L.E. LABUSCHAGNE

DSI-NRF CoE in Mathematics and Statistics Science, Focus Area for PAA,Internal Box 209, School of Mathematics and Statistics Science NWU, PVT. BAG X6001, 2520 Potchefstroom, South Africa

E-mail: Louis.Labuschagne@nwu.ac.za

W.A. MAJEWSKI

Focus Area for PAA, North-West-University, Potchefstroom, South Africa

E-mail: fizwam@univ.gda.pl

Abstract Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [26]where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spacessince this framework gives a better description of regular observables, and also allows for a well-defined entropy function. In the present paper we “complete” the picture by addressing the issue of the dynamics of such a system,as described by a Markov semigroup corresponding to some Dirichlet form (see [4, 13, 14]).Specifically, we show that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation strategy for extending the action of such maps to the appropriate intermediate spaces of the pairAs a consequence, we obtain that completely positive quantum Markov dynamics naturally extends to the context proposed in [26].

Key words detailed balance; Orlicz space; Markov semigroup; completely positive

1 Introduction

This paper completes a sequence of ideas detailing the utility of Orlicz spaces for quantum physics, with the primary thesis being that the pair of Orlicz spacesoffers a more natural home for regular observables and states with “good” entropy, than the more common pair ofWe pause to trace this evolution of ideas, before highlighting the significance of the present work.

The origins of a quantity representing something like entropy may be found in the work of Ludwig Boltzmann. In his study of the dynamics of rarefied gases, Boltzmann formulated the so-called spatially homogeneous Boltzmann equation as far back as 1872,namely

The natural Lyapunov-type functional for this equation is the so-called Boltzmann H-function, which is

where f is a postulated solution of the Boltzmann equation. The connection to entropy may be seen in the fact that the classical description of continuous entropy S differs from the functional H only by sign. Hence Boltzmann’s H-functional may be viewed as the first formalisation of the concept of entropy. The reader will have noticed that the definition of H+requires the existence of a solution of Boltzmann’s equation, as well as convergence of the integral defining H+. Consequently a suitable technique for handling these problems needed to be found. Lions and DiPerna were the first to rigorously demonstrate the existence of solutions to Boltzmann’s equation. (Lions later received the Fields medal for his work on nonlinear partial differential equations.) Their solution was for the density of colliding hard spheres, given general initial data (see for example [5, 6]for a sampling of this work). Villani subsequently announced, see [42], Chapter 2, Theorem 9, that for particular cross sections (collision kernels in Villani’s terminology) weak solutions of Boltzmann equation are actually in L log(L+1).So one consequence of the work by these authors was to give a strong indication that the space L log(L+1) is the appropriate framework for studying entropy-like quantities like the Boltzmann H-functional.

The identification of the space Lcosh?1as the home for “regular observables”,stems from a paper of Pistone and Sempi[33]in which they identified a class of moment-generating functions characterised by regular behaviour of their Laplace transforms in a neighbourhood of 0–the socalled regular random variables. These regular random variables have many useful properties,but the one aspect which is important from a philosophical point of view,was the demonstration of Pistone and Sempi that these regular random variables form a closed subspace of the Orlicz space Lcosh?1. The analysis of Lcosh?1as the home for regular observables,was then extended to the context of von Neumann algebras M equipped with a faithful normal semifinite trace in[21].

An aspect of great philosophical significance is the fact that(up to a Banach space isomorphism) the space Lcosh?1is the Kthe dual of L log(L+1). This strongly suggests that states with “good” entropy (which by Lions’ work one expects to find in L log(L+1)), are somehow dual to regular observables. This aspect was investigated in some detail in[26],where we argued that statistical physics of regular systems, both classical and quantum, should be based on the pair of Orlicz spaceswith these spaces providing a natural home for regular observables and states with “good” entropy. In support of this claim, we point to the fact that in the case of a von Neumann algebra M equipped with a faithful normal semifinite trace, the entropy-like quantity τ(f log(f)) was in [26]shown to be well-defined for all f in the positive cone of (L1∩ L log(L+1))+(M,τ).

However, any mathematical formalism which aspires to be the “natural” framework for quantum physics must be able to handle the most general quantum systems imaginable. Thus a theory which only works for semifinite algebras is unacceptable. Faced with this challenge,the construction of Orlicz spaces for possibly non-semifinite algebras, was pioneered in [20].Then in a very recent follow-up paper [28]to the above three, this newly developed technology was used to for the very first time introduce a notion of continuous entropy for a single state valid for general possibly non-semifinite algebras. Importantly this notion was shown to be a canonical extension of the trace-based semifinite theory,and also to harmonise with the theory of relative entropy for general algebras pioneered by Araki [1, 2]. Significantly, it was once again the geometry of the space L log(L+1) that played a crucial role in this construction; a fact which clearly demonstrates that the conclusions of [26]extend to general von Neumann algebras.

Observables clearly need not be essentially bounded. This fact poses a challenge to a formalism based on(where formally L∞is regarded as the “home” for observables)since observables which are not essentially bounded are not actually in L∞as such,but at best merely “affiliated” to L∞. By contrast in the case of non-atomic measures there is room for unbounded observables in the space Lcosh?1. (For non-atomic measure spaces,the space L∞is a proper subspace of Lcosh?1.) Thus a formalism based on the pairhas the advantage of making room for unbounded observables in the space Lcosh?1which here is considered to be the “home” for observables. This fact proves to be crucial when applications to the local algebras of Quantum Field Theory are considered. Specifically, the field operators of Quantum Field Theory are necessarily unbounded,and hence to have any hope of embedding these operators into a space of observables,such a space would have to be able to accommodate unbounded operators. Significantly, in a recently completed preprint(see[23]), we showed that a large number of quantum field theoretic models do indeed admit a canonical embedding of the field operators into the associated Lcosh?1space.

Although at first sight passing from a quantum theory modeled by the pairto one modeled by the pairmay be regarded as restrictive, the important point to note here is that the standard formulation of classical theory based on the pair of Banach spaces(respectively the pairin the case of elementary quantum mechanics)do already satisfy the above regularity requirements in cases where time evolution is purely deterministic, and also in the case of important classes of Lvy processes such as Wiener and Poisson processes.

However compelling the above sequence of ideas may be, to get a fully-fledged theory, a description of dynamics should be provided. In particular, one wants to describe dynamical semigroups within the proposed scheme based on the pair of Orlicz spacesThis then is the topic of the present paper, the success of which will complete the picture described above. A very common approach for showing that Markov dynamics behaves well on all Lpspaces (1 ≤ p ≤ ∞) is to use interpolation theory to extend good Markov dynamics on L1and L∞to intermediate spaces. The main difficulty in carrying out such a strategy in the present context is that in a type III setting one is forced to resort to complex interpolation,which at this point in time cannot accommodate Orlicz spaces. Hence some means must be found to adapt the standard interpolation theory to a theory powerful enough to handle a large enough class of non-commutative Orlicz spaces. We will do this in the sections devoted to the study of quantum maps on the distinguished Orlicz spaces.

The paper is organized as follows: We start by revising relevant material from [26]for the sake of context, in the process also making our exposition more self-contained. We then give a brief exposition of the aspects of crossed products relevant to the exposition. Section 4 is devoted to the study of criteria which allow for the extension of Markov maps on the underlying von Neumann algebra to a large class of Orlicz spaces. Along the way,we will also compare the various strategies of achieving this extension. To illustrate the utility of the theory, a class of quantum dynamical maps which fit naturally with the developed techniques will be considered in Section 5. The last section contains some conclusions and remarks.

2 Notation, Terminology, and Previous Results

We follow notation used in [26]and [27]. Let (X,μ) be a measure space. We denote L1(X,μ)={f :X|f|dμ<∞},while L∞(X,μ)stands for the essentially bounded,measurable functions on X. Their non-commutative analogues are: FT(H) - the trace class operators on a Hilbert space H, and B(H) – all linear bounded operators on H. We remind the reader that Lp(X,Σ,m) spaces (1 ≤ p < ∞), (X,Σ,m) a measure space, may be regarded as spaces of measurable functions conditioned by the functionsThe more general category of Orlicz spaces is defined as spaces of measurable functions conditioned by a more general class of convex functions; the so-called Young’s functions. A function Φ : [0,∞) →[0,∞]is a Young’s function if Φ is convex,and Φ is non-constant on (0,∞). It is worth pointing out that such functions have a nice integral representation, for details see [26]and the references given there.

Let L0be the space of measurable functions on some σ-finite measure space (X,Σ,μ). We will always assume that the considered measures are σ-finite.

Definition 2.1The Orlicz space LΨ(being a Banach space)associated with Ψ is defined to be the set

Orlicz spaces may be normed in one of two (ultimately equivalent) ways,namely either by the so-called Luxemburg-Nakano norm

or the equivalent Orlicz norm, given by the formula

where Ψ?stands for the complementary Orlicz function defined byFor further details regarding Ψ?and its relation to the duality theory of Orlicz spaces, see the discussion preceding Definition 4.2. Following convention, we will write LΨwhen the Luxemburg-Nakano norm is used, and LΨif the Orlicz norm is used.

The basic Orlicz spaces used in this paper are L log(L+1),and Lcosh?1defined by Young’s functions:respectively.

A natural question that arises,is what can be said about uniqueness of the correspondence:Young’s functionspace. To answer this question one needs the concept of equivalent Young’s functions. To define this concept we firstly introduce an order relation on Young’s functions by saying that F1? F2if and only if F1(bx) ≥ F2(x) for x ≥ 0 and some b > 0, and then say that the functions F1and F2are equivalent, F1≈ F2, if F1? F2and F1?F2. One has (see [34])

Theorem 2.2Let Φi, i=1,2 be a pair of equivalent Young’s function. Then as Banach spaces, LΦ1=LΦ2.

Consequently, on condition that equivalence is preserved, one can “replace” complicated Young’s function’s with simpler versions!

Our main results concerning classical statistical physics,stated and proved in [26](see also[27]) indicate the following:

Principle 2.3The dual pairprovides a rigorous mathematical formalism for a description of a general, classical regular system.

Turning to the quantum case, as a first step one should define the quantum counterpart of measurable functions L0. In the quantum world there is no known space that is a direct analogue of L0. But one is able to define a quantum analogue of the space of all measurable functions which are bounded,except on a set of finite measure. This space turns out to be more than adequate for our purposes. So to this end let M ?B(H) be a semifinite von Neumann algebra equipped with an fns(faithful normal semifinite)trace τ. The space of all τ-measurable operators is defined as follows. Let a be a densely defined closed operator on H with domain D(a)and let a=u|a|be its polar decomposition. One says that a is affiliated with M(denoted aηM) if u and all the spectral projections of |a| belong to M. Then a is τ-measurable if aηM,and for each δ > 0, there exists a projection e ∈ M such that eH ? D(a) and τ(1 ? e) ≤ δ.We denote bythe set of all τ-measurable operators. The algebra(equipped with the topology of convergence in measure) is a substitute for L0in the quantum world (for details see [30, 36, 39]).

Following the Dodds, Dodds, de Pagter approach [7]we need the concept of generalized singular values. Namely, given an element f ∈and t ∈[0,∞),the generalized singular valueμt(f) is defined by μt(f) = inf{s ≥ 0 : τ(I ? es(|f|)) ≤ t} where es(|f|) s ∈ R is the spectral resolution of |f|. The function t → μt(f) will generally be denoted by μ(f). For details on the generalized singular values see[9]. Here,we note only that this directly extends classical notions where for any f ∈ L0, the function (0,∞) → [0,∞]: t → μt(f) is known as the decreasing rearrangement of f.

The key ingredient of the Dodds, Dodds, de Pagter approach is the concept of a Banach Function Space. To define this concept, let L0(0,∞) stand for almost everywhere finite measurable functions on (0,∞) anddenote {f ∈ L0(0,∞):f ≥ 0}. A function norm

ρ on L0(0,∞) is defined to be a mapping ρ :→ [0,∞]satisfying

? ρ(f)=0 iff f =0 a.e.

? ρ(f +g)≤ ρ(f)+ ρ(g) for all f,g ∈.

? f ≤ g implies ρ(f)≤ ρ(g) for all f,g ∈.

Such a ρ may be extended to all of L0by setting ρ(f)= ρ(|f|),in which case we may then define Lρ(0,∞)={f ∈ L0(0,∞):ρ(f)< ∞}. If now Lρ(0,∞) turns out to be a Banach space when equipped with the norm ρ(·), we refer to it as a Banach Function space. If ρ(f) ≤ lim infnρ(fn)whenever (fn) ? L0converges almost everywhere to f ∈ L0, we say that ρ has the Fatou Property. If less generally this implication only holds for (fn)∪ {f} ? Lρ, we say that ρ is lower semi-continuous. If further the situation f ∈ Lρ, g ∈ L0and μt(f)= μt(g) for all t > 0,forces g ∈ Lρand ρ(g)= ρ(f), we call Lρrearrangement invariant (or symmetric).

By employing generalized singular values and Banach Function Spaces, Dodds, Dodds and de Pagter [7]formally defined the noncommutative space≡Lρ(M) to be

and showed that if ρ is lower semicontinuous and Lρ(0,∞) rearrangement-invariant,is a Banach space when equipped with the norm fρ= ρ(μ(f)).

In the context of semifinite algebras, we then obtained the following quantized version of the principle 2.3 (see [26]and [27]for details).

Principle 2.4The dual pair of quantum Orlicz spacesprovides a rigorous mathematical formalism for a description of a general, quantum regular system.

3 Crossed Products

In contexts where one has to deal with von Neumann algebras which do not have a trace,one of course does not have access to the elegant theory of Dodds, Dodds, and de Pagter. In such cases one needs to follow the philosophy of Haagerup and Terp, which makes essential use of the notion of crossed products of von Neumann algebras to posit quantum Lp-spaces. For the sake of the reader we briefly review some of the essential facts regarding continuous crossed products, before going on to analyse the behaviour of quantum dynamical maps with respect to these crossed products.

Let M be a von Neumann algebra acting on the Hilbert space H, and let ν be a fixed faithful normal semifinite weight on M. With σ = σνdenoting the modular automorphism group of the weight ν, the crossed product algebra M=M ?σR is then defined to be the von Neumann algebra, acting on L2(R,H), which is generated by operators π(x) and λ(t), defined by (cf.[18, 41])

where x ∈ M, t ∈ R, and ξ ∈ L2(R,H) - the space of essentially separably valued squareintegrable functions f :R → H (specifically ξ ∈ L2(R,H) ifOne may then define a dual action of R on M in the form of a one-parameter group of automorphisms (θs),by means of the prescription

It turns out that M ?σR is a semifinite von Neumann algebra which admits a faithful normal semifinite trace satisfying the condition that τ ? θs=e?sτ for all s ∈ R. (See [15, Lemma 5.2],[38, Lemma 8.2]for details.)

On noting that the λ(t)’s are unitaries(whence λ?(t)= λ(?t)),it follows from the definition of the crossed product that

1. λ(t)π(x)λ?(t)= π(σt(x)) for x ∈ M and t ∈ R.

2. π(x)λ(t)π(y)λ(s) = π(xσt(y))λ(ts),

3. (π(x)λ(t))?= π(σ?t(x?))λ(?t),

4. M?σR is the σ-weak closure of the?-algebra of linear combinations of products λ(s)π(x)with x ∈ M and s ∈ R.

4 Completely Positive Maps on Quantum Orlicz Spaces

The basic aim of this section is to provide conditions under which Markov(positive normal)maps on von Neumann algebras, allow for an extension to a large class of Orlicz spaces; in particular to the Orlicz space Lcosh?1, which is a natural home for the regular observables(see[21, 26, 27]). Ultimately we wish to show that an important class of quantum maps originally defined on the von Neumann algebra generated by bounded observables,also gives well defined time evolution of regular observables on this Orlicz space.

In achieving this extension, an appropriate extension of some form of interpolation scheme to the setting of type III von Neumann algebras will be needed, but the question that arises is exactly which mode of interpolation will be suited to the task? To clarify the picture (and justify some of the subsequent investigations),we will, for the reader’s convenience,make some preliminary observations:

Observation 4.11. Classically the exact interpolation spaces for the couplecoincide with the rearrangement invariant spaces on R andis a Calderón couple (for all details see Chapter 26, Interpolation of Banach spaces by N. Kalton,S. Montgomery-Smith in [17]).

2. Dodds, Dodds, de Pagter have shown(see Theorem 3.2 in [8]) that the classical interpolation scheme for fully symmetric Banach function spaces on R+can be extended to the context of semifinite von Neumann algebras equipped with a faithful normal semifinite trace.

3. Whilst complex interpolation has a proven track record in constructing and studying Lp-spaces corresponding to type III algebras, there is as yet no complex interpolative way to construct Orlicz spaces.

4. We therefore need a scheme built on the philosophy of the first point above, but which also takes note of the specific structure of Orlicz spaces corresponding to type III algebras.Some sort of hybrid of the two approaches therefore seems to be appropriate. This objective will be finally achieved in Theorem 4.11. (For a description of the structure of the pairin the general setting, see [20].)

Before proceeding with the study of interpolation in the non-commutative setting, let us pause to clarify the description of the relevant quantum spaces.

When starting with a general von Neumann algebra M,we gain access to non-commutative measurability, by passing to the (much larger) semifinite von Neumann algebra(see Sections 2 and 3). We emphasize that no information is lost in this transition, since the original algebra M, can be canonically embedded in the larger algebra M, see Remark 6.1 in[26]. Within this context Haagerup then defined the spaces Lp(M) (p > 0), by the simple prescription

To describe the manner in which Orlicz spaces are constructed for type III algebras, we need some preliminaries. Let M be a von Neumann algebra with fns weight ν. Further, letwhereis the dual weight of ν on the crossed product M and τ ≡ τMis its canonical trace. We will write nνfor {a ∈ M;ν(a?a)< ∞}. For any Orlicz space LΨ(R), the associated fundamental function on [0,∞), is defined by t → χEΨwhere E is a measurable subset of(R,λ) for which λ(E)= t. To distinguish the two cases, we will write ?Ψfor the fundamental function induced by it’s Luxemburg norm, andfor the fundamental function corresponding to the Orlicz norm. Recall that a complementary Orlicz function Ψ?may be defined by settingA crucial fact underlying the definition given below is that classically the space LΨ?(R) is the Kthe dual of LΨ(R) with=t for all t ≥0 [3].

Definition 4.2(see [20]) With [a]denoting the minimal closure of a closable operator a,we define the Orlicz space LΨ(M) to be

It is shown in [20], that in the case Ψ(t)=tp(p ≥ 1), the above prescription yields exactly the Haagerup Lpspace Lp(M).

In the case where M is semifinite and the canonical weight a trace, it is common to rather denote the associated spaces by either of LΨ(M,τ), orIf Haagerup’s strategy is used,we write LΨ(M) as above.

As noted earlier,if we are primarily interested in demonstrating the existence of a quantum map on the Lp(M)-spaces (1 ≤ p ≤ ∞), then theorems like Theorem 5.1 of [16](which is built on the technology of complex interpolation) will suffice. These results seem to be the current state of the art as far as the interpolation of positive maps between M and L1(M)is concerned.For the sake of contrast with the alternative approach we shall develop shortly, we briefly review the essentials of the above approach. Let T be a positive normal map on M which in this case need not be completely positive. For the sake of simplicity of exposition, we will here assume that M is σ-finite, with ν a faithful normal state. Here we will follow the exposition of[13, 14, 19], and [39]. There is an operator h ∈L1(M) (whereis the dual weight of ω andassociated with the state ν. One may then use this operator to define embeddings of M into Lp(M) as follows:

Further, let T also be unital in its action on M. Define T(p)by

for a ∈ M. Note that ιp(M) is dense in Lp(M) (see Lemma 1.6 in [13]). Therefore, T(p)is densely defined. (Similar conclusions hold in the general non-σ-finite case where we have a weight instead of a state. But in that case the embedding ιpshould be defined on the subalgebra span{y?x:x,y ∈ M;ν(x?x)< ∞,ν(y?y)< ∞} rather than the full algebra.)

Moreover by Theorem 5.1 in [16](see also [13, Proposition 2.2]), whenever ν ? T ≤ γν for some γ >0, the map T(p)will extend to a positive bounded map on Lp(M).

However,we are interested in demonstrating the existence of quantum maps on spaces quite different from Lp-spaces (here and subsequently in this section Lpstands for Haagerup’s Lpspace). We therefore present an alternative to the above process. In proving that we do have good behaviour of such maps on the space of regular observables, the primary difficulty we need to overcome is that the current versions of the complex method only really work for Lpspaces, thereby excluding Lcosh?1. So some ingenuity is needed if we are to be successful. The assumption that seems to help to bridge this gap is the requirement that T also be completely positive. We pause to point out that all of the theory developed in this section holds true for general von Neumann algebras, not just σ-finite ones. So unless otherwise stated, we will for the remainder of this section assume that M is a possibly non-σ-finite algebra equipped with a faithful normal semifinite weight ν. We will writefor the modular group associated with the weight ν.

The first step in our alternative approach to extending T, is to find a way to extend T to M ?σνR. One may formally define such an extension on a dense subspace of M ?σνR by means of the prescriptionfor all x ∈M and s ∈R, and then attempt to extend this map to all of M ?σνR by continuity. However, the well-definiteness of this extension then becomes an issue.

Remark 4.3Assuming T to be a completely positive map satisfyingT for all t ∈R, we investigate the well-definiteness of the mapproposed above, in the case where M is a σ-finite von Neumann algebra in standard form,and ν a state with cyclic and separating vector ?. To have a well defined linear mapon M ?σR one wishes to have

for si∈ R and xi∈ M. To this end let us consider (4.4) in detail. Namely, note that (4.4)implies=0 for any ξ ∈ L2(R,H). Taking ξ(t) to be of the form ξ(t)=f(t)?with f ∈L2(R), one has

Further note that on setting ai,j≡one obtains the positive definite matrix ai,j. As f ∈L2(R) is an arbitrary function one can then expect that ai,jis an arbitrary positive definite matrix. As the matrix a is positive, a ≡ {ai,j} ≥ 0, it should then be of the form a=b?b, see Lemma 3.1 in Chapter IV [37]. Therefore

Let us now take a basis in L2(R), for example consisting of Hn(t)-Hermite polynomials. In this case we may select b so thatfor each pair (n,i). To see this,note that then

When combined with our choice of the vector ξ, the fact that=0 leads to

But, ? is cyclic and separating, and ν therefore a faithful state. Thus, one gets

This equation can of course be rewritten as

for any f ∈L2(R) and any n.

The finer properties of this extended map were investigated by Haagerup, Junge and Xu,who demonstrated the following:

Theorem 4.4([16, Theorem 4.1]) Let M be as before, and assume that T : M →M is a completely bounded normal map such that

Then T admits a unique bounded normal extensionon M ?σνR such thatand

1. Let B be the von Neumann subalgebra on L2(R,H) generated by all λ(s), s ∈ R. Then

3. If T is positive, then so is.

4. Assume in addition that ν ?T ≤ ν. Then

In addition to the properties noted by Haagerup, Junge and Xu, the extensionalso satisfies the following requirement. The establishment of this additional property proves to be crucial, as it is this fact which will give us access to real interpolation at the crossed product level.

Proposition 4.5Let T andbe as before. If each of(1)–(4)holds,thenwhere τ is the canonical trace on M=M ?σνR.

ProofLetBy equation (1.1) of [16]and page 2130 of [16], the action ofis induced by a →hitah?it. In the language of [32],is then of the form(see[32, Theorem 5.12]). So by [32, Proposition 4.3], we have thatFrom the proof of Theorem 7.4 of [32]considered alongside the discussion following Proposition 4.1 of [32],it is clear that this means that for any a ∈ (M ?σνR)+, have that?h?1)?1/2]a[h?1/2(I + ?h?1)?1/2]). By the Borel functional calculus for affiliated operators,we have that h?1(I + ?h?1)?1= (h+ ?I)?1for each ? > 0. So this formula becomes τ(a) =We also have that the von Neumann algebra B generated by the λ(t)’s, agrees with the commutative von Neumann algebra generated by h. Now for any ?>0, (?I +h)?1/2≡ (?+h)?1/2is bounded, and so belongs to B. So it is a simple matter to use parts (1) and (4) of Theorem 4.4 to see that

Letting ? decrease to zero, now yieldsas required.

With the above corollary at our disposal we may at this point appeal to Yeadon’s ergodic theorem for positive maps [43](recently extended and significantly sharpened by Haagerup,Junge and Xu in the form of [16, Theorem 5.1]), to see thatextends canonically to a bounded map on L1(M,τ). Since M is semifinite, the full power of the Dodds, Dodds and de Pagter approach is therefore in this context at our disposal (see [8]). One crucial piece of information we may glean from this theory is that in this context the noncommutative analogue Lρ(M,τ) of any Banach function space Lρ(0,∞), must be an exact interpolation space of the pair (L1(M,τ),M), whenever Lρ(0,∞) is an exact interpolation space of the pair (L1(0,∞),L∞(0,∞)) [8, Theorem 3.4]). In particular, any noncommutative Orlicz space LΨ(M,τ), is an exact interpolation space of the pair (L1(M,τ),M). This means that if the action of a bounded map S on M extends to a bounded action on L1(M,τ),then S also extends to a bounded action on LΨ(M,τ), with the norm on LΨmajorised by max( S∞, S1),where S∞and S1are respectively the norms of the action of S on M and L1(M,τ). Specifically,canonically induces an action on each Orlicz space LΦ(M,τ). Whilst this fact is worthy of noting, we are here interested in the action ofon spaces associated with M, not M. For this purpose we need the following information regarding the extension to the specific Orlicz space(L∞+L1)(M,τ).

Proposition 4.6Let T be a completely positive map on M satisfying ν ? T ≤ ν andfor each t ∈R,and letbe its completely positive extension to M=M?σνR.Thencanonically induces a map on the space (L∞+L1)(M,τ).

ProofIf we apply [16, Theorem 5.1]to Proposition 4.5, it is clear thatcanonically induces a map on L1(M,τ). So this claim follows from the theory of Dodds, Dodds and de Pagter [8].

The final phase in our construction,is to extract the information we need regarding LΨ(M),from what we’ve already shown for the action ofWe pass to investigating this point.

Proposition 4.7Let T be a completely positive map on M satisfying ν ? T ≤ ν andfor each t ∈R, and letbe its completely positive extension to M=M?σνR.For the sake of simplicity we will also writefor the extension to (L∞+L1)(M,τ). In its action on (L∞+L1)(M,τ), it satisfies the condition thatfor all a ∈B and all b ∈ (L∞+L1)(M,τ). (Here B is von Neumann subalgebra generated by all λ(s), s ∈ R.)

ProofThe stated property is known to hold for M. Since M ∩ L1(M,τM) is normdense in L1(M,τM), the continuity ofon L1(M,τM), then ensures that it also holds for L1(M,τM).

The reason for proving the above,is that all the type III Orlicz spaces with upper fundamental index less that 1, live inside (L∞+L1)(M,τ). We first proceed to define the fundamental indices of an Orlicz space. These were introduced by Zippin [44].

Definition 4.8Let LΨ(M) be an Orlicz space, and let ?ψbe the fundamental function of the space LΨ(0,∞). (Here we consider the Luxemburg norm,but the case of the Orlicz norm is completely analogous.) Let(If we use the fundamental function of the space LΨ(0,∞) equipped with the Orlicz norm, we will writefor this function.) Then the lower and upper fundamental indices of LΨ(M) are defined to be

respectively.

The following result is a variant of [20, Theorem 3.13], where the Boyd indices were used to prove a similar theorem.

Proposition 4.9Let LΨ(M) be an Orlicz space with upper fundamental index strictly less than 1. Then Lψ(M)? (L∞+L1)(M,τM) where M=M ?σνR. Moreover the canonical topology on Lψ(M)then agrees with the subspace topology inherited from(L∞+L1)(M,τM).

ProofIt is clear from exercise 14 of [3, Chapter 3](see also Corollary 8.15, page 275 in[3])that 0So given any n ∈N,there must exist 1 ≥t0>0 such that

for all 0

It is an exercise to see that

On selecting st<0 so that est=t, this in turn ensures that

Let x ∈ LΨ(M) be given. We remind the reader that thenand tμt(x) =for any 0 < t ≤ 1 (see [20, Theorem 3.10]and its proof). For 0 < t ≤ t0we then clearly have thatFinally use the fact that t → μt(x) is decreasing, to see that for n>1

What we still need is an alternative criterion to the one given in Definition 4.2 for identifying the elements of(L∞+L1)(M,τ)that belong to some LΨ(M). The following lemma succeeds in this regard, and also greatly simplifies the prescription for constructing Orlicz spaces for type III algebras.

Lemma 4.10Let M=M ?σ?R, let θsbe the dual action of R on M, and letGiven some Young’s function Ψ, the maps vs= ?Ψ(e?sh)?Ψ(h)?1are bounded for each s ∈ R.Moreoverbelongs to LΨ(M)if and only if for any s ∈R we have that

ProofWe show that the mapsare bounded for each s ∈R,withbelonging to LΨ(M)if and only iffor any s ∈R. The lemma will then follow from the observation that for any s ∈ R, we have that vs=e?sds. To see this,observe that it follows from[3,Theorem II.5.2&Corollary IV.8.15]thatfor any t>0.

To see the claim regarding boundedness, observe that if s < 0, we may conclude from [3,Corollary II.5.3]thatfor all t>0, or equivalently thatThe map dswill then clearly be contractive. If on the other hand s ≥0, then again by [3, Corollary II.5.3], we have thatfor all t>0, or equivalently thatSo in this case

Next let a ∈ LΨ(M)be given. It was shown in the proof of[20,3.10],that we then have thatfor any s ≥ 0. So for s ≥ 0, we must then have that a = θ?s(θs(a)) =or equivalently thatNow observe thatIn the case s ≥0, we therefore also have thatThis proves the “only if” part of the equivalence.

For the converse assume that the stated condition regarding the action of θsholds for someNote that the action of the θ’s extends to operators affiliated to M. So for any projection e ∈M of finite weight, we know thatis closable with τM-dense domain. It is easy to conclude that

for all s ∈ R. Hence by definition a ∈ LΨ(M) (see [20]).

We are finally ready to prove that CP Markov dynamics on M canonically extends to a large class of quantum Orlicz spaces. We remind the reader that a map T on M, is called sub-Markov, if the situation 0 ≤ a ≤ I (a ∈ M), ensures that 0 ≤ T(a) ≤ I. In the case where M is σ-finite and ν a state, the densityis actually an element of L1(M), and hence ?Ψ(h) ∈ LΨ(M). In this setting we say that a map S on LΨ(M) is sub-LΨ-Markov, if the situation 0 ≤ a ≤ ?Ψ(h) (a ∈ LΨ(M)), ensures that 0 ≤ S(a)≤ ?Ψ(h).

Theorem 4.11Let M = M ?σνR, and let Ψ be a Young’s function. Let T : M → M be a completely positive normal map such that ν ? T ≤ ν, and

? In the case where M is σ-finite and ν a state, the map TΨwill in it’s action on LΨ(M), map elements of the form ?Ψ(h)1/2a?Ψ(h)1/2(a ∈ M) onto elements of the form ?Ψ(h)1/2T(a)?Ψ(h)1/2. In particular, TΨwill be sub-LΨ-Markov whenever T is sub-Markov.Moreover,for any sub-LΨ-Markov map S on LΨ(M), we may then find a sub-Markov map S∞on M such that S(?Ψ(h)1/2a?Ψ(h)1/2)= ?Ψ(h)1/2S∞(a)?Ψ(h)1/2for all a ∈ M.

ProofIfthen LΨ(M) lives inside (L∞+L1)(M,τ), and also gets its topology from (L∞+L1)(M,τ). So all one needs to do to prove the first claim, is to note

? that if a ∈ (L∞+L1)(M,τ), then also

? and then simply apply the Lemma.

For the second claim, note that by the preceding proposition, LΨ(M) lives inside (L1+L∞)(M). Given a ∈ M,we will then have that ?Ψ(h)1/2a?Ψ(h)1/2∈ LΨ(M)? (L1+L∞)(M).On applying Proposition 4.7 to, it follows thatfor each n ∈N. But since [?Ψ(h)1/2χ[0,n](h)]∈B, it follows from the definition ofthat

Hence for each n, we have

This is enough to prove the second claim.

We proceed with the proof of the third claim. To this end let a ∈ LΨ(M) be given with 0 ≤ a ≤ ?Ψ(h). It then follows from [35, Lemma 2.2 d]that there exists a contractive element x ∈ M+such that a= ?Ψ(h)1/2x?Ψ(h)1/2. For any s ∈ R we may then apply θsto both sides.Recall that in Lemma 4.10 we had that vs= ?Ψ(e?sh)?Ψ(h)?1. So by that Lemma, we will then have that

Equivalently x = θs(x) for each s ∈ R, which ensures that x ∈ M. Since x is positive and contractive,we have that 0 ≤ x ≤ I and hence that 0 ≤ T(x) ≤ I. But then by the second part of the proof, TΨ(a)= ?Ψ(h)1/2T(x)?Ψ(h)1/2≤ ?Ψ(h), as required.

It remains to prove the final claim. The proof uses the fact verified above that if we are given a ∈ LΨ(M) with 0 ≤ a ≤ ?Ψ(h), then a = ?Ψ(h)1/2x?Ψ(h)1/2for some contractive element of x ∈M+, and is to all intents and purposes a minor modification of the second part of the proof of [13, Proposition 2.5]. The one fact we need to verify for that proof to adapt to the present context is that if for some b ∈ M we have that 0 = ?Ψ(h)1/2b?Ψ(h)1/2,then b = 0. This can be seen to follow by noting that if 0 = ?Ψ(h)1/2b?Ψ(h)1/2, then 0 =and applying [13, Lemma 1.3].

Example 4.12The upper fundamental index of the space Lcosh?1(0,∞)isIsomorphic Orlicz spaces share the same indices. So it is sufficient to prove this for a space isomorphic to Lcosh?1(0,∞). We show how to construct such a space before proving the claim. It is easy to see that the graphs of etandare tangent at t=2. This fact ensures that

is a Young’s function. Using Maclaurin series it is easy to see thatSince we also have thatit is clear that Ψe≈ cosh ?1. (To see this note that the limit formulae ensure that we may find 0<α<β <∞so thaton [0,α], andSince the functionhas a both a minimum and maximum on the interval [α,β], a combination of these facts ensures that we can find positive constants 0

It remains to compute the fundamental indices of LΨe(0,∞). We will use the formulas in Remark 2.3 of [44]to compute these indices. We will assume that LΨe(0,∞) is equipped with the Luxemburg norm. Since

it now follows from [3, 4.8.17]that the fundamental function of LΨe(0,∞) is given by

We proceed to compute the functionIn computing this function, we first consider the case where 0

for any t>0. Since we also have that

it is clear that MΨe(s)=1 in this case.

Now let s be given with s>1. We then have that

It is not too difficult to see that the functionhas a maximum of s1/2at t = e?2on the interval (0,1). So forthe supremum of the above quotient is s1/2. Finally consider the function

It is easy to see that

as claimed. Similarly

Corollary 4.13If T is a completely positive map on M satisfying ν ?T ≤ ν andfor each t ∈R, the extension T then canonically induces an action on Lcosh?1(M).

Of course the question now arises as to how the maps T(p)on Lp(M) (defined earlier)compare to the extension ofto Lp(M) by means of the above process, and ultimately also how the work of Goldstein and Lindsay, and Haagerup, Junge and Xu, compare to ours. This relationship is clarified by the following corollary to Theorem 4.11.

Corollary 4.14In the case Ψ(t) = tp(p > 1), the maps induced byon Lp(M), are exactly the maps T(p)constructed in [16, Theorem 5.1].

ProofFor the sake of clarity of exposition, we restrict attention to the σ-finite case,assuming that ν is in fact a state. The claim may then be proven by simply replacing ?Ψ(h)1/2with h1/(2p)in the proof of the second claim of Theorem 4.11.

5 Utility of the Theory for Mathematical Physics

In mathematical physics the set of observables of a given quantum system will for the most part lead to a σ-finite von Neumann algebra M on a separable Hilbert space,and hence in this second part of the paper, we will restrict to this context. Our primary goal in this section is to demonstrate the utility of the preceding theory for mathematical physics,by showing that a large class of natural quantum maps fulfil the criteria of the preceding section,and hence allow for an extension to the space Lcosh?1(M). But what exactly is a “natural quantum map”?

In standard quantum statistical mechanics the starting point is a triple(A,Tt,ω)consisting of a C?-algebra, a family of dynamical maps {Tt}, and a time-invariant faithful state ω. The dynamics of the family can also be expressed at the Hilbert space level. Specifically, on passing to the GNS-construction(πω,Hω,?), it is easy to see that the action of each Tton A,in a very natural way induces an action ?Tton the dense set πω(A)? ?Hωwhich is defined by

The question arises as to whetheris bounded or even closable. The additional assumption of complete positivity leads to an easy proof of the contractivity of. There are however alternative conditions similarly ensuring the contractivity of. A condition of this type which arises from physical considerations is the so-called detailed balance condition (DBC for short).At a physical level DBC may also be thought of as a condition corresponding to the microscopic reversibility of the system. This condition too ensures the contractivity of. In fact it does more; the assumption of DBC also ensures that eachcommutes with the modular operator and that at least one of the conditions in Theorem 4.4 is then automatically satisfied. Thus the assumption that the quantum maps {Tt} satisfy DBC, ensures that at the Hilbert space level,the maps {} induce very regular dynamics. From the perspective of Physics, this elegant duality reflects the duality between the Heisenberg and Schrdinger pictures in the description of quantum systems. In this regard we point out that since the publication of Tolman’s book[40]and the paper of Glauber [11], DBC is frequently used in statistical physics. (See for example [10]and [29], and the references therein.)

5.1 Detailed Balance Condition

Recall that for us a positive, normal, unital map T : M →M is deemed to be a Markov map. The class of Markov maps seems to be too general to describe the most interesting genuine dynamics. Hence, to select more regular maps we define:

Definition 5.1A Markov map T satisfies the Detailed Balance Condition (for brevity DBC) with respect to a state ω on M if there exists a reversing operation Θ, i.e., an antilinear Jordan morphism on M such that Θ2= identity map and ω(Θ(xy)) = ω(Θ(x)Θ(y)) for any x,y ∈M, for which the following conditions are satisfied (see [24, 25])

for any x,y ∈M.

DBC implies that (see [25]):

and that

defines a bounded operator on H which commutes with the modular operator ?. Moreover,it is an easy observation to make that

To see this note that for any x ∈ M and any y′∈ M′(M′stands for the commutator of M)one has:

which proves the claim.

Before proceeding further let us pause to make some important remarks on the DBC.

Remark 5.21. There are various versions of DBC. For example, one can use the more general form of DBC which was given in [29]. However, the form given here has a more“transparent” physical interpretation. In particular, to the best of our knowledge, only the form of DBC given in Definition 5.1 leads to a one-to-one correspondence between dynamical semigroups on the set of observables and semigroups on the Hilbert space of (state) vectors respectively (see [25]).

2. Frequently, DBC is related to KMS symmetry. However it is important to note that only DBC forces the map T to commute with the automorphism group, and this property is essential in our analysis.

3. But,in general,tensor product structure is not respected by DBC.Namely,if a(positive)map T :M → M satisfies DBC then T ?id:M?N → M?N,where N is a?-algebra,does not need to be a positive map. Therefore,one can not expect that an extension of a positive map T on the tensor product structure will satisfy DBC. Consequently, to get well defined dynamical maps on the crossed products,a further selection of positive maps should be done. To this end,we will additionally assume complete positivity.

4. For a recent account on DBC we refer the reader to [10].

5.2 Extension of dynamical maps to crossed products and their corresponding algebra of τ-measurable operators

In our study of quantum maps, we need to canonically extend a dynamical map T defined on a von Neumann algebra M to a corresponding map which is defined on a certain noncommutative Orlicz space. As a first step we have to extend T to the corresponding crossed product.

Definition 5.3Let T :M →M be a positive,unital map satisfying DBC with respect to a faithful, normal state ω(·)=(?,· ?). As in the previous section, we construct the extensionof T by first definingon a weak* dense subspace of the crossed product by means of the prescription

for t ∈ R, and x ∈ M.

To formulate and then to prove results concerning, we need some preliminaries. Firstly note that if x ∈M then we can define an operatoron L2(R,H) byfor ξ ∈ Cc(R,H) (see [41]). The important point to make here is that the form of ξ = ξ0? f with ξ0∈ H and f ∈ Cc(R) leads to

Turning to measurable operators,we recall that M ≡ M?σR is a semifinite von Neumann algebra equipped with a canonical normal faithful semifinite trace τ (cf. Section 3). It is worth pointing out that invariance of τ with respect to the extensionof a natural physical map indicates that the action ofcan be lifted to the τ-measurable operators. We are now in a position to formulate and prove the main result of this section,which is that in this framework we don’t just have that τ ?≤ τ, but in fact have equality.

Theorem 5.4Let T :M →M be a completely positive,unital map satisfying DBC with respect to a faithful, normal state ω(·)=(?,· ?). Then

ProofWe remind the reader that any completely positive map T is also completely bounded andstands for the cb-norm, see Proposition 3.6 in [31]. Moreover, all finite linear combinations of λ(s)π(x), s ∈ R x ∈ M form a?-dense involutive subalgebra of M ?σR. Thus, the statement thatis a well-defined map on M ?σR follows from Theorem 4.1 in [16]and this proves the first claim.

To prove the second claim, we need some facts from the book of van Daele [41].

1. L2(R,H) can be canonically identified with H ?L2(R) by

for any ξ0∈ H and f ∈ Cc(R) (Cc(R) – the space of continuous complex valued functions on R with compact supports), see Proposition 2.2 in [41].

2. Let λtdenote the left translation by ?t in L2(R). Then, see Proposition 2.8 in [41],

3. M ?σR is spatially isomorphic to the von Neumann algebra on H ? L2(R) generated by the operators {x ? I,?is? λs;x ∈ M,s ∈ R}, see Proposition 2.12 in [41].

The canonical faithful semi-finite trace on (M ?σR)+, can in this case then be defined by means of the formula

where τK(a)=(ξK,aξK), a ∈ M ?σR. Here ξK= ? ? F?fK, whereand F stands for the Fourier transform on L2(R). K denotes a compact subset in R.

Furthermore, see Lemma 3.3 in [41], if f ∈Cc(R) with support in the compact set K, then

where λ(f)=I ? λf, λf=F?mfF, and mfis the multiplication operator by f in L2(R).

So for f = χK(χKstands for the indicator function of a compact subset K ? R) we have by (5.11) that

where we have used the invariance of ω with respect to the modular automorphism and the formula (5.11). The last equality follows from

Secondly

where we have used Definition 5.3, (5.12), and the invariance of ω with respect to T. Thirdly,we note that

and hence as λ(t)π(x)λ(t)?= π(σt(x)):

for any s,t ∈R and x,y ∈M. Defineas in Remark 4.3, i.e.,

where we have used (5.13). But since τKincreases to the trace τ over M ?σR as K increases,the trace τ is also invariant with respect to.

6 Conclusions and Final Remarks

By focusing attention on observables, algebras and states, we proposed a new formalism for Statistical Mechanics, both classical and quantum, in [26]and [27]. It is based on two distinguished Orlicz spaces Lcosh?1and L log(L+1),and proves to be a canonical extension of the traditional formalism for elementary quantum mechanics; for details see [27].

However, in general, physical systems are dynamical, i.e. they evolve in time. So a state(respectively an observable) can exhibit changes brought about by the passage of time. With this aim in mind, we have in this paper defined and examined quantum maps which are able to describe dynamical processes within this same formalism.

(1) The following axiomatic framework for the study of(quantum)statistical analysis now emerges from the foregoing analysis,

? A von Neumann algebra M and an associated faithful normal semifinite weight ν corresponding to the given quantum system.

? The pair of spaces L log(L+1)(M),Lcosh?1(M) as respective homes for good states and observables. [26]

In particular,it was shown that a large class of physical maps satisfy the basic requirements necessary for their extension to and examination as maps on quantum Orlicz spaces.

(2) In closing we remind the reader that Dirichlet forms are a natural source of Markov semigroups. The extension of this theory to von Neumann algebras therefore provides an alternative source of quantum maps on noncommutative Orlicz spaces. However,this topic and the role of such quantum maps in Statistical Physics exceed the scope of this paper. For a recent account of this topic, we refer the reader to [22].