Abstract

We prove a version of the Jacobs-de Leeuw-Glicksberg splitting theorem for weak* continuous one-parameter semigroups on dual Banach spaces. This result is applied to give sufficient conditions for a quantum dynamical semigroup to display decoherence. The underlying notion of decoherence is that introduced by Blanchard and Olkiewicz (2003). We discuss this notion in some detail.

1. Introduction

The theory of environmental decoherence starts from the question of why macroscopic physical systems obey the laws of classical physics, despite the fact that our most fundamental physical theoryβ€”quantum theoryβ€”results in contradictions when directly applied to these objects. The infamous SchrΓΆdinger cat is a well-known illustration of this problem. This is an embarrassing situation since, from its inception in the 1920s until today, quantum theory has seen a remarkable success and an ever increasing range of applicability. Thus the question of how to reconcile quantum theory with classical physics is a fundamental one, and efforts to find answers to it persisted throughout its history. At present, the most promising and most widely discussed answer is the notion of environmental decoherence. The starting point is the contention that quantum theory is universally valid, in particular in the macroscopic domain, but that one has to take into account the fact that macroscopic objects are strongly interacting with their environment, and that precisely this interaction is the origin of classicality in the physical world. Thus classicality is a dynamically emergent phenomenon due to the essential openness of macroscopic quantum systems, that is, their interaction with other quantum systems surrounding them leads to an effective restriction of the superposition principle and results in a state space with properties different from the pure quantum case.

In order to clarify the status of decoherence and to provide a rigorous definition, Ph. Blanchard and R. Olkiewicz suggested a notion of decoherence formulated in the algebraic framework [1, 2] of quantum physics in [3], drawing on earlier work in [4]. The algebraic framework is especially useful for the discussion of decoherence, since it is able to accommodate classical systems, provides an elegant formulation of superselection rules, and can even describe systems with infinitely many degrees of freedom in a rigorous way. This is why it is becoming increasingly popular in the discussion of foundational and philosophical problems of quantum physics [5, 6].

In the present paper, we assume that the algebra of observables of the system under study is a von Neumann algebra, and due to its openness the time evolution is irreversible and hence given by a family {𝑇𝑑}𝑑β‰₯0 of normal completely positive and unital linear maps on the von Neumann algebra [7, 8]. In the Markovian approximation, the family {𝑇𝑑}𝑑β‰₯0 becomes a so-called quantum dynamical semigroup. It is our purpose to discuss the Blanchard-Olkiewicz notion of decoherence for quantum dynamical semigroups. To this end we study a weak* version of the so-called Jacobs-de Leeuw-Glicksberg splitting for one-parameter semigroups on dual Banach spaces. In the Markovian case, the Blanchard-Olkiewicz notion of decoherence relies on the so-called isometric-sweeping splitting, which is similar to the Jacobs-de Leeuw-Glicksberg splitting, and we will be able to prove a new criterion for the appearance of decoherence in the case of uniformly continuous quantum dynamical semigroups by examining the connection between the two asymptotic splittings.

The paper is organized as follows. In Section 2 we establish the Jacobs-de Leeuw-Glicksberg splitting for weak* continuous contractive one-parameter semigroups on dual Banach spaces. We provide a sufficient condition which ensures that the semigroup is weak* stable on the stable subspace of the splitting (Proposition 2.3). In Section 3 we turn to the study of quantum dynamical semigroups on von Neumann algebras. We begin by applying the results of Section 2 in the von Neumann algebra setting (Proposition 3.3). As a complement to Proposition 2.3, we prove Proposition 3.6, which gives another condition for weak* stability on the stable subspace of the splitting. In Section 4, we discuss a notion of decoherence which is very close to that given in [3] and establish some mathematical results related to it. In the final Section 4.2, we use the previous results to give a sufficient condition that a uniformly continuous quantum dynamical semigroup having a faithful normal invariant state displays decoherence.

2. The Jacobs-de Leeuw-Glicksberg Splitting

Suppose that 𝖷 is a Banach space and assume that it has a predual space denoted by π–·βˆ—, that is, (π–·βˆ—)βˆ—β‰…π–·. If π‘₯βˆˆπ–· and πœ‘βˆˆπ–·βˆ—, we will denote the evaluation of π‘₯ at πœ‘ by ⟨π‘₯,πœ‘βŸ© and consider this as a dual pairing between 𝖷 and π–·βˆ—. The set of all bounded linear operators from 𝖷 to 𝖷, endowed with the operator norm, will be denoted by L(𝖷), and its unit ball by L(𝖷)1={π‘‡βˆˆL(𝖷)βˆΆβ€–π‘‡β€–β‰€1}. Operators from L(𝖷)1 are called contractive. We consider the algebraic tensor product π–·βŠ™π–·βˆ— and endow it with the projective cross norm 𝛾; the completion of π–·βŠ™π–·βˆ— with respect to 𝛾 is a Banach space which will be denoted by π–·βŠ—π›Ύπ–·βˆ—. Then its dual space (π–·βŠ—π›Ύπ–·βˆ—)βˆ— is isometrically isomorphic in a canonical way with L(𝖷,(π–·βˆ—)βˆ—)=L(𝑋): If πœ“βˆˆ(π–·βŠ—π›Ύπ–·βˆ—)βˆ—, we define Ξ¦(πœ“)∈L(𝖷) by⟨π‘₯βŠ—πœ‘,πœ“βŸ©=⟨Φ(πœ“)(π‘₯),πœ‘βŸ©(2.1) for all π‘₯βˆˆπ–· and πœ‘βˆˆπ–·βˆ—. It can now be shown that πœ“β†¦Ξ¦(πœ“) extends to an isometric isomorphism, and we can thus write (π–·βŠ—π›Ύπ–·βˆ—)βˆ—β‰…L(𝖷).

We now introduce the pointwise weak* topology on L(𝖷). Let π‘₯βˆˆπ–·, πœ‘βˆˆπ–·βˆ—, and define the seminorm L(𝖷)βˆ‹π‘‡β†¦π‘π‘₯,πœ‘(𝑇)=|βŸ¨π‘‡(π‘₯),πœ‘βŸ©|. The pointwise weak* topology is the locally convex topology on L(𝑋) induced by the family {𝑝π‘₯,πœ‘βˆΆπ‘₯βˆˆπ–·,πœ‘βˆˆπ–·βˆ—} of seminorms. If 𝑇=Ξ¦(πœ“)∈L(𝖷), we see that 𝑝π‘₯,πœ‘(𝑇)=|βŸ¨π‘‡(π‘₯),πœ‘βŸ©|=|⟨π‘₯βŠ—πœ‘,πœ“βŸ©|; thus the pointwise weak* topology coalesces with the 𝜎(L(𝖷),π–·βŠ—π›Ύπ–·βˆ—) topology on L(𝖷), that is, the pointwise weak* topology is a weak* topology as well. Thus we can conclude from Alaoglu's theorem that L(𝖷)1 is compact in the pointwise weak* topology.

A linear operator π‘‡βˆˆL(𝖷) will be called normal provided it is a continuous map from 𝖷 to 𝖷 when 𝖷 is endowed with the weak* topology. We denote the set of all normal operators by Ln(𝖷). We can consider the set of all normal contractive operators Ln(𝖷)1 as a semigroup under multiplication of operators, that is, if 𝑇1,𝑇2∈Ln(𝖷)1, then 𝑇1βˆ˜π‘‡2 is normal and contractive; moreover, the multiplication is associative. The semigroup Ln(𝖷)1 is semitopological when endowed with the pointwise weak* topology, that is, the multiplication is separately continuous. This means that the maps π‘‡β†¦π‘‡βˆ˜π‘† and π‘†β†¦π‘‡βˆ˜π‘† are both continuous with respect to the pointwise weak* topology. Finally we remark that it is important to note that Ln(𝖷)1 is not closed in L(𝖷)1 with respect to the pointwise weak* topology. Moreover, recall that an operator π‘‡βˆˆL(𝖷) is normal if and only if there exists a (unique) predual operator π‘‡βˆ— from π–·βˆ— into π–·βˆ—, defined by βŸ¨π‘‡(π‘₯),πœ‘βŸ©=⟨π‘₯,π‘‡βˆ—(πœ‘)⟩, π‘₯βˆˆπ–·, πœ‘βˆˆπ–·βˆ—.

In this section, our goal is to study one-parameter semigroups on dual Banach spaces. A contractive one-parameter semigroup [9, 10] is a family {𝑇𝑑}𝑑β‰₯0 of linear and contractive operators on 𝖷, such that π‘‡π‘ βˆ˜π‘‡π‘‘=𝑇𝑠+𝑑 for all 𝑠,𝑑β‰₯0 and 𝑇0=id𝖷. The semigroup is called weak* continuous provided each 𝑇𝑑 is a normal operator and [0,∞[βˆ‹π‘‘β†¦π‘‡π‘‘(π‘₯) is weak* continuous for any π‘₯βˆˆπ–·. For a weak* continuous semigroup there exists the following concept of a weak* generator 𝑍:𝑍π‘₯=lim𝑑↓0𝑇𝑑(π‘₯)βˆ’π‘₯𝑑intheweakβˆ—topology,(2.2)dom𝑍={π‘₯βˆˆπ–·βˆΆthelimitin(2.2)exists}.(2.3) The predual semigroup {𝑇𝑑,βˆ—}𝑑β‰₯0 of a weak* continuous semigroup {𝑇𝑑}𝑑β‰₯0 is strongly continuous, and the adjoint of its generator π‘βˆ— is equal to the weak* generator 𝑍.

Suppose now that {𝑇𝑑}𝑑β‰₯0 is a weak* continuous contractive semigroup on 𝖷, and write S0={π‘‡π‘‘βˆΆπ‘‘β‰₯0}βŠ†Ln(𝖷)1. In the following, we assume that the closure of S0 in L(𝖷)1 with respect to the pointwise weak* topology consists of normal operators, that is, we assume that S=S0βŠ†Ln(𝖷)1, where the bar denotes closure in the pointwise weak* topology. Then S is a compact commutative semitopological subsemigroup of Ln(𝖷)1. We now use the fact that every compact commutative semitopological semigroup has a unique minimal ideal GβŠ†S, the so-called Sushkevich kernel [9], which is given byG=π‘…βˆˆSπ‘…βˆ˜S,(2.4) and π‘„βˆˆG will denote the unit of G. We then have G=π‘„βˆ˜S. By compactness of G, it follows that G is, in fact, a commutative topological group. In the following, we will simplify our notation by writing 𝑇1𝑇2 instead of 𝑇1βˆ˜π‘‡2.

We are now able to prove a weak* version of the Jacobs-de Leeuw-Glicksberg splitting theorem, originally going back to Jacobs [11] and de Leeuw and Glicksberg [12, 13], see also [14]. The present proof mimics the one given in [9] for weakly almost periodic one-parameter semigroups.

Theorem 2.1. Let S0={𝑇𝑑}𝑑β‰₯0 be a weak* continuous contractive one-parameter semigroup with generator 𝑍. Assume that S=S0 consists of normal operators. Then there exist weak* closed subspaces 𝖷s,𝖷r of 𝖷 invariant under all operators 𝑇𝑑, 𝑑β‰₯0, such that 𝖷=𝖷sβŠ•π–·r, and 𝖷s=ξ‚»π‘₯βˆˆπ–·βˆΆ0βˆˆξ€½π‘‡π‘‘ξ€Ύ(π‘₯)βˆΆπ‘‘β‰₯0wβˆ—ξ‚Όπ–·,(2.5)r=lin{π‘₯∈domπ‘βˆΆβˆƒπ›Όβˆˆβ„suchthat𝑍π‘₯=i𝛼π‘₯}wβˆ—=ξ€½linπ‘₯∈XβˆΆβˆƒπ›Όβˆˆβ„suchthat𝑇𝑑(π‘₯)=ei𝛼𝑑π‘₯βˆ€π‘‘β‰₯0wβˆ—.(2.6)

Proof. Since 𝑄2=𝑄, the unit 𝑄 is a normal projection such that [𝑄,𝑇𝑑]=0 for all 𝑑β‰₯0. The theorem will be established once we prove that 𝖷s=ker𝑄 and 𝖷r=ran𝑄.
Let π‘₯∈ker𝑄. Since π‘„βˆˆS, there is a net {𝑇𝑖}π‘–βˆˆπΌβŠ†S0 such that 𝑇𝑖→𝑄 relative to the pointwise weak* topology; hence 𝑇𝑖(π‘₯)→𝑄π‘₯=0, so 0∈{𝑇𝑑(π‘₯)βˆΆπ‘‘β‰₯0}wβˆ—. Conversely, assume 0∈{𝑇𝑑(π‘₯)βˆΆπ‘‘β‰₯0}wβˆ— for some π‘₯. Then there is a net {𝑇𝑖}π‘–βˆˆπΌβŠ†S0 such that 𝑇𝑖(π‘₯)β†’0 relative to the weak* topology. By compactness of S, there is a subnet {𝑇𝑗}π‘—βˆˆπ½βŠ†{𝑇𝑖}π‘–βˆˆπΌ with 𝑇𝑗→𝑅 relative to the pointwise weak* topology for some π‘…βˆˆS, and it follows that 𝑅π‘₯=0. Hence 𝑅′𝑄𝑅π‘₯=0 for all π‘…β€²βˆˆS. Choosing 𝑅′ to be the inverse of 𝑄𝑅 in G, we get 𝑄π‘₯=0, hence π‘₯∈ker𝑄. We have thus proved that 𝖷s=ker𝑄.
Let G be the character group of G. For each Gπœ’βˆˆ define the operator π–·βˆ‹π‘₯βŸΌπ‘ƒπœ’ξ€œπ‘₯=Gπœ’(𝑆)𝑆π‘₯dπœ‡(𝑆),(2.7) where πœ‡ is the normalized Haar measure of G. The integral is to be understood as a weak* integral, thus π‘ƒπœ’ is a well-defined bounded operator in L(𝖷) with β€–π‘ƒπœ’β€–β‰€1. Then for all π‘…βˆˆG we get π‘…π‘ƒπœ’ξ€œπ‘₯=Gξ€œπœ’(𝑆)𝑅𝑆π‘₯dπœ‡(𝑆)=πœ’(𝑅)Gξ€œπœ’(𝑅𝑆)𝑅𝑆π‘₯dπœ‡(𝑆)=πœ’(𝑅)Gπœ’(𝑆)𝑆π‘₯dπœ‡(𝑆)=πœ’(𝑅)π‘ƒπœ’π‘₯,(2.8) in particular π‘„π‘ƒπœ’=π‘ƒπœ’; therefore, π‘‡π‘‘π‘ƒπœ’=π‘‡π‘‘π‘„π‘ƒπœ’=πœ’(𝑇𝑑𝑄)π‘ƒπœ’ for all 𝑑β‰₯0. Since π‘‘β†¦πœ’(𝑇𝑑𝑄) is continuous and satisfies the functional equation πœ’ξ€·π‘‡π‘‘π‘„ξ€Έξ€·π‘‡β‹…πœ’π‘ π‘„ξ€Έξ€·π‘‡=πœ’π‘‘+π‘ π‘„ξ€Έβˆˆ{π‘§βˆˆβ„‚βˆΆ|𝑧|=1}(2.9) for all 𝑑,𝑠β‰₯0, we have πœ’(𝑇𝑑𝑄)=ei𝛼𝑑 for some π›Όβˆˆβ„. Thus π‘‡π‘‘π‘ƒπœ’=eiπ›Όπ‘‘π‘ƒπœ’, hence π‘ƒπœ’π–·βŠ†dom𝑍 and π‘π‘ƒπœ’=iπ›Όπ‘ƒπœ’ for all Gπœ’βˆˆ. We next define the subspace 𝑀=⎧βŽͺ⎨βŽͺ⎩linπœ’βˆˆξGπ‘ƒπœ’π–·βŽ«βŽͺ⎬βŽͺ⎭wβˆ—βŠ†π–·r.(2.10) We prove that ranπ‘„βŠ†π‘€βŠ†π–·r. Let πœ‘βˆˆπ‘€βŸ‚={πœ‘βˆˆπ–·βˆ—βˆΆβŸ¨π‘₯,πœ‘βŸ©=0βˆ€π‘₯βˆˆπ‘€}. Then βŸ¨π‘ƒπœ’π‘₯,πœ‘βŸ©=0 for all π‘₯βˆˆπ–·, Gπœ’βˆˆ, that is, ξ€œξGπœ’(𝑆)βŸ¨π‘†π‘₯,πœ‘βŸ©dπœ‡(𝑆)=0(2.11) for all π‘₯βˆˆπ–·, Gπœ’βˆˆ. Since the character group G is total in 𝐿2(𝐺,πœ‡) by the Stone-Weierstraß theorem and since π‘†β†¦βŸ¨π‘†π‘₯,πœ‘βŸ© is continuous it follows that βŸ¨π‘†π‘₯,πœ‘βŸ©=0 for all π‘†βˆˆG and π‘₯βˆˆπ–·. Take 𝑆=𝑄, then we obtain πœ‘βˆˆranπ‘„βŸ‚ and thus π‘€βŸ‚βŠ†ranπ‘„βŸ‚. By the bipolar theorem we obtain ranπ‘„βŠ†co𝑀=𝑀, since ran𝑄 is a weak* closed subspace. Conversely, let π‘₯∈dom𝑍 with 𝑍π‘₯=i𝛼π‘₯ for some π›Όβˆˆβ„. It follows that 𝑇𝑑(π‘₯)=ei𝛼𝑑π‘₯ for all 𝑑β‰₯0 and consequently 𝑅π‘₯=ei𝛼π‘₯ for π‘…βˆˆS. Thus there exists π›½βˆˆβ„ such that 𝑄π‘₯=ei𝛽π‘₯=𝑄2π‘₯. Consequently, we must have 𝛽=0 which implies 𝑄π‘₯=π‘₯∈ran𝑄, hence 𝖷rβŠ†ran𝑄, and the proof is finished.

Corollary 2.2. Under the hypothesis of Theorem 2.1, there exists a weak* continuous one-parameter group {𝛼𝑑}π‘‘βˆˆβ„ of isometries on 𝖷r such that 𝛼𝑑=𝑇𝑑↾𝖷rfor 𝑑β‰₯0.

Proof. Let π‘‡βˆˆS, then π‘„π‘‡βˆˆG, and let 𝑅 be the inverse of 𝑄𝑇 in G, that is, 𝑅(𝑄𝑇)=𝑄. Then for all π‘₯βˆˆπ–·r, we have 𝑅𝑇π‘₯=𝑅𝑄𝑇π‘₯=𝑄π‘₯=π‘₯. Now write 𝛼𝑑=𝑄𝑇𝑑 for 𝑑β‰₯0 and let π›Όβˆ’π‘‘ be the inverse of 𝛼𝑑 in G. The foregoing calculation shows that {𝛼𝑑}π‘‘βˆˆβ„ is a one-parameter group on 𝖷r. Moreover, it is clear that it is weak* continuous and contractive. Now assume that there is π‘₯βˆˆπ–·r and 𝑑β‰₯0 such that ‖𝛼𝑑(π‘₯)β€–<β€–π‘₯β€–. Then it follows that β€–π›Όβˆ’π‘‘β€–>1, contradiction; thus {𝛼𝑑}π‘‘βˆˆβ„ is isometric.

The subspace 𝖷r is called the reversible subspace and 𝖷s is called the stable subspace; its elements are sometimes called flight vectors.

In applications it is sometimes desirable to have a stronger characterization of the subspace 𝖷s, namely, we are interested in a stronger stability property of the elements in 𝖷s. In particular, this is relevant in the applications to decoherence we discuss in Section 4. The next result provides a sufficient condition for weak* stability to hold on 𝖷s based on the boundary spectrum specπ‘βˆ©iℝ of the generator 𝑍.

Proposition 2.3. Assume that the hypothesis of Theorem 2.1 is satisfied and additionally that specπ‘βˆ©iℝ is at most countable. Then the stable subspace (2.5) is given by 𝖷s=ξ‚»π‘₯βˆˆπ–·βˆΆlimπ‘‘β†’βˆžπ‘‡π‘‘(π‘₯)=0relativetotheweakβˆ—ξ‚Όtopology.(2.12) Moreover, the convergence in (2.12) is uniform for π‘₯ in 𝖷sβˆ©π–·1.

Proof. Consider the predual semigroup {𝑇𝑑,βˆ—}𝑑β‰₯0 with generator π‘βˆ—; as already remarked, (π‘βˆ—)βˆ— is 𝑍. The predual π‘„βˆ— of 𝑄 is a projection and induces a splitting π–·βˆ—=𝖷r,βˆ—βŠ•π–·s,βˆ— by way of 𝖷r,βˆ—=ranπ‘„βˆ— and 𝖷s,βˆ—=kerπ‘„βˆ—. Let {𝑇𝑠𝑑,βˆ—}𝑑β‰₯0 be the restriction of {𝑇𝑑,βˆ—}𝑑β‰₯0 to 𝖷s,βˆ—. Since 𝖷s,βˆ— is closed; the generator π‘π‘ βˆ— of {𝑇s𝑑,βˆ—}𝑑β‰₯0 is given by the restriction 𝑍sβˆ—=π‘βˆ—β†Ύπ–·s,βˆ—, dom𝑍sβˆ—=domZβˆ—βˆ©π–·s,βˆ—. A similar construction applies to the reversible subspace 𝖷r. We check that spec𝑍sβˆ—βŠ†specπ‘βˆ—. Let πœ†βˆˆπœŒ(π‘βˆ—), that is, the map (πœ†πŸ™βˆ’π‘βˆ—)∢domπ‘βˆ—β†’π–·βˆ— is bijective. Then clearly the map (πœ†πŸ™βˆ’π‘sβˆ—)=(πœ†πŸ™βˆ’π‘βˆ—)↾𝖷𝑠,βˆ—βˆΆdomπ‘βˆ—βˆ©π–·s,βˆ—β†’π–·s,βˆ— is injective. It is also surjective: let πœ‘βˆˆπ–·s,βˆ—, then there is πœ“=πœ“sβŠ•πœ“r∈domπ‘βˆ— such that (πœ†πŸ™βˆ’π‘βˆ—)πœ“=πœ‘. Now ξ€·πœ†πŸ™βˆ’π‘βˆ—ξ€Έξ€·πœ“=πœ†πŸ™βˆ’π‘βˆ—πœ“ξ€Έξ€·sβŠ•πœ“rξ€Έ=πœ‘βŠ•0,(2.13) so (πœ†πŸ™βˆ’π‘βˆ—)πœ“r=0 and πœ“r=0 by injectivity. Thus (πœ†πŸ™βˆ’π‘sβˆ—) is bijective and πœ†βˆˆπœŒ(𝑍sβˆ—). In particular, using spec𝑍=specπ‘βˆ— we find that spec𝑍sβˆ—βˆ©iℝiscountable.(2.14) We now see that specp𝑍s∩iℝ=βˆ…,(2.15) for if iπœ†βˆˆspecp𝑍s, πœ†βˆˆβ„, then the corresponding eigenvector π‘₯∈dom𝑍sβŠ†π–·s satisfying 𝑍π‘₯=iπœ†π‘₯ must lie in 𝖷r by (2.6), hence π‘₯=0, contradiction.
From (2.14) and (2.15), it follows by the Arendt-Batty-Lyubich-VΕ© theorem [15, 16], see also [9], that the semigroup {𝑇s𝑑,βˆ—}𝑑β‰₯0 is strongly stable, that is, for all π‘₯βˆ—βˆˆπ–·s,βˆ— we have limπ‘‘β†’βˆžβ€–π‘‡s𝑑,βˆ—(π‘₯βˆ—)β€–=0. Thus if π‘₯βˆˆπ–·s and π‘₯βˆ—βˆˆπ–·βˆ— it follows that ||βŸ¨π‘‡π‘‘(π‘₯),π‘₯βˆ—βŸ©||=||ξ«π‘„βŸ‚ξ€·π‘‡π‘‘(ξ€Έπ‘₯),π‘₯βˆ—ξ¬||=||π‘₯,𝑇𝑑,βˆ—ξ€·π‘„βŸ‚βˆ—ξ€·π‘₯βˆ—||≀‖π‘₯‖⋅‖𝑇s𝑑,βˆ—ξ€·π‘„βŸ‚βˆ—ξ€·π‘₯βˆ—ξ€Έξ€Έβ€–βŸΆ0(2.16) as π‘‘β†’βˆž uniformly for π‘₯βˆˆπ–·sβˆ©π–·1, where π‘„βŸ‚=idπ–·βˆ’π‘„ denotes the projection onto 𝖷s.

3. Semigroups on von Neumann Algebras

The results of the previous section apply to the case of Neumann algebras. Let β„‹ be a Hilbert space. A von Neumann algebra is a *-subalgebra of the Banach-*-algebra L(β„‹) of all bounded linear operators acting on β„‹, which is additionally closed in the weak (or equivalently strong) operator topology. The identity operator will be denoted by πŸ™, and we will always assume that πŸ™βˆˆπ”. The ultraweak topology on 𝔐 is defined by the seminorms π‘πœŒ(π‘₯)=|tr(𝜌π‘₯)|, where 𝜌 runs through the trace class operators on β„‹, it agrees with the weak operator topology on bounded portions of 𝔐. The set of all ultraweakly continuous linear functionals on 𝔐 forms a Banach space, and this Banach space is the unique (up to isomorphism) predual space of 𝔐, for this reason we denote it by π”βˆ—. The ultraweak topology on 𝔐 can be shown to be equivalent to the 𝜎(𝔐,π”βˆ—) (i.e., weak*) topology. Hence the setup of the previous section applies to this case. The set of all positive operators of 𝔐 will be denoted by 𝔐+. A functional πœ‘ in π”βˆ— which is positive (i.e., πœ‘(π‘₯)β‰₯0 provided π‘₯βˆˆπ”+) and normalized (i.e., β€–πœ‘β€–=1, equivalently πœ‘(πŸ™)=1) will be called a normal state. A state is called faithful if π‘₯βˆˆπ”+ and πœ”(π‘₯)=0 implies π‘₯=0. For proofs of these results, we refer to [10, 17].

Let π‘‡βˆˆL(𝔐). Then 𝑇 is called positive if 𝑇(𝔐+)βŠ†π”+. A positive operator is normal (i.e., weak* continuous) if and only if for every uniformly bounded increasing net {π‘₯𝑖}π‘–βˆˆπΌβŠ†π”+ we have sup𝑖𝑇(π‘₯𝑖)=𝑇(sup𝑖π‘₯𝑖). Furthermore, 𝑇 is called strongly positive whenever it satisfies Kadison's inequality, that is, ‖𝑇(πŸ™)‖𝑇(π‘₯βˆ—π‘₯)β‰₯𝑇(π‘₯)βˆ—π‘‡(π‘₯) for any π‘₯βˆˆπ”. Clearly strong positivity implies positivity. An even stronger notion of positivity is complete positivity: π‘‡βˆˆL(𝔐) is called completely positive whenever βˆ‘π‘›π‘–,𝑗=1π‘¦βˆ—π‘–π‘‡(π‘₯βˆ—π‘–π‘₯𝑗)𝑦𝑗β‰₯0 for all π‘›βˆˆβ„• and all π‘₯1,…,π‘₯𝑛 and 𝑦1,…,𝑦𝑛 from 𝔐. The map 𝑇 is called unital if 𝑇(πŸ™)=πŸ™; a positive unital map is automatically contractive, that is, ‖𝑇(π‘₯)‖≀‖π‘₯β€– for all π‘₯βˆˆπ”.

The following result has been established in [18].

Proposition 3.1. Suppose that π‘†βŠ†Ln(𝔐)1 is a subset of normal contractive linear operators. Then the following assertions are equivalent. (1)The set {π‘‡βˆ—(πœ‘)βˆΆπ‘‡βˆˆπ‘†}βŠ†π”βˆ— is relatively weakly compact for every πœ‘βˆˆπ”βˆ—. (2)The set 𝑆 is equicontinuous when 𝔐 is endowed with the Mackey topology (i.e., the 𝜏(𝔐,π”βˆ—) topology). (3)The pointwise weak* closure of 𝑆 consists of normal operators: π‘†βŠ†Ln(𝔐)1. Moreover, these conditions are satisfied whenever there is a faithful normal state πœ” on 𝔐 such that πœ”ξ€·π‘‡ξ€·π‘₯βˆ—ξ€Έπ‘‡ξ€Έξ€·π‘₯(π‘₯)β‰€πœ”βˆ—π‘₯ξ€Έforanyπ‘‡βˆˆπ‘†,π‘₯βˆˆπ”.(3.1) In particular, if each element in 𝑆 is strongly positive we conclude that (3.1) can be rewritten as πœ”(𝑇(π‘₯))β‰€πœ”(π‘₯) for all π‘₯βˆˆπ”+, π‘‡βˆˆπ‘†, or briefly πœ”βˆ˜π‘‡β‰€πœ”, for all π‘‡βˆˆπ‘†.

If π‘†βŠ†Ln(𝔐)1 is a subset of normal contractive linear operators and πœ” a normal state, we call πœ” an invariant state under 𝑆 provided πœ”(𝑇(π‘₯))=πœ”(π‘₯) for all π‘₯βˆˆπ” and π‘‡βˆˆπ‘†. We now apply the results of Section 2 to weak* continuous semigroups on von Neumann algebras. This gives us the following result.

Corollary 3.2. Suppose that {𝑇𝑑}𝑑β‰₯0 is a weak* continuous contractive strongly positive one-parameter semigroup on a von Neumann algebra 𝔐 with ultraweak generator 𝑍, and suppose that there exists a faithful normal invariant state πœ”. Then there exist weak* closed and 𝑇𝑑-invariant subspaces 𝔐r and 𝔐s of 𝔐, given by (2.5) and (2.6), such that 𝔐=𝔐rβŠ•π”s.

Proof. By Kadison’s inequality, (3.1) holds; thus Proposition 3.1 implies that the pointwise weak* closure S=S0, with S0={𝑇𝑑}𝑑β‰₯0, consists of normal operators. Hence Theorem 2.1 applies.

It is worth pointing out that a similar result was recently established in [19] for general semigroups acting on a W*-algebra and possessing a faithful family of subinvariant states.

We now prove that 𝔐r is actually a von Neumann subalgebra. Recall that a conditional expectation 𝑄 from a C*-algebra 𝔄 onto a C*-subalgebra π”…βŠ†π”„ is a completely positive contraction with 𝑄(π‘₯)=π‘₯ for π‘₯βˆˆπ”… and 𝑄(π‘₯𝑦π‘₯)=π‘₯𝑄(𝑦)π‘₯ for π‘₯βˆˆπ”…, π‘¦βˆˆπ”„.

Proposition 3.3. Let {𝑇𝑑}𝑑β‰₯0 be a weak* continuous semigroup of strongly positive unital operators and suppose there exists a faithful normal invariant state πœ”. Then 𝔐r is a von Neumann subalgebra of 𝔐 and there exists a group of *-automorphisms {𝛼𝑑}π‘‘βˆˆβ„ on 𝔐r such that 𝑇𝑑↾𝔐r=𝛼𝑑 for all 𝑑β‰₯0. Moreover, there exists a normal conditional expectation 𝑄 from 𝔐 onto 𝔐r such that πœ”βˆ˜π‘„=πœ”. Finally, 𝔐s is *-invariant.

Proof. Since each 𝑇𝑑 is a contraction; Corollary 3.2 applies. Let 𝑀0={π‘₯βˆˆπ”βˆΆβˆƒπ›Όβˆˆβ„suchthat𝑇𝑑(π‘₯)=ei𝛼𝑑π‘₯βˆ€π‘‘β‰₯0}, that is, we have lin𝑀0wβˆ—=𝔐r. As in [20] we define the sesquilinear map π·βˆΆπ”Γ—π”β†’π” by 𝐷(π‘₯,𝑦)=𝑇𝑑(π‘₯βˆ—π‘¦)βˆ’π‘‡π‘‘(π‘₯)βˆ—π‘‡π‘‘(𝑦) for some fixed 𝑑β‰₯0. By Kadison's inequality, the sesquilinear form πœ‘βˆ˜π· is positive-definite for any πœ‘βˆˆπ”+βˆ—, so by the Cauchy-Schwarz inequality, 𝐷(π‘₯,π‘₯)=0 if and only if 𝐷(π‘₯,𝑦)=0 for all π‘¦βˆˆπ”. Now let π‘₯βˆˆπ‘€0, then 𝑇𝑑(π‘₯βˆ—π‘₯)β‰₯𝑇𝑑(π‘₯)βˆ—π‘‡π‘‘(π‘₯)=eβˆ’i𝛼𝑑e+i𝛼𝑑π‘₯βˆ—π‘₯=π‘₯βˆ—π‘₯. Thus 0β‰€πœ”(𝑇𝑑(π‘₯βˆ—π‘₯)βˆ’π‘₯βˆ—π‘₯)β‰€πœ”(π‘₯βˆ—π‘₯βˆ’π‘₯βˆ—π‘₯)=0, and by faithfulness 𝑇𝑑(π‘₯βˆ—π‘₯)=π‘₯βˆ—π‘₯, hence 𝐷(π‘₯,π‘₯)=0 for all π‘₯βˆˆπ‘€0. So 𝐷(π‘₯,𝑦)=0 for all π‘₯,π‘¦βˆˆπ‘€0, that is, 𝑇𝑑(π‘₯βˆ—π‘¦)=𝑇𝑑(π‘₯)βˆ—π‘‡π‘‘(𝑦)=ei(𝛼2βˆ’π›Ό1)𝑑π‘₯βˆ—π‘¦, and we conclude that π‘₯π‘¦βˆˆπ‘€0 whenever π‘₯,π‘¦βˆˆπ‘€0. It follows that lin𝑀0 is a *-subalgebra of 𝔐 (containing πŸ™) and consequently 𝔐r is a von Neumann subalgebra, and 𝑇𝑑(π‘₯βˆ—π‘¦)=𝑇𝑑(π‘₯βˆ—)𝑇𝑑(𝑦) for all π‘₯,π‘¦βˆˆπ”r. By Corollary 2.2, the restriction of 𝑇𝑑 to 𝔐r extends to a one-parameter group {𝛼𝑑}π‘‘βˆˆβ„ of isometries and the above argument shows that 𝛼𝑑 must be a *-homomorphism. Let 𝑄 be the Sushkevich kernel of the semigroup π‘†βŠ†Ln(𝔐)1. Since 𝑄 is a projection and ‖𝑄‖=1 it follows from Tomiyama’s theorem [21] that 𝑄 is a conditional expectation; since π‘„βˆˆS; it is also clear that πœ”βˆ˜π‘„=πœ”. The last assertion is clear as well.

In the following, we will be interested in the stronger characterization of 𝔐s by a stability property as in (2.12). We start by quoting the following result.

Lemma 3.4. Suppose that {𝑇𝑑}𝑑β‰₯0 is a weak* continuous one-parameter semigroup of strongly positive unital operators on the von Neumann algebra 𝔐 with a faithful normal invariant state πœ”. Introduce the subsets 𝑀=π‘₯βˆˆπ”βˆΆπ‘‡π‘‘ξ€·π‘₯βˆ—π‘₯ξ€Έ=𝑇𝑑(π‘₯)βˆ—π‘‡π‘‘ξ€Ύ,𝑀(π‘₯)βˆ€π‘‘β‰₯0βˆ—=ξ€½π‘₯βˆˆπ”βˆΆπ‘‡π‘‘ξ€·π‘₯π‘₯βˆ—ξ€Έ=𝑇𝑑(π‘₯)𝑇𝑑(π‘₯)βˆ—ξ€Ύ,π”βˆ€π‘‘β‰₯01=π‘€βˆ©π‘€βˆ—.(3.2) Then 𝔐1 is a 𝑇𝑑-invariant von Neumann subalgebra of 𝔐, and there exists a group of *-automorphisms {𝛼𝑑}π‘‘βˆˆβ„ on 𝔐1 such that 𝑇𝑑↾𝔐1=𝛼𝑑 for 𝑑β‰₯0. Moreover, 𝔐1 is a maximal (in the sense of not being properly contained in a larger von Neumann subalgebra) von Neumann subalgebra on which the restriction of {𝑇𝑑}𝑑β‰₯0 is given by a group of *-automorphisms.

A proof can be found in [22] (see the proof of Proposition 2). It is easy to see that we always have 𝔐rβŠ†π”1.

Lemma 3.5. Under the assumptions of Lemma 3.4, for every π‘₯βˆˆπ” the weak* limit points of the net {𝑇𝑑(π‘₯)}π‘‘βˆˆβ„+ lie in 𝔐1.

A proof of this statement is contained in the proof of Theorem 3.1 of [23].

We can now establish the following result.

Proposition 3.6. Let {𝑇𝑑}𝑑β‰₯0 be a weak* continuous semigroup of strongly positive unital operators on the von Neumann algebra 𝔐 with a faithful normal invariant state πœ”. If 𝔐r=𝔐1, it follows that 𝔐s=ξ‚»π‘₯βˆˆπ”βˆΆlimπ‘‘β†’βˆžπ‘‡π‘‘(π‘₯)=0intheweakβˆ—ξ‚Όtopology.(3.3)

Proof. Let π‘₯βˆˆπ”s and assume without loss of generality that β€–π‘₯‖≀1. By Alaoglu's theorem the net {𝑇𝑑(π‘₯)}π‘‘βˆˆβ„+ contained in the unit ball of 𝔐 has a limit point π‘₯0 for π‘‘β†’βˆž. Then using Lemma 3.5, we find that π‘₯0βˆˆπ”1=π”π‘Ÿ. But since π‘₯βˆˆπ”s, it follows that also π‘₯0βˆˆπ”s, that is, π‘₯0βˆˆπ”sβˆ©π”r={0}; hence π‘₯0=0. This proves that any limit point of the net {𝑇𝑑(π‘₯)}π‘‘βˆˆβ„+ is equal to 0; therefore we conclude that limπ‘‘β†’βˆžπ‘‡π‘‘(π‘₯)=0 in the weak* topology for all π‘₯βˆˆπ”s.

Moreover, let us remark the following: suppose that {𝑇𝑑}𝑑β‰₯0 is a weak* continuous semigroup of strongly positive unital operators with generator 𝑍 having a faithful normal invariant state πœ”, and assume that the peripheral spectrum specπ‘βˆ©iℝ is at most countable. Then by using Proposition 2.3 the conclusion of Proposition 3.6 holds. These results will be used in the next section when we discuss the notion of decoherence for uniformly continuous quantum dynamical semigroups.

4. Applications to Decoherence

4.1. The Notion of Decoherence in the Algebraic Framework

Consider a closed quantum system whose algebra of observables is a von Neumann algebra 𝔑, and its reversible time evolution is given by a weak* continuous group of *-automorphisms {𝛽𝑑}π‘‘βˆˆβ„ on 𝔑. A subsystem can be described by a von Neumann subalgebra π”βŠ†π”‘ containing the observables belonging to the subsystem. We will assume that there exists a normal conditional expectation 𝐸 from 𝔑 onto 𝔐. In this situation, we can define the reduced dynamics as follows:𝑇𝑑(π‘₯)=πΈβˆ˜π›½π‘‘(π‘₯),π‘₯βˆˆπ”,𝑑β‰₯0.(4.1) This is the Heisenberg picture time evolution an observer whose experimental capabilities are limited to the system described by 𝔐 would witness. Since it is the time evolution of an open system it is, in general, irreversible. From (4.1) we can isolate some mathematical properties of the reduced dynamics. (1){𝑇𝑑}𝑑β‰₯0 is a family of completely positive and normal linear operators on 𝔐.(2)𝑇𝑑(πŸ™)=πŸ™ for all 𝑑β‰₯0, in particular; each 𝑇𝑑 is contractive. (3)𝑑↦𝑇𝑑(π‘₯) is weak* continuous for any π‘₯βˆˆπ”.

In general the reduced dynamics {𝑇𝑑}𝑑β‰₯0 is not Markovian, that is, memory-free, and hence the operators {𝑇𝑑}𝑑β‰₯0 do not form a one-parameter semigroup. However, in many physically relevant situations it is a good approximation to describe the reduced dynamics by a semigroup satisfying the above properties (1)–(3), that is, a weak* continuous semigroup of completely positive unital maps on the von Neumann algebra 𝔐. Such a semigroup is called a quantum dynamical semigroup. We remark that in many physically relevant models we have the following structure: 𝔑=π”βŠ—π”0, acting on a tensor product β„‹βŠ—β„‹0 of two Hilbert spaces, where 𝔐0 describes the environment of the system (e.g., a heat bath). The time evolution of the system and environment is Hamiltonian, that is, 𝛽𝑑(π‘₯)=ei𝑑𝐻π‘₯eβˆ’i𝑑𝐻 with 𝐻=𝐻1βŠ—πŸ™+πŸ™βŠ—π»2+𝐻int, where 𝐻1 and 𝐻2 are the Hamiltonians belonging to the system and its environment, and 𝐻int is an interaction term. Moreover, the conditional expectation πΈπœ” is given with respect to a reference state πœ” of the environment, that is, πœ‘βŠ—πœ”(π‘₯)=πœ‘(πΈπœ”(π‘₯)) for all π‘₯βˆˆπ”‘ and πœ‘βˆˆπ”βˆ—. In this situation, the predual time evolution is given by the familiar formula𝑇𝑑,βˆ—(πœ‘)=tr2ξ€Ίeβˆ’i𝑑𝐻(πœ‘βŠ—πœ”)ei𝑑𝐻,(4.2) where πœ‘ is a normal state on 𝔐 and tr2 denotes the partial trace with respect to the degrees of freedom of the environment.

Since the reduced dynamics is in general not reversible, new phenomena like the approach to equilibrium can appear. In this paper, we are particularly interested in an effect called decoherence. The following general and mathematically rigorous characterization of decoherence in the algebraic framework was introduced by Blanchard and Olkiewicz [3, 4]. Its present form is taken from [24]; the relation of this form and that given in [3] is discussed in [25].

Definition 4.1. We say that the reduced dynamics {𝑇𝑑}𝑑β‰₯0 displays decoherence if the following assertions are satisfied: there exists a 𝑇𝑑-invariant von Neumann subalgebra 𝔐1 of 𝔐 and an weak* continuous group {𝛼𝑑}π‘‘βˆˆβ„ of *-automorphisms on 𝔐1 such that 𝑇𝑑↾𝔐1=𝛼𝑑 for 𝑑β‰₯0, and a 𝑇𝑑-invariant and *-invariant weak* closed subspace 𝔐2 of 𝔐 such that 𝔐=𝔐1βŠ•π”2,(4.3)limπ‘‘β†’βˆžπ‘‡π‘‘(π‘₯)=0intheweakβˆ—topologyforanyπ‘₯βˆˆπ”2.(4.4) Moreover, we require that 𝔐1 is a maximal von Neumann subalgebra of 𝔐 (in the sense of not being properly contained in any larger von Neumann subalgebra) on which {𝑇𝑑}𝑑β‰₯0 extends to a group of *-automorphisms. We call 𝔐1 the algebra of effective observables.

The physical interpretation of this definition is rather clear: if decoherence takes place there is a (maximal) von Neumann subalgebra on which the reduced dynamics is reversible, that is, given by an automorphism group, and a complementary subspace on which the expectation values with respect to any normal state of all its elements tend to zero in time. Thus any observable π‘₯βˆˆπ” has a decomposition π‘₯=π‘₯1+π‘₯2, where π‘₯π‘–βˆˆπ”π‘–, such that πœ‘(𝑇𝑑(π‘₯2))β†’0 as π‘‘β†’βˆž for any normal state πœ‘ on 𝔐. Hence after a sufficiently long time the system behaves effectively like a closed system described by 𝔐1 and {𝛼𝑑}π‘‘βˆˆβ„. By analyzing the structure of the algebra of effective observables 𝔐1 and the reversible dynamics {𝛼𝑑}π‘‘βˆˆβ„ various physically relevant and well-known phenomena of decoherence can be identified, like the appearance of pointer states, environment-induced superselection rules, and classical dynamical systems. In this way it is possible to obtain an exhaustive classification of possible decoherence scenarios, see [3] for a thorough discussion of this point. Particularly interesting is the case when 𝔐1 is a factor, that is, after decoherence we still have a system of pure quantum character. This is of interest in the context of quantum computation since in this way one may obtain a system which retains its quantum character despite decoherence.

According to Definition 4.1, if {𝑇𝑑}𝑑β‰₯0 is a group of automorphisms, decoherence takes place and the splitting (4.3) is trivial with 𝔐2={0}. However, we will keep this slightly unfortunate terminology since it simplifies the statements of theorems, keeping in mind that physically decoherence corresponds to the case when 𝔐2β‰ {0}. This can only happen if {𝑇𝑑}𝑑β‰₯0 is irreversible.

We remark that the algebra 𝔐1 has been studied in [26] and explicit representations of 𝔐1 are obtained for quantum dynamical semigroups with unbounded generators. Moreover, in [27] a different notion of decoherence for quantum dynamical semigroups is introduced in a mathematically rigorous way, and its connection to the Blanchard-Olkiewicz notion is briefly discussed in [28]. In [29] an asymptotic property similar to (4.3) and (4.4) is discussed under the designation β€œlimited relaxation.”

In connection with Definition 4.1, the following question arises: if there exists a maximal von Neumann subalgebra of 𝔐 on which {𝑇𝑑}𝑑β‰₯0 extends to a group of automorphisms, is this subalgebra necessarily unique? The following theorem answers this question.

Theorem 4.2. Let {𝑇𝑑}tβ‰₯0 be a quantum dynamical semigroup and suppose it has a faithful normal invariant state πœ”. Then there exists a unique maximal von Neumann subalgebra on which {𝑇𝑑}𝑑β‰₯0 extends to a group of automorphisms.

Proof. Let 𝔐𝑖, where π‘–βˆˆπΌ is some index set, be a collection of von Neumann subalgebras on each of which {𝑇𝑑}𝑑β‰₯0 extends to a group of automorphisms. Then we put ξ˜π”…=π‘–βˆˆπΌπ”π‘–=ξ€½π‘₯lin𝑖1β‹―π‘₯π‘–π‘˜βˆΆπ‘–1,…,π‘–π‘˜βˆˆπΌ,π‘˜βˆˆβ„•,π‘₯π‘–β„“βˆˆπ”π‘–β„“ξ€Ύ,(4.5) where the closure is taken in the ultraweak topology. We proceed as in the proof of Proposition 3.3 and introduce 𝐷(π‘₯,𝑦)=𝑇𝑑(π‘₯βˆ—π‘¦)βˆ’π‘‡π‘‘(π‘₯)βˆ—π‘‡π‘‘(𝑦) for a fixed 𝑑β‰₯0 and π‘₯,π‘¦βˆˆπ”. Then 𝐷(π‘₯,π‘₯)=0 if and only if 𝐷(π‘₯,𝑦)=0 for all π‘¦βˆˆπ”. Now if π‘₯βˆˆπ”π‘–, we get 𝐷(π‘₯,π‘₯)=0, thus 𝐷(π‘₯,𝑦)=0 for π‘¦βˆˆπ”, where π‘–βˆˆπΌ, that is, 𝑇𝑑(π‘₯𝑦)=𝑇𝑑(π‘₯)𝑇𝑑(𝑦). Proceeding inductively we have 𝑇𝑑π‘₯𝑖1β‹―π‘₯π‘–π‘˜ξ€Έ=𝑇𝑑π‘₯𝑖1⋯𝑇𝑑π‘₯π‘–π‘˜ξ€Έ,(4.6) for an arbitrary monomial π‘₯𝑖1β‹―π‘₯π‘–π‘˜, where π‘₯π‘–β„“βˆˆπ”π‘–β„“, 𝑖1,…,π‘–π‘˜βˆˆπΌ. Therefore, if 𝐡0=lin{π‘₯𝑖1β‹―π‘₯π‘–π‘˜βˆΆπ‘–1,…,π‘–π‘˜βˆˆπΌ,π‘˜βˆˆβ„•}, then π‘‡π‘‘βˆΆπ΅0→𝐡0 is a *-homomorphism of the *-subalgebra 𝐡0, and π‘‡π‘‘βˆΆπ”…β†’π”… is a *-homomorphism as well.
Next we note that 𝑇𝑑↾𝔅 is injective. Namely, if π‘₯∈ker(𝑇𝑑↾𝔅), we get πœ”(π‘₯βˆ—π‘₯)=πœ”(𝑇𝑑(π‘₯βˆ—π‘₯))=πœ”(𝑇𝑑(π‘₯)βˆ—π‘‡π‘‘(π‘₯))=0, thus π‘₯=0 by faithfulness of πœ” and so ker(𝑇𝑑↾𝔅)={0}.
We now establish that π‘‡π‘‘βˆΆπ”…β†’π”… is surjective. Since 𝑇𝑑 is an injective *-homomorphism on the C*-algebra 𝔅, it follows that ‖𝑇𝑑(π‘₯)β€–=β€–π‘₯β€– for all π‘₯βˆˆπ”…. Moreover, notice that π‘‡π‘‘βˆΆπ΅0→𝐡0 is invertible since each restriction 𝑇𝑑↾𝔐𝑖 is invertible. This implies β€–π‘‡π‘‘βˆ’1(π‘₯)β€–=‖𝑇𝑑(π‘‡π‘‘βˆ’1(π‘₯))β€–=β€–π‘₯β€– for any π‘₯∈𝐡0. Now for the proof of surjectivity choose π‘¦βˆˆπ”…. By the Kaplansky density theorem, there exists a net {𝑦𝑗}π‘—βˆˆπ½βŠ†π΅0 such that ‖𝑦𝑗‖≀𝐢 for all π‘—βˆˆπ½, and such that lim𝑦𝑗=𝑦. Put π‘₯𝑗=π‘‡π‘‘βˆ’1(𝑦𝑗). Then β€–π‘₯𝑗‖=‖𝑦𝑗‖≀𝐢, and by Alaoglu's theorem the net {π‘₯𝑗}π‘—βˆˆπ½ has an ultraweak limit point π‘₯0. Thus there is a subnet {π‘₯𝑗ℓ}βŠ†{π‘₯𝑗}π‘—βˆˆπ½ such that limπ‘₯𝑗ℓ=π‘₯0, hence 𝑦𝑗ℓ=𝑇𝑑π‘₯π‘—β„“ξ€ΈβŸΆπ‘¦=𝑇𝑑π‘₯0ξ€Έ.(4.7) We conclude by injectivity that any ultraweak limit point of the net {π‘₯𝑗}π‘—βˆˆπ½ is equal to π‘₯0, hence this net is convergent to π‘₯0, and 𝑇𝑑(π‘₯0)=𝑦, establishing surjectivity. We have thus proved that 𝑇𝑑↾𝔅 is a *-automorphism.
To finish the proof, we have to choose the collection {π”π‘–βˆΆπ‘–βˆˆπΌ} in (4.5) to consist of all von Neumann subalgebras on which {𝑇𝑑}𝑑β‰₯0 extends to a group of automorphisms.

We next prove that in certain cases the splitting (4.3) is always given by a conditional expectation.

Proposition 4.3. Suppose that the (not necessarily Markovian) reduced dynamics {𝑇𝑑}𝑑β‰₯0 on the von Neumann algebra 𝔐 displays decoherence. If 𝑇𝑑↾𝔐1=id𝔐1, then there exists a normal conditional expectation 𝐸 from 𝔐 onto 𝔐1.

Proof. Let π‘₯βˆˆπ” and write π‘₯=π‘₯1+π‘₯2 with π‘₯π‘–βˆˆπ”π‘–, 𝑖=1,2, and define πΈβˆΆπ”β†’π”1 by 𝐸(π‘₯)=π‘₯1. Then 𝐸(𝔐)=𝔐1 and 𝐸2=𝐸. Since 𝔐2 is *-invariant, 𝐸(π‘₯βˆ—)=𝐸(π‘₯βˆ—1+π‘₯βˆ—2)=π‘₯βˆ—1=𝐸(π‘₯)βˆ—, hence 𝐸(𝔐sa)βŠ†π”sa. Now let π‘₯βˆˆπ”+, consider the decomposition π‘₯=π‘₯1+π‘₯2, π‘₯π‘–βˆˆπ”π‘–, 𝑖=1,2, and suppose that π‘₯1βˆˆπ”sa is not positive. Then there exists πœ‘βˆˆπ”+βˆ— such that πœ‘(π‘₯1)<0. This implies 𝑇0β‰€πœ‘π‘‘ξ€Έξ€·π‘‡(π‘₯)=πœ‘π‘‘ξ€·π‘₯1𝑇+πœ‘π‘‘ξ€·π‘₯2ξ€·π‘₯ξ€Έξ€Έ=πœ‘1𝑇+πœ‘π‘‘ξ€·π‘₯2ξ€Έξ€Έ,(4.8) and letting π‘‘β†’βˆž yields πœ‘(𝑇𝑑(π‘₯2))β†’0, a contradiction. Thus 𝐸(𝔐+)βŠ†π”+, and since 𝐸(πŸ™)=πŸ™ it follows that ‖𝐸‖≀1. From 𝐸2=𝐸, we get ‖𝐸‖=1, so 𝐸 is a projection of norm 1 and hence by Tomiyama's theorem a conditional expectation. Since ker𝐸={π‘₯βˆˆπ”βˆΆπΈ(π‘₯)=π‘₯1=0}=𝔐2 is ultraweakly closed, we obtain by a theorem of Tomiyama [30] that 𝐸 is normal.

Given a reduced dynamics, the question arises of under what conditions decoherence will occur. In the Markovian case, sufficient conditions for the appearance of decoherence in the sense of Definition 4.1 have been formulated in [31]. In fact, we have the following theorem.

Theorem 4.4. Let {𝑇𝑑}𝑑β‰₯0 be a weak* continuous one-parameter semigroup with a faithful normal state πœ”. Assume that the following conditions are satisfied. (1)Each 𝑇𝑑, 𝑑β‰₯0, is strongly positive and unital. (2)Let {πœŽπœ”π‘‘}π‘‘βˆˆβ„ denote the modular group (see, e.g., [10]) corresponding to the state πœ”. Assume that [𝑇𝑑,πœŽπœ”π‘ ]=0 for all π‘ βˆˆβ„ and 𝑑β‰₯0. Then {𝑇𝑑}𝑑β‰₯0 displays decoherence and there exists a normal conditional expectation 𝐸 from 𝔐 onto 𝔐1 such that [𝑇𝑑,𝐸]=0 for all 𝑑β‰₯0 and πœ”βˆ˜πΈ=πœ”.

The splitting 𝔐=𝔐1βŠ•π”2 of {𝑇𝑑}𝑑β‰₯0 provided by this theorem is called the isometric-sweeping splitting. In [31] the theorem was proved under the more general hypothesis that πœ” is only a faithful semifinite normal weight. Then some additional technical assumptions about {𝑇𝑑}𝑑β‰₯0 are necessary. A simpler proof for the theorem as stated above has been given in [24].

The Jacobs-de Leeuw-Glicksberg splitting for weak* continuous semigroups on von Neumann algebras established in Corollary 3.2 can now be applied to establish decoherence.

Corollary 4.5. Let {𝑇𝑑}𝑑β‰₯0 be a weak* continuous one-parameter semigroup of strongly positive unital operators with a faithful normal state πœ”. Assume that 𝔐r=𝔐1. Then {𝑇𝑑}𝑑β‰₯0 displays decoherence.

Proof. According to Corollary 3.2, the Jacobs-de Leeuw-Glicksberg splitting exists, and by Proposition 3.6 we conclude that the requirements of Definition 4.1 are satisfied.

Remark 4.6. Whenever a conditional expectation 𝐸 from a von Neumann algebra 𝔐 onto a von Neumann subalgebra 𝔐1 satisfies πœ”βˆ˜πΈ=πœ” for a faithful normal state πœ”, it is uniquely determined by these conditions [17, Corollary II.6.10]. Since in case of the isometric-sweeping splitting as given by Theorem 4.4 we have 𝔐1=𝐸𝔐 and 𝔐2=(πŸ™βˆ’πΈ)𝔐, and in the Jacobs-de Leeuw-Glicksberg splitting as given by Corollary 3.2 we have 𝔐r=𝑄𝔐 and 𝔐s=(πŸ™βˆ’π‘„)𝔐, it follows that the isometric-sweeping and Jacobs-de Leeuw-Glicksberg splittings agree whenever 𝔐r=𝔐1.

4.2. Uniformly Continuous Semigroups

The purpose of this section is to show how the Jacobs-de Leeuw-Glicksberg splitting can be applied to establish decoherence of quantum dynamical semigroups in the sense of Definition 4.1. To avoid complications arising from unbounded generators, we will concentrate on the case of uniformly continuous quantum dynamical semigroups. We will arrive at a result which avoids assumption (2) in Theorem 4.4.

Let 𝔐 be a von Neumann algebra acting on a separable Hilbert space β„‹, and let {𝑇𝑑}𝑑β‰₯0 be a quantum dynamical semigroup such that 𝑑↦𝑇𝑑 is continuous in the uniform topology. Then by [32] the generator 𝑍 of {𝑇𝑑}𝑑β‰₯0, which is a bounded operator on 𝔐, is given by𝑍π‘₯=πΊβˆ—π‘₯+π‘₯𝐺+Ξ¦(π‘₯),(4.9) where 𝐺∈L(β„‹) and Ξ¦βˆΆπ”β†’L(β„‹) is a normal completely positive map. Since we have 𝑇𝑑(πŸ™)=πŸ™ for all 𝑑β‰₯0, it follows that π‘πŸ™=0 which forces πΊβˆ—=βˆ’πΊβˆ’Ξ¦(πŸ™). Upon introducing the operator 𝐻=i𝐺+(1/2)iΞ¦(πŸ™), it is seen that 𝐻 is a bounded selfadjoint operator on β„‹ and 𝑍 may be written as[]βˆ’1𝑍π‘₯=i𝐻,π‘₯2{Ξ¦(πŸ™),π‘₯}+Ξ¦(π‘₯),(4.10) where {β‹…,β‹…} denotes the anticommutator. Let us suppose now that Ξ¦ has a Kraus decompositionΞ¦(π‘₯)=βˆžξ“π‘›=1π΄βˆ—π‘›π‘₯𝐴𝑛,(4.11) where {𝐴𝑛}π‘›βˆˆβ„• is a sequence of bounded linear operators on β„‹ and the series converges in the weak* topology. This is always the case if 𝔐 is injective or equivalently, by the Connes theorem [33] and separability of β„‹, that 𝔐 is hyperfinite (this includes the case 𝔐=L(β„‹)). The preadjoint operator of 𝑍 on π”βˆ— then has the familiar Lindblad form [34]π‘βˆ—[]βˆ’1𝜌=βˆ’i𝐻,𝜌2βˆžξ“π‘›=1ξ€·πœŒπ΄βˆ—π‘›π΄π‘›+π΄βˆ—π‘›π΄π‘›πœŒξ€Έ+βˆžξ“π‘›=1π΄π‘›πœŒπ΄βˆ—π‘›,(4.12) where πœŒβˆˆπ”βˆ—. We are now able to prove the following theorem.

Theorem 4.7. Let {𝑇𝑑}𝑑β‰₯0 be as above. Assume that there is a faithful normal state πœ” such that πœ”βˆ˜π‘‡π‘‘=πœ” for all 𝑑β‰₯0 and that 𝐻 in (4.10) has pure point spectrum. Then {𝑇𝑑}𝑑β‰₯0 displays decoherence, and for the effective subalgebra 𝔐1 we have 𝔐1βŠ†ξ€½π΄π‘›,π΄βˆ—π‘›ξ€ΎβˆΆπ‘›βˆˆβ„•ξ…žβˆ©π”,(4.13) where the prime denotes the commutant. Moreover, there exists a normal conditional expectation 𝑄 from 𝔐 onto 𝔐1 such that πœ”βˆ˜π‘„=πœ”. If the derivation π‘₯↦i[𝐻,π‘₯] leaves the subalgebra {𝐴𝑛,π΄βˆ—π‘›βˆΆπ‘›βˆˆβ„•}β€²βˆ©π” invariant, equality holds in (4.13).

Proof. First note that the assumptions of Proposition 3.3 are satisfied, that is, π”π‘Ÿ is a von Neumann subalgebra. Consider the subalgebra 𝔐1 defined in Lemma 3.4, then {𝑇𝑑}𝑑β‰₯0 restricted to 𝔐1 extends to a group of automorphisms. We start by proving (4.13). By a simple calculation as in [20], one obtains 𝑍π‘₯βˆ—π‘₯ξ€Έξ€·π‘₯βˆ’π‘βˆ—ξ€Έπ‘₯βˆ’π‘₯βˆ—π‘(π‘₯)=π‘₯βˆ—Ξ¦ξ€·π‘₯(πŸ™)π‘₯+Ξ¦βˆ—π‘₯ξ€Έξ€·π‘₯βˆ’Ξ¦βˆ—ξ€Έπ‘₯βˆ’π‘₯βˆ—Ξ¦=(π‘₯)βˆžξ“π‘›=1𝐴𝑛,π‘₯βˆ—ξ€Ίπ΄π‘›ξ€».,π‘₯(4.14) The generator 𝑍, when restricted to 𝔐1, is a *-derivation; thus if π‘₯βˆˆπ”1, then ξ€·π‘₯0=π‘βˆ—π‘₯ξ€Έβˆ’π‘₯βˆ—ξ€·π‘₯𝑍(π‘₯)βˆ’π‘βˆ—ξ€Έπ‘₯=βˆžξ“π‘›=1𝐴𝑛,π‘₯βˆ—ξ€Ίπ΄π‘›ξ€»,π‘₯,(4.15) that is, [𝐴𝑛,π‘₯]=0 for all π‘›βˆˆβ„•, and, moreover, [π΄βˆ—π‘›,π‘₯]=0 for all π‘›βˆˆβ„• since 𝔐1 is a *-subalgebra. This proves that 𝔐1βŠ†{𝐴𝑛,π΄βˆ—π‘›βˆΆπ‘›βˆˆβ„•}β€²βˆ©π”. Conversely, under the assumption that i[𝐻,β‹…] leaves the right-hand side of (4.13) invariant, we have 𝑍π‘₯=i[𝐻,π‘₯] or 𝑇𝑑(π‘₯)=ei𝑑𝐻π‘₯eβˆ’i𝑑𝐻 on {𝐴𝑛,π΄βˆ—π‘›βˆΆπ‘›βˆˆβ„•}β€²βˆ©π”, which implies equality in (4.13).
Now let π‘₯βˆˆπ”sβˆ©π”1, π‘₯β‰ 0. Then there exist eigenvectors πœ‰,πœ‚βˆˆβ„‹ of 𝐻 with corresponding eigenvalues πΈπœ‰ and πΈπœ‚ such that βŸ¨πœ‰,π‘₯πœ‚βŸ©β‰ 0, thus βŸ¨πœ‰,𝑇𝑑e(π‘₯)πœ‚βŸ©=βˆ’iπ‘‘π»πœ‰,π‘₯eβˆ’iπ‘‘π»πœ‚ξ¬=ei𝑑(πΈπœ‰βˆ’πΈπœ‚)βŸ¨πœ‰,π‘₯πœ‚βŸ©(4.16) is bounded away from 0, so 0 is not a weak* limit point of {𝑇𝑑(π‘₯)βˆΆπ‘‘β‰₯0}. Sinceπ‘₯βˆˆπ”π‘ , this is a contradiction in view of (2.5), hence 𝔐sβˆ©π”1={0}. Now let π‘₯βˆˆπ”1 and write π‘₯=π‘₯s+π‘₯rβˆˆπ”sβŠ•π”r. Since 𝔐rβŠ†π”1, we have π‘₯rβˆˆπ”1, and π‘₯s=π‘₯βˆ’π‘₯rβˆˆπ”sβˆ©π”1={0}, thus π‘₯βˆˆπ”r. This proves 𝔐1=𝔐r and it follows by Proposition 3.6 that 𝔐s has the property (3.3). So we conclude that {𝑇𝑑}𝑑β‰₯0 displays decoherence. The last assertion is clear from Proposition 3.3.

We remark that in [29, equation (34)], a class of generators has been given for which equality in (4.13) always holds. As a corollary, we obtain the following result which is similar to the one proved in [35] and is also contained in [36].

Corollary 4.8. Let {𝑇𝑑}𝑑β‰₯0 be a uniformly continuous semigroup on 𝔐 consisting of normal completely positive and contractive operators, and suppose it has a faithful normal invariant state πœ”. If {𝐴𝑛,π΄βˆ—π‘›βˆΆπ‘›βˆˆβ„•}ξ…ž=β„‚πŸ™, then limπ‘‘β†’βˆžπ‘‡π‘‘(π‘₯)=πœ”(π‘₯)πŸ™intheweakβˆ—topology(4.17) for any π‘₯βˆˆπ”.

Thus if 𝔐1 is trivial the semigroup describes the approach to equilibrium.

We remark that the last theorem can be generalized to certain cases when the semigroup {𝑇𝑑}𝑑β‰₯0 is not uniformly continuous but has an unbounded generator of the form (4.10).

The existence of a faithful normal invariant state of a quantum dynamical semigroup as required by Theorems 4.7 and 4.4 has been discussed in the literature. It is particularly simple in the case of a finite-dimensional von Neumann algebra, that is, a matrix algebra. Suppose 𝔐=𝑀ℂ(𝑑) is the 𝑑×𝑑-matrix algebra and consider a quantum dynamical semigroup {𝑇𝑑}𝑑β‰₯0 on 𝔐, then its generator is given by (4.10) and its preadjoint by (4.12), thus if the 𝐴𝑛 are normal, 𝜌0=(1/𝑑)πŸ™ is a faithful normal invariant state for {𝑇𝑑}𝑑β‰₯0. Such generators arise, for example, in the singular coupling limit of 𝑁-level systems, see [8].

Acknowledgments

Thanks are due to Professor Ph. Blanchard (Bielefeld) for various discussions and to the anonymous referees for a number of remarks which helped to improve the paper and for pointing out an error in it.