 ( id − P ) ( a ) = 0 a ∈ R a \in\mathcal{R} a ∈ R □ \square □
Corollary 6.5. Under the assumptions of Theorem 6.1 , the following are equivalent:
(i) R = C 1 \mathcal{R} = \mathbb{C}1 R = C 1
(ii) Fix ( T ) = C 1 \operatorname{Fix}(T) = \mathbb{C}1 Fix ( T ) = C 1 A λ = { 0 } \mathcal{A}_{\lambda} = \{0\} A λ = { 0 } λ ≠ 0 \lambda\ne0 λ = 0
(iii) there is a unique invariant state π \pi π
∥ T t ( a ) − π ( a ) 1 ∥ ⟶ 0 ( a ∈ A ) . \lVert T_t(a) - \pi(a)1\rVert\longrightarrow0 \qquad(a \in\mathcal{A}). ∥ T t ( a ) − π ( a ) 1 ∥ ⟶ 0 ( a ∈ A ) . Proof. The equivalence of (i) and (ii) follows from Theorem 6.4 . If R = C 1 \mathcal{R} = \mathbb{C}1 R = C 1 P ( a ) = π ( a ) 1 P(a) = \pi(a)1 P ( a ) = π ( a ) 1 π \pi π P P P Theorem 6.3 , so it equals π \pi π r ∈ R r \in\mathcal{R} r ∈ R T t ∣ R T_t|_{\mathcal{R}} T t ∣ R C ∗ C^* C ∗ π \pi π
∥ r − π ( r ) 1 ∥ = ∥ T t ( r − π ( r ) 1 ) ∥ ⟶ 0. \lVert r-\pi(r)1\rVert=\lVert T_t(r-\pi(r)1)\rVert\longrightarrow0. ∥ r − π ( r ) 1 ∥ = ∥ T t ( r − π ( r ) 1 )∥ ⟶ 0. Thus r = π ( r ) 1 r=\pi(r)1 r = π ( r ) 1 R = C 1 \mathcal{R}=\mathbb{C}1 R = C 1
The multiplicative reversible core
For a unital completely positive map P : A → A P:\mathcal{A}\to\mathcal{A} P : A → A
MD ( P ) : = { a ∈ A : P ( a ∗ a ) = P ( a ) ∗ P ( a ) , \operatorname{MD}(P):=\{a\in\mathcal{A}:P(a^*a)=P(a)^*P(a), MD ( P ) := { a ∈ A : P ( a ∗ a ) = P ( a ) ∗ P ( a ) , P ( a a ∗ ) = P ( a ) P ( a ) ∗ } . P(aa^*)=P(a)P(a)^*\}. P ( a a ∗ ) = P ( a ) P ( a ) ∗ } . It is a C ∗ C^* C ∗
P ( a x ) = P ( a ) P ( x ) , P ( x a ) = P ( x ) P ( a ) P(ax)=P(a)P(x),\qquad P(xa)=P(x)P(a) P ( a x ) = P ( a ) P ( x ) , P ( x a ) = P ( x ) P ( a ) whenever a ∈ MD ( P ) a\in\operatorname{MD}(P) a ∈ MD ( P ) x ∈ A x\in\mathcal{A} x ∈ A [13 , 20 ] . The next result isolates the part of a completely positive range on which the Choi–Effros and ambient products already agree.
Theorem 7.1. Let P : A → A P:\mathcal{A}\to\mathcal{A} P : A → A R = P ( A ) \mathcal{R}=P(\mathcal{A}) R = P ( A )
R a m b : = R ∩ MD ( P ) = { a ∈ R : a ∗ a , a a ∗ ∈ R } \mathcal{R}_{\mathrm{amb}}:=\mathcal{R}\cap\operatorname{MD}(P)=\{a\in\mathcal{R}:a^*a,aa^*\in\mathcal{R}\} R amb := R ∩ MD ( P ) = { a ∈ R : a ∗ a , a a ∗ ∈ R } is a unital C ∗ C^* C ∗ A \mathcal{A} A C ∗ C^* C ∗ R \mathcal{R} R P P P R a m b \mathcal{R}_{\mathrm{amb}} R amb
(i) R a m b = R \mathcal{R}_{\mathrm{amb}}=\mathcal{R} R amb = R
(ii) R \mathcal{R} R C ∗ C^* C ∗ A \mathcal{A} A
(iii) P P P R \mathcal{R} R
Proof. For a ∈ R a\in\mathcal{R} a ∈ R P ( a ) = a P(a)=a P ( a ) = a
a ∈ MD ( P ) ⟺ P ( a ∗ a ) = a ∗ a and P ( a a ∗ ) = a a ∗ , a\in\operatorname{MD}(P)\quad\Longleftrightarrow\quad P(a^*a)=a^*a\ \text{and}\ P(aa^*)=aa^*, a ∈ MD ( P ) ⟺ P ( a ∗ a ) = a ∗ a and P ( a a ∗ ) = a a ∗ , which is the displayed characterisation. The intersection of the operator system R \mathcal{R} R C ∗ C^* C ∗ MD ( P ) \operatorname{MD}(P) MD ( P ) a , b ∈ R a m b a,b\in\mathcal{R}_{\mathrm{amb}} a , b ∈ R amb a b ∈ MD ( P ) ab\in\operatorname{MD}(P) ab ∈ MD ( P )
P ( a b ) = P ( a ) P ( b ) = a b , P(ab)=P(a)P(b)=ab, P ( ab ) = P ( a ) P ( b ) = ab , so a b ∈ R a m b ab\in\mathcal{R}_{\mathrm{amb}} ab ∈ R amb C ∗ C^* C ∗
Let B ⊆ R \mathcal{B}\subseteq\mathcal{R} B ⊆ R C ∗ C^* C ∗ b ∈ B b\in\mathcal{B} b ∈ B b ∗ b , b b ∗ ∈ B ⊆ R b^*b,bb^*\in\mathcal{B}\subseteq\mathcal{R} b ∗ b , b b ∗ ∈ B ⊆ R b ∈ MD ( P ) b\in\operatorname{MD}(P) b ∈ MD ( P ) B ⊆ R a m b \mathcal{B}\subseteq\mathcal{R}_{\mathrm{amb}} B ⊆ R amb
The equivalence of (i) and (ii) follows from maximality. If (ii) holds, P P P C ∗ C^* C ∗ [47 ] . The converse is immediate. □
Corollary 7.2. Let P P P Theorem 6.1 . The following are equivalent:
(i) the invariant states form a faithful family;
(ii) P P P x ∈ A + x \in\mathcal{A}_{+} x ∈ A + P ( x ) = 0 P(x) = 0 P ( x ) = 0 x = 0 x = 0 x = 0
If these conditions hold, then
R a m b = R , \mathcal{R}_{\mathrm{amb}} = \mathcal{R}, R amb = R , so R \mathcal{R} R C ∗ C^* C ∗ P P P
Proof. Assume (i) and let x ∈ A + x \in\mathcal{A}_{+} x ∈ A + P ( x ) = 0 P(x) = 0 P ( x ) = 0 φ \varphi φ φ = φ ∘ P \varphi= \varphi\circ P φ = φ ∘ P Theorem 6.3 , so φ ( x ) = 0 \varphi(x) = 0 φ ( x ) = 0 x = 0 x = 0 x = 0 P P P 0 ≠ x ∈ A + 0 \ne x \in\mathcal{A}_{+} 0 = x ∈ A + P ( x ) > 0 P(x) > 0 P ( x ) > 0 R \mathcal{R} R ψ 0 ∈ S ( R ) \psi_0 \in S(\mathcal{R}) ψ 0 ∈ S ( R ) ψ 0 ( P ( x ) ) > 0 \psi_0(P(x)) > 0 ψ 0 ( P ( x )) > 0
ψ : = ∫ G ψ 0 ∘ g d m G ( g ) . \psi:= \int_G \psi_0 \circ g \, \mathrm{d}m_G(g). ψ := ∫ G ψ 0 ∘ g d m G ( g ) . The integrand is non-negative on P ( x ) P(x) P ( x ) ψ ( P ( x ) ) > 0 \psi(P(x)) > 0 ψ ( P ( x )) > 0 φ : = ψ ∘ P \varphi:= \psi\circ P φ := ψ ∘ P Theorem 6.3 , and φ ( x ) > 0 \varphi(x) > 0 φ ( x ) > 0 a ∈ R a \in\mathcal{R} a ∈ R
d : = P ( a ∗ a ) − a ∗ a d := P(a^*a) - a^*a d := P ( a ∗ a ) − a ∗ a is positive. Every invariant state annihilates d d d φ ∘ P = φ \varphi\circ P = \varphi φ ∘ P = φ d = 0 d = 0 d = 0 a ∗ a^* a ∗ P ( a a ∗ ) = a a ∗ P(aa^*) = aa^* P ( a a ∗ ) = a a ∗ a ∈ MD ( P ) a \in\operatorname{MD}(P) a ∈ MD ( P ) Theorem 7.1 completes the proof. □
Example 7.4. Let B \mathcal{B} B C ∗ C^* C ∗ M n ( B ) M_n(\mathcal{B}) M n ( B ) M n ( C ) ⊗ B M_n(\mathbb{C}) \otimes\mathcal{B} M n ( C ) ⊗ B
E 2 : M 2 ( B ) ⟶ B , E 2 ( ( b i j ) ) : = b 11 + b 22 2 . E_2: M_2(\mathcal{B}) \longrightarrow\mathcal{B}, \qquad E_2((b_{ij})) := \frac{b_{11}+b_{22}}{2}. E 2 : M 2 ( B ) ⟶ B , E 2 (( b ij )) := 2 b 11 + b 22 . Let q : M 3 ( B ) → M 2 ( B ) q: M_3(\mathcal{B}) \to M_2(\mathcal{B}) q : M 3 ( B ) → M 2 ( B )
j ( Y ) : = ( Y 0 0 E 2 ( Y ) ) ( Y ∈ M 2 ( B ) ) . j(Y) := \begin{pmatrix} Y & 0 \\ 0 & E_2(Y) \end{pmatrix} \qquad(Y \in M_2(\mathcal{B})). j ( Y ) := ( Y 0 0 E 2 ( Y ) ) ( Y ∈ M 2 ( B )) . Both maps are unital and completely positive, and q j = id qj = \operatorname{id} q j = id
P : = j q : M 3 ( B ) ⟶ M 3 ( B ) P := jq: M_3(\mathcal{B}) \longrightarrow M_3(\mathcal{B}) P := j q : M 3 ( B ) ⟶ M 3 ( B ) is a unital completely positive projection. The semigroup
T t : = e − t id + ( 1 − e − t ) P T_t := e^{-t}\operatorname{id} + (1-e^{-t})P T t := e − t id + ( 1 − e − t ) P has compact orbits and reversible projection P P P ∗ * ∗ M 2 ( B ) M_2(\mathcal{B}) M 2 ( B )
P ( M 3 ( B ) ) ∩ MD ( P ) = 1 3 ⊗ B . P(M_3(\mathcal{B})) \cap\operatorname{MD}(P) = 1_3 \otimes\mathcal{B}. P ( M 3 ( B )) ∩ MD ( P ) = 1 3 ⊗ B . No invariant state is faithful. In particular, choosing B \mathcal{B} B
Proof. The semigroup law follows from P 2 = P P^2=P P 2 = P T t \mathcal{T}_t T t T t ( x ) → P ( x ) \mathcal{T}_t(x)\to P(x) T t ( x ) → P ( x ) Y , Z ∈ M 2 ( B ) Y,Z\in M_2(\mathcal{B}) Y , Z ∈ M 2 ( B )
j ( Y ) ∘ j ( Z ) = P ( j ( Y ) j ( Z ) ) = j ( Y Z ) . j(Y)\circ j(Z)=P(j(Y)j(Z))=j(YZ). j ( Y ) ∘ j ( Z ) = P ( j ( Y ) j ( Z )) = j ( Y Z ) . so j j j M 2 ( B ) M_2(\mathcal{B}) M 2 ( B ) Y = e 11 ⊗ 1 B Y=e_{11}\otimes1_{\mathcal{B}} Y = e 11 ⊗ 1 B
j ( Y ) 2 = ( Y 0 0 1 4 1 B ) ≠ j ( Y 2 ) . j(Y)^2=\begin{pmatrix}Y&0\\0&\frac{1}{4}1_{\mathcal{B}}\end{pmatrix}\ne j(Y^2). j ( Y ) 2 = ( Y 0 0 4 1 1 B ) = j ( Y 2 ) . Let x = j ( Y ) x=j(Y) x = j ( Y )
P ( x ∗ x ) = j ( Y ∗ Y ) , x ∗ x = ( Y ∗ Y 0 0 E 2 ( Y ) ∗ E 2 ( Y ) ) . P(x^*x)=j(Y^*Y),\qquad x^*x=\begin{pmatrix}Y^*Y&0\\0&E_2(Y)^*E_2(Y)\end{pmatrix}. P ( x ∗ x ) = j ( Y ∗ Y ) , x ∗ x = ( Y ∗ Y 0 0 E 2 ( Y ) ∗ E 2 ( Y ) ) . Thus P ( x ∗ x ) = x ∗ x P(x^*x)=x^*x P ( x ∗ x ) = x ∗ x
E 2 ( Y ∗ Y ) = E 2 ( Y ) ∗ E 2 ( Y ) . E_2(Y^*Y)=E_2(Y)^*E_2(Y). E 2 ( Y ∗ Y ) = E 2 ( Y ) ∗ E 2 ( Y ) . and the analogous condition for x x ∗ xx^* x x ∗ Y Y Y Y ∗ Y^* Y ∗ E 2 E_2 E 2 B \mathcal{B} B
Z : = Y − 1 2 ⊗ E 2 ( Y ) , Z:=Y-1_2\otimes E_2(Y), Z := Y − 1 2 ⊗ E 2 ( Y ) , then E 2 ( Z ∗ Z ) = 0 E_2(Z^*Z)=0 E 2 ( Z ∗ Z ) = 0 Z = 0 Z=0 Z = 0 Y = 1 2 ⊗ b Y=1_2\otimes b Y = 1 2 ⊗ b
P ( M 3 ( B ) ) ∩ MD ( P ) = j ( 1 2 ⊗ B ) = 1 3 ⊗ B . P(M_3(\mathcal{B}))\cap\operatorname{MD}(P)=j(1_2\otimes\mathcal{B})=1_3\otimes\mathcal{B}. P ( M 3 ( B )) ∩ MD ( P ) = j ( 1 2 ⊗ B ) = 1 3 ⊗ B . Finally, an invariant state φ \varphi φ φ = φ ∘ P \varphi=\varphi\circ P φ = φ ∘ P
P ( e 33 ⊗ 1 B ) = 0 , P(e_{33}\otimes1_{\mathcal{B}})=0, P ( e 33 ⊗ 1 B ) = 0 , every invariant state vanishes on the non-zero positive element e 33 ⊗ 1 B e_{33}\otimes1_{\mathcal{B}} e 33 ⊗ 1 B
Lip-norm criteria and metric consequences
Definition 8.1. Let A \mathcal{A} A C ∗ C^* C ∗ L ⊆ A s a \mathcal{L}\subseteq\mathcal{A}_{\mathrm{sa}} L ⊆ A sa L : L → [ 0 , ∞ ) L:\mathcal{L}\to[0,\infty) L : L → [ 0 , ∞ ) Lip-norm if:
(i) L ( a ) = 0 L(a)=0 L ( a ) = 0 a ∈ R 1 a\in\mathbb{R}1 a ∈ R 1
(ii) the Monge–Kantorovich metric
d L ( φ , ψ ) : = sup { ∣ φ ( a ) − ψ ( a ) ∣ : a ∈ L , L ( a ) ≤ 1 } d_L(\varphi,\psi):=\sup\{|\varphi(a)-\psi(a)|:a\in\mathcal{L},\ L(a)\leq1\} d L ( φ , ψ ) := sup { ∣ φ ( a ) − ψ ( a ) ∣ : a ∈ L , L ( a ) ≤ 1 } induces the weak-* topology on S ( A ) S(\mathcal{A}) S ( A )
Remark 8.2. This is the standard compact quantum metric-space framework of Rieffel [42 ] . The only point we need is that weak-* neighbourhoods of a state may be described by small d L d_L d L
Lemma 8.3. Let L L L L \mathcal{L} L a ∈ L a \in\mathcal{L} a ∈ L φ , ψ ∈ S ( A ) \varphi,\psi\in S(\mathcal{A}) φ , ψ ∈ S ( A )
∣ φ ( a ) − ψ ( a ) ∣ ≤ L ( a ) d L ( φ , ψ ) . \lvert\varphi(a)-\psi(a)\rvert\leq L(a)\,d_L(\varphi,\psi). ∣ φ ( a ) − ψ ( a )∣ ≤ L ( a ) d L ( φ , ψ ) . Proof. If L ( a ) = 0 L(a)=0 L ( a ) = 0 a a a L ( a ) > 0 L(a)>0 L ( a ) > 0 d L d_L d L a / L ( a ) a/L(a) a / L ( a )
Proposition 8.4. Let A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M ( T t ) (T_t) ( T t ) L L L L \mathcal{L} L T t ( L ) ⊆ L T_t(\mathcal{L})\subseteq\mathcal{L} T t ( L ) ⊆ L t ≥ 0 t\geq0 t ≥ 0 C ≥ 0 C\geq0 C ≥ 0
L ( T t ( a ) ) ≤ C L ( a ) , a ∈ L , t ≥ 0 , (6) L(T_t(a))\leq C L(a),\qquad a\in\mathcal{L},\quad t\geq0, \tag*{(6)} L ( T t ( a )) ≤ C L ( a ) , a ∈ L , t ≥ 0 , (6) then ( T t ) (T_t) ( T t ) e e e L \mathcal{L} L
Proof. Fix a ∈ L a\in\mathcal{L} a ∈ L φ , ψ ∈ S ( A ) \varphi,\psi\in S(\mathcal{A}) φ , ψ ∈ S ( A ) Lemma 8.3 and (6) ,
∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ ≤ L ( T t ( a ) ) d L ( ψ , φ ) ≤ C L ( a ) d L ( ψ , φ ) \lvert\psi(T_t(a))-\varphi(T_t(a))\rvert\leq L(T_t(a))d_L(\psi,\varphi)\leq C L(a)d_L(\psi,\varphi) ∣ ψ ( T t ( a )) − φ ( T t ( a ))∣ ≤ L ( T t ( a )) d L ( ψ , φ ) ≤ C L ( a ) d L ( ψ , φ ) for all t ≥ 0 t\geq0 t ≥ 0 d L d_L d L
Corollary 8.5. Let A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M ( T t ) (T_t) ( T t ) L L L L \mathcal{L} L T t ( L ) ⊆ L T_t(\mathcal{L})\subseteq\mathcal{L} T t ( L ) ⊆ L t ≥ 0 t\geq0 t ≥ 0 t 0 ≥ 0 t_0\geq0 t 0 ≥ 0 C ≥ 0 C\geq0 C ≥ 0
L ( T t ( a ) ) ≤ C L ( a ) , a ∈ L , t ≥ t 0 , L(T_t(a))\leq C L(a),\qquad a\in\mathcal{L},\quad t\geq t_0, L ( T t ( a )) ≤ C L ( a ) , a ∈ L , t ≥ t 0 , then ( T t ) (T_t) ( T t ) e e e L \mathcal{L} L e e e L \mathcal{L} L
Proof. The first claim is proved exactly as in Proposition 8.4, but only for t ≥ t 0 t\geq t_0 t ≥ t 0 Theorem 3.2 . □
Corollary 8.6. Let A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M ( T t ) (T_t) ( T t ) δ : dom ( δ ) → E \delta:\operatorname{dom}(\delta)\to E δ : dom ( δ ) → E ∗ \ast ∗ ∗ \ast ∗ dom ( δ ) ⊆ A \operatorname{dom}(\delta)\subseteq\mathcal{A} dom ( δ ) ⊆ A A \mathcal{A} A E E E T t ( dom ( δ ) ) ⊆ dom ( δ ) T_t(\operatorname{dom}(\delta))\subseteq\operatorname{dom}(\delta) T t ( dom ( δ )) ⊆ dom ( δ ) t ≥ 0 t\geq0 t ≥ 0
L ( a ) : = ∥ δ ( a ) ∥ E , a ∈ dom ( δ ) s a , L(a):=\lVert\delta(a)\rVert_E,\qquad a\in\operatorname{dom}(\delta)_{\mathrm{sa}}, L ( a ) := ∥ δ ( a ) ∥ E , a ∈ dom ( δ ) sa , and assume that L L L dom ( δ ) s a \operatorname{dom}(\delta)_{\mathrm{sa}} dom ( δ ) sa
∥ δ ( T t ( a ) ) ∥ E ≤ C ∥ δ ( a ) ∥ E , a ∈ dom ( δ ) , t ≥ 0 , (7) \lVert\delta(T_t(a))\rVert_E\leq C\lVert\delta(a)\rVert_E,\qquad a\in\operatorname{dom}(\delta),\quad t\geq0, \tag*{(7)} ∥ δ ( T t ( a )) ∥ E ≤ C ∥ δ ( a ) ∥ E , a ∈ dom ( δ ) , t ≥ 0 , (7) then ( T t ) (T_t) ( T t ) e e e dom ( δ ) s a \operatorname{dom}(\delta)_{\mathrm{sa}} dom ( δ ) sa
More generally, if (7) is assumed only for t ≥ t 0 t\geq t_0 t ≥ t 0 ( T t ) (T_t) ( T t ) e e e e e e
Proof. Apply Proposition 8.4 and Corollary 8.5 with L ( a ) = ∥ δ ( a ) ∥ E L(a)=\lVert\delta(a)\rVert_E L ( a ) = ∥ δ ( a ) ∥ E
Remark 8.7. Here “closed” means that the graph of δ \delta δ A × E \mathcal{A}\times E A × E dom ( δ ) \operatorname{dom}(\delta) dom ( δ )
∥ x ∥ δ : = ∥ x ∥ + ∥ δ ( x ) ∥ E . \lVert x\rVert_\delta:=\lVert x\rVert+\lVert\delta(x)\rVert_E. ∥ x ∥ δ := ∥ x ∥ + ∥ δ ( x ) ∥ E . Theorem 8.8. Let A ⊆ M \mathcal{A} \subseteq\mathcal{M} A ⊆ M Feller algebra for ( T t ) (T_t) ( T t ) L L L Lip-norm on L \mathcal{L} L T t ( L ) ⊆ L T_t(\mathcal{L}) \subseteq\mathcal{L} T t ( L ) ⊆ L t ≥ 0 t \ge0 t ≥ 0 t 0 > 0 t_0 > 0 t 0 > 0 κ ∈ [ 0 , 1 ) \kappa\in[0,1) κ ∈ [ 0 , 1 )
L ( T t 0 ( a ) ) ≤ κ L ( a ) , a ∈ L . (8) L(T_{t_0}(a)) \le\kappa L(a), \qquad a \in\mathcal{L}. \tag*{(8)} L ( T t 0 ( a )) ≤ κ L ( a ) , a ∈ L . (8) Then the map Φ : S ( A ) → S ( A ) \Phi: S(\mathcal{A}) \to S(\mathcal{A}) Φ : S ( A ) → S ( A ) Φ ( φ ) = φ ∘ T t 0 \Phi(\varphi) = \varphi\circ T_{t_0} Φ ( φ ) = φ ∘ T t 0 ( S ( A ) , d L ) (S(\mathcal{A}),d_L) ( S ( A ) , d L ) φ ∗ ∈ S ( A ) \varphi_* \in S(\mathcal{A}) φ ∗ ∈ S ( A ) φ ∗ ∘ T t 0 = φ ∗ \varphi_* \circ T_{t_0} = \varphi_* φ ∗ ∘ T t 0 = φ ∗ φ ∗ \varphi_* φ ∗ φ ∈ S ( A ) \varphi\in S(\mathcal{A}) φ ∈ S ( A ) n ∈ N n \in\mathbb{N} n ∈ N
d L ( φ ∘ T n t 0 , φ ∗ ) ≤ κ n d L ( φ , φ ∗ ) . d_L(\varphi\circ T_{nt_0},\varphi_*) \le\kappa^n d_L(\varphi,\varphi_*). d L ( φ ∘ T n t 0 , φ ∗ ) ≤ κ n d L ( φ , φ ∗ ) . If, in addition, there exists C ≥ 0 C \ge0 C ≥ 0 L ( T t ( a ) ) ≤ C L ( a ) L(T_t(a)) \le C L(a) L ( T t ( a )) ≤ C L ( a ) a ∈ L a \in\mathcal{L} a ∈ L 0 ≤ t ≤ t 0 0 \le t \le t_0 0 ≤ t ≤ t 0 t ≥ 0 t \ge0 t ≥ 0
d L ( φ ∘ T t , φ ∗ ) ≤ C κ ⌊ t / t 0 ⌋ d L ( φ , φ ∗ ) . d_L(\varphi\circ T_t,\varphi_*) \le C\kappa^{\lfloor t/t_0\rfloor}d_L(\varphi,\varphi_*). d L ( φ ∘ T t , φ ∗ ) ≤ C κ ⌊ t / t 0 ⌋ d L ( φ , φ ∗ ) . Proof. For φ , ψ ∈ S ( A ) \varphi,\psi\in S(\mathcal{A}) φ , ψ ∈ S ( A ) a ∈ L a \in\mathcal{L} a ∈ L L ( a ) ≤ 1 L(a) \le1 L ( a ) ≤ 1 (8) give ∣ φ ( T t 0 ( a ) ) − ψ ( T t 0 ( a ) ) ∣ ≤ L ( T t 0 ( a ) ) d L ( φ , ψ ) ≤ κ d L ( φ , ψ ) |\varphi(T_{t_0}(a))-\psi(T_{t_0}(a))| \le L(T_{t_0}(a))d_L(\varphi,\psi) \le\kappa d_L(\varphi,\psi) ∣ φ ( T t 0 ( a )) − ψ ( T t 0 ( a )) ∣ ≤ L ( T t 0 ( a )) d L ( φ , ψ ) ≤ κ d L ( φ , ψ ) a a a
d L ( Φ ( φ ) , Φ ( ψ ) ) ≤ κ d L ( φ , ψ ) . d_L(\Phi(\varphi),\Phi(\psi)) \le\kappa d_L(\varphi,\psi). d L ( Φ ( φ ) , Φ ( ψ )) ≤ κ d L ( φ , ψ ) . Since ( S ( A ) , d L ) (S(\mathcal{A}),d_L) ( S ( A ) , d L ) φ ∗ \varphi_* φ ∗ Φ \Phi Φ
To prove (ii), fix s ≥ 0 s \ge0 s ≥ 0
( φ ∗ ∘ T s ) ∘ T t 0 = φ ∗ ∘ T s + t 0 = φ ∗ ∘ T t 0 + s = ( φ ∗ ∘ T t 0 ) ∘ T s = φ ∗ ∘ T s . (\varphi_* \circ T_s) \circ T_{t_0} = \varphi_* \circ T_{s+t_0} = \varphi_* \circ T_{t_0+s} = (\varphi_* \circ T_{t_0}) \circ T_s = \varphi_* \circ T_s. ( φ ∗ ∘ T s ) ∘ T t 0 = φ ∗ ∘ T s + t 0 = φ ∗ ∘ T t 0 + s = ( φ ∗ ∘ T t 0 ) ∘ T s = φ ∗ ∘ T s . Thus φ ∗ ∘ T s \varphi_* \circ T_s φ ∗ ∘ T s Φ \Phi Φ φ ∗ \varphi_* φ ∗
For the final estimate, write t = n t 0 + r t=nt_0+r t = n t 0 + r n = ⌊ t / t 0 ⌋ n=\lfloor t/t_0\rfloor n = ⌊ t / t 0 ⌋ 0 ≤ r < t 0 0 \le r<t_0 0 ≤ r < t 0
d L ( φ ∘ T t , φ ∗ ) = d L ( ( φ ∘ T r ) ∘ T n t 0 , φ ∗ ) ≤ κ n d L ( φ ∘ T r , φ ∗ ) = κ n d L ( φ ∘ T r , φ ∗ ∘ T r ) ≤ C κ n d L ( φ , φ ∗ ) , \begin{aligned}
d_L(\varphi\circ T_t,\varphi_*) = d_L((\varphi\circ T_r)\circ T_{nt_0},\varphi_*) \le\kappa^n d_L(\varphi\circ T_r,\varphi_*) \\
&= \kappa^n d_L(\varphi\circ T_r,\varphi_* \circ T_r) \le C\kappa^n d_L(\varphi,\varphi_*),
\end{aligned} d L ( φ ∘ T t , φ ∗ ) = d L (( φ ∘ T r ) ∘ T n t 0 , φ ∗ ) ≤ κ n d L ( φ ∘ T r , φ ∗ ) = κ n d L ( φ ∘ T r , φ ∗ ∘ T r ) ≤ C κ n d L ( φ , φ ∗ ) , as claimed. □
Remark 8.9. Theorem 8.8 is the metric-state-space version of the classical Wasserstein contraction principle. It is also one of the cleanest places where the commutative and the non-commutative theories genuinely coincide.
Corollary 8.10. Assume the hypotheses of Theorem 8.8. Suppose also that
diam ( S ( A ) , d L ) < ∞ \operatorname{diam}(S(\mathcal{A}),d_L)<\infty diam ( S ( A ) , d L ) < ∞ and that
L ( T t ( a ) ) ≤ C L ( a ) ( a ∈ L , 0 ≤ t ≤ t 0 ) . L(T_t(a)) \le C L(a) \qquad(a \in\mathcal{L},\ 0 \le t \le t_0). L ( T t ( a )) ≤ C L ( a ) ( a ∈ L , 0 ≤ t ≤ t 0 ) . Then, for the unique invariant state φ ∗ \varphi_* φ ∗ a ∈ L a \in\mathcal{L} a ∈ L
∥ T t ( a ) − φ ∗ ( a ) 1 ∥ ≤ C diam ( S ( A ) , d L ) κ ⌊ t / t 0 ⌋ L ( a ) . \lVert T_t(a)-\varphi_*(a)1\rVert\le C\operatorname{diam}(S(\mathcal{A}),d_L)\kappa^{\lfloor t/t_0\rfloor}L(a). ∥ T t ( a ) − φ ∗ ( a ) 1 ∥ ≤ C diam ( S ( A ) , d L ) κ ⌊ t / t 0 ⌋ L ( a ) . :::`
Proof. Use the dual contraction estimate from Theorem 8.8 , test it on a / L ( a ) a/L(a) a / L ( a )
Doeblin minorisation and strict-contraction maps
The global classical Doeblin condition says that one step of the dynamics dominates a fixed reference probability measure. In the present setting it is a strong sufficient condition for the e-property and belongs beside non-commutative Dobrushin and quantum Doeblin coefficients [26 , 27 ] . In operator-algebraic language the exact rank-one analogue is domination by the unital positive rank-one map a ↦ σ ( a ) 1 a \mapsto\sigma(a)1 a ↦ σ ( a ) 1
Theorem 9.1. Let T : A → A T:\mathcal{A}\to\mathcal{A} T : A → A A \mathcal{A} A Ψ : A → A \Psi:\mathcal{A}\to\mathcal{A} Ψ : A → A λ ∈ ( 0 , 1 ] \lambda\in(0,1] λ ∈ ( 0 , 1 ] β ∈ [ 0 , 1 ) \beta\in[0,1) β ∈ [ 0 , 1 )
T ( a ) ≥ λ Ψ ( a ) , a ∈ A + , (9) T(a)\ge\lambda\Psi(a),\qquad a\in\mathcal{A}_{+}, \tag*{(9)} T ( a ) ≥ λ Ψ ( a ) , a ∈ A + , (9) and
∥ φ ∘ Ψ − ψ ∘ Ψ ∥ ≤ β ∥ φ − ψ ∥ , φ , ψ ∈ S ( A ) . (10) \|\varphi\circ\Psi-\psi\circ\Psi\|\le\beta\|\varphi-\psi\|,\qquad\varphi,\psi\in S(\mathcal{A}). \tag*{(10)} ∥ φ ∘ Ψ − ψ ∘ Ψ∥ ≤ β ∥ φ − ψ ∥ , φ , ψ ∈ S ( A ) . (10) Put
q : = 1 − λ ( 1 − β ) ∈ [ 0 , 1 ) . q:=1-\lambda(1-\beta)\in[0,1). q := 1 − λ ( 1 − β ) ∈ [ 0 , 1 ) . Then:
(i) the dual map T ∗ : S ( A ) → S ( A ) T^*:S(\mathcal{A})\to S(\mathcal{A}) T ∗ : S ( A ) → S ( A ) q q q A ∗ \mathcal{A}^* A ∗
(ii) there exists a unique invariant state φ ∗ ∈ S ( A ) \varphi_*\in S(\mathcal{A}) φ ∗ ∈ S ( A ) T T T
(iii) for every φ , ψ ∈ S ( A ) \varphi,\psi\in S(\mathcal{A}) φ , ψ ∈ S ( A ) n ∈ N n\in\mathbb{N} n ∈ N
∥ φ ∘ T n − ψ ∘ T n ∥ ≤ q n ∥ φ − ψ ∥ , \|\varphi\circ T^n-\psi\circ T^n\|\le q^n\|\varphi-\psi\|, ∥ φ ∘ T n − ψ ∘ T n ∥ ≤ q n ∥ φ − ψ ∥ , (iv) for every a ∈ A s a a\in\mathcal{A}_{\mathrm{sa}} a ∈ A sa n ∈ N n\in\mathbb{N} n ∈ N
sup φ , ψ ∈ S ( A ) ∣ φ ( T n ( a ) ) − ψ ( T n ( a ) ) ∣ ≤ 2 ∥ a ∥ q n . \sup_{\varphi,\psi\in S(\mathcal{A})}\left|\varphi(T^n(a))-\psi(T^n(a))\right|\le2\|a\|q^n. φ , ψ ∈ S ( A ) sup ∣ φ ( T n ( a )) − ψ ( T n ( a )) ∣ ≤ 2∥ a ∥ q n . Proof. If λ = 1 \lambda=1 λ = 1 T ( a ) ≥ Ψ ( a ) T(a)\ge\Psi(a) T ( a ) ≥ Ψ ( a ) a ≥ 0 a\ge0 a ≥ 0 1 − a 1-a 1 − a 0 ≤ a ≤ 1 0\le a\le1 0 ≤ a ≤ 1
T ( 1 − a ) ≥ Ψ ( 1 − a ) , T(1-a)\ge\Psi(1-a), T ( 1 − a ) ≥ Ψ ( 1 − a ) , that is,
− T ( a ) ≥ − Ψ ( a ) . -T(a)\ge-\Psi(a). − T ( a ) ≥ − Ψ ( a ) . Hence T ( a ) ≤ Ψ ( a ) T(a)\le\Psi(a) T ( a ) ≤ Ψ ( a ) T = Ψ T=\Psi T = Ψ 0 < λ < 1 0<\lambda<1 0 < λ < 1
Q : = T − λ Ψ 1 − λ . Q:=\frac{T-\lambda\Psi}{1-\lambda}. Q := 1 − λ T − λ Ψ . By (9) , the map Q Q Q A + \mathcal{A}_{+} A + T ( 1 ) = Ψ ( 1 ) = 1 T(1)=\Psi(1)=1 T ( 1 ) = Ψ ( 1 ) = 1 Q ( 1 ) = 1 Q(1)=1 Q ( 1 ) = 1 Q Q Q Q ∗ Q^{*} Q ∗ φ , ψ \varphi,\psi φ , ψ
φ ∘ T − ψ ∘ T = λ ( φ ∘ Ψ − ψ ∘ Ψ ) + ( 1 − λ ) ( φ ∘ Q − ψ ∘ Q ) . \varphi\circ T-\psi\circ T=\lambda(\varphi\circ\Psi-\psi\circ\Psi)+(1-\lambda)(\varphi\circ Q-\psi\circ Q). φ ∘ T − ψ ∘ T = λ ( φ ∘ Ψ − ψ ∘ Ψ ) + ( 1 − λ ) ( φ ∘ Q − ψ ∘ Q ) . Using (10) and the contractivity of Q ∗ Q^{*} Q ∗
∥ φ ∘ T − ψ ∘ T ∥ ≤ λ ∥ φ ∘ Ψ − ψ ∘ Ψ ∥ + ( 1 − λ ) ∥ φ ∘ Q − ψ ∘ Q ∥ ≤ λ β ∥ φ − ψ ∥ + ( 1 − λ ) ∥ φ − ψ ∥ = q ∥ φ − ψ ∥ . \begin{aligned}
\|\varphi\circ T-\psi\circ T\| &\leq\lambda\|\varphi\circ\Psi-\psi\circ\Psi\|+(1-\lambda)\|\varphi\circ Q-\psi\circ Q\|\\
&\leq\lambda\beta\|\varphi-\psi\|+(1-\lambda)\|\varphi-\psi\|\\
&=q\|\varphi-\psi\|.
\end{aligned} ∥ φ ∘ T − ψ ∘ T ∥ ≤ λ ∥ φ ∘ Ψ − ψ ∘ Ψ∥ + ( 1 − λ ) ∥ φ ∘ Q − ψ ∘ Q ∥ ≤ λ β ∥ φ − ψ ∥ + ( 1 − λ ) ∥ φ − ψ ∥ = q ∥ φ − ψ ∥. This proves (i).
The state space is norm closed in A ∗ \mathcal{A}^{*} A ∗ φ ∗ \varphi_{*} φ ∗ T ∗ T^{*} T ∗ a ∈ A s a a\in\mathcal{A}_{\mathrm{sa}} a ∈ A sa
∣ φ ( T n ( a ) ) − ψ ( T n ( a ) ) ∣ ≤ ∥ a ∥ ∥ φ ∘ T n − ψ ∘ T n ∥ ≤ ∥ a ∥ q n ∥ φ − ψ ∥ \begin{aligned}
\left|\varphi(T^{n}(a))-\psi(T^{n}(a))\right|\leq\|a\|\|\varphi\circ T^{n}-\psi\circ T^{n}\| \\
&\leq\|a\|q^{n}\|\varphi-\psi\|
\end{aligned} ∣ φ ( T n ( a )) − ψ ( T n ( a )) ∣ ≤ ∥ a ∥∥ φ ∘ T n − ψ ∘ T n ∥ ≤ ∥ a ∥ q n ∥ φ − ψ ∥ ≤ 2 ∥ a ∥ q n . \leq2\|a\|q^{n}. ≤ 2∥ a ∥ q n . This proves (iv). □
Corollary 9.2. In Theorem 9.1 , if Ψ ( a ) = σ ( a ) 1 \Psi(a)=\sigma(a)1 Ψ ( a ) = σ ( a ) 1 σ ∈ S ( A ) \sigma\in S(\mathcal{A}) σ ∈ S ( A ) β = 0 \beta=0 β = 0 q = 1 − λ q=1-\lambda q = 1 − λ
Proof. For the rank-one map Ψ ( a ) = σ ( a ) 1 \Psi(a)=\sigma(a)1 Ψ ( a ) = σ ( a ) 1 φ ∘ Ψ = σ = ψ ∘ Ψ \varphi\circ\Psi=\sigma=\psi\circ\Psi φ ∘ Ψ = σ = ψ ∘ Ψ φ , ψ \varphi,\psi φ , ψ (10) holds with β = 0 \beta=0 β = 0 Theorem 9.1 . □
Corollary 9.3. Let ( T t ) t ≥ 0 (T_{t})_{t\geq0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M t 0 > 0 t_{0}>0 t 0 > 0 Ψ : A → A \Psi:\mathcal{A}\to\mathcal{A} Ψ : A → A λ ∈ ( 0 , 1 ] \lambda\in(0,1] λ ∈ ( 0 , 1 ] β ∈ [ 0 , 1 ) \beta\in[0,1) β ∈ [ 0 , 1 )
T t 0 ( a ) ≥ λ Ψ ( a ) , a ∈ A + , T_{t_{0}}(a)\geq\lambda\Psi(a),\qquad a\in\mathcal{A}_{+}, T t 0 ( a ) ≥ λ Ψ ( a ) , a ∈ A + , and (10) holds for Ψ \Psi Ψ q = 1 − λ ( 1 − β ) q=1-\lambda(1-\beta) q = 1 − λ ( 1 − β )
(i) there exists a unique state φ ∗ ∈ S ( A ) \varphi_{*}\in S(\mathcal{A}) φ ∗ ∈ S ( A )
(ii) for every φ , ψ ∈ S ( A ) \varphi,\psi\in S(\mathcal{A}) φ , ψ ∈ S ( A ) t ≥ 0 t\geq0 t ≥ 0
∥ φ ∘ T t − ψ ∘ T t ∥ ≤ q ⌊ t / t 0 ⌋ ∥ φ − ψ ∥ , \|\varphi\circ T_{t}-\psi\circ T_{t}\|\leq q^{\lfloor t/t_{0}\rfloor}\|\varphi-\psi\|, ∥ φ ∘ T t − ψ ∘ T t ∥ ≤ q ⌊ t / t 0 ⌋ ∥ φ − ψ ∥ , (iii) ( T t ) (T_{t}) ( T t ) e e e A s a \mathcal{A}_{\mathrm{sa}} A sa U = S ( A ) U=S(\mathcal{A}) U = S ( A ) e e e A s a \mathcal{A}_{\mathrm{sa}} A sa
Proof. Apply Theorem 9.1 to the skeleton map T : = T t 0 ∣ A T:=T_{t_{0}}|_{\mathcal{A}} T := T t 0 ∣ A φ ∗ \varphi_{*} φ ∗ φ ∗ ∘ T t 0 = φ ∗ \varphi_{*}\circ T_{t_{0}}=\varphi_{*} φ ∗ ∘ T t 0 = φ ∗
( φ ∗ ∘ T s ) ∘ T t 0 = φ ∗ ∘ T s + t 0 = φ ∗ ∘ T t 0 + s = ( φ ∗ ∘ T t 0 ) ∘ T s = φ ∗ ∘ T s , (\varphi_{*}\circ T_{s})\circ T_{t_{0}}=\varphi_{*}\circ T_{s+t_{0}}=\varphi_{*}\circ T_{t_{0}+s}=(\varphi_{*}\circ T_{t_{0}})\circ T_{s}=\varphi_{*}\circ T_{s}, ( φ ∗ ∘ T s ) ∘ T t 0 = φ ∗ ∘ T s + t 0 = φ ∗ ∘ T t 0 + s = ( φ ∗ ∘ T t 0 ) ∘ T s = φ ∗ ∘ T s , so uniqueness of the fixed point of T t 0 ∗ T_{t_0}^* T t 0 ∗ φ ∗ ∘ T s = φ ∗ \varphi_* \circ T_s = \varphi_* φ ∗ ∘ T s = φ ∗ s ≥ 0 s \ge0 s ≥ 0
For (ii), write t = n t 0 + r t=nt_0+r t = n t 0 + r n = ⌊ t / t 0 ⌋ n=\lfloor t/t_0\rfloor n = ⌊ t / t 0 ⌋ 0 ≤ r < t 0 0 \le r < t_0 0 ≤ r < t 0
∥ φ ∘ T t − ψ ∘ T t ∥ = ∥ ( φ ∘ T r ) ∘ T n t 0 − ( ψ ∘ T r ) ∘ T n t 0 ∥ ≤ q n ∥ φ ∘ T r − ψ ∘ T r ∥ ≤ q n ∥ φ − ψ ∥ , \begin{aligned}
\|\varphi\circ T_t-\psi\circ T_t\| &= \|(\varphi\circ T_r)\circ T_{nt_0}-(\psi\circ T_r)\circ T_{nt_0}\|\\
&\le q^n\|\varphi\circ T_r-\psi\circ T_r\|\\
&\le q^n\|\varphi-\psi\|,
\end{aligned} ∥ φ ∘ T t − ψ ∘ T t ∥ = ∥ ( φ ∘ T r ) ∘ T n t 0 − ( ψ ∘ T r ) ∘ T n t 0 ∥ ≤ q n ∥ φ ∘ T r − ψ ∘ T r ∥ ≤ q n ∥ φ − ψ ∥ , because the dual action of every unital positive map is norm contractive in the dual norm.
To prove (iii), fix a ∈ A s a a \in\mathcal{A}_{\mathrm{sa}} a ∈ A sa ε > 0 \varepsilon>0 ε > 0 N ∈ N N \in\mathbb{N} N ∈ N 2 ∥ a ∥ q N < ε 2\|a\|q^N<\varepsilon 2∥ a ∥ q N < ε τ : = N t 0 \tau:=Nt_0 τ := N t 0 φ , ψ ∈ S ( A ) \varphi,\psi\in S(\mathcal{A}) φ , ψ ∈ S ( A ) t ≥ τ t\ge\tau t ≥ τ
∣ φ ( T t ( a ) ) − ψ ( T t ( a ) ) ∣ ≤ ∥ a ∥ ∥ φ ∘ T t − ψ ∘ T t ∥ ≤ 2 ∥ a ∥ q ⌊ t / t 0 ⌋ ≤ 2 ∥ a ∥ q N < ε . |\varphi(T_t(a))-\psi(T_t(a))|\le\|a\|\|\varphi\circ T_t-\psi\circ T_t\|\le2\|a\|q^{\lfloor t/t_0\rfloor}\le2\|a\|q^N<\varepsilon. ∣ φ ( T t ( a )) − ψ ( T t ( a )) ∣ ≤ ∥ a ∥∥ φ ∘ T t − ψ ∘ T t ∥ ≤ 2∥ a ∥ q ⌊ t / t 0 ⌋ ≤ 2∥ a ∥ q N < ε . Thus the eventual e e e Theorem 3.2 now gives the e e e
Corollary 9.4. Let η : A → M m ( C ) \eta:\mathcal{A}\to M_m(\mathbb{C}) η : A → M m ( C ) θ : M m ( C ) → A \theta:M_m(\mathbb{C})\to\mathcal{A} θ : M m ( C ) → A R : M m ( C ) → M m ( C ) R:M_m(\mathbb{C})\to M_m(\mathbb{C}) R : M m ( C ) → M m ( C )
Ψ : = θ ∘ R ∘ η . \Psi:=\theta\circ R\circ\eta. Ψ := θ ∘ R ∘ η . Assume that there exists β ∈ [ 0 , 1 ) \beta\in[0,1) β ∈ [ 0 , 1 )
∥ ρ ∘ R − ω ∘ R ∥ ≤ β ∥ ρ − ω ∥ , ρ , ω ∈ S ( M m ( C ) ) . \|\rho\circ R-\omega\circ R\|\le\beta\|\rho-\omega\|,\qquad\rho,\omega\in S(M_m(\mathbb{C})). ∥ ρ ∘ R − ω ∘ R ∥ ≤ β ∥ ρ − ω ∥ , ρ , ω ∈ S ( M m ( C )) . Then Ψ \Psi Ψ (10) with the same constant β \beta β
T t 0 ( a ) ≥ λ Ψ ( a ) , a ∈ A + , T_{t_0}(a)\ge\lambda\Psi(a),\qquad a\in\mathcal{A}_+, T t 0 ( a ) ≥ λ Ψ ( a ) , a ∈ A + , falls under Corollary 9.3 .
Proof. Let φ , ψ ∈ S ( A ) \varphi,\psi\in S(\mathcal{A}) φ , ψ ∈ S ( A ) η \eta η θ \theta θ φ ∘ θ \varphi\circ\theta φ ∘ θ ψ ∘ θ \psi\circ\theta ψ ∘ θ M m ( C ) M_m(\mathbb{C}) M m ( C ) R R R
∥ φ ∘ Ψ − ψ ∘ Ψ ∥ = ∥ ( φ ∘ θ ) ∘ R ∘ η − ( ψ ∘ θ ) ∘ R ∘ η ∥ ≤ ∥ ( φ ∘ θ ) ∘ R − ( ψ ∘ θ ) ∘ R ∥ ≤ β ∥ φ ∘ θ − ψ ∘ θ ∥ ≤ β ∥ φ − ψ ∥ . \begin{aligned}
\|\varphi\circ\Psi-\psi\circ\Psi\|&=\|(\varphi\circ\theta)\circ R\circ\eta-(\psi\circ\theta)\circ R\circ\eta\|\\
&\le\|(\varphi\circ\theta)\circ R-(\psi\circ\theta)\circ R\|\\
&\le\beta\|\varphi\circ\theta-\psi\circ\theta\|\\
&\le\beta\|\varphi-\psi\|.
\end{aligned} ∥ φ ∘ Ψ − ψ ∘ Ψ∥ = ∥ ( φ ∘ θ ) ∘ R ∘ η − ( ψ ∘ θ ) ∘ R ∘ η ∥ ≤ ∥ ( φ ∘ θ ) ∘ R − ( ψ ∘ θ ) ∘ R ∥ ≤ β ∥ φ ∘ θ − ψ ∘ θ ∥ ≤ β ∥ φ − ψ ∥. Hence (10) holds for Ψ \Psi Ψ
Proposition 9.5. Let S , T : A → A S,T:\mathcal{A}\to\mathcal{A} S , T : A → A Ψ , Φ : A → A \Psi,\Phi:\mathcal{A}\to\mathcal{A} Ψ , Φ : A → A λ , κ ∈ ( 0 , 1 ] \lambda,\kappa\in(0,1] λ , κ ∈ ( 0 , 1 ] β ∈ [ 0 , 1 ) \beta\in[0,1) β ∈ [ 0 , 1 )
T ( a ) ≥ λ Ψ ( a ) , a ∈ A + , (11) T(a)\ge\lambda\Psi(a),\qquad a\in\mathcal{A}_+, \tag*{(11)} T ( a ) ≥ λ Ψ ( a ) , a ∈ A + , (11) S ( Ψ ( a ) ) ≥ κ Φ ( a ) , a ∈ A + , (12) S(\Psi(a))\ge\kappa\Phi(a),\qquad a\in\mathcal{A}_+, \tag*{(12)} S ( Ψ ( a )) ≥ κ Φ ( a ) , a ∈ A + , (12) and
(9.5)
∥ φ ∘ Φ − ψ ∘ Φ ∥ ≤ β ∥ φ − ψ ∥ , φ , ψ ∈ S ( A ) . (13) \lVert\varphi\circ\Phi- \psi\circ\Phi\rVert\le\beta\lVert\varphi- \psi\rVert,\qquad\varphi,\psi\in\mathcal{S}(\mathcal{A}). \tag*{(13)} ∥ φ ∘ Φ − ψ ∘ Φ ∥ ≤ β ∥ φ − ψ ∥ , φ , ψ ∈ S ( A ) . (13) If, moreover, Φ \Phi Φ
S T ( a ) ≥ λ κ Φ ( a ) , a ∈ A + , ST(a) \ge\lambda\kappa\Phi(a),\qquad a \in\mathcal{A}_{+}, S T ( a ) ≥ λκ Φ ( a ) , a ∈ A + , and therefore S T ST S T Theorem 9.1 with the same constant β \beta β
Proof. If a ∈ A + a \in\mathcal{A}_{+} a ∈ A + S S S (11) ,
S T ( a ) = S ( T ( a ) ) ≥ λ S ( Ψ ( a ) ) . ST(a)=S(T(a))\ge\lambda S(\Psi(a)). S T ( a ) = S ( T ( a )) ≥ λ S ( Ψ ( a )) . Using (12) , we obtain
S T ( a ) ≥ λ κ Φ ( a ) . ST(a)\ge\lambda\kappa\Phi(a). S T ( a ) ≥ λκ Φ ( a ) . Since Φ \Phi Φ (13) , Theorem 9.1 applies to the map S T ST S T
Corollary 9.6. Let ( T t ) t ≥ 0 (T_t)_{t\ge0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊂ M \mathcal{A}\subset\mathcal{M} A ⊂ M h ∈ A + h\in\mathcal{A}_{+} h ∈ A + σ ∈ S ( A ) \sigma\in\mathcal{S}(\mathcal{A}) σ ∈ S ( A ) t 0 , s > 0 t_0,s>0 t 0 , s > 0 λ , κ ∈ ( 0 , 1 ] \lambda,\kappa\in(0,1] λ , κ ∈ ( 0 , 1 ]
T t 0 ( a ) ≥ λ σ ( a ) h , a ∈ A + , (14) T_{t_0}(a)\ge\lambda\sigma(a)h,\qquad a\in\mathcal{A}_{+}, \tag*{(14)} T t 0 ( a ) ≥ λσ ( a ) h , a ∈ A + , (14) and
T s ( h ) ≥ κ 1. (15) T_s(h)\ge\kappa1. \tag*{(15)} T s ( h ) ≥ κ 1. (15) Then
T s + t 0 ( a ) ≥ λ κ σ ( a ) 1 , a ∈ A + . T_{s+t_0}(a)\ge\lambda\kappa\sigma(a)1,\qquad a\in\mathcal{A}_{+}. T s + t 0 ( a ) ≥ λκσ ( a ) 1 , a ∈ A + . Consequently ( T t ) (T_t) ( T t ) Corollary 9.3 , hence has a unique invariant state on A \mathcal{A} A e e e A s a \mathcal{A}_{\mathrm{sa}} A sa
Proof. Apply Proposition 9.5 with
T : = T t 0 ∣ A , S : = T s ∣ A , Ψ ( a ) : = σ ( a ) h , Φ ( a ) : = σ ( a ) 1. T:=T_{t_0}|_{\mathcal{A}},\qquad S:=T_s|_{\mathcal{A}},\qquad\Psi(a):=\sigma(a)h,\qquad\Phi(a):=\sigma(a)1. T := T t 0 ∣ A , S := T s ∣ A , Ψ ( a ) := σ ( a ) h , Φ ( a ) := σ ( a ) 1. The map Φ \Phi Φ (13) with β = 0 \beta=0 β = 0 (12) is exactly the consequence of (15) ,
S ( Ψ ( a ) ) = σ ( a ) T s ( h ) ≥ κ σ ( a ) 1 = κ Φ ( a ) . S(\Psi(a))=\sigma(a)T_s(h)\ge\kappa\sigma(a)1=\kappa\Phi(a). S ( Ψ ( a )) = σ ( a ) T s ( h ) ≥ κσ ( a ) 1 = κ Φ ( a ) . Hence
T s + t 0 ( a ) = S T ( a ) ≥ λ κ σ ( a ) 1 , a ∈ A + , T_{s+t_0}(a)=ST(a)\ge\lambda\kappa\sigma(a)1,\qquad a\in\mathcal{A}_{+}, T s + t 0 ( a ) = S T ( a ) ≥ λκσ ( a ) 1 , a ∈ A + , as claimed. □
Corollary 9.7. Let ( T t ) t ≥ 0 (T_t)_{t\ge0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊂ M \mathcal{A}\subset\mathcal{M} A ⊂ M p ∈ A p\in\mathcal{A} p ∈ A σ ∈ S ( A ) \sigma\in\mathcal{S}(\mathcal{A}) σ ∈ S ( A ) t 0 , s > 0 t_0,s>0 t 0 , s > 0 λ , κ ∈ ( 0 , 1 ] \lambda,\kappa\in(0,1] λ , κ ∈ ( 0 , 1 ]
p T t 0 ( a ) p ≥ λ σ ( a ) p , a ∈ A + , pT_{t_0}(a)p\ge\lambda\sigma(a)p,\qquad a\in\mathcal{A}_{+}, p T t 0 ( a ) p ≥ λσ ( a ) p , a ∈ A + , T s ( p ) ≥ κ 1 , T_s(p)\ge\kappa1, T s ( p ) ≥ κ 1 , and, in addition,
[ p , T t 0 ( a ) ] = 0 ( a ∈ A + ) . [p,T_{t_0}(a)]=0\qquad(a\in\mathcal{A}_{+}). [ p , T t 0 ( a )] = 0 ( a ∈ A + ) . Then the conclusion of Corollary 9.6 holds. In particular, the conclusion holds whenever p ∈ Z ( A ) p \in Z(\mathcal{A}) p ∈ Z ( A )
Proof. If x ∈ A + x \in\mathcal{A}_{+} x ∈ A + p p p
x = p x p + ( 1 − p ) x ( 1 − p ) ≥ p x p . x=pxp+(1-p)x(1-p)\ge pxp. x = p x p + ( 1 − p ) x ( 1 − p ) ≥ p x p . Applying this to x = T t 0 ( a ) x=\mathcal{T}_{t_{0}}(a) x = T t 0 ( a )
T t 0 ( a ) ≥ p T t 0 ( a ) p ≥ λ σ ( a ) p , a ∈ A + . \mathcal{T}_{t_{0}}(a)\ge p\mathcal{T}_{t_{0}}(a)p\ge\lambda\sigma(a)p,\qquad a\in\mathcal{A}_{+}. T t 0 ( a ) ≥ p T t 0 ( a ) p ≥ λσ ( a ) p , a ∈ A + . Now apply Corollary 9.6. □ \square □
Corollary 9.9. Let ( T t ) t ≥ 0 (\mathcal{T}_{t})_{t\ge0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M h ∈ A + h\in\mathcal{A}_{+} h ∈ A + σ ∈ S ( A ) \sigma\in\mathcal{S}(\mathcal{A}) σ ∈ S ( A ) t 0 > 0 t_{0}>0 t 0 > 0 S > 0 S>0 S > 0 λ , κ ∈ ( 0 , 1 ] \lambda,\kappa\in(0,1] λ , κ ∈ ( 0 , 1 ]
T t 0 ( a ) ≥ λ σ ( a ) h , a ∈ A + , \mathcal{T}_{t_{0}}(a)\ge\lambda\sigma(a)h,\qquad a\in\mathcal{A}_{+}, T t 0 ( a ) ≥ λσ ( a ) h , a ∈ A + , and
1 S ∫ 0 S T s ( h ) d s ≥ κ 1. (16) \frac{1}{S}\int_{0}^{S}\mathcal{T}_{s}(h)\,\mathrm{d}s\ge\kappa1. \tag*{(16)} S 1 ∫ 0 S T s ( h ) d s ≥ κ 1. (16) Then the averaged map
R : = M S ∘ T t 0 : A → A R:=M_{S}\circ\mathcal{T}_{t_{0}}:\mathcal{A}\to\mathcal{A} R := M S ∘ T t 0 : A → A satisfies the global rank-one Doeblin minorisation
R ( a ) ≥ λ κ σ ( a ) 1 , a ∈ A + . R(a)\ge\lambda\kappa\sigma(a)1,\qquad a\in\mathcal{A}_{+}. R ( a ) ≥ λκσ ( a ) 1 , a ∈ A + . Consequently R R R φ ∗ ∈ S ( A ) \varphi_{*}\in\mathcal{S}(\mathcal{A}) φ ∗ ∈ S ( A ) ( T t ) (\mathcal{T}_{t}) ( T t ) R R R a ∈ A s a a\in\mathcal{A}_{\mathrm{sa}} a ∈ A sa n ∈ N n\in\mathbb{N} n ∈ N
∥ R n ( a ) − φ ∗ ( a ) 1 ∥ ≤ 2 ∥ a ∥ ( 1 − λ κ ) n . \lVert R^{n}(a)-\varphi_{*}(a)1\rVert\le2\lVert a\rVert(1-\lambda\kappa)^{n}. ∥ R n ( a ) − φ ∗ ( a ) 1 ∥ ≤ 2 ∥ a ∥ ( 1 − λκ ) n . Proof. If a ∈ A + a\in\mathcal{A}_{+} a ∈ A +
R ( a ) = M S ( T t 0 ( a ) ) ≥ λ σ ( a ) M S ( h ) ≥ λ κ σ ( a ) 1. R(a)=M_{S}(\mathcal{T}_{t_{0}}(a))\ge\lambda\sigma(a)M_{S}(h)\ge\lambda\kappa\sigma(a)1. R ( a ) = M S ( T t 0 ( a )) ≥ λσ ( a ) M S ( h ) ≥ λκσ ( a ) 1. Thus R R R Corollary 9.2 . Let φ ∗ \varphi_{*} φ ∗
For every u ≥ 0 u\ge0 u ≥ 0 T u \mathcal{T}_{u} T u M S M_{S} M S T t 0 \mathcal{T}_{t_{0}} T t 0
( φ ∗ ∘ T u ) ∘ R = ( φ ∗ ∘ R ) ∘ T u = φ ∗ ∘ T u . (\varphi_{*}\circ\mathcal{T}_{u})\circ R=(\varphi_{*}\circ R)\circ\mathcal{T}_{u}=\varphi_{*}\circ\mathcal{T}_{u}. ( φ ∗ ∘ T u ) ∘ R = ( φ ∗ ∘ R ) ∘ T u = φ ∗ ∘ T u . Thus φ ∗ ∘ T u \varphi_{*}\circ\mathcal{T}_{u} φ ∗ ∘ T u R R R φ ∗ ∘ T u = φ ∗ \varphi_{*}\circ\mathcal{T}_{u}=\varphi_{*} φ ∗ ∘ T u = φ ∗
Finally, if a ∈ A s a a\in\mathcal{A}_{\mathrm{sa}} a ∈ A sa R n ( a ) − φ ∗ ( a ) 1 R^{n}(a)-\varphi_{*}(a)1 R n ( a ) − φ ∗ ( a ) 1 φ ∗ \varphi_{*} φ ∗ R R R
∥ R n ( a ) − φ ∗ ( a ) 1 ∥ = sup ψ ∈ S ( A ) ∣ ψ ( R n ( a ) ) − φ ∗ ( R n ( a ) ) ∣ ≤ 2 ∥ a ∥ ( 1 − λ κ ) n \lVert R^{n}(a)-\varphi_{*}(a)1\rVert
=\sup_{\psi\in\mathcal{S}(\mathcal{A})}\left|\psi(R^{n}(a))-\varphi_{*}(R^{n}(a))\right|
\le2\lVert a\rVert(1-\lambda\kappa)^{n} ∥ R n ( a ) − φ ∗ ( a ) 1 ∥ = ψ ∈ S ( A ) sup ∣ ψ ( R n ( a )) − φ ∗ ( R n ( a )) ∣ ≤ 2 ∥ a ∥ ( 1 − λκ ) n by Theorem 9.1 . □ \square □
Remark 9.10. A corner estimate of the form
p T t 0 ( a ) p ≥ λ σ ( a ) p pT_{t_0}(a)p \geq\lambda\sigma(a)p p T t 0 ( a ) p ≥ λσ ( a ) p does not by itself imply the global lower bound
T t 0 ( a ) ≥ λ σ ( a ) p , T_{t_0}(a) \geq\lambda\sigma(a)p, T t 0 ( a ) ≥ λσ ( a ) p , because for non-central projections one cannot infer x ≥ p x p x \geq pxp x ≥ p x p x x x [32 ] is needed as the genuine local non-commutative substitute for the classical small-set peeling argument. By contrast, the global Doeblin mechanism above only needs a positive return profile h ∈ A + h \in\mathcal{A}_{+} h ∈ A + (14) and (15) ; projections enter there only as a particularly important special case.
Remark 9.11. Corollary 9.9 is the natural Cesàro version of Corollary 9.6: mean return to the positive profile h h h R = M S ∘ T t 0 R = M_{S} \circ T_{t_0} R = M S ∘ T t 0 § 5 re-enters.
Remark 9.12. Corollary 9.4 explains the possible rôle of nuclearity or injectivity. Those approximation properties furnish only approximate finite-dimensional unital completely positive factorisations of the identity map, and therefore a plentiful supply of candidate finite-dimensional smoothing channels Ψ \Psi Ψ [7 Ch. 2] . What they do not provide by themselves is either an exact factorisation of the identity through a finite-dimensional algebra or the order lower bound T t 0 ≥ λ Ψ T_{t_0} \geq\lambda\Psi T t 0 ≥ λ Ψ
Proposition 9.13. Let T : M n ( C ) → M n ( C ) T \colon M_n(\mathbb{C}) \to M_n(\mathbb{C}) T : M n ( C ) → M n ( C ) unital completely positive map . Assume that T ( a ) T(a) T ( a ) a ∈ M n ( C ) a \in M_n(\mathbb{C}) a ∈ M n ( C ) λ > 0 \lambda> 0 λ > 0
T ( a ) ≥ λ tr n ( a ) 1 , a ∈ M n ( C ) + , T(a) \geq\lambda\operatorname{tr}_n(a)1, \qquad a \in M_n(\mathbb{C})_{+}, T ( a ) ≥ λ tr n ( a ) 1 , a ∈ M n ( C ) + , where tr n \operatorname{tr}_n tr n M n ( C ) M_n(\mathbb{C}) M n ( C ) T T T
Proof. Let
K : = { ρ ∈ M n ( C ) + : tr n ( ρ ) = 1 } . K := \{\rho\in M_n(\mathbb{C})_{+} : \operatorname{tr}_n(\rho) = 1\}. K := { ρ ∈ M n ( C ) + : tr n ( ρ ) = 1 } . This is a compact subset of M n ( C ) M_n(\mathbb{C}) M n ( C )
ρ ⟼ λ min ( T ( ρ ) ) \rho\longmapsto\lambda_{\min}(T(\rho)) ρ ⟼ λ m i n ( T ( ρ )) is continuous on K K K λ min \lambda_{\min} λ m i n K K K λ \lambda λ
T ( ρ ) ≥ λ 1 , ρ ∈ K . T(\rho) \geq\lambda1, \qquad\rho\in K. T ( ρ ) ≥ λ 1 , ρ ∈ K . If a ∈ M n ( C ) + a \in M_n(\mathbb{C})_+ a ∈ M n ( C ) + a = tr n ( a ) ρ a=\operatorname{tr}_n(a)\rho a = tr n ( a ) ρ ρ ∈ K \rho\in K ρ ∈ K
T ( a ) = tr n ( a ) T ( ρ ) ≥ λ tr n ( a ) 1. T(a)=\operatorname{tr}_n(a)T(\rho)\ge\lambda\operatorname{tr}_n(a)1. T ( a ) = tr n ( a ) T ( ρ ) ≥ λ tr n ( a ) 1. The last assertion follows from Corollary 9.2 . □
Corollary 9.14. Let η : A → M m ( C ) \eta:\mathcal{A}\to M_m(\mathbb{C}) η : A → M m ( C ) θ : M m ( C ) → A \theta:M_m(\mathbb{C})\to\mathcal{A} θ : M m ( C ) → A R : M m ( C ) → M m ( C ) R:M_m(\mathbb{C})\to M_m(\mathbb{C}) R : M m ( C ) → M m ( C ) Proposition 9.13 , and put
Ψ : = θ ∘ R ∘ η . \Psi:=\theta\circ R\circ\eta. Ψ := θ ∘ R ∘ η . Then there exists λ R > 0 \lambda_R>0 λ R > 0
Ψ ( a ) ≥ λ R ( tr m ∘ η ) ( a ) 1 , a ∈ A + . \Psi(a)\ge\lambda_R(\operatorname{tr}_m\circ\eta)(a)1,\qquad a\in\mathcal{A}_+. Ψ ( a ) ≥ λ R ( tr m ∘ η ) ( a ) 1 , a ∈ A + . Consequently, every minorisation of the form
T t 0 ( a ) ≥ μ Ψ ( a ) , a ∈ A + , T_{t_0}(a)\ge\mu\Psi(a),\qquad a\in\mathcal{A}_+, T t 0 ( a ) ≥ μ Ψ ( a ) , a ∈ A + , automatically improves to the rank-one Doeblin bound
T t 0 ( a ) ≥ μ λ R ( tr m ∘ η ) ( a ) 1 , a ∈ A + . T_{t_0}(a)\ge\mu\lambda_R(\operatorname{tr}_m\circ\eta)(a)1,\qquad a\in\mathcal{A}_+. T t 0 ( a ) ≥ μ λ R ( tr m ∘ η ) ( a ) 1 , a ∈ A + . In particular, Corollary 9.3 applies.
Proof. By Proposition 9.13 , there exists λ R > 0 \lambda_R>0 λ R > 0
R ( b ) ≥ λ R tr m ( b ) 1 , b ∈ M m ( C ) + . R(b)\ge\lambda_R\operatorname{tr}_m(b)1,\qquad b\in M_m(\mathbb{C})_+. R ( b ) ≥ λ R tr m ( b ) 1 , b ∈ M m ( C ) + . Therefore, for a ∈ A + a\in\mathcal{A}_+ a ∈ A +
Ψ ( a ) = θ ( R ( η ( a ) ) ) ≥ λ R tr m ( η ( a ) ) θ ( 1 ) = λ R ( tr m ∘ η ) ( a ) 1. \Psi(a)=\theta(R(\eta(a)))\ge\lambda_R\operatorname{tr}_m(\eta(a))\theta(1)=\lambda_R(\operatorname{tr}_m\circ\eta)(a)1. Ψ ( a ) = θ ( R ( η ( a ))) ≥ λ R tr m ( η ( a )) θ ( 1 ) = λ R ( tr m ∘ η ) ( a ) 1. The final assertion is immediate. □
Proposition 9.15. Let A \mathcal{A} A C ∗ C^* C ∗ σ 1 , … , σ m ∈ S ( A ) \sigma_1,\ldots,\sigma_m\in S(\mathcal{A}) σ 1 , … , σ m ∈ S ( A ) h 1 , … , h m ∈ A + h_1,\ldots,h_m\in\mathcal{A}_+ h 1 , … , h m ∈ A +
∑ j = 1 m h j = 1. \sum_{j=1}^{m}h_j=1. j = 1 ∑ m h j = 1. Define the unital positive measure-prepare map
Ψ ( a ) : = ∑ j = 1 m σ j ( a ) h j , a ∈ A . \Psi(a):=\sum_{j=1}^{m}\sigma_j(a)h_j,\qquad a\in\mathcal{A}. Ψ ( a ) := j = 1 ∑ m σ j ( a ) h j , a ∈ A . Assume that there are constants r j ≥ 0 r_j\ge0 r j ≥ 0
h j ≥ r j 1 ( 1 ≤ j ≤ m ) . h_j\ge r_j1\qquad(1\le j\le m). h j ≥ r j 1 ( 1 ≤ j ≤ m ) . Put r : = ∑ j = 1 m r j r:=\sum_{j=1}^{m}r_j r := ∑ j = 1 m r j
σ : = 1 r ∑ j = 1 m r j σ j . \sigma:=\frac{1}{r}\sum_{j=1}^{m}r_j\sigma_j. σ := r 1 j = 1 ∑ m r j σ j . Then
Ψ ( a ) ≥ r σ ( a ) 1 , a ∈ A + . \Psi(a)\ge r\sigma(a)1,\qquad a\in\mathcal{A}_+. Ψ ( a ) ≥ r σ ( a ) 1 , a ∈ A + . Consequently, every lower bound
T t 0 ( a ) ≥ μ Ψ ( a ) , a ∈ A + , \mathcal{T}_{t_0}(a) \ge\mu\Psi(a), \qquad a \in\mathcal{A}_+, T t 0 ( a ) ≥ μ Ψ ( a ) , a ∈ A + , upgrades to the rank-one Doeblin bound
T t 0 ( a ) ≥ μ r σ ( a ) 1 , a ∈ A + . \mathcal{T}_{t_0}(a) \ge\mu r\sigma(a)1, \qquad a \in\mathcal{A}_+. T t 0 ( a ) ≥ μ r σ ( a ) 1 , a ∈ A + . Proof. For a ∈ A + a \in\mathcal{A}_+ a ∈ A +
Ψ ( a ) = ∑ j = 1 m σ j ( a ) h j ≥ ∑ j = 1 m r j σ j ( a ) 1 = r σ ( a ) 1. \Psi(a)=\sum_{j=1}^{m}\sigma_j(a)h_j\ge\sum_{j=1}^{m}r_j\sigma_j(a)1=r\sigma(a)1. Ψ ( a ) = j = 1 ∑ m σ j ( a ) h j ≥ j = 1 ∑ m r j σ j ( a ) 1 = r σ ( a ) 1. The final assertion is immediate. Notice that r ≤ 1 r \le1 r ≤ 1 1 = ∑ j h j ≥ r 1 1=\sum_j h_j\ge r1 1 = ∑ j h j ≥ r 1
Remark 9.16. In finite-dimensional quantum information language, maps of the form in Proposition 9.15 are the basic entanglement-breaking, or measure-prepare, channels. The proposition says that an exact lower bound by such a channel becomes a genuine rank-one Doeblin bound as soon as the prepared positive elements have a uniform scalar part. This is stronger than what nuclearity or injectivity provide by themselves: those approximation properties give approximate finite-dimensional factorisations of the identity, whereas the proposition requires an exact minorising map with a quantitative order lower bound.
Remark 9.17. Proposition 9.13 gives the simplest finite-dimensional model for the non-commutative Doeblin condition. Corollary 9.14 shows that exact finite-dimensional factorisations of a minorising channel, with a positivity-improving middle channel, upgrade factorised lower bounds to genuine rank-one Doeblin bounds. This clarifies the approximation-property discussion: exact factorisations of minorising channels can solve the lower-bound problem, while approximate factorisations coming from nuclearity or injectivity alone only produce candidate channels and not the required order inequalities. For standard finite-dimensional quantum-channel terminology and background, see for instance [48 , 49 ] .
Quantitative transfer and noiseless dynamics
For a bounded norm-continuous trajectory x : [ 0 , ∞ ) → A s a x\colon[0,\infty)\to\mathcal{A}_{\mathrm{sa}} x : [ 0 , ∞ ) → A sa χ ( x ) \chi(x) χ ( x ) e ( x ) e(x) e ( x ) { x ~ ( t ) : t ≥ 0 } \{\widetilde{x}(t):t\ge0\} { x ( t ) : t ≥ 0 } S ( A ) S(\mathcal{A}) S ( A )
Theorem 10.1. Let x , y : [ 0 , ∞ ) → A s a x,y\colon[0,\infty)\to\mathcal{A}_{\mathrm{sa}} x , y : [ 0 , ∞ ) → A sa
η : = lim sup t → ∞ ∥ x ( t ) − y ( t ) ∥ . \eta:=\limsup_{t\to\infty}\|x(t)-y(t)\|. η := t → ∞ lim sup ∥ x ( t ) − y ( t ) ∥. Then
∣ χ ( x ) − χ ( y ) ∣ ≤ η , ∣ e ( x ) − e ( y ) ∣ ≤ 2 η . |\chi(x)-\chi(y)|\le\eta,\qquad|e(x)-e(y)|\le2\eta. ∣ χ ( x ) − χ ( y ) ∣ ≤ η , ∣ e ( x ) − e ( y ) ∣ ≤ 2 η . If η = 0 \eta=0 η = 0 x x x y y y
Proof. Fix ε > 0 \varepsilon> 0 ε > 0 T ≥ 0 T \ge0 T ≥ 0
∥ x ( t ) − y ( t ) ∥ ≤ η + ε ( t ≥ T ) . \lVert x(t)-y(t)\rVert\le\eta+\varepsilon\qquad(t \ge T). ∥ x ( t ) − y ( t )∥ ≤ η + ε ( t ≥ T ) . The two tail ranges are therefore at Hausdorff distance at most η + ε \eta+\varepsilon η + ε η + ε \eta+\varepsilon η + ε
For the defect, on every U ⊆ S ( A ) U \subseteq S(A) U ⊆ S ( A ) t ≥ T t \ge T t ≥ T
osc U ( x ~ ( t ) ) ≤ osc U ( y ~ ( t ) ) + 2 ( η + ε ) . \operatorname{osc}_{U}(\widetilde{x}(t)) \le\operatorname{osc}_{U}(\widetilde{y}(t)) + 2(\eta+\varepsilon). osc U ( x ( t )) ≤ osc U ( y ( t )) + 2 ( η + ε ) . Taking the relevant suprema and infima, and then interchanging x x x y y y ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0
If η = 0 \eta=0 η = 0 t i → ∞ t_i \to\infty t i → ∞ x ( t i ) → z x(t_i) \to z x ( t i ) → z y ( t i ) → z y(t_i) \to z y ( t i ) → z
Corollary 10.2. Let B ⊂ A \mathcal{B} \subset\mathcal{A} B ⊂ A C ∗ C^* C ∗ E : A → B E:\mathcal{A} \to\mathcal{B} E : A → B a = a ∗ ∈ A a=a^* \in\mathcal{A} a = a ∗ ∈ A
η ( a ) : = lim sup t → ∞ ∥ T t ( a − E ( a ) ) ∥ . \eta(a):=\limsup_{t\to\infty}\lVert\mathcal{T}_t(a-E(a))\rVert. η ( a ) := t → ∞ lim sup ∥ T t ( a − E ( a ))∥ . Then
∣ χ T ( a ) − χ T ∣ B ( E ( a ) ) ∣ ≤ η ( a ) , \left|\chi_{\mathcal{T}}(a)-\chi_{\mathcal{T}|_{\mathcal{B}}}(E(a))\right|\le\eta(a), χ T ( a ) − χ T ∣ B ( E ( a )) ≤ η ( a ) , ∣ e T ( a ) − e T ∣ B ( E ( a ) ) ∣ ≤ 2 η ( a ) . \left|e_{\mathcal{T}}(a)-e_{\mathcal{T}|_{\mathcal{B}}}(E(a))\right|\le2\eta(a). e T ( a ) − e T ∣ B ( E ( a )) ≤ 2 η ( a ) . If η ( a ) = 0 \eta(a)=0 η ( a ) = 0 e e e
Proof. Apply Theorem 10.1 to
x ( t ) = T t ( a ) , y ( t ) = T t ( E ( a ) ) . x(t)=\mathcal{T}_t(a), \qquad y(t)=\mathcal{T}_t(E(a)). x ( t ) = T t ( a ) , y ( t ) = T t ( E ( a )) . Their difference is T t ( a − E ( a ) ) \mathcal{T}_t(a-E(a)) T t ( a − E ( a )) B \mathcal{B} B A \mathcal{A} A
r : S ( A ) → S ( B ) , r ( ρ ) = ρ ∣ B , r:S(\mathcal{A})\to S(\mathcal{B}), \qquad r(\rho)=\rho|_{\mathcal{B}}, r : S ( A ) → S ( B ) , r ( ρ ) = ρ ∣ B , and
i : S ( B ) → S ( A ) , i ( θ ) = θ ∘ E . i:S(\mathcal{B})\to S(\mathcal{A}), \qquad i(\theta)=\theta\circ E. i : S ( B ) → S ( A ) , i ( θ ) = θ ∘ E . Then r r r r ∘ i = id r\circ i=\operatorname{id} r ∘ i = id y y y S ( A ) S(\mathcal{A}) S ( A ) r r r S ( B ) S(\mathcal{B}) S ( B ) i ( S ( B ) ) i(S(\mathcal{B})) i ( S ( B ))
Example 10.3. Let ω 1 , ω 2 ∈ R \omega_1,\omega_2\in\mathbb{R} ω 1 , ω 2 ∈ R ω 1 / ω 2 ∉ Q \omega_1/\omega_2\notin\mathbb{Q} ω 1 / ω 2 ∈ / Q T 2 \mathbb{T}^2 T 2
θ t ( z 1 , z 2 ) = ( e i ω 1 t z 1 , e i ω 2 t z 2 ) . \theta_t(z_1,z_2)=\left(e^{i\omega_1t}z_1,e^{i\omega_2t}z_2\right). θ t ( z 1 , z 2 ) = ( e i ω 1 t z 1 , e i ω 2 t z 2 ) . Let m m m T 2 \mathbb{T}^2 T 2
M = L ∞ ( T 2 , B ( T 2 ) , m ) ⊗ ‾ M 2 ( C ) , A = C ( T 2 , M 2 ( C ) ) . \mathcal{M}=L^\infty(\mathbb{T}^2,\mathcal{B}(\mathbb{T}^2),m)\,\overline{\otimes}\,M_2(\mathbb{C}), \qquad\mathcal{A}=C(\mathbb{T}^2,M_2(\mathbb{C})). M = L ∞ ( T 2 , B ( T 2 ) , m ) ⊗ M 2 ( C ) , A = C ( T 2 , M 2 ( C )) . Let D 2 ⊆ M 2 ( C ) \mathcal{D}_2 \subseteq M_2(\mathbb{C}) D 2 ⊆ M 2 ( C ) E D 2 E_{\mathcal{D}_2} E D 2
D t = E D 2 + e − t ( id − E D 2 ) . D_t = E_{\mathcal{D}_2} + e^{-t}(\operatorname{id} - E_{\mathcal{D}_2}). D t = E D 2 + e − t ( id − E D 2 ) . Define
( T t f ) ( z ) : = D t ( f ( θ t z ) ) . (T_t f)(z) := D_t(f(\theta_t z)). ( T t f ) ( z ) := D t ( f ( θ t z )) . Then the formula extends to a weak-* continuous normal quantum Markov semigroup on M \mathcal{M} M A \mathcal{A} A e e e A s a \mathcal{A}_{\mathrm{sa}} A sa
E : C ( T 2 , M 2 ( C ) ) ⟶ C ( T 2 , D 2 ) , E : C(\mathbb{T}^2, M_2(\mathbb{C})) \longrightarrow C(\mathbb{T}^2, \mathcal{D}_2), E : C ( T 2 , M 2 ( C )) ⟶ C ( T 2 , D 2 ) , and its stable sector is the off-diagonal kernel of E E E
Proof. The base maps form a strongly continuous group of -automorphisms on A \mathcal{A} A continuous normal group on L ∞ ( T 2 , m ) L^\infty(\mathbb{T}^2,m) L ∞ ( T 2 , m ) ( D t ) (D_t) ( D t ) E D 2 E_{\mathcal{D}_2} E D 2 B = C ( T 2 , D 2 ) \mathcal{B}=C(\mathbb{T}^2,\mathcal{D}_2) B = C ( T 2 , D 2 ) ( e i ω 1 t , e i ω 2 t ) (e^{i\omega_1t},e^{i\omega_2t}) ( e i ω 1 t , e i ω 2 t ) B \mathcal{B} B
∥ T t ( f − E f ) ∥ = e − t ∥ f − E f ∥ . \|T_t(f-Ef)\|=e^{-t}\|f-Ef\|. ∥ T t ( f − E f ) ∥ = e − t ∥ f − E f ∥. Corollary 10.2 yields the e e e
For example,
b ( z 1 , z 2 ) = ( Re z 1 0 0 Re z 2 ) . b(z_1,z_2)=
\begin{pmatrix}
\operatorname{Re} z_1 & 0\\
0 & \operatorname{Re} z_2
\end{pmatrix}. b ( z 1 , z 2 ) = ( Re z 1 0 0 Re z 2 ) . has a non-periodic compact orbit. Indeed, a period τ > 0 \tau>0 τ > 0 e i ω 1 τ = 1 e^{i\omega_1\tau}=1 e i ω 1 τ = 1 e i ω 2 τ = 1 e^{i\omega_2\tau}=1 e i ω 2 τ = 1
For z ∈ T 2 z\in\mathbb{T}^2 z ∈ T 2 ρ z ( f ) = tr 2 ( f ( z ) ) \rho_z(f)=\operatorname{tr}_2(f(z)) ρ z ( f ) = tr 2 ( f ( z )) D t D_t D t
ρ z ∘ T t = ρ θ t z . \rho_z\circ T_t=\rho_{\theta_tz}. ρ z ∘ T t = ρ θ t z . Distinct points remain distinct under the flow, and a scalar function separating them shows that the dual-norm distance of the corresponding states remains two. Hence no time slice has a strict dual-norm contraction coefficient. □ \square □
Expected subalgebras and decoherence-free decompositions
We now turn to another source of equicontinuity: the existence of an invariant expected subalgebra. The guiding situation is that a conditional expectation E : A → B E:\mathcal{A}\to\mathcal{B} E : A → B ker E \ker E ker E e e e B \mathcal{B} B
Definition 11.1. Let A \mathcal{A} A D ⊆ A \mathcal{D}\subseteq\mathcal{A} D ⊆ A Cartan masa if:
D \mathcal{D} D A \mathcal{A} A
the normaliser
N A ( D ) : = { n ∈ A : n D n ∗ ⊆ D and n ∗ D n ⊆ D } \mathcal{N}_{\mathcal{A}}(\mathcal{D}):=\{n\in\mathcal{A}:n\mathcal{D}n^*\subseteq\mathcal{D}\text{ and }n^*\mathcal{D}n\subseteq\mathcal{D}\} N A ( D ) := { n ∈ A : n D n ∗ ⊆ D and n ∗ D n ⊆ D } generates A \mathcal{A} A C ∗ C^* C ∗
there exists a faithful conditional expectation E D : A → D E_{\mathcal{D}}:\mathcal{A}\to\mathcal{D} E D : A → D
This is the C ∗ C^* C ∗ C ∗ C^* C ∗ [36 , 41 ] .
Theorem 11.2. Let ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A}\subseteq\mathcal{M} A ⊆ M B ⊆ A \mathcal{B}\subseteq\mathcal{A} B ⊆ A C ∗ C^* C ∗ E : A → B E:\mathcal{A}\to\mathcal{B} E : A → B T t ( B ) ⊆ B \mathcal{T}_t(\mathcal{B})\subseteq\mathcal{B} T t ( B ) ⊆ B t ≥ 0 t\geq0 t ≥ 0 L ⊆ A s a \mathcal{L}\subseteq\mathcal{A}_{\mathrm{sa}} L ⊆ A sa L B ⊆ B s a \mathcal{L}_{\mathcal{B}}\subseteq\mathcal{B}_{\mathrm{sa}} L B ⊆ B sa
E ( L ) ⊆ L B . E(\mathcal{L})\subseteq\mathcal{L}_{\mathcal{B}}. E ( L ) ⊆ L B . Assume further that:
the restricted semigroup ( T t ∣ B ) (\mathcal{T}_t|_{\mathcal{B}}) ( T t ∣ B ) e e e L B \mathcal{L}_{\mathcal{B}} L B
for every a ∈ L a\in\mathcal{L} a ∈ L
∥ T t ( a − E ( a ) ) ∥ ⟶ 0 ( t → ∞ ) . \|\mathcal{T}_t(a-E(a))\|\longrightarrow0\qquad(t\to\infty). ∥ T t ( a − E ( a )) ∥ ⟶ 0 ( t → ∞ ) . Then ( T t ) (\mathcal{T}_t) ( T t ) e e e L \mathcal{L} L e e e L \mathcal{L} L
Proof. Fix a ∈ L a\in\mathcal{L} a ∈ L φ ∈ S ( A ) \varphi\in\mathcal{S}(\mathcal{A}) φ ∈ S ( A ) ε > 0 \varepsilon>0 ε > 0
d : = E ( a ) ∈ L B , b : = a − d . d:=E(a)\in\mathcal{L}_{\mathcal{B}},\qquad b:=a-d. d := E ( a ) ∈ L B , b := a − d . Let
r : S ( A ) → S ( B ) , r ( ω ) : = ω ∣ B , r:\mathcal{S}(\mathcal{A})\to\mathcal{S}(\mathcal{B}),\qquad r(\omega):=\omega|_{\mathcal{B}}, r : S ( A ) → S ( B ) , r ( ω ) := ω ∣ B , be the restriction map. Since B ⊆ A \mathcal{B}\subseteq\mathcal{A} B ⊆ A r r r
Because ( T t ∣ B ) (\mathcal{T}_t|_{\mathcal{B}}) ( T t ∣ B ) e e e L B \mathcal{L}_{\mathcal{B}} L B V V V r ( φ ) r(\varphi) r ( φ ) S ( B ) \mathcal{S}(\mathcal{B}) S ( B ) T 0 ≥ 0 T_0\geq0 T 0 ≥ 0
sup ρ ∈ V sup t ≥ T 0 ∣ ρ ( T t ( d ) ) − r ( φ ) ( T t ( d ) ) ∣ < ε / 2. \sup_{\rho\in V}\sup_{t\geq T_0}|\rho(\mathcal{T}_t(d))-r(\varphi)(\mathcal{T}_t(d))|<\varepsilon/2. ρ ∈ V sup t ≥ T 0 sup ∣ ρ ( T t ( d )) − r ( φ ) ( T t ( d )) ∣ < ε /2. Put
U : = r − 1 ( V ) , U:=r^{-1}(V), U := r − 1 ( V ) , which is a weak-* neighbourhood of φ \varphi φ S ( A ) \mathcal{S}(\mathcal{A}) S ( A )
Choose T 1 ≥ 0 T_1\geq0 T 1 ≥ 0
∥ T t ( b ) ∥ < ε / 4 , t ≥ T 1 . \|\mathcal{T}_t(b)\|<\varepsilon/4,\qquad t\geq T_1. ∥ T t ( b ) ∥ < ε /4 , t ≥ T 1 . Then for every ψ ∈ U \psi\in U ψ ∈ U t ≥ max { T 0 , T 1 } t\geq\max\{T_0,T_1\} t ≥ max { T 0 , T 1 }
∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ ≤ ∣ ψ ( T t ( d ) ) − φ ( T t ( d ) ) ∣ + ∣ ψ ( T t ( b ) ) ∣ + ∣ φ ( T t ( b ) ) ∣ = ∣ r ( ψ ) ( T t ( d ) ) − r ( φ ) ( T t ( d ) ) ∣ + ∣ ψ ( T t ( b ) ) ∣ + ∣ φ ( T t ( b ) ) ∣ < ε / 2 + 2 ∥ T t ( b ) ∥ < ε . \begin{aligned}
|\psi(\mathcal{T}_t(a))-\varphi(\mathcal{T}_t(a))|
&\leq|\psi(\mathcal{T}_t(d))-\varphi(\mathcal{T}_t(d))|+|\psi(\mathcal{T}_t(b))|+|\varphi(\mathcal{T}_t(b))|\\
&=|r(\psi)(\mathcal{T}_t(d))-r(\varphi)(\mathcal{T}_t(d))|+|\psi(\mathcal{T}_t(b))|+|\varphi(\mathcal{T}_t(b))|\\
&<\varepsilon/2+2\|\mathcal{T}_t(b)\|\\
&<\varepsilon.
\end{aligned} ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ ≤ ∣ ψ ( T t ( d )) − φ ( T t ( d )) ∣ + ∣ ψ ( T t ( b )) ∣ + ∣ φ ( T t ( b )) ∣ = ∣ r ( ψ ) ( T t ( d )) − r ( φ ) ( T t ( d )) ∣ + ∣ ψ ( T t ( b )) ∣ + ∣ φ ( T t ( b )) ∣ < ε /2 + 2∥ T t ( b ) ∥ < ε . Thus ( T t ) (\mathcal{T}_t) ( T t ) e e e φ \varphi φ a a a a ∈ L a \in\mathcal{L} a ∈ L φ ∈ S ( A ) \varphi\in\mathcal{S}(\mathcal{A}) φ ∈ S ( A ) e e e L \mathcal{L} L Theorem 3.2 now yields the e e e L \mathcal{L} L □ \square □
Corollary 11.6. Let A \mathcal{A} A C ∗ \mathrm{C}^{*} C ∗ B ⊆ A \mathcal{B}\subseteq\mathcal{A} B ⊆ A C ∗ \mathrm{C}^{*} C ∗ E : A → B E:\mathcal{A}\to\mathcal{B} E : A → B
T t : = e − t id + ( 1 − e − t ) E , t ≥ 0 , \mathcal{T}_t:=e^{-t}\operatorname{id}+(1-e^{-t})E,\qquad t\geq0, T t := e − t id + ( 1 − e − t ) E , t ≥ 0 , is a norm-continuous semigroup of unital completely positive maps on A \mathcal{A} A T t \mathcal{T}_t T t e e e A s a \mathcal{A}_{\mathrm{sa}} A sa
In particular, Cartan masas provide one important special case.
Proof. Since E E E
T s T t = e − ( s + t ) id + ( 1 − e − ( s + t ) ) E = T s + t \mathcal{T}_s\mathcal{T}_t=e^{-(s+t)}\operatorname{id}+(1-e^{-(s+t)})E=\mathcal{T}_{s+t} T s T t = e − ( s + t ) id + ( 1 − e − ( s + t ) ) E = T s + t is immediate. Norm continuity in t t t
Let L B = B s a \mathcal{L}_{\mathcal{B}}=\mathcal{B}_{\mathrm{sa}} L B = B sa L = A s a \mathcal{L}=\mathcal{A}_{\mathrm{sa}} L = A sa ( T t ) (\mathcal{T}_t) ( T t ) B \mathcal{B} B e e e d ∈ B s a d\in\mathcal{B}_{\mathrm{sa}} d ∈ B sa S ( B ) ∋ ρ ↦ ρ ( d ) \mathcal{S}(\mathcal{B})\ni\rho\mapsto\rho(d) S ( B ) ∋ ρ ↦ ρ ( d ) ∗ * ∗
T t ( a − E ( a ) ) = e − t ( a − E ( a ) ) ( a ∈ A s a ) , \mathcal{T}_t(a-E(a))=e^{-t}(a-E(a))\qquad(a\in\mathcal{A}_{\mathrm{sa}}), T t ( a − E ( a )) = e − t ( a − E ( a )) ( a ∈ A sa ) , so the complementary part decays exponentially in norm. Theorem 11.2 therefore gives the e e e A s a \mathcal{A}_{\mathrm{sa}} A sa □ \square □
Proposition 11.7. Let A \mathcal{A} A C ∗ \mathrm{C}^{*} C ∗ ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0 A \mathcal{A} A ω \omega ω B ⊆ A \mathcal{B}\subseteq\mathcal{A} B ⊆ A C ∗ \mathrm{C}^{*} C ∗ E : A → B E:\mathcal{A}\to\mathcal{B} E : A → B ω \omega ω
E T t = T t E = E E\mathcal{T}_t=\mathcal{T}_tE=E E T t = T t E = E t ≥ 0 t\geq0 t ≥ 0
(ii) there exists γ > 0 \gamma> 0 γ > 0 ∥ x ∥ 2 , ω : = ω ( x ∗ x ) 1 / 2 \lVert x\rVert_{2,\omega} := \omega(x^*x)^{1/2} ∥ x ∥ 2 , ω := ω ( x ∗ x ) 1/2
∥ T t ( b ) ∥ 2 , ω ≤ e − γ t ∥ b ∥ 2 , ω , b ∈ ker E , t ≥ 0. \lVert T_t(b)\rVert_{2,\omega} \le e^{-\gamma t}\lVert b\rVert_{2,\omega}, \qquad b \in\ker E,\quad t \ge0. ∥ T t ( b ) ∥ 2 , ω ≤ e − γ t ∥ b ∥ 2 , ω , b ∈ ker E , t ≥ 0. Then ( T t ) (T_t) ( T t ) e e e A s a \mathcal{A}_{\mathrm{sa}} A sa
Proof. Hypothesis (i) already forces the restricted dynamics on B \mathcal{B} B d ∈ B d \in\mathcal{B} d ∈ B d = E ( d ) d = E(d) d = E ( d )
T t ( d ) = T t E ( d ) = E ( d ) = d , t ≥ 0. T_t(d) = T_tE(d) = E(d) = d,\qquad t \ge0. T t ( d ) = T t E ( d ) = E ( d ) = d , t ≥ 0. Thus T t ∣ B = id B T_t|_{\mathcal{B}} = \operatorname{id}_{\mathcal{B}} T t ∣ B = id B e e e B s a \mathcal{B}_{\mathrm{sa}} B sa
Since A \mathcal{A} A ω \omega ω C ∗ C^* C ∗ C ω > 0 C_\omega> 0 C ω > 0
∥ x ∥ ≤ C ω ∥ x ∥ 2 , ω , x ∈ A . \lVert x\rVert\le C_\omega\lVert x\rVert_{2,\omega},\qquad x \in\mathcal{A}. ∥ x ∥ ≤ C ω ∥ x ∥ 2 , ω , x ∈ A . If a ∈ A s a a \in\mathcal{A}_{\mathrm{sa}} a ∈ A sa a − E ( a ) ∈ ker E a - E(a) \in\ker E a − E ( a ) ∈ ker E
∥ T t ( a − E ( a ) ) ∥ ≤ C ω e − γ t ∥ a − E ( a ) ∥ 2 , ω ⟶ 0. \lVert T_t(a-E(a))\rVert\le C_\omega e^{-\gamma t}\lVert a-E(a)\rVert_{2,\omega} \longrightarrow0. ∥ T t ( a − E ( a ))∥ ≤ C ω e − γ t ∥ a − E ( a ) ∥ 2 , ω ⟶ 0. The conclusion follows from Theorem 11.2 , with L = A s a \mathcal{L} = \mathcal{A}_{\mathrm{sa}} L = A sa L B = B s a \mathcal{L}_{\mathcal{B}} = \mathcal{B}_{\mathrm{sa}} L B = B sa
Remark 11.8. Detailed balance or KMS-symmetry is a standard way of producing the hypotheses of Proposition 11.7 : in finite dimensions the generator becomes self-adjoint on L 2 ( A , ω ) L^2(\mathcal{A},\omega) L 2 ( A , ω ) L 2 L^2 L 2 L 2 L^2 L 2 [34 , 33 , 12 , 6 ] .
Theorem 11.9. Let ( T t ) t ≥ 0 (T_t)_{t\ge0} ( T t ) t ≥ 0 M \mathcal{M} M A ⊆ M \mathcal{A} \subseteq\mathcal{M} A ⊆ M N ⊆ A \mathcal{N} \subseteq\mathcal{A} N ⊆ A C ∗ C^* C ∗ E N : A → N E_{\mathcal{N}}:\mathcal{A}\to\mathcal{N} E N : A → N
(i) there exists a strongly continuous one-parameter group ( α t ) t ∈ R (\alpha_t)_{t\in\mathbb{R}} ( α t ) t ∈ R ∗ ^* ∗ N \mathcal{N} N
T t ∣ N = α t ( t ≥ 0 ) , T_t|_{\mathcal{N}} = \alpha_t \qquad(t \ge0), T t ∣ N = α t ( t ≥ 0 ) , (ii) there exists a unital real linear test space L ⊆ A s a \mathcal{L} \subseteq\mathcal{A}_{\mathrm{sa}} L ⊆ A sa
E N ( L ) ⊆ L E_{\mathcal{N}}(\mathcal{L}) \subseteq\mathcal{L} E N ( L ) ⊆ L and
∥ T t ( a − E N ( a ) ) ∥ ⟶ 0 ( t → ∞ ) \lVert T_t(a-E_{\mathcal{N}}(a))\rVert\longrightarrow0 \qquad(t\to\infty) ∥ T t ( a − E N ( a ))∥ ⟶ 0 ( t → ∞ ) for every a ∈ L a \in\mathcal{L} a ∈ L
Then the following are equivalent:
(a) ( T t ) (T_t) ( T t ) e e e L \mathcal{L} L
(b) ( T t ) (T_t) ( T t ) e e e L \mathcal{L} L
(c) the semigroup ( α t ) t ≥ 0 (\alpha_t)_{t\ge0} ( α t ) t ≥ 0 e e e E N ( L ) ⊆ N s a E_{\mathcal{N}}(\mathcal{L}) \subseteq\mathcal{N}_{\mathrm{sa}} E N ( L ) ⊆ N sa
Proof. The implication (a)⇒ \Rightarrow ⇒ Theorem 3.2 . We prove (c)⇒ \Rightarrow ⇒ a ∈ L a \in\mathcal{L} a ∈ L φ ∈ S ( A ) \varphi\in\mathcal{S}(\mathcal{A}) φ ∈ S ( A ) ε > 0 \varepsilon> 0 ε > 0
d : = E N ( a ) ∈ E N ( L ) ⊆ L , b : = a − d . d := E_{\mathcal{N}}(a) \in E_{\mathcal{N}}(\mathcal{L}) \subseteq\mathcal{L}, \qquad b := a-d. d := E N ( a ) ∈ E N ( L ) ⊆ L , b := a − d . Because the semigroup ( α t ) t ≥ 0 (\alpha_t)_{t \ge0} ( α t ) t ≥ 0 e e e E N ( L ) E_{\mathcal{N}}(\mathcal{L}) E N ( L ) V V V φ ∣ N \varphi|_{\mathcal{N}} φ ∣ N S ( N ) \mathcal{S}(\mathcal{N}) S ( N )
sup ρ ∈ V sup t ≥ 0 ∣ ρ ( α t ( d ) ) − ( φ ∣ N ) ( α t ( d ) ) ∣ < ε / 2. \sup_{\rho\in V}\sup_{t \ge0} \left|\rho(\alpha_t(d))-(\varphi|_{\mathcal{N}})(\alpha_t(d))\right| < \varepsilon/2. ρ ∈ V sup t ≥ 0 sup ∣ ρ ( α t ( d )) − ( φ ∣ N ) ( α t ( d )) ∣ < ε /2. Let
r : S ( A ) ⟶ S ( N ) , r ( ω ) = ω ∣ N , r:\mathcal{S}(\mathcal{A}) \longrightarrow\mathcal{S}(\mathcal{N}), \qquad r(\omega)=\omega|_{\mathcal{N}}, r : S ( A ) ⟶ S ( N ) , r ( ω ) = ω ∣ N , which is weak-* continuous. Then U : = r − 1 ( V ) U:=r^{-1}(V) U := r − 1 ( V ) φ \varphi φ S ( A ) \mathcal{S}(\mathcal{A}) S ( A ) T 0 ≥ 0 T_0 \ge0 T 0 ≥ 0
∥ T t ( b ) ∥ < ε / 4 , t ≥ T 0 . \|\mathcal{T}_t(b)\| < \varepsilon/4, \qquad t \ge T_0. ∥ T t ( b ) ∥ < ε /4 , t ≥ T 0 . For ψ ∈ U \psi\in U ψ ∈ U t ≥ T 0 t \ge T_0 t ≥ T 0
∣ ψ ( T t ( a ) ) − φ ( T t ( a ) ) ∣ ≤ ∣ ψ ( T t ( d ) ) − φ ( T t ( d ) ) ∣ + ∣ ψ ( T t ( b ) ) ∣ + ∣ φ ( T t ( b ) ) ∣ = ∣ r ( ψ ) ( α t ( d ) ) − ( φ ∣ N ) ( α t ( d ) ) ∣ + ∣ ψ ( T t ( b ) ) ∣ + ∣ φ ( T t ( b ) ) ∣ < ε / 2 + 2 ∥ T t ( b ) ∥ < ε . \begin{aligned}
|\psi(\mathcal{T}_t(a))-\varphi(\mathcal{T}_t(a))|
&\leq|\psi(\mathcal{T}_t(d))-\varphi(\mathcal{T}_t(d))|+|\psi(\mathcal{T}_t(b))|+|\varphi(\mathcal{T}_t(b))| \\
&= |r(\psi)(\alpha_t(d))-(\varphi|_{\mathcal{N}})(\alpha_t(d))|+|\psi(\mathcal{T}_t(b))|+|\varphi(\mathcal{T}_t(b))| \\
&< \varepsilon/2+2\|\mathcal{T}_t(b)\| \\
&< \varepsilon.
\end{aligned} ∣ ψ ( T t ( a )) − φ ( T t ( a )) ∣ ≤ ∣ ψ ( T t ( d )) − φ ( T t ( d )) ∣ + ∣ ψ ( T t ( b )) ∣ + ∣ φ ( T t ( b )) ∣ = ∣ r ( ψ ) ( α t ( d )) − ( φ ∣ N ) ( α t ( d )) ∣ + ∣ ψ ( T t ( b )) ∣ + ∣ φ ( T t ( b )) ∣ < ε /2 + 2∥ T t ( b ) ∥ < ε . Thus ( T t ) (\mathcal{T}_t) ( T t ) e e e L \mathcal{L} L
It remains to prove (b)⇒ \Rightarrow ⇒ ( T t ) (\mathcal{T}_t) ( T t ) e e e L \mathcal{L} L d ∈ E N ( L ) d \in E_{\mathcal{N}}(\mathcal{L}) d ∈ E N ( L ) ρ ∈ S ( N ) \rho\in\mathcal{S}(\mathcal{N}) ρ ∈ S ( N ) ε > 0 \varepsilon> 0 ε > 0
i : S ( N ) ⟶ S ( A ) , i ( ρ ) : = ρ ∘ E N . i:\mathcal{S}(\mathcal{N}) \longrightarrow\mathcal{S}(\mathcal{A}), \qquad i(\rho):=\rho\circ E_{\mathcal{N}}. i : S ( N ) ⟶ S ( A ) , i ( ρ ) := ρ ∘ E N . Since E N E_{\mathcal{N}} E N i i i e e e ( T t ) (\mathcal{T}_t) ( T t ) i ( ρ ) i(\rho) i ( ρ ) d ∈ L d \in\mathcal{L} d ∈ L U U U i ( ρ ) i(\rho) i ( ρ ) S ( A ) \mathcal{S}(\mathcal{A}) S ( A )
sup ψ ∈ U sup t ≥ 0 ∣ ψ ( T t ( d ) ) − i ( ρ ) ( T t ( d ) ) ∣ < ε . \sup_{\psi\in U}\sup_{t \ge0} |\psi(\mathcal{T}_t(d))-i(\rho)(\mathcal{T}_t(d))| < \varepsilon. ψ ∈ U sup t ≥ 0 sup ∣ ψ ( T t ( d )) − i ( ρ ) ( T t ( d )) ∣ < ε . Put
V : = i − 1 ( U ) , V:=i^{-1}(U), V := i − 1 ( U ) , a weak-* neighbourhood of ρ \rho ρ S ( N ) \mathcal{S}(\mathcal{N}) S ( N ) ω ∈ V \omega\in V ω ∈ V i ( ω ) ∈ U i(\omega) \in U i ( ω ) ∈ U d ∈ N d \in\mathcal{N} d ∈ N T t ∣ N = α t \mathcal{T}_t|_{\mathcal{N}}=\alpha_t T t ∣ N = α t t ≥ 0 t \ge0 t ≥ 0
∣ ω ( α t ( d ) ) − ρ ( α t ( d ) ) ∣ = ∣ i ( ω ) ( T t ( d ) ) − i ( ρ ) ( T t ( d ) ) ∣ < ε ( t ≥ 0 ) . \begin{aligned}
|\omega(\alpha_t(d))-\rho(\alpha_t(d))|=|i(\omega)(\mathcal{T}_t(d))-i(\rho)(\mathcal{T}_t(d))| \\
&<\varepsilon\qquad(t \ge0).
\end{aligned} ∣ ω ( α t ( d )) − ρ ( α t ( d )) ∣ = ∣ i ( ω ) ( T t ( d )) − i ( ρ ) ( T t ( d )) ∣ < ε ( t ≥ 0 ) . Hence the semigroup ( α t ) t ≥ 0 (\alpha_t)_{t \ge0} ( α t ) t ≥ 0 e e e E N ( L ) E_{\mathcal{N}}(\mathcal{L}) E N ( L )
Remark 11.10. For a strongly continuous automorphism group ( α t ) t ∈ R (\alpha_t)_{t\in\mathbb{R}} ( α t ) t ∈ R C ∗ C^* C ∗ N \mathcal{N} N L 0 ⊆ N s a \mathcal{L}_0 \subseteq\mathcal{N}_{\mathrm{sa}} L 0 ⊆ N sa ( α t ) t ≥ 0 (\alpha_t)_{t\geq0} ( α t ) t ≥ 0 e e e L 0 \mathcal{L}_0 L 0
{ α t ( d ) : t ≥ 0 } ( d ∈ L 0 ) \{\alpha_t(d):t\geq0\}\qquad(d\in\mathcal{L}_0) { α t ( d ) : t ≥ 0 } ( d ∈ L 0 ) is norm relatively compact. Indeed, norm relative compactness of the positive orbit gives the required weak-∗ ^* ∗ Lemma 3.1 . Conversely, if ( α t ) t ≥ 0 (\alpha_t)_{t\geq0} ( α t ) t ≥ 0 e e e L 0 \mathcal{L}_0 L 0
α t ( d ) ^ : S ( N ) → R , α t ( d ) ^ ( ρ ) : = ρ ( α t ( d ) ) , \widehat{\alpha_t(d)}:\mathcal{S}(\mathcal{N})\to\mathbb{R},\qquad\widehat{\alpha_t(d)}(\rho):=\rho(\alpha_t(d)), α t ( d ) : S ( N ) → R , α t ( d ) ( ρ ) := ρ ( α t ( d )) , indexed by t ≥ 0 t\geq0 t ≥ 0 ∗ ^* ∗ C ( S ( N ) , R ) C(\mathcal{S}(\mathcal{N}),\mathbb{R}) C ( S ( N ) , R )
∥ x ^ − y ^ ∥ ∞ = ∥ x − y ∥ ( x , y ∈ N s a ) , \|\widehat{x}-\widehat{y}\|_\infty=\|x-y\|\qquad(x,y\in\mathcal{N}_{\mathrm{sa}}), ∥ x − y ∥ ∞ = ∥ x − y ∥ ( x , y ∈ N sa ) , the positive orbit { α t ( d ) : t ≥ 0 } \{\alpha_t(d):t\geq0\} { α t ( d ) : t ≥ 0 }
If one strengthens this to norm relative compactness of the full two-sided orbit { α t ( d ) : t ∈ R } \{\alpha_t(d):t\in\mathbb{R}\} { α t ( d ) : t ∈ R } t ↦ α t ( d ) t\mapsto\alpha_t(d) t ↦ α t ( d ) N \mathcal{N} N Theorem 11.9 recovers the familiar slogan that decoherence towards a decoherence-free subalgebra reduces the e e e
Remark 11.11. If A \mathcal{A} A ∗ ^* ∗ S ( A ) \mathcal{S}(\mathcal{A}) S ( A ) A \mathcal{A} A e e e e e e
Example 11.12. Let A = M n ( C ) \mathcal{A}=M_n(\mathbb{C}) A = M n ( C ) p 1 , … , p r p_1,\ldots,p_r p 1 , … , p r ∑ j = 1 r p j = 1 \sum_{j=1}^{r}p_j=1 ∑ j = 1 r p j = 1
N : = ⨁ j = 1 r p j M n ( C ) p j , E N ( a ) : = ∑ j = 1 r p j a p j . \mathcal{N}:=\bigoplus_{j=1}^{r}p_jM_n(\mathbb{C})p_j,\qquad E_{\mathcal{N}}(a):=\sum_{j=1}^{r}p_jap_j. N := j = 1 ⨁ r p j M n ( C ) p j , E N ( a ) := j = 1 ∑ r p j a p j . Assume that at least one block p j M n ( C ) p j p_jM_n(\mathbb{C})p_j p j M n ( C ) p j N \mathcal{N} N H = H ∗ ∈ N H=H^*\in\mathcal{N} H = H ∗ ∈ N
α t ( a ) : = e i t H a e − i t H , a ∈ N , \alpha_t(a):=e^{itH}ae^{-itH},\qquad a\in\mathcal{N}, α t ( a ) := e i t H a e − i t H , a ∈ N , and define
T t ( a ) : = e i t H E N ( a ) e − i t H + e − t e i t H ( a − E N ( a ) ) e − i t H , a ∈ M n ( C ) , t ≥ 0. T_t(a):=e^{itH}E_{\mathcal{N}}(a)e^{-itH}+e^{-t}e^{itH}(a-E_{\mathcal{N}}(a))e^{-itH},\qquad a\in M_n(\mathbb{C}),\ t\geq0. T t ( a ) := e i t H E N ( a ) e − i t H + e − t e i t H ( a − E N ( a )) e − i t H , a ∈ M n ( C ) , t ≥ 0. Then ( T t ) t ≥ 0 (T_t)_{t\geq0} ( T t ) t ≥ 0 M n ( C ) M_n(\mathbb{C}) M n ( C ) E N ∘ T t = α t ∘ E N E_{\mathcal{N}}\circ T_t=\alpha_t\circ E_{\mathcal{N}} E N ∘ T t = α t ∘ E N T t T_t T t N \mathcal{N} N ( α t ) (\alpha_t) ( α t )
∥ T t ( a − E N ( a ) ) ∥ = e − t ∥ a − E N ( a ) ∥ ( a ∈ M n ( C ) ) . \|T_t(a-E_{\mathcal{N}}(a))\|=e^{-t}\|a-E_{\mathcal{N}}(a)\|\qquad(a\in M_n(\mathbb{C})). ∥ T t ( a − E N ( a )) ∥ = e − t ∥ a − E N ( a ) ∥ ( a ∈ M n ( C )) . Since N \mathcal{N} N ( α t ) (\alpha_t) ( α t ) ( α t ) t ≥ 0 (\alpha_t)_{t\geq0} ( α t ) t ≥ 0 e e e N s a \mathcal{N}_{\mathrm{sa}} N sa Theorem 11.9 yields the e e e ( T t ) (\mathcal{T}_t) ( T t ) M n ( C ) s a M_n(\mathbb{C})_{\mathrm{sa}} M n ( C ) sa
Proof. The semigroup law follows from the decomposition
T t = Ad ( e i t H ) ∘ ( E N + e − t ( id − E N ) ) \mathcal{T}_t=\operatorname{Ad}(e^{itH})\circ\left(E_{\mathcal{N}}+e^{-t}(\operatorname{id}-E_{\mathcal{N}})\right) T t = Ad ( e i t H ) ∘ ( E N + e − t ( id − E N ) ) and the fact that Ad ( e i t H ) \operatorname{Ad}(e^{itH}) Ad ( e i t H ) E N E_{\mathcal{N}} E N H ∈ N H\in\mathcal{N} H ∈ N E N E_{\mathcal{N}} E N N \mathcal{N} N
Theorem 11.13. Let A \mathcal{A} A C ∗ C^* C ∗ B ⊆ A \mathcal{B}\subseteq\mathcal{A} B ⊆ A C ∗ C^* C ∗ E : A → B E:\mathcal{A}\to\mathcal{B} E : A → B P : A → A P:\mathcal{A}\to\mathcal{A} P : A → A
(i) P ( B ) ⊆ B P(\mathcal{B})\subseteq\mathcal{B} P ( B ) ⊆ B P ∣ B = id B P|_{\mathcal{B}}=\operatorname{id}_{\mathcal{B}} P ∣ B = id B
(ii) P E = E P = E PE=EP=E P E = E P = E
(iii) there exists θ ∈ [ 0 , 1 ) \theta\in[0,1) θ ∈ [ 0 , 1 )
∥ P ( b ) ∥ ≤ θ ∥ b ∥ , b ∈ ker E . \|P(b)\|\leq\theta\|b\|,\qquad b\in\ker E. ∥ P ( b ) ∥ ≤ θ ∥ b ∥ , b ∈ ker E . Let ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0 P − id P-\operatorname{id} P − id
T t = e t ( P − id ) = e − t ∑ n = 0 ∞ t n n ! P n . \mathcal{T}_t=e^{t(P-\operatorname{id})}=e^{-t}\sum_{n=0}^{\infty}\frac{t^n}{n!}P^n. T t = e t ( P − id ) = e − t n = 0 ∑ ∞ n ! t n P n . Then:
(i) T t ( B ) = B \mathcal{T}_t(\mathcal{B})=\mathcal{B} T t ( B ) = B E ∘ T t = T t ∘ E = E E\circ\mathcal{T}_t=\mathcal{T}_t\circ E=E E ∘ T t = T t ∘ E = E t ≥ 0 t\geq0 t ≥ 0
(ii)
∥ T t ( a − E ( a ) ) ∥ ≤ e − t ( 1 − θ ) ∥ a − E ( a ) ∥ , a ∈ A , \|\mathcal{T}_t(a-E(a))\|\leq e^{-t(1-\theta)}\|a-E(a)\|,\qquad a\in\mathcal{A}, ∥ T t ( a − E ( a )) ∥ ≤ e − t ( 1 − θ ) ∥ a − E ( a ) ∥ , a ∈ A , (iii) ( T t ) (\mathcal{T}_t) ( T t ) e e e A s a \mathcal{A}_{\mathrm{sa}} A sa
Proof. Because P P P P n P^n P n T t \mathcal{T}_t T t t ≥ 0 t\geq0 t ≥ 0 T t \mathcal{T}_t T t
If d ∈ B d\in\mathcal{B} d ∈ B P ( d ) = d P(d)=d P ( d ) = d T t ( d ) = d \mathcal{T}_t(d)=d T t ( d ) = d t ≥ 0 t\geq0 t ≥ 0 T t ( B ) = B \mathcal{T}_t(\mathcal{B})=\mathcal{B} T t ( B ) = B E E E P P P P P P T t \mathcal{T}_t T t E T t ∣ B = id B E\mathcal{T}_t|_{\mathcal{B}}=\operatorname{id}_{\mathcal{B}} E T t ∣ B = id B E ∘ T t = E E\circ\mathcal{T}_t=E E ∘ T t = E
Now fix b ∈ ker E b\in\ker E b ∈ ker E P ( ker E ) ⊆ ker E P(\ker E)\subseteq\ker E P ( ker E ) ⊆ ker E
∥ P n ( b ) ∥ ≤ θ n ∥ b ∥ , n ∈ N . \|P^n(b)\|\leq\theta^n\|b\|,\qquad n\in\mathbb{N}. ∥ P n ( b ) ∥ ≤ θ n ∥ b ∥ , n ∈ N . Therefore
∥ T t ( b ) ∥ ≤ e − t ∑ n = 0 ∞ t n n ! ∥ P n ( b ) ∥ ≤ e − t ∑ n = 0 ∞ ( θ t ) n n ! ∥ b ∥ = e − t ( 1 − θ ) ∥ b ∥ . \|\mathcal{T}_t(b)\|\leq e^{-t}\sum_{n=0}^{\infty}\frac{t^n}{n!}\|P^n(b)\|\leq e^{-t}\sum_{n=0}^{\infty}\frac{(\theta t)^n}{n!}\|b\|=e^{-t(1-\theta)}\|b\|. ∥ T t ( b ) ∥ ≤ e − t n = 0 ∑ ∞ n ! t n ∥ P n ( b ) ∥ ≤ e − t n = 0 ∑ ∞ n ! ( θ t ) n ∥ b ∥ = e − t ( 1 − θ ) ∥ b ∥. This proves (ii). Since the restriction of ( T t ) (\mathcal{T}_t) ( T t ) B \mathcal{B} B e e e B s a \mathcal{B}_{\mathrm{sa}} B sa Theorem 11.2 now applies with L B = B s a \mathcal{L}_{\mathcal{B}}=\mathcal{B}_{\mathrm{sa}} L B = B sa L = A s a \mathcal{L}=\mathcal{A}_{\mathrm{sa}} L = A sa
Corollary 11.14. Let G G G C ∗ C^* C ∗ A \mathcal{A} A ∗ * ∗ α g \alpha_g α g
B : = A G : = { a ∈ A : α g ( a ) = a for all g ∈ G } \mathcal{B}:=\mathcal{A}^{G}:=\{a\in\mathcal{A}:\alpha_g(a)=a\text{ for all }g\in G\} B := A G := { a ∈ A : α g ( a ) = a for all g ∈ G } admits the canonical conditional expectation
E ( a ) : = ∫ G α g ( a ) d g . E(a):=\int_G\alpha_g(a)\,dg. E ( a ) := ∫ G α g ( a ) d g . Let μ \mu μ G G G
P ( a ) : = ∫ G α g ( a ) d μ ( g ) , a ∈ A , P(a):=\int_G\alpha_g(a)\,d\mu(g),\qquad a\in\mathcal{A}, P ( a ) := ∫ G α g ( a ) d μ ( g ) , a ∈ A , and assume that there exists θ ∈ [ 0 , 1 ) \theta\in[0,1) θ ∈ [ 0 , 1 )
∥ P ( b ) ∥ ≤ θ ∥ b ∥ , b ∈ ker E . \lVert P(b)\rVert\leq\theta\lVert b\rVert,\qquad b\in\ker E. ∥ P ( b )∥ ≤ θ ∥ b ∥ , b ∈ ker E . Then the semigroup generated by P − id P-\operatorname{id} P − id Theorem 11.13 . If, in addition, B \mathcal{B} B
Proof. The map E E E B \mathcal{B} B P E = E P = E PE=EP=E P E = E P = E P ∣ B = id B P|_{\mathcal{B}}=\operatorname{id}_{\mathcal{B}} P ∣ B = id B B \mathcal{B} B α g \alpha_g α g Theorem 11.13 . □
Example 11.15. Let A = M n ( C ) \mathcal{A}=M_n(\mathbb{C}) A = M n ( C ) D ⊆ M n ( C ) \mathcal{D}\subseteq M_n(\mathbb{C}) D ⊆ M n ( C ) ( e i j ) 1 ≤ i , j ≤ n (e_{ij})_{1\leq i,j\leq n} ( e ij ) 1 ≤ i , j ≤ n E D E_{\mathcal{D}} E D
H = diag ( h 1 , … , h n ) , V r = diag ( v r , 1 , … , v r , n ) ( 1 ≤ r ≤ m ) , H=\operatorname{diag}(h_1,\ldots,h_n),\qquad V_r=\operatorname{diag}(v_{r,1},\ldots,v_{r,n})\quad(1\leq r\leq m), H = diag ( h 1 , … , h n ) , V r = diag ( v r , 1 , … , v r , n ) ( 1 ≤ r ≤ m ) , and let ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0
G ( a ) = i [ H , a ] + ∑ r = 1 m ( V r a V r − 1 2 ( V r 2 a + a V r 2 ) ) . \mathcal{G}(a)=i[H,a]+\sum_{r=1}^{m}\left(V_raV_r-\frac{1}{2}\left(V_r^2a+aV_r^2\right)\right). G ( a ) = i [ H , a ] + r = 1 ∑ m ( V r a V r − 2 1 ( V r 2 a + a V r 2 ) ) . If
γ : = 1 2 min i ≠ j ∑ r = 1 m ( v r , i − v r , j ) 2 > 0 , \gamma:=\frac{1}{2}\min_{i\ne j}\sum_{r=1}^{m}(v_{r,i}-v_{r,j})^2>0, γ := 2 1 i = j min r = 1 ∑ m ( v r , i − v r , j ) 2 > 0 , then
∥ T t ( a − E D ( a ) ) ∥ ≤ n e − γ t ∥ a − E D ( a ) ∥ , a ∈ M n ( C ) , \lVert\mathcal{T}_t(a-E_{\mathcal{D}}(a))\rVert\leq\sqrt{n}e^{-\gamma t}\lVert a-E_{\mathcal{D}}(a)\rVert,\qquad a\in M_n(\mathbb{C}), ∥ T t ( a − E D ( a ))∥ ≤ n e − γ t ∥ a − E D ( a )∥ , a ∈ M n ( C ) , and consequently ( T t ) (\mathcal{T}_t) ( T t ) e e e M n ( C ) s a M_n(\mathbb{C})_{\mathrm{sa}} M n ( C ) sa
Proof. For the matrix units one computes
G ( e i j ) = ( i ( h i − h j ) − 1 2 ∑ r = 1 m ( v r , i − v r , j ) 2 ) e i j . \mathcal{G}(e_{ij})=\left(i(h_i-h_j)-\frac{1}{2}\sum_{r=1}^{m}(v_{r,i}-v_{r,j})^2\right)e_{ij}. G ( e ij ) = ( i ( h i − h j ) − 2 1 r = 1 ∑ m ( v r , i − v r , j ) 2 ) e ij . Hence
T t ( e i j ) = exp ( t i ( h i − h j ) − t 2 ∑ r = 1 m ( v r , i − v r , j ) 2 ) e i j . \mathcal{T}_t(e_{ij})=\exp\left(t\,i(h_i-h_j)-\frac{t}{2}\sum_{r=1}^{m}(v_{r,i}-v_{r,j})^2\right)e_{ij}. T t ( e ij ) = exp ( t i ( h i − h j ) − 2 t r = 1 ∑ m ( v r , i − v r , j ) 2 ) e ij . In particular, T t ( e i i ) = e i i \mathcal{T}_t(e_{ii})=e_{ii} T t ( e ii ) = e ii T t \mathcal{T}_t T t D \mathcal{D} D E D E_{\mathcal{D}} E D
b = a − E D ( a ) = ∑ i ≠ j a i j e i j , b=a-E_{\mathcal{D}}(a)=\sum_{i\ne j}a_{ij}e_{ij}, b = a − E D ( a ) = i = j ∑ a ij e ij , then
T t ( b ) = ∑ i ≠ j c i j ( t ) a i j e i j , ∣ c i j ( t ) ∣ ≤ e − γ t . \mathcal{T}_t(b)=\sum_{i\ne j}c_{ij}(t)a_{ij}e_{ij},\qquad|c_{ij}(t)|\le e^{-\gamma t}. T t ( b ) = i = j ∑ c ij ( t ) a ij e ij , ∣ c ij ( t ) ∣ ≤ e − γ t . Therefore the Hilbert–Schmidt norm satisfies
∥ T t ( b ) ∥ 2 2 = ∑ i ≠ j ∣ c i j ( t ) ∣ 2 ∣ a i j ∣ 2 ≤ e − 2 γ t ∑ i ≠ j ∣ a i j ∣ 2 = e − 2 γ t ∥ b ∥ 2 2 . \|\mathcal{T}_t(b)\|_2^2=\sum_{i\ne j}|c_{ij}(t)|^2|a_{ij}|^2\le e^{-2\gamma t}\sum_{i\ne j}|a_{ij}|^2=e^{-2\gamma t}\|b\|_2^2. ∥ T t ( b ) ∥ 2 2 = i = j ∑ ∣ c ij ( t ) ∣ 2 ∣ a ij ∣ 2 ≤ e − 2 γ t i = j ∑ ∣ a ij ∣ 2 = e − 2 γ t ∥ b ∥ 2 2 . Since ∥ x ∥ ≤ ∥ x ∥ 2 ≤ n ∥ x ∥ \|x\|\le\|x\|_2\le\sqrt{n}\|x\| ∥ x ∥ ≤ ∥ x ∥ 2 ≤ n ∥ x ∥ M n ( C ) M_n(\mathbb{C}) M n ( C )
∥ T t ( b ) ∥ ≤ ∥ T t ( b ) ∥ 2 ≤ e − γ t ∥ b ∥ 2 ≤ n e − γ t ∥ b ∥ . \|\mathcal{T}_t(b)\|\le\|\mathcal{T}_t(b)\|_2\le e^{-\gamma t}\|b\|_2\le\sqrt{n}e^{-\gamma t}\|b\|. ∥ T t ( b ) ∥ ≤ ∥ T t ( b ) ∥ 2 ≤ e − γ t ∥ b ∥ 2 ≤ n e − γ t ∥ b ∥. The conclusion now follows from Theorem 11.2 . This example is the simplest Cartan-masa instance of a dephasing Lindbladian, compare the qualitative finite-dimensional analyses of decoherence in [3 , 40 , 10 , 11 ] . □
Further questions
Problem 12.1. Determine when approximation properties such as nuclearity, injectivity or finite nuclear dimension can be combined with positivity-improving estimates, approximate equivariance or functional inequalities to produce an exact Doeblin order lower bound. Approximate finite-dimensional factorisations alone do not yield the required inequality [7 , 12 , 34 , 33 ] .
Problem 12.2. While Theorem 4.10 and Proposition 5.8 settle the full-state-space hierarchy among the ordinary, eventual, Cesàro and pointwise eventual equicontinuity properties, it remains open to classify when the Cesàro e e e e e e
Problem 12.3. Find broad infinite-dimensional hypotheses under which decay in an L 2 L^2 L 2
Use of artificial intelligence
This paper was prepared in 2024–2025 with some incremental additions in 2026. ChatGPT was used to polish the presentation. The authors take full responsibility for the content of the paper.
References
(T. Kania) Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic and Institute of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
Email address: kania@math.cas.cz, tomasz.marcin.kania@gmail.com
(M. Niwiński) Doctoral School of Exact and Natural Sciences, Faculty of Mathematics and Computer Science, Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
Email address: mariusz.niwinski@doctoral.uj.edu.pl
