Reversible part of quantum dynamical systems: A review

CONFLUENTES MATHEMATICI

Carlo PANDISCIA

Reversible part of quantum dynamical systems:A review Tome 10, n^o2 (2018), p. 51-74.

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REVERSIBLE PART OF QUANTUM DYNAMICAL SYSTEMS:

A REVIEW

CARLO PANDISCIA

Abstract. In this work a quantum dynamical system (M,Φ, ϕ) is constituted by a von Neumann algebraM, a unital Schwartz map Φ :M→Mand a Φ-invariant normal faithful stateϕonM. We will prove that the ergodic properties of a quantum dynamical system are determined by its reversible part (D∞,Φ∞, ϕ∞); i.e. by a von Neumann sub-algebraD∞of M, with an automorphism Φ∞and a normal stateϕ∞, as the restrictions onD∞. Moreover, ifD∞is a trivial algebra, then the quantum dynamical system is ergodic. Furthermore, we will show some properties of reversible part of the quantum dynamical system, finally we will study its relations with the canonical decomposition of Nagy-Fojas of linear contraction related to a quantum dynamical system.

1. Introduction

This exploratory paper deals with the main properties of the reversible part of a discrete quantum dynamical system.

It is well-known that in the axiomatic approach of von Neumann-Segal-Mackey [10, 11], a physical system is characterized by a pair (A,S) constituted by a *- algebra with unitAand by a convex subsetSof the dual space of algebra A.

According to Kadison [20], the temporal evolution of a dynamical system is described by an affine map Φ^◦:S→S which induces a positive map Φ :A→A such thatω(Φ(a)) = Φ^◦(ω)(a) for allω∈Sanda∈A. This passage from Φ^◦to Φ is referred to, in the physics literature, as the passage from the Heisenberg pictures to the Schrodinger pictures [11].

Physically the loss of pure states of system, during its temporal evolution is a reversibility index of the map Φ^◦.

Indeed, if the affine map Φ^◦ is bijective (and therefore maps pure states into pure states), we obtain that Φ is a Jordan automorphism of the algebra of the observables [20]. in this case the physical system will be called reversible; however in the general case the loss of pure states during temporal evolution is unavoidable therefore the affine map Φ^◦ is not bijective. Is it possible to outline a part of the observables algebra of physical system which is reversible along its temporary evolution? And, if this is not trivial, what are its main properties?

In this brief work we study the problem when the algebra of the observables is described by a von Neumann algebra which admits a normal stationary state for our temporal evolution.

We therefore consider a pair (M,Φ) composed of a von Neumann algebraMand a unital Schwartz map Φ :M→M, i.e. a normal map with Φ(1) = 1 which satisfies the inequality:

06Φ(a^∗)Φ(a)6Φ(a^∗a), a∈M. (1.1) The pair (M,Φ) is called (discrete) quantum process and Φ its dynamics.

Math. classification:46L07, 46L55, 81S22.

Keywords: Quantum dynamical system, Multiplicative core, Algebra of effective observables.

51

A quantum process is called reversible if its dynamics is an automorphism of von Neumann algebras.

A quantum process (R,Φ) is a sub-process of (M,ˇ Φ) if there is an injective homomorphism of von Neumann algebrasi:R→Mand a conditional expectation E :M→i(R) such that ˇΦⁿ=E ◦Φⁿ◦i for all natural numbers n.

The reversible part of a quantum process is its maximal reversible sub-process.

We denote withB(H) the C*-algebra of the bounded linear operators on Hilbert spaceHand with sandσ respectively, the ultrastrong operator topology and the ultraweakly operator topology on von Neumann algebra M, while with M∗ its predual. Furthermore, a normal map or a normal state areσ-continuous maps [7].

We recall that a normal state ϕ on M is called either a stationary state for the quantum process (M,Φ) ifϕ(Φ(a)) = ϕ(a) for all a ∈ M or an asymptotic equilibrium if Φⁿ(a)→ϕ(a)1 asn→ ∞in σ-topology.

Let (M,Φ) be a quantum process and ϕ its stationary state, the dynamics Φ admits aϕ-adjoint if there is a normal unital Schwartz map Φ^]:M→Msuch that ϕ(bΦ(a)) =ϕ(Φ^\(b)a) for all a, b∈M.

The conditions for the existence of a ϕ-adjointness of dynamics of quantum process can be found in [1] and [28].

In what follow we will call a quantum dynamical system (QDS) a triple (M,Φ, ϕ) constituted by a quantum processes (M,Φ) and normal faithful stationary stateϕ where the dynamics Φ admits a ϕ-adjoint Φ^].

We have tried to keep the exposition as ’self-contained’ as possible, recalling and proving some well known results of dynamical systems theory [14, 15] and of decoherence theory [4, 9].

This paper is organized as follows:

In section 2 we recall briefly some properties of multiplicative domain of Schwartz maps.

In section 3 we introduce the decomposition theorem for an algebraic probability space (M, ϕ), constituted by a von Neumann algebra Mand by one of its normal faithful statesϕ. Afterwards we study its connection with the multiplicative domain of dynamics of QDS.

In section 4 we study the connection between the decomposition theorem and the canonical decomposition of Nagy-Fojas of linear contraction on Hilbert space [27].

In section 5 we list some simple properties of the reversible part of a QDS in this case we prove that the ergodic properties of a QDS depend on the ergodic properties of its reversible part. Furthermore the decomposition th allows to define a new algebraic structure onMof *-Banach algebra such that its dynamics Φ is a

*-homomorphism.

In section 6 by using some well known results on the Cesaro mean, we study under what conditions the QDS is completely irreversible, i.e. the reversible part is a trivial algebra.

2. Multiplicative domains

The multiplicative domain DΦ of a Schwartz map Φ : M → M, is defined as follows [30, 34]:

D_Φ={a∈M: Φ(a^∗a) = Φ(a^∗)Φ(a), Φ(aa^∗) = Φ(a)Φ(a^∗)} (2.1)

and an elementa∈DΦif and only if Φ(ax) = Φ(a)Φ(x) and Φ(xa) = Φ(x)Φ(a) for allx∈M.

It follows thatDΦis a von Neumann algebra, since it is a unital *-algebra closed in the σ-topology, that is,DΦ is the largest sub-algebra ofMon which Φ behaves multiplicatively.

A consequence of Schwartz’s inequality is the following remark:

If Φ :M→Mis a unital Schwartz map which admits an inverse Φ⁻¹:M→M (i.e. a unital Schwartz map such that Φ(Φ⁻¹(a)) = Φ⁻¹(Φ(a)) =afor alla∈M), then Φ is an automorphism.

IfD⁺_∞is the following von Neumann algebra:

D⁺_∞= \

n∈N

D_Φn (2.2)

then we have that Φ(D⁺_∞) ⊂D⁺_∞ and Φ restricted to D⁺_∞ is a *-homomorphism, but it is not surjective map.

We observe that

D⁺_∞={a∈D_Φ: Φⁿ(a)∈D_Φfor alln∈N}. Defining the multiplicative coreof Φ [33] as

CΦ= \

n∈N

Φⁿ(D⁺_∞)⊂D⁺_∞ (2.3)

we obtain that Φ(CΦ)⊂ CΦ.

Indeed for eachn>0 we have Φⁿ⁺¹(D⁺_∞)⊂Φⁿ(D⁺_∞) and Φ(\

n∈N

Φⁿ(D⁺_∞))⊂ \

n∈N

Φ(Φⁿ(D⁺_∞)) = \

n∈N

Φⁿ⁺¹(D⁺_∞) = \

n∈N

Φⁿ(D⁺_∞).

It is clear that the restriction of Φ toC_Φ is a *-homomorphism.

Since Φ is a normal map and its restriction toD⁺_∞is a *-homomorphism, the set Φⁿ(D⁺_∞) is a von Neumann algebra [7] thereforeCφ is a von Neumann algebra.

Proposition 2.1. — If Φ is an injective map on D⁺_∞ then we obtain that Φ(CΦ) =CΦand its restriction to theCΦis a *-automorphism.

Proof. — Let y ∈ CΦ, by the definition of CΦ for each natural number n > 0 there exists an element x_n∈D⁺_∞ such thaty= Φⁿ(x_n) for alln>0.

We have Φ(y) = Φⁿ⁺¹(x_n) = Φⁿ⁺¹(x_n+1) for alln>0 and by the injectivity of Φ we obtain thatx_n =x_n+1 for alln∈Nhencey= Φ(x₀).

Letϕbe a stationary state for the quantum process (M,Φ) and (H_ϕ, π_ϕ,Ω_ϕ) its GNS representation. It is well-known [28] that there is a unique linear contraction UΦ,ϕ ofB(Hϕ) such that for each a∈Awe have:

U_Φ,ϕπ_ϕ(a)Ω_ϕ=π_ϕ(Φ(a))Ω_ϕ. (2.4) Furthermore, if ϕ is a faithful state, then there is a unital Schwartz map Φ• : π_ϕ(M)→π_ϕ(M) such that

Φ•(A)Ωϕ=UΦ,ϕAΩϕ , A∈πϕ(M). (2.5) Proposition 2.2. — Let(M,Φ)be a quantum process andϕits faithful sta- tionary state, we have:

1. For each d∈D_Φ results U_Φ,ϕπ_ϕ(d) =π_ϕ(Φ(d))U_Φ,ϕ .

2. If UΦ,ϕπϕ(a) = πϕ(Φ(a))UΦ,ϕ then for each x∈ M result suchΦ(ax) = Φ(a)Φ(x).

3. U_Φ,ϕ^∗ UΦ,ϕ∈πϕ(DΦ)⁰ while UΦ,ϕU_Φ,ϕ^∗ ∈πϕ(Φ(DΦ))⁰ .

4. The elementdbelongs toDΦif, and only if||UΦ,ϕπϕ(d)Ωϕ||=||πϕ(d)Ωϕ||

and ||UΦ,ϕπ_ϕ(d^∗)Ωϕ||=||πϕ(d^∗)Ωϕ||.

Proof. — It is straightforward.

By Schwartz’s inequality and by the existence of a faithful stationary state for a quantum process (M,Φ) we obtain the following inclusions:

· · ·DΦⁿ ⊂DΦⁿ⁻¹ ⊂ · · ·DΦ² ⊂DΦ⊂M (2.6) for all natural numbersn∈N.

In this case, the map Φ is injective onD⁺_∞, since for eacha∈D⁺_∞with Φ(a) = 0, we have:

ϕ(Φ(a^∗)Φ(a)) =ϕ(Φ(a^∗a)) =ϕ(a^∗a) = 0 hence its restricted toC_Φ is a *-automorphism.

An example of a Schwartz map whose restriction to the multiplicative core C_Φ is not an automorphism is described in Example 2.3.

Example 2.3. — Let H be a Hilbert space and V an isometry of B(H). We consider the unital Schwartz map:

Φ(A) =V^∗AV , A∈B(H)

and we assume that there exists an element Ω∈ Hsuch thatVΩ = Ω.

For each natural numberk we have

D_Φk={Ek}⁰ , Ek =V^kV^∗k. Indeed, if A∈ {Ek}⁰ thenAalso belongs toD_Φk, since

Φ^k(A^∗A) =V^∗kA^∗AV^k=V^∗kA^∗AV^kV^∗kV^k=V^∗A^∗V^kV^∗kAV^k = Φ(A^∗)Φ(A).

LetAbelongs toD_Φk, we have:

V^∗kAE_kBΩ =V^∗kAV^kV^∗kBΩ = Φ^k(AB)Ω =V^∗kABΩ It follows that for eachA∈D_Φk we obtain:

V^∗kAEk =V^∗kA =⇒ EkAEk=EkA.

and hence

D⁺_∞= \

k∈N

{E_k}⁰.

Let us notice that the algebraD⁺_∞is not trivial since one-rank projection|Ω><Ω|:

|Ω><Ω|ξ=hΩ, ξiΩ, ξ∈ H belongs to it.

Let Θ :B(H)→B(H) be any unital Schwartz map, we define Ψ(A) =V AV^∗+FVΘ(A)FV , A∈B(H) whereFV =I−V V^∗.

The unital Schwartz map Ψ has the following properties:

Φ(Ψ(A) =A , A∈B(H)

and for eachA, B∈B(H)

<Ω, AΦ(B)Ω>=<Ω,Ψ(A)BΩ> . SinceF_VE_k = 0 for allk∈N, we have:

Ψ(T)∈ {E_k}⁰ , T ∈ {E_k}⁰. Therefore the multiplicative core CΦ=D⁺_∞.

3. Decomposition theorem

It is well known that any Hilbert space decomposes into direct sum of one its subspaces and its orthogonal complement. We have a similar result in von Neumann algebras theory.

Indeed, given a faithful normal state of a von Neumann algebra, for any its sub-algebra that satisfies a modular property we can decompose the von Neumann algebra in direct sum of vector spaces where a vector space is our sub-algebra and the other is its orthogonal related to the faithful state. We take as sub-algebra the multiplicative core of our unital Schwartz map consequently its orthogonal space is constituted by observables subject to dissipation. The temporal evolution is not a bijective map for these observables.

In this section we will study some properties of this algebraic decomposition and its relations with the orthogonal complement of Hilbert space of GNS representation obtained by our faithful normal state.

We consider a von Neumann algebra Mand its faithful normal stateϕand we denote by (Hϕ, πϕ,Ωϕ) the GNS representation ofϕand denoting by{σ_t^ϕ}_t∈Rits modular automorphism group.

LetRbe a von Neumann sub-algebra ofM, we recall [22] that theϕ-orthogonal ofR is the set:

R^⊥ϕ={a∈M:ϕ(a^∗x) = 0 for all x∈R}. (3.1) The setR^⊥ϕ is a closed linear space in the σ-topology withR^⊥ϕ∩R={0}.

We notice thatR^⊥ϕ ⊂kerϕ and if R =CI thenR^⊥ϕ = kerϕ, where kerϕ = {a∈M: ϕ(a) = 0}.

Moreover if y∈Randd_⊥∈R^⊥ϕ thenyd_⊥∈R^⊥ϕ, since ϕ((yd⊥)^∗x) =ϕ(d^∗_⊥y^∗x) = 0, x∈R.

We notice sinceϕis a faithful state, the representationπ_ϕ:M→B(Hϕ) is faithful normal map andπ_ϕ(M) a von Neumann algebra isomorphic toM.

Theorem 3.1. — The von Neumann algebra R is invariant under modular automorphism group σ_t^ϕif and only if both conditions are fulfilled:

a. The setR^⊥ϕ is closed under the involution operation.

b. For each a∈Mthere is a unique a_k∈Randa_⊥∈R^⊥ϕ such that a=a_k+a_⊥.

In other words we have the following algebraic decomposition:

M=R⊕R^⊥ϕ. (3.2)

Proof. — From Takesaki [35] we have σ_t^ϕ(πϕ(R))⊂πϕ(R) for allt ∈R if and only if there exists a normal conditional expectationE :M→Rsuch thatϕ◦E=ϕ.

Assuming R be invariant under modular automorphism group σ_t^ϕ is simple to prove that

R^⊥ϕ={a∈M:E(a) = 0}. (3.3) ThereforeR^⊥ϕ is closed under the involution operation.

For each a∈Mwe set a⊥=a− E(a) and

ϕ(a^∗_⊥x) =ϕ((a^∗− E(a^∗))x) =ϕ(a^∗x)−ϕ(E(a^∗)x) =ϕ(a^∗x)−ϕ(E(a^∗x)) = 0 for allx∈R, hencea_⊥∈R^⊥ϕ.

So for eacha∈Mexists a uniquea_k∈Randa_⊥∈R^⊥ϕ such thata=a_k+a_⊥ where we have seta_k=E(a).

The uniqueness follows since if a= 0 thena_k=a_⊥ = 0.

Indeed, sincea^∗_ka_⊥ anda^∗_⊥a_kbelongs toR^⊥ϕ, we haveϕ(a^∗_ka_⊥,) =ϕ(a^∗_⊥a_k) = 0 and

ϕ(a^∗a) =ϕ(a^∗_ka_k) +ϕ(a^∗_⊥a_⊥) = 0 withϕfaithful state.

Vice-versa, if the set R^⊥ϕ is closed under the involution operation and M = R⊕R^⊥ϕ then for each a ∈M exists a unique a_k ∈ R and a_⊥ ∈R^⊥ϕ such that a=a_k+a_⊥ .

The map a∈M→a_k∈R is a projection of norm one (i.e. it satisfies (1)_k = 1 and ((a)_k)_k =a_k for alla∈M) and for Tomiyama [36] it is a normal conditional

expectation withϕ(a) =ϕ(a_k) for all a∈M.

Remark 3.2. — If the set R^⊥ϕ is a *-algebra (without unit) then R^⊥ϕ = {0}, sinceϕis a faithful state.

Moreover, ifpis an orthogonal projector ofMthenp /∈R^⊥ϕ.

Remark 3.3. — Ifa∈Mwitha=a_k+a_⊥ wherea_k∈Randa_⊥ ∈R^⊥ϕ, then

||πϕ(a)Ωϕ||²=||πϕ(a_k)Ωϕ||²+||πϕ(a_⊥)Ωϕ||². (3.4) Proposition 3.4. — LetRbe a von Neumann algebra, invariant under modu- lar automorphism groupσ^ϕ_t . IfHoandKoare respictively the closure of the linear spaceπ_ϕ(R)Ω_ϕ andπ_ϕ(R^⊥ϕ)Ω_ϕ then

Hϕ=Ho⊕ Ko .

Moreover, the orthogonal projectionPoon Hilbert space Ho belongs toπϕ(R)⁰. Proof. — We have thatKo⊂ H^⊥_o, since for eachr⊥∈R^⊥ϕ andψo∈ Hothen

hπϕ(r_⊥)Ωϕ, ψoi= lim

α→∞hπϕ(r_⊥)Ωϕ, πϕ(rα)Ωϕi= lim

α→∞ϕ(r^∗_⊥rα) = 0 whereψ_o= lim

α→∞π_ϕ(r_α)Ω_ϕ with{r_α}_αnet belongs toR.

Ifψ∈ Hϕ we can write:

ψ= lim

α→∞πϕ(mα)Ωϕ= lim

α→∞(πϕ(rα)Ωϕ+πϕ((r_α⊥)Ωϕ) wherem_α=r_α+r_α⊥ for allα.

The net{π_ϕ(r_α)Ω_ϕ}has limit, since of the relation (3.4) for each>0 there is a indexν such that forα>ν andβ >ν, we have the Cauchy relation:

||πϕ(r_α)Ω_ϕ−π_ϕ(r_β)Ω_ϕ||6||πϕ(m_α)Ω_ϕ−π_ϕ(m_β)Ω_ϕ||6

It follows that there areψ_k∈ Hoandψ_⊥∈ Kosuch that:

ψ= lim

α→∞πϕ(rα)Ωϕ+ lim

α→∞πϕ((rα⊥)Ωϕ=ψ_k+ψ⊥∈ Ho⊕ Ko .

It is simple to prove thatπϕ(R)Ho⊂ Ho, soPo∈πϕ(R)⁰. Proposition 3.5. — Let(M,Φ)be a quantum process andϕa normal faithful state on M. For each natural numbernwe have:

M=DΦⁿ⊕D^⊥_Φ^ϕn (3.5)

and

M=D⁺_∞⊕D^+⊥_∞ ^ϕ . (3.6)

Furthermore, ifϕis a stationary state for Φthen

M=CΦ⊕ C_Φ^⊥ϕ (3.7)

and the restriction of Φto CΦ is a *-automorphism withΦ(C_Φ^⊥ϕ)⊂ C_Φ^⊥ϕ. Proof. — For eachd∈DΦⁿ and natural numbernwe get:

Φⁿ_•(σ_t^ϕ(πϕ(d)^∗)σ^ϕ_t(πϕ(d))) = Φⁿ_•(σ_t^ϕ(πϕ(d)^∗))Φⁿ_•(σ_t^ϕ(πϕ(d))) because Φ commutes with that modular automorphism group σ_t^ϕ.

It follows thatσ_t^ϕ(πϕ(DΦⁿ)) is intoπϕ(DΦⁿ) for all n∈Nandt∈R. Ifb∈ CΦthenσ_t^ϕ(πϕ(b))∈πϕ(CΦ) for all real numbert.

Indeed for each natural numbernthere is axn∈D⁺_∞ such thatb= Φⁿ(xn), so we obtain:

σ_t^ϕ(πϕ(b)) =σ_t^ϕ(πϕ(Φⁿ(xn)) = Φⁿ_•(σ^ϕ_t(πϕ(xn)) and consequentlyσ_t^ϕ(πϕ(xn))∈πϕ(D⁺_∞) for all natural numbern.

Thereforeσ_t^ϕ(πϕ(b))∈πϕ(Φⁿ(D⁺_∞) for all natural number n.

Lety belongs toC_Φ^⊥ϕ, since Φ(CΦ) =CΦ, we have that for each c∈ CΦ

ϕ(Φ(y)c) =ϕ(Φ(y)Φ(co)) =ϕ(yco) = 0

wherec= Φ(c_o), withc_o∈ CΦ⊂D⁺_∞.

Now we deal with a QDS (M,Φ, ϕ) withϕ-adjoint Φ^].

We define D_∞ (or with D_∞(Φ) when we have to highlight the map Φ) the following von Neumann algebra:

D_∞= \

k∈Z

DΦ_k (3.8)

where for each integerk we denote:

Φ_k=

Φ^k k>0 Φ^]|k| k <0

and DΦ_k the von Neumann algebra of the multiplicative domain of the dynamics Φk.

Following Robinson [32] for eacha, b∈Mand integer numberkwe define:

S_k(a, b) = Φ_k(a^∗b)−Φ_k(a^∗)Φ_k(b)∈M (3.9) and we have these simple relations:

a. S_k(a, a)>0 for alla∈Mand integerk.

b. S_k(a, b)^∗=S_k(b, a) for all a, b∈Mand integerk.

c. d∈D_∞if and only if Sk(d, d) =Sk(d^∗, d^∗) = 0 for all integerk.

d. d∈D∞if and only if ϕ(Sk(d, d)) =ϕ(Sk(d^∗, d^∗)) = 0 for all integerk.

e. The map a, b ∈M →ϕ(Sk(a, b)) for all integer k, is a sesquilinear form, hence

|ϕ(Sk(a, b))|²6ϕ(Sk(a, a))ϕ(Sk(b, b)), a, b∈M. Note that Φ(D_∞)⊂D_∞and Φ^](D_∞)⊂D_∞.

Indeed, for each element d∈D_∞ and integerkwe get:

ϕ(S_k(Φ(d),Φ(d)) =ϕ(S_k+1(d, d) = 0 and

ϕ(S_k(Φ^](d),Φ^](d)) =ϕ(S_k−1(d, d) = 0. Assuming thatd^∗ belongs toD∞ we have

ϕ(Sk(Φ(d)^∗,Φ(d)^∗) =ϕ(Sk(Φ^](d)^∗,Φ^](d)^∗) = 0.

It follows that restriction of the map Φ to the von Neumann algebra D_∞ is a

*-automorphism with Φ(Φ^](d)) = Φ^](Φ(d)) =dfor alld∈D_∞. To sum up we have the following statement:

Proposition 3.6. — Let (M,Φ, ϕ) be a QDS. The restriction of Φ to D_∞ denoted withΦ_∞ is a *-automorphism of the von Neumann algebraD_∞.

IfB is a von Neumann sub-algebra ofMsuch that the restriction ofΦto Bis a *-automorphism thenB⊂D∞.

We have a (maximal) reversible QDS(D∞,Φ∞, ϕ∞)where the normal stateϕ∞

and the ϕ∞-adjoint Φ^]_∞ are respectively the restriction of ϕ and Φ^] to the von Neumann algebraD∞.

Proof. — We prove that if the restriction of Φ toB is an automorphism then B⊂D∞.

As B⊂D_Φn for all natural numbernand if Ψ :B→Bis the map such that Ψ(Φ(b)) = Φ(Ψ(b)) =b for allb∈Bthen Ψ(b) = Φ^](b) since

ϕ(aΨ(b)) =ϕ(Φ(aΨ(b))) =ϕ(Φ(a)Φ(Ψ(b)) =ϕ(Φ(a)b)) =ϕ(aΦ^](b)) for alla∈M.

It follows that B is also Φ^]-invariant, hence B ⊂D_Φn] for all natural number

n.

As it is clear thatD∞ is Φk-invariant for all integer numberkand is invariant under automorphism groupσ_t^ϕ, by the previous decomposition theorem follows:

Proposition 3.7. — Let(M,Φ, ϕ)be a QDS, there is a conditional expectation E∞:M→D_∞such that

a. ϕ◦ E∞=ϕ.

b. D^⊥_∞^ϕ= kerE_∞. c. M=D_∞⊕D^⊥_∞^ϕ.

d. Φ_k(D^⊥_∞^ϕ)⊂D^⊥_∞^ϕ for all integer numberk.

e. E∞(Φ_k(a)) = Φ_k(E∞(a))for alla∈Mand integer numberk.

f. H_ϕ=H_∞⊕K_∞whereH_∞andK_∞denotes the linear closure ofπ_ϕ(D_∞)Ω_ϕ and ofπ_ϕ(D^⊥_∞^ϕ)Ω_ϕ respectively.

Proof. — The statements (a), (b) and (c) are simple consequences of theorem 3.1.

For the statement (d), if d_⊥ ∈D^⊥_∞^ϕ then for each integer number k andx∈D_∞, we have ϕ(Φ_k(d_⊥)^∗x) =ϕ(d^∗_⊥Φ_−k(x)) = 0 since Φ_−k(x)∈D_∞.

For the statement (e), for eacha, b∈Mwe obtain:

ϕ(bE∞(Φk(a))) = ϕ((b_k+b_⊥)E∞(Φk(a))) =ϕ(b_kE∞(Φk(a)) =

= ϕ(E_∞(bkΦk(a))) =ϕ(bkΦk(a))) =ϕ(Φ_−k(b_k)a) =

= ϕ(E∞(Φ−k(b_k)a)) =ϕ(E∞(Φ−k(b_k)a)) =

= ϕ(Φ_−k(b_k)E_∞(a)) =ϕ(b_kΦk(E_∞(a)) =

= ϕ((b_k+b_⊥)Φk(E∞(a)) =ϕ(bΦ_k(E∞(a))

where we have writtenb=b_k+b⊥ withb_k=E∞(b).

The QDS (D_∞,Φ_∞, ϕ_∞) is called the reversible part of the QDS (M,Φ, ϕ).

Furthermore, a QDS is calledcompletely irreversible ifD_∞=C1.

In this case for eacha∈Mwe obtaina=ϕ(a)1 +a_⊥ and we can write:

M=C1⊕kerϕ .

In the decoherence theory, the set D_∞ is calledalgebra of effective observables of our QDS (see e.g. [4, 9]) and we highlight that the previous theorem is a particular case of a more general theorem that can be found in [5, 24].

We notice also that for all natural number nwe obtain Φ^]n(Φⁿ(d)) =d , d∈D⁺_∞ and

Φⁿ(D⁺_∞)⊂D_Φ]n . We conclude this section with some simple remarks.

Remark 3.8. — The algebra of effective observables is independent by the sta- tionary stateϕsince

D_∞=C_Φ . (3.10)

Indeed we have thatD_∞⊂ T

n∈N

Φⁿ(D⁺_∞) sinceD_∞⊂D⁺_∞andCΦ⊂D_∞by theorem 3.6.

Summarizing, if the quantum process (M,Φ) admits a stationary state ϕ and the dynamic Φ admits aϕ-adjoint then its reversible part is (CΦ,Φ_∞).

Remark 3.9. — Let (M,Φ, ϕ) be a QDS, we denote withA(P) the von Neumann algebra generated by the set of all orthogonal projectionsp∈Msuch that Φk(p) = Φk(p)² for all integer numberk. It is easily proved thatD_∞=A(P) (see e.g [8]).

Remark 3.10. — If an orthogonal projectionP ∈Msatisfies the relationϕ(P)− ϕ(P)²= 0 thenP ∈D∞.

Indeed by faithfulness of ϕ and by Schwartz inequality (1.1) we have Φk(P)− Φk(P)² = 0 for all integer number k. Therefore, if our QDS is completely irre- versible then we have ϕ(P)−ϕ(P)²>0 for all no trivial orthogonal projectionP ofM.

In section 6 we will find the conditions to have D∞ =C1 (see also [9] for the continuous case andD⁺_∞=C1).

4. Decomposition theorem and linear contractions

We are going to study the relations between canonical decomposition of Nagy- Fojas of linear contraction UΦ,ϕ [27] and the decomposition theorem 3.7, therefore recalling the main statements of these topics.

A contractionT on the Hilbert spaceHis called completely non-unitary (c.n.u.) if for no non zero, reducing subspace K for T, is T_|K a unitary operator, where T_|K is the restriction of contractionT on the Hilbert space K. We set withD_T =

√I−T^∗T the defect operator of the contractionT and it is well-known that T DT =DT^∗T .

Moreover,||T ψ|||=||ψ|| if, and only ifDTψ= 0.

We consider the following Hilbert subspace ofH:

H0={ψ∈ H:||Tⁿψ||=||ψ||=||T^∗nψ||f or all n∈N}. (4.1) It is trivial to show that TⁿH₀=H₀ andT^∗nH₀=H₀ for all natural numbern.

We then obtain the following canonical decomposition [27]:

Theorem 4.1(Sz-Nagy and Fojas). — For every contractionT onHthere is a uniquely decomposition ofHinto an orthogonal sum of two subspace reducingT; we sayH=H0⊕H1, such thatT0=T_|H0 is unitary andT1=T_|H1 is c.n.u., where

H0= \

k∈Z

ker (DT_k) and H1=H^⊥₀ with

T_k =

T^k k>0 T^∗−k k <0 .

We underline that the linear operators, T₋ = so− lim

n→+∞T^∗nTⁿ and T₊ = so− lim

n→+∞TⁿT^∗n, exist in sense of strong operator (so) convergence (see [27] pag.

40) .

We now focus our attention on the quantum dynamical systems (M,Φ, ϕ).

We setV₋ =so− lim

n→+∞U_Φ,ϕ^∗n U_Φ,ϕⁿ and V+ =so− lim

n→+∞U_Φ,ϕⁿ U_Φ,ϕ^∗n , whereUΦ,ϕ

that is the contraction as defined in (2.4).

It follows that for eacha, b∈M, we obtain:

n→±∞lim ϕ(Sn(a, b)) =hπϕ(a)Ωϕ,(I−V±)πϕ(b)Ωϕi (4.2) whereS_n(a, b) is given by (3.9).

We recall that, by the proposition 3.7 we getH_ϕ=H_∞⊕ K_∞withU_kH_∞=H_∞ andU_kK_∞⊂ K_∞, where for every integerk

Uk=

U_Φ,ϕ^k k>0 U_Φ,ϕ^∗−k k <0 .

A simple consequence of proposition 2.2 is the following remark:

Remark 4.2. — For each integers k, we obtain that a ∈ DΦ_k if and only if πϕ(a)Ωϕ∈ker(DU_k) andπϕ(a^∗)Ωϕ∈ker(DU_k).

Therefore,H_∞⊂ H0 because πϕ(D_∞)Ωϕ⊂ \

k∈Z

πϕ(D_Φk)Ωϕ⊂ \

k∈Z

ker(DU_k).

We have shown that for eacha, b∈ Mand natural number kfollows that (see [14] theorem 3.1):

n→+∞lim ϕ(Sk(Φⁿ(a), b)) = 0. (4.3) Indeed, for each natural numberskandn

ϕ(Sk(Φⁿ(a),Φⁿ(b)) =ϕ(Sk+n(a, b))−ϕ(Sn(a, b)), and by the relation (4.2) results that lim

n→+∞(ϕ(S_k+n(a, b))−ϕ(S_n(a, b))) = 0.

Furthermore, for each natural numberkand a, b∈M

|ϕ(S_k(Φⁿ(a), b))|²6ϕ(S_k(Φⁿ(a),Φⁿ(a))ϕ(S_k(b, b)). It follows that lim

n→+∞ϕ(Sk(Φⁿ(a), b) = 0.

We get a well-known statement [17, 24, 33]:

Proposition 4.3. — For alla∈M, everyσ-limit point of the set {Φ^k(a)}_k∈N belongs to the von Neumann algebraD_∞.

Moreover, for eachd_⊥ in D^⊥∞^ϕ we have:

lim

k→+∞Φ^k(d_⊥) = 0 and lim

k→+∞Φ^]k(d_⊥) = 0 where the limits are inσ-topology.

Proof. — Ify is aσ-limit point of{Φⁿ(a)}_n∈N, then there is a net{Φ^nj(a)}_j∈N such thaty= lim

j→+∞Φ^nj(a) inσ-topology. Furthermore, for eachb∈M S_k(y, b) =σ− lim

j→+∞[ Φ^k(Φ^nj(a)b)−Φ^k(Φ^nj(a))Φ^k(b) ] =σ− lim

j→+∞S_k(Φ^nj(a), b) then from (4.3) we obtain lim

j→+∞ϕ(Sk(Φ^nj(a), b)) = 0 henceϕ(Sk(y, b)) = 0 and in the end follows that ϕ(S_k(y, y)) = 0 andS_k(y, y) = 0.

As that the adjoint is σ-continuous, then we have y^∗ = lim

j→+∞Φ^nj(a^∗) and re- peating the previous steps we find thatSk(y^∗, y^∗) = 0 hencey∈D_∞.

Following to the last statement, we see that ||Φ^k(d⊥)|| 6 ||d⊥|| for all natural numberkand because the unit ball of the von Neumann algebraMisσ- compact, we have proved that there is a subnet such that Φ^kα(d_⊥)→y∈D^⊥_∞^ϕinσ-topology.

From the previous lemma, we can affirm that y∈D_∞∩D^⊥∞^ϕ as y= 0, then it can be only lim

k→+∞Φ^k(d_⊥) = 0 inσ-topology.

We have understood that the Hilbert space H∞, which is the linear closure of π_ϕ(D_∞)Ω_ϕ, is contained in H0. The next step is to understand when these two Hilbert spaces are equal.

Let (M,Φ, ϕ) be a QDS, so we can define for each integerk, the unital Schwartz mapτk:M→Mas

τ_k = Φ_−k◦Φ_k , k∈Z. (4.4)

For every integerk 1. ϕ◦τk=ϕ;

2. τ_k=τ_k^], whereτ_k^] is theϕ-adjoint ofτ_k.

Then for every integerk, we obtain the dynamical system{M, τk, ϕ} with D_∞(τk) = \

j>0

D(τ_k^j), whereD(τ_k^j) denotes the multiplicative domains of mapτ_k^j.

From decomposition theorem 3.1, for every integerk M=D∞(τk)⊕D∞(τk)^⊥ϕ and following the proposition 3.4 we obtain that

H_ϕ=H_(k)⊕ K_(k)

where H(k) and K(k) are the Hilbert spaces, respectively the closure of the linear spaceπϕ(D_∞(τk))Ωϕ and ofπϕ(D_∞(τk))^⊥ϕ)Ωϕ.

Proposition 4.4. — If π_ϕ(D_Φk)Ω_ϕ denotes the closure of the linear space π_ϕ(D_Φk)Ω_ϕ, it follows that

H0= \

k∈Z

πϕ(DΦ_k)Ωϕ,

where theH0is the Hilbert space of Nagy decomposition of theorem 4.1.

Furthermore, for eacha∈M,ξ₀∈ H0and integerk

U_Φ,ϕ^k πϕ(a)ξ0=πϕ(Φ^k(a))U_Φ,ϕ^k ξ0. (4.5) Proof. — We assume thatD(τ_k)⊂D_Φk for all integersk.

In fact, ifa∈D(τ_k), then

ϕ(Φ_k(a^∗a)) = ϕ(a^∗a) =ϕ(τ_k(a^∗a)) =ϕ(τ_k(a^∗)τ_k(a)) =

= ϕ(Φ_−k(Φk(a)^∗)Φ_−k(Φk(a)))6

6 ϕ(Φ_−k(Φ_k(a)^∗Φ_k(a))) =ϕ(Φ_k(a^∗)Φ_k(a))6 6 ϕ(Φk(a^∗a)).

This proves that ϕ(Sk(a, a)) = 0 and, in the same way, ϕ(Sk(a^∗, a^∗)) = 0 for all integers k.

We have proved that

D_∞(τ_k) = \

j∈N

D_τj

k ⊂D_τk⊂D_Φk. Ifξ0∈ H0, then for eachkinteger and natural numbern

(U_Φ,ϕ,^∗k U_Φ,ϕ^k )ⁿξ₀=ξ₀ and, for eachr⊥∈D∞(τk)^⊥ϕ, we can write that

hπϕ(r⊥)Ωϕ, ξ0i=

(U_Φ,ϕ,^∗k U_Φ,ϕ^k )ⁿ πϕ(r⊥)Ωϕ, ξ0

=hπϕ(τ_kⁿ(r⊥)Ωϕ, ξ0i and for each integer number kresult:

n→+∞lim hπ_ϕ(τ_kⁿ(r_⊥)Ω_ϕ, ξ₀i= 0 becauseτ_kⁿ(r_⊥)−→0 asn→ ∞inσ-topology.

It follows thatH0⊂[π_ϕ(D_∞(τ_k)^⊥ϕ)Ω_ϕ]^⊥= [Kk]^⊥.

Therefore, for every integersk, we obtain:

H0⊂ H_(k)⊂πϕ(DΦ_k)Ωϕ =⇒ H0⊂ \

k∈Z

πϕ(DΦ_k)Ωϕ. Let assumeξ0 ∈T

k∈Zπϕ(DΦ_k)Ωϕ, for every integersk, we have a net dα,k∈DΦ_k

such thatπϕ(dα,k)Ωϕ→ξ0as α→ ∞and, for k>0, we can obtain U_Φ,ϕ^∗k U_Φ,ϕ^k ξ0 = U_Φ,ϕ^∗k U_Φ,ϕ^k lim

α πϕ(dα,k)Ωϕ= lim

α U_Φ,ϕ^∗k U_Φ,ϕ^k πϕ(dα,k)Ωϕ=

= lim

α π_ϕ(d_α,k)Ωϕ=ξ₀ .

In the same way, fork>0, we have gotU_Φ,ϕ^k U_Φ,ϕ^∗k ξ0=ξ0. It follows that

\

k∈Z

πϕ(DΦ_k)Ωϕ⊂ H0.

The relationship (4.5) is a straightforward.

We note that, for eacha∈Mandd_⊥∈D^⊥_∞^ϕ

n→∞limϕ(a^∗Φ_n(d_⊥)a) = 0 (4.6) because, for eachd⊥∈D^⊥∞^ϕ, we obtain Φn(d⊥)→0 asn→ ∞inσ-topology.

The polarization identity allows to say that

n→∞limϕ(aΦ_n(d_⊥)b) = 0, a, b∈M, d_⊥∈D^⊥_∞^ϕ

so becauseUϕ is a contraction, it follows that for eachξ∈ Hϕandψ∈ K∞ n→∞limhξ, U_Φ,ϕⁿ ψi= 0 and lim

n→∞hξ, U_Φ,ϕ^∗n ψi= 0. (4.7) The proposition 4.5 is a simple statement on the Hilbert spacesH∞ andH0.

Proposition 4.5. — IfΦⁿ(d_⊥)→0 [Φ^]n(d_⊥)→0] as n→ ∞in s-topology for alld_⊥∈D^⊥∞^ϕ; thenH∞=H0 andV₊=P_∞ [V₋=P_∞].

Proof. — We observe that, for everyψ∈ K_∞, ||U_Φ,ϕⁿ ψ|| →0 asn→ ∞, because for eachk∈Nthere isd^⊥_k ∈D^⊥∞^ϕ such that ||ψ−πϕ(d^⊥_k)Ωϕ||<1/k; asU_Φ,ϕⁿ is a linear contraction, for all natural numbern, we obtain:

||U_Φ,ϕⁿ ψ||< 1

k+ϕ(Φⁿ(d^⊥_k)^∗Φⁿ(d^⊥_k)).

Ifξ0∈ H0 thenξ0=ξ_k+ξ⊥ withξ_k∈ H∞andξ⊥∈ K∞. Asξ⊥=ξ0−ξ_k∈ H0

||ξ_k||+||ξ⊥||=||U_Φ,ϕⁿ ξ0||=||U_Φ,ϕⁿ ξ_k+U_Φ,ϕⁿ ξ⊥||=||ξ_k||+||U_Φ,ϕⁿ ξ⊥||

for all natural numbersnthen follows thatξ_⊥ = 0.

Moreover, for each ξ∈ Hϕ, we obtain that U_Φ,ϕ^n∗ U_Φ,ϕⁿ ξ=ξ₀+U_Φ,ϕ^n∗ U_Φ,ϕⁿ ξ₁ with ξ_i∈ H_i fori= 1,2 andV₊ξ=ξ₀, because ||U_Φ,ϕⁿ ξ₁|| →0 asn→ ∞.

Remark 4.6. — We recall that a QDS{M,Φ, ω}is mixing, if

n→∞lim ϕ(aΦⁿ(b)) =ϕ(a)ϕ(b), a, b∈M. (4.8) By the relation (4.6), we can obtain that {M,Φ, ω} is mixing if and only if its reversible part (D_∞,Φ_∞, ϕ_∞) is mixing.

Furthermore, let {M,Φ, ω} be a mixing Abelian QDS, then there is a measur- able dynamics space (X,A, µ, T) such that D∞ is isomorphic to the von Neumann algebra L^∞(X,A, µ) of the measurable bounded function onX. If the setX is a metric space and ϕ∞ is the unique stationary state ofD∞ for the dynamics Φ∞, then by the corollary 4.3 of [13] follows thatD∞=C1.

5. Some properties of reversible part of a QDS In this section we give some consequences of the previous propositions.

5.1. Ergodicity properties. We prove that the ergodic properties of a quantum dynamical system depends on its reversible part, determined from the algebra of the effective observablesD∞.

Let assume a QDS (M,Φ, ϕ) withϕ-adjoint Φ^]; it is ergodic if:

lim

N→∞

1 N+ 1

N

X

k=0

ϕ(aΦ^k(b))−ϕ(a)ϕ(b)

= 0, a, b∈M, (5.1) while it is weakly mixing if:

Nlim→∞

1 N+ 1

N

X

k=0

ϕ(aΦ^k(b))−ϕ(a)ϕ(b)

= 0 , a, b∈M. (5.2) We will use again the following notations a_k =E_∞(a) while a_⊥ = a−a_k for all a ∈ M, where E_∞ : M → D_∞ is the conditional expectation of decomposition theorem 3.7.

We have the following:

Proposition 5.1. — The QDS(M,Φ, ϕ)is ergodic [weakly mixing] if and only if the reversible QDS (D_∞,Φ_∞, ϕ_∞)is ergodic [weakly mixing].

Proof. — For eacha, b∈Mwe have got:

ϕ(aΦ^k(b))−ϕ(a)ϕ(b) =ϕ(aΦ^k(b_k)) +ϕ(aΦ^k(b_⊥))−ϕ(a_k)ϕ(b_k). Moreover, lim

N→∞

1 N+ 1

N

P

k=0

ϕ(aΦ^k(b_⊥)) = 0, because by relation (4.6) for everya∈ Mwe have lim

k→∞ϕ(aΦ^k(b_⊥)) = 0, hence lim

N→∞

1 N+ 1

N

X

k=0

ϕ(aΦ^k(b))−ϕ(a)ϕ(b)

=

= lim

N→∞

1 N+ 1

N

X

k=0

ϕ(aΦ^k(b_k))−ϕ(a_k)ϕ(b_k)