Gibbs states, algebraic dynamics and generalized Riesz systems

In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita-Takesaki theory in our context.


Introduction
In the past 25 years or so it has become clearer and clearer that the role of self-adjointness of the observables of some given microscopic system can be, sometimes, relaxed, without modifying the essential benefits of dealing with, for instance, a self-adjoint Hamiltonian. In fact, we can still find real eigenvalues, a unitary time evolution and a preserved probability even if the requirement of the Hamiltonian being self-adjoint is replaced by some milder assumption, like in PT-or in pseudo-hermitian quantum mechanics. We refer to [1]- [5] for some references on these approaches, both from a more physical point of view and from their mathematical consequences.
Considering a non-selfadjoint Hamiltonian H = H * may lead to the appearance of new and often unpleasant features; for instance, the set {ϕ n } of eigenstates of H, if any, in general is no longer an orthonormal system, but this set {ϕ n } and the set {ψ n } of the eigenstates of H * turn out to be biorthogonal i.e., (ϕ n |ψ m ) = δ n,m . Also, in concrete examples they are not bases for the Hilbert space H where the model is defined, but they may still be complete in H. This is the reason why the notion of D-quasi bases was proposed in [6].
This concept can be thought as a suitable extension of Riesz biorthogonal bases, and similar biorthogonal sets are found in several concrete physical applications, playing often the role that in the traditional setup is played by orthonormal bases (ONB). In recent papers many other extensions of Riesz bases, mostly involving unbounded operators, have also been considered. In particular we mention generalized Riesz systems introduced by one of us (H.I) and analyzed in a series of papers [13,14,15,19,20,21,22,23]). For other studies on extensions of Riesz bases or on generalizations to different environments (Krein spaces, Rigged Hilbert spaces) we refer to [11,12,17].
In [16] the role of similar biorthogonal sets, in particular Riesz bases, in the analysis of Gibbs states, KMS condition and algebraic Heisenberg dynamics was first considered. More recently a similar analysis has been carried out by other authors (see, e.g. [7]). Here we want to give our contribution to this line of research, by using the biorthogonal sets originated by generalized Riesz systems.
The paper is organized as follows: in the next section we give some preliminaries. In Section 3 we propose our definition of Gibbs state defined by generalized Riesz systems, when the dynamics is driven by a self-adjoint operator H 0 . The natural settings which we will adopt is the O * -algebra L † (D), where D is a dense subspace of H, [8,9,10]. This will appear to be a good choice, due to the fact that the operators appearing in our analysis are mostly unbounded.
In Section 4 we will consider possible definitions of the algebraic dynamics for non self-adjoint Hamiltonians, and then we will consider how these dynamics are related to the generalized Gibbs states introduced first, and the KMS-like relations which arise from this construction. In Section 5 we will propose a preliminary analysis of the Tomita-Takesaki modular theory in our context, while our conclusions are given in Section 6.

Preliminaries
In this section we review the basic definitions and results on generalized Riesz systems and O * -algebras needed in this paper.
is called a constructing pair for {ϕ n } and T is called a constructing operator for {ϕ n }.
Let D be a dense subspace in H. We denote by L † (D, H) the set of all closable linear operators X in H such that D(X) = D and D(X * ) ⊃ D. As usual we put, for X ∈ L † (D, H), Then L(D) is an algebra with the usual operations: X + Y , αX and XY , and L † (D) is a * -algebra with the involution X → X † := X * ⌈ D , inherited by L † (D, H). A * -subalgebra M of L † (D) is said to be an O * -algebra on D in H. Here we assume that M has the identity operator I. A locally convex topology defined by a family { · X ; X ∈ M} of seminorms: ξ X := Xξ , ξ ∈ D is called the graph topology on D and denoted by t M . If the locally convex space D[t M ] is complete, then M is called closed and it is shown that M is closed if and only if D = X∈M D(X). If D = X∈M D(X * ), then M is called self-adjoint. Next we define a weak commutant of M as follows: for all X ∈ M and ξ, η ∈ D}, where B(H) is the C * -algebra of all bounded linear operators on H. Then M ′ w is a weakly closed * -invariant subspace of B(H), but it is not necessarily an algebra. If M is self-adjoint, then M ′ w is a von Neumann algebra on H satisfying M ′ w D ⊂ D. Furthermore, we see that L † (D) ′ w = CI. We define some topologies on M. For any ξ, η ∈ D we put p ξ,η (X) := |(Xξ|η)|, p ξ (X) := Xξ , X ∈ L † (D). The locally convex topology on L † (D) defined by the family {p ξ,η (·); ξ, η ∈ D} (resp. {p ξ (·); ξ ∈ D}) of seminorms on L † (D) is called the weak (resp. strong) topology, and the induced topology of the weak (resp. strong) topology on M is called the weak (resp. strong) topology on M. For any Y ∈ M and ξ ∈ D we define a seminorm on M by The locally convex topology on M defined by the family {P ξ,Y (·); ξ ∈ D, Y ∈ M} is called the quasi-strong topology on M. A linear functional ω on M is called positive if ω(X † X) ≧ 0 for all X ∈ M, and a positive linear functional ω on M is a state if ω(I) = 1. A ( * )-isomorphism of M onto M is called a ( * )-automorphism of M and {α t } t∈R is called a one-parameter group of ( * )-automorphisms of M if α 0 (X) = X and α s+t (X) = α s (α t (X)) for all X ∈ M. A one-parameter group {α t } t∈R of automorphisms of M is weakly (resp. strongly, quasi-strongly) continuous if lim t→0 α t (X) = X for any X ∈ M under the weak (resp. strong, quasi-strong) topology. An operator H in L † (D) is called a weak (resp. strong, quasi-strong) generator for {α t } t∈R if lim t→0 αt(X)−X t = i[H, X] under the weak (resp. strong, quasi-strong) topology. For O * -algebras refer to [27].

Gibbs states defined by generalized Riesz systems
Throughout this section let {ϕ n } be a generalized Riesz system in a Hilbert space H with a constructing pair ({f n }, T ) and λ n > 0, n = 0, 1, · · · , such that {e − β 2 λn } ∈ ℓ 1 , for some β > 0. In this section we shall define and investigate a Gibbs state ω β ϕ defined through {ϕ n } on the maximal O * -algebra L † (D) on a dense subspace D in H. We put where, for x, y ∈ H, the operator x ⊗ȳ is defined by Then H 0 is a non-singular positive self-adjoint operator in H such that and it is called a standard hamiltonian for {f n }.
Before entering in the main matter of the paper some comments are in order. Once H 0 and a generalized Riesz system {ϕ n } with constructing pair ({f n }, T ) are given, one can define an operator H on the linear span D ϕ of {ϕ n } by putting Hϕ n = λ n ϕ n ; n ∈ N 0 and extending by linearity to D ϕ . Since D ϕ needs not be dense in H, it is natural to consider H as an operator acting in H ϕ , the closure of D ϕ in H. It is then natural to write Hϕ n = HT f n = λ n ϕ n = T H 0 f n , n ∈ N, which looks like an intertwining (or, better, when T is invertible, a similarity) condition for H and H 0 , as discussed in [16] for Riesz bases. Similarity is a quite strong condition in particular when considering the spectrum of the involved operators or trying to get a functional calculus. We will not pursue this approach here because it doesn't fit with the general situation we are considering.
Then Xe −βH 0 is trace class on H, for all X ∈ L † (D, H).

Proof. Take an arbitrary
Thus, Xe − β 2 H 0 is an everywhere defined operator on H and it is simple to show that it is closable. Therefore Xe − β 2 H 0 is a closed operator in H. By the closed graph theorem Xe − β 2 H 0 is a bounded operator on H and since we have Xe −βH 0 is trace class. This completes the proof. ✷

We remark that any subspace
for all X ∈ L † (D), and hence ω β f is a state on L † (D), and it is called a Gibbs state on L † (D) for the ONB {f n }. We formally define a Gibbs state ω β ϕ on L † (D) for the generalized Riesz system {ϕ n } by where Z ϕ := ∞ n=0 e −βλn ϕ n 2 . Conditions for that are discussed in [16]. In what follows we will consider only generalized Riesz system {ϕ n } for which Z ϕ < ∞. We do not know whether ω β ϕ is a state on L † (D), namely, in particular, |ω β ϕ (X)| < ∞ for all X ∈ L † (D). For that, we assume that a constructing pair ({f n }, T ) for a generalized Riesz system {ϕ n } satisfies the following In the rest of the paper we will use the same symbol for T , e − β 2 H 0 and e −βH 0 , and for their restrictions to D. Then we have the following and Proof. Take an arbitrary X ∈ L † (D). Then, by Assumption 1, T * XT ∈ L † (D, H) and by Lemma 3.1 (T * XT )e −βH 0 is trace class, which implies that for all X ∈ L † (D). Hence ω β ϕ (X) = 1 Zϕ tr T * XT e −βH 0 and it is a state on L † (D). Since T and T * are non-singular; that is, T −1 and (T * ) −1 exist, we see that ω β ϕ is faithful. Furthermore, we have Then {ψ n } is a generalized Riesz system with a constructing pair ({f n }, (T −1 ) * ) and {ϕ n } and {ψ n } are biorthogonal sequences. For the constructing operator (T −1 ) * for {ψ n }, we assume the following, which is completely analogous to what stated in Assumption 1 above.
As before, we use the same symbol for the operators (T −1 ) * , e − β 2 H 0 and e −βH 0 and for their restrictions to E. Now we define put where Z ψ := ∞ n=0 e −βλn ψ n 2 , which is assumed to exist finite, see [16]. Then we have the following Theorem 3.3. Under Assumption 2, ω β ψ is a faithful state on L † (E) and Proof. It is proved similarly to Theorem 3.2.    Proof. First, we show that {α 0 t } is strongly continuous. Take arbitrary X ∈ L † (D) and ξ ∈ D. Then by assumption we have Thus, {α 0 t } t∈R is strongly continuous. Next, we show that H 0 is a weak generator of {α 0 t } t∈R . Take arbitrary X ∈ L † (D) and ξ, η ∈ D. Then it follows from our assumptions that which yields that H 0 is a weak generator of {α 0 t } t∈R . Let D = D ∞ (H 0 ) and t H 0 be a locally convex topology on D defined by a sequence { · H n 0 ; n ∈ N 0 } of norms on D. Since H n 0 ∈ L † (D), for all n ∈ N, we have t H 0 ≺ t L † (D) . Conversely we show that t L † (D) ≺ t H 0 . Take an arbitrary X ∈ L † (D). Since the identity ι is a closed map of the Fréchet space D[t H 0 ] into the Hilbert space D( · X ) with the graph norm · X := · + X · , it follows from the closed graph theorem that it is continuous, which implies that t L † (D) ≺ t H 0 . Thus we have and for any X ∈ L † (D) there exist n ∈ N and γ > 0 such that Then, for any X, Y ∈ L † (D) and ξ ∈ D it follows from H n 0 X ∈ L † (D) and by our assumptions that Then, it follows from H m 0 X ∈ L † (D) and (4.3) that

The Heisenberg time evolution for generalized Riesz systems
Let {ϕ n } be a generalized Riesz system with a constructing pair ({f n }, T ). We assume the following Henceforth we denote an operator A⌈ D ∈ L † (D) by A for simplicity. Then, we have ϕ n , ψ n := (T † ) −1 f n ∈ D, for all n ∈ N 0 , and we can define a non-self-adjoint operator H by H := T H 0 T −1 . Then H ∈ L † (D) with H † = (T † ) −1 H 0 T † and Hϕ n = λ n ϕ n and H † ψ n = λ n ψ n , n ∈ N 0 (we notice that (iii) implies that (T † ) −1 = (T * ) −1 ⌈ D and then (T † ) −1 = (T −1 ) † ). Hence H and H † can be considered as non-self-adjoint hamiltonians for {ϕ n } and {ψ n }, respectively. Furthermore, take arbitrary ξ, η ∈ D and t ∈ R. By Assumption 3, (iii) there exists a element ζ ∈ D such that ξ = T ζ. Then it follows that Thus, it is natural to define e itH and e itH † by e itH := T e itH 0 T −1 and e itH † := (T † ) −1 e itH 0 T † t ∈ R. (4.5) Then we have the following Proof. By (4.5) it is immediately shown that {e itH } and {e itH † } are one-parameter groups of L † (D) satisfying (e itH ) † = e −itH † , for all t ∈ R. We show that they are quasi-strongly continuous. Indeed, it follows from Assumption 3, (iv) that for any X ∈ L † (D) and ξ ∈ D Similarly, we can show that {e itH † } is quasi-strongly continuous. ✷ We now define what we call the Heisenberg time evolution for {ϕ n } and {ψ n } as follows: t was defined before. This is in complete agreement with what originally proposed in [16]. Analogously,  Proof. By Lemma 4.2.1, {α ϕ t } t∈R and {α ψ t } t∈R are one-parameter groups of automorphisms of L † (D) satisfying α ϕ t (X) † = α ψ t (X † ), for all X ∈ L † (D) and t ∈ R. Let us now show that {α ϕ t } t∈R and {α ψ t } t∈R are weakly continuous. Take arbitrary X ∈ L † (D) and ξ, η ∈ D. Since Thus, H is a weak generator of {α ϕ t } t∈R . Similarly we can show that {α ψ t } t∈R is weakly continuous and its weak generator is H † . Finally, we show that if T ∈ B(H), then {α ϕ t } t∈R is strongly continuous. Take arbitrary X ∈ L † (D) and ξ ∈ D. Then, as usual, there exists an element ζ ∈ D such that ξ = T ζ and by Assumption 3, (iii) we have Similarly, if T −1 ∈ B(H), then we can show that {α ψ t } t∈R is strongly continuous. This completes the proof. ✷ Next, let us consider the case of D = D ∞ (H 0 ). Then, Assumption 3, (i) and (ii) hold automatically, and (iv) holds from (4.1.2). Therefore, the following result easily follows. Proof. Since t L † (D) = t H 0 by (4.2), for any X ∈ L † (D) there exist n ∈ N and r > 0 such that Xξ ≦ r ξ H n 0 for all ξ ∈ D. (4.6) For any X, Y ∈ L † (D) and ξ ∈ D with ξ = T ζ for some ζ ∈ D, it follows from (4.6) and Assumption 3, (iv) that Thus {α ϕ t } is quasi-strongly continuous. We show that the quasi-strong generator of {α ϕ t } equals H. Indeed, take arbitrary X, Y ∈ L † (D) and ξ ∈ D. Then ξ = T ζ for some ζ ∈ D and by Lemma 4.1.1 the generator of {α 0 t } equals H 0 , which yields that Thus, the quasi-strong generator of {α ϕ t }is H. Similarly, we can show that {α ψ t } is quasistrongly continuous and its quasi-strong generator of {α ψ t } is H † . This completes the proof. ✷

Few words on generalized von Neumann entropy
In this section we briefly show how what is done with the dynamics can be repeated for the von Neumann entropy. We work here under a slightly generalized version of Assumption 3. In particular, we assume (i) and (iii) hold as in Assumption 3, and that t in (ii) can be complexvalued, t = t r + it i , with t i > 0. More explicitly we assume that e itH 0 D ⊂ D, for all t ∈ C, with Im t > 0. Assumption (3.iv) is not relevant for us here, and will not be considered. Our original assumption on the eigenvalues λ n , ∞ n=0 e − 1 2 λn < ∞, is here replaced by the stronger assumptions for all γ > 0. Therefore, in particular we have Z 0 (β) = ∞ n=0 e −βλn < ∞. To simplify our treatment, from now on we will assume the following normalization: Z 0 (β) = 1. Here β is just a positive parameter which, in the following section, will acquire an explicit physical meaning, the inverse temperature of a given system.
The von Neumann entropy connected to the self-adjoint Hamiltonian H 0 is defined as where, with our normalization, ρ 0 = e −βH 0 . A straightforward computation of S ρ 0 produces S ρ 0 = β ∞ n=0 λ n e −γλn , which is finite because of our assumption (4.7). With the same steps as in the definition of e itH and e itH † , using our stronger assumptions, we conclude that n This suggests to define, in analogy with (4.5), Notice now that (ρ − 1 1) k = T (ρ 0 − 1 1) k T −1 , for all k = 0, 1, 2, . . .. Therefore, using the same argument bringing to definitions (4.5), we can check that n k=1 (−1) k−1 1 k (ρ − 1 1) k converges weakly to T log(ρ 0 )T −1 on D, and n k=0 (−1) k−1 1 k (ρ † −1 1) k converges weakly to (T † ) −1 log(ρ 0 )T † on D. Hence we put and we define a new von Neumann-like entropy as follows: under our working assumptions, and in particular the fact that ρ † ψ n ∈ D and (log ρ)ϕ n ∈ D, we easily conclude that S ρ = S ρ 0 .
Remark. It is worth pointing out that even in the cases when H 0 D ⊂ D, we cannot say (see 4.3 in [26]). Then for any z 0 ∈ S β with 0 < Im z 0 < β there exists a constant δ > 0 such that Im z 0 < β − δ. By (4.11), f X,Y is analytic at z 0 . Thus f X,Y is analytic on S β with 0 < Im z < β. Furthermore, by (4.9) we have This completes the proof. ✷ Thus ω β ϕ does not satisfy the KMS-condition with respect to {α ϕ t }, but still it satisfies the KMS-like condition with respect to {α ϕ t }, as Theorem 4.4.1 shows. Furthermore, we have a similar result for the Gibbs state ω β ψ as follows: For any X, Y ∈ L † (D) there exists a bounded continuous function F X,Y on the strip S β in C which is analytic on 0 < Im z < β such that for all t ∈ R.
Remark. We do not know whether Theorem 4.4.1 and Theorem 4.4.2 hold for a general subspace D satisfying Assumption 3. This is because we do not know whether (4.10) holds for unbounded operators T † Xα ϕ z (Y )T .
5 Gibbs states and unbounded Tomita-Takesaki theory

Conclusions
In this paper we have discussed how to generalize the standard notions of Heisenberg dynamics, Gibbs states, KMS-condition and Tomita-Takesaki theory to the case in which the dynamics is driven by a non self-adjoint Hamiltonian, as it often happens in PT-and in pseudo-hermitian quantum mechanics and we have chosen to consider observables as elements of L † (D). We have also seen how generalized Riesz systems can be used in this context, and how the results deduced here differ from the standard ones. We have also discussed some preliminary results on entropy and on the Tomita-Takesaki theory in our settings. Of course, many other aspects could be considered in future, from the use of Gibbs states defined by generalized Riesz systems in the analysis of concrete physical systems to more mathematical aspects. For instance, since it is often difficult or even impossible to find a common invariant dense domain D for the observables, one could try to enlarge the setting to some other relevant subset of L † (D, H). We plan to work on these and other aspects of our framework soon.